Research ArticleGeometric Assortative Growth Model for Small-World Networks

Yilun Shang12

1 Singapore University of Technology and Design Singapore 1386822 Institute for Cyber Security University of Texas at San Antonio TX 78249 USA

Correspondence should be addressed to Yilun Shang shylmathhotmailcom

Received 8 August 2013 Accepted 21 October 2013 Published 23 January 2014

Academic Editors H M Chamberlin and Y Zhang

Copyright copy 2014 Yilun Shang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It has been shown that both humanly constructed and natural networks are often characterized by small-world phenomenon andassortative mixing In this paper we propose a geometrically growing model for small-world networks The model displays bothtunable small-world phenomenon and tunable assortativity We obtain analytical solutions of relevant topological properties suchas order size degree distribution degree correlation clustering transitivity and diameter It is also worth noting that the modelcan be viewed as a generalization for an iterative construction of Farey graphs

1 Introduction

Recent studies of networked systems have led to the con-struction of models to explore their relevant properties asone of the fundamental steps to understanding real-worldphenomena of many kinds Among them small-world effectand network transitivity (or clustering) have attracted greatresearch attention [1 2] Many real-life systems such associal networks food webs World Wide Web and airportnetworks show both a high level of local clustering similarto a regular lattice and a relatively small average distanceor diameter namely small-world effect similar to a randomgraphNetworkswith these twodistinguishing characteristicsare often said to be small-world networks

The first and seminal model of small-world networkis the Watts-Strogatz rewiring model [1] which inducedan avalanche of works on studying small-world effect ofcomplex networks and setting up variant models to expoundthe mechanism of small-world phenomenon A variety ofmodels of small-world networks have been studied includingstochastic ones modeled by adding randomness to regulargraphs [1ndash7] and deterministic ones by making use of graphconstruction on some specific graphs such as planar latticesand Cayley graphs [8ndash11]

In this paper we study a geometric growth model119866(119898 119905) for small-world networks controlled by a tunable

parameter119898 Our model is constructed in a deterministicand recursive fashion At each step a multiple of 119898 ver-tices will be added into the network as per some simplegeometric structure Compared with probabilistic methodsour model has some remarkable features First the modelevolves through time which mimics the network growthin many real-world systems Second the simple generationmethod yields to analytical treatment of relevant topologicalproperties include order size degree distribution and cor-relations clustering transitivity and diameter Finally themodel shows assortative mixing on the degrees which isobserved in varied social networks and has profound impli-cations for network resilience [12 13] Many of the importantproperties studied in this paper (as mentioned above) aretunable by adjusting the parameter 119898 in the model Forexample we show that the level of assortativity increases with119898 in terms of Pearson correlation coefficient while clusteringas well as transitivity coefficients decrease with 119898 This givesinteresting characterization of a family of social networkmodels since both properties (ie assortative mixing andlocal clustering) are prevalent in social networks Moreoveralthough the diameter always grows proportionally to thelogarithm of the number of nodes in the network (hencedisplaying the small-world effect) it is shown to have distinctvalues for 119898 = 1 and 119898 gt 1 Table 1 summarizes the maincontributions

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 759391 8 pageshttpdxdoiorg1011552014759391

2 The Scientific World Journal

Table 1 Properties of model 119866(119898 119905)

119898 = 1a

119898 ge 2

Cumulative degree distribution 119875cum(120575) (2119898)minus1205752119898 for large 119905

Average neighbor degree 119896nn(120575)1198982

120575

2(2119898 minus 1)for large 119905

Pearson correlation coefficient 119903(119905) Increase with119898 as 2(2119898)2

+ 2 minus (9 (2119898 minus 1))

3(2119898)2

+ 1 minus (9 (2119898 minus 1))for large 119905

Clustering coefficient 119888(119905) Decrease with119898 as (2119898 minus 1) (2119898)1119898

21198982ln(2119898) for large 119905

Transitivity coefficient 1198881015840(119905) Decrease with119898 as 31198982

2 (2119898 minus 1) ((2119898)2

+ 1)

for large 119905

Diameter diam(119905) 119905 119905 + 1

a the properties for 119866(1 119905) were obtained in [14]

Here we should mention prior work that is conceptuallyor spiritually relevant The 119898 = 1 case of our model wasproposed in [14] as an alternative construction for Fareygraphs Farey graphs have many interesting properties suchas minimally 3-colorable uniquely Hamiltonian maximallyouterplanar and perfect see for example [15] Randomconstructions of Farey graph were explored in [16 17] wherean edge is removed with some probability 119902 and 119902(119905) at eachstep respectively Also for a different purpose Dorogovtsevet al [18] used a similar deterministic iteration process togenerate pseudofractal scale-free networks (see also [19])They have relevant but distinct properties with respect to ourmodel

The rest of the paper is organized as follows In Section 2we present our growth model for small-world networks Wereport the structure properties of the model in Section 3 Weconclude the paper in Section 4 with open problems

2 The Network Model 119866(119898119905)

In this section we introduce the geometric assortativegrowth model for small-world networks in a deterministicmanner and we denote the network graph by 119866(119898 119905) =

(119881(119898 119905) 119864(119898 119905)) with vertex set119881(119898 119905) and edge set 119864(119898 119905)

after 119905 iteration steps The construction algorithm of themodel is the following (i) for 119905 = 0 119866(119898 0) contains twoinitial vertices and an edge joining them namely 119870

2

(ii) for119905 ge 1 119866(119898 119905) is obtained from 119866(119898 119905 minus 1) by adding 119898 newvertices for each edge introduced at step 119905 minus 1 and attachingthem to two end vertices of this edge As such we will call anedge a generating edge if it is used to introduce new verticesin the next iteration step The first three steps of generationprocess of the growth model are shown in Figure 1

In what follows wewill oftenwrite119866(119905)119881(119905)119864(119905) and soforth suppressing the variable 119898 if we do not emphasize thespecific value of119898 We denote the two initial vertices in 119866(0)

by V0

and V1

and the number of new vertices and edges addedat step 119905 by 119871

119881

(119905) and 119871119864

(119905) respectively Therefore we have

t = 0

t = 1 t = 2

Figure 1 A depiction of graphs 119866(119898 119905) produced at iterations 119905 =

0 1 2 with119898 = 2

119871119881

(0) = 2 and 119871119864

(0) = 1 From the above construction it iseasy to see that 119871

119881

(119905) = 119898119871119864

(119905 minus 1) and 119871119864

(119905) = 2119898119871119864

(119905 minus 1)which give rise to 119871

119864

(119905) = (2119898)119905 and 119871

119881

(119905) = 119898(2119898)119905minus1 for

any 119905 ge 1 We have the following result

Proposition 1 The order and size of the graph 119866(119905) are

|119881 (119905)| =119898(2119898)

119905

+ 3119898 minus 2

2119898 minus 1 |119864 (119905)| =

(2119898)119905+1

minus 1

2119898 minus 1

(1)

respectively Moreover the average degree of 119866(119905) is

120575 (119905) = 4 minus12119898 minus 6

119898(2119898)119905

+ 3119898 minus 2 (2)

Proof They can be directly checked by |119881(119905)| = sum119905

119894=0

119871119881

(119894)|119864(119905)| = sum

119905

119894=0

119871119864

(119894) and 120575(119905) = 2|119864(119905)||119881(119905)|

Note that the average degree tends to 4 as 119905 rarr infin

irrespective of 119898 This kind of sparse networks are commonin both humanly constructed and natural networks [20 21]Some more sophisticated properties will be addressed inthe following We will for example improve the one-pointaverage-degree characterization of a network by consideringassortativity a two-point correlation quantity

The Scientific World Journal 3

3 Topological Properties of 119866(119898119905)

Thanks to the deterministic nature of the graphs 119866(119898 119905)in this section we will derive analytically some main topo-logical properties namely the degree distribution degreecorrelations clustering coefficient transitivity coefficientand diameter

31 Degree Distribution A fundamental quantity character-izing the structure and driving the behavior of a large networkis the probability distribution function 119875(120575) of vertex degree120575 It is the probability that a randomly chosen vertex has120575 direct neighbors It is often convenient to consider thecumulative degree distribution [17 21 22]

119875cum (120575) =

infin

sum

120575

1015840=120575

119875 (1205751015840

) (3)

which indicates the proportion of the vertices whose degreeis greater than or equal to 120575 An appealing property of thecumulative distribution is Networks with exponential degreedistribution namely 119875(120575) sim 119890

minus120572120575 also have exponentialcumulative distribution with the same exponent Indeed

119875cum (120575) =

infin

sum

120575

1015840=120575

119875 (1205751015840

) asymp

infin

sum

120575

1015840=120575

119890minus120572120575

1015840

= (119890120572

119890120572 minus 1) 119890minus120572120575

(4)

The Watts-Strogatz small-world model [1] also has an expo-nential degree distribution as we will study hereWementionthat there are some other geometric growthmodels proposedin the literature which follow another ubiquitous degreedistribution scale-free distributions see for example [18 2324]

Proposition 2 The cumulative degree distribution of 119866(119905)follows an exponential distribution 119875

119888119906119898

(120575) sim (2119898)minus(1205752119898) for

large 119905

Proof Let 120575V(119905) denote the degree of vertex V in 119866(119905) Let119905119894V be the step at which a vertex V is added to the graphFrom the construction all the vertices in the graph (excepttwo initial vertices V

0

and V1

) are always connected to twogenerating edges and will increase their degrees by 2119898 at thenext iteration

At 119905 = 0 the graph has two initial vertices V0

and V1

with degree 1 that is 120575V0

(0) = 120575V1

(0) = 1 For 119905 ge 1 byconstruction we have

120575V0

(119905) = 120575V1

(119905) = 1 + 119898 + (119905 minus 1)1198982

(5)

For other vertices we have 120575V(119905119894V) = 2 and 120575V(119905 + 1) = 120575V(119905) +

2119898 Thus

120575V (119905) = 2 (119898 (119905 minus 119905119894V) + 1) (6)

for 119905 ge 119905119894V Hence the degree distribution of the graph

119866(119905) is as follows The number of vertices of degree 2 sdot

1 2 sdot (119898 + 1) 2 sdot (2119898 + 1) 2 sdot (119898(119905 minus 1) + 1) equals119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 respectively and thedegrees of two initial vertices are given by (5)

Using (6) we have 119875cum(120575) = 119875(1199051015840

le 120591 = 119905 minus (120575 minus 2)2119898)Thus by exploiting Proposition 1 we obtain

119875cum (120575) asymp

120591

sum

119905

1015840=0

119871119881

(1199051015840

)

|119881 (119905)|

=2 (2119898 minus 1)

119898(2119898)119905

+ 3119898 minus 2

+

120591

sum

119905

1015840=1

119898(2119898)119905

1015840minus1

(2119898 minus 1)

119898(2119898)119905

+ 3119898 minus 2

=119898(2119898)

120591

+ 3119898 minus 2

119898(2119898)119905

+ 3119898 minus 2sim (2119898)

minus(1205752119898)

(7)

for large graphs (ie 119905 rarr infin)

We will make use of the exact degree distribution of 119866(119905)obtained in the above proof to study the clustering coefficientin the sequel

32 Degree Correlations (Average Neighbor Degree) Touncover correlations between the degrees of connected ver-tices the average neighbor degree 119896

119899119899

(120575) for vertices ofdegree 120575 is defined as the average degree of nearest neighborsof vertices with degree 120575 as a function of this degree value[25 26] If 119896

119899119899

(120575) is an increasing function of 120575 verticeswith high-degree have a larger probability to be connectedwith large degree vertices In this case the graph is saidto be assortative and this property is referred to in socialsciences as assortative mixing [12] Generally assortativity isthe tendency of entities to seek out and group with thoseother entities that exhibit similar characteristics In contrasta decreasing behavior of 119896

119899119899

(120575) defines a disassortative graphin the sense that high-degree vertices have a majority ofneighbors with low-degree whereas the opposite holds forlow-degree vertices In the absence of degree correlations119896119899119899

(120575) is a constantWe remark here that the concept of 119896119899119899

(120575)

is related to the groupie in graphs (see eg [27 28])

Proposition 3 The average neighbor degree for 119866(119905) isrespectively

(i)

119896119899119899

(1205750

) =1199051205750

+ 119898119905 + 21198982

119905 + 2119898 minus 119905

1 + 119898 + (119905 minus 1)1198982 (8)

where 1205750

= 1+119898+(119905minus1)1198982 (cf (5)) is the degree of two initial

vertices V0

and V1

(ii)

119896119899119899

(120575) =1198982

120575

2 (2119898 minus 1)+

119898(2119898)3

(2119898 minus 1)2

120575+

1198982

2119898 minus 1

minus119898 (119905 + 3) (2119898)

(1+(1205752))119898

(2119898)119905

120575

(9)

where 120575 = 2(119898(119905minus119905119894

)+1) (cf (6)) is the degree of other verticesadded to the network at step 119905

119894

ge 1

4 The Scientific World Journal

Proof We first show (9) It is clear that all vertices introducedat the same iteration step have the same degree No vertices(except V

0

and V1

) added to the network at the same step willbe connected to each other When a new vertex is added tothe network it connects vertices with larger degrees and itwill connect vertices with smaller degrees in the subsequentsteps From (6) for vertices introduced to the network at step119905119894

ge 1 they have the same degree 120575 = 2(119898(119905 minus 119905119894

) + 1)Let 120575(119905

119894

119905) represent the degree at step 119905 of a vertex thatwas generated at step 119905

119894

Thus 120575(119905119894

119905) = 2(119898(119905 minus 119905119894

) + 1) Wehave

119896119899119899

(120575) =1

119871119881

(119905119894

) 120575 (119905119894

119905)

times (2119898

119905

119894minus1

sum

119905

1015840

119894=1

119871119881

(1199051015840

119894

) 120575 (1199051015840

119894

119905)

+ 2119898

119905

sum

119905

1015840

119894=119905

119894+1

119871119881

(1199051015840

119894

) 120575 (1199051015840

119894

119905)

+119871119881

(0) 120575 (0 119905))

(10)

The first sum on the left-hand side of (10) accounts for theadjacencies made to vertices with larger degree namely 1 le

1199051015840

119894

lt 119905119894

and the second sum represents the edges introducedto vertices with a smaller degree at each step 119905

1015840

119894

gt 119905119894

The lastterm in (10) accounts for the adjacencies made to the initialvertices V

0

and V1

From (10) we derive that

119896119899119899

(120575) =1

(2119898)119905

119894 (119898 (119905 minus 119905119894

) + 1)

times (

2119898 (2119898 + 21198982

119905) ((2119898)119905

119894minus1

minus 1)

2119898 minus 1

minus

41198983

((2119898)119905

119894minus1

minus 1 + 2119898 (119905119894

minus 1))

2119898 minus 1

+

41198983

((2119898)119905

119894minus1

minus 1)

(2119898 minus 1)2

+

(2119898)119905

119894+1

(2119898 + 21198982

119905) ((2119898)119905minus119905

119894 minus 1)

2119898 minus 1

minus ((41198983

((119905119894

+ 1) (2119898)119905

119894 ((2119898)119905minus119905

119894 minus 1)

+ (2119898)119905

(119905 minus 119905119894

)))

times (2119898 minus 1)minus1

)

+41198983

(2119898)119905

119894+1

((2119898)119905minus119905

119894 minus 1)

(2119898 minus 1)2

+2 (1 + 119898 + (119905 minus 1)1198982

))

(11)

Feed 120575 = 2(119898(119905 minus 119905119894

) + 1) into the above expression eliminate119905119894

and simplify the consequential expression giving rise to (9)finally

Next for the two initial vertices with degree 1205750

= 1 +119898+

(119905 minus 1)1198982 we obtain

119896119899119899

(1205750

) =119898

1205750

119905

sum

119905

1015840

119894=0

120575 (1199051015840

119894

119905) =2119898 (119905 + 1) + 119898

2

119905 (119905 + 1)

1 + 119898 + (119905 minus 1)1198982 (12)

which yields to (8) as desired

Note that as 119905 tends to infinity (8) is tantamount to(1205750

+ 119898 + 21198982

minus 1)1198982 and the last term on the right-

hand side of (9) is vanishing Therefore we conclude that119896119899119899

(120575) is approximately a linear function of 120575 for large 119905which implies that our model 119866(119905) undergoes assortativegrowth

To find the impact of parameter 119898 we note that (8)decreases with119898 while (9) increases with119898 for large 119905 Sincethe contribution to the degree correlation of the two initialvertices of 119866(119905) is small we can safely think of 119896

119899119899

(120575) asan increasing function with respect to 119898 for large graphsmeaning that 119866(119898 119905) shows more significant assortativemixing for larger119898 This fact will be even clearer drawing onthe correlation coefficient (see below)

33 Degree Correlations (Pearson Correlation Coefficient)Another quantity often used to probe the assortativity is thePearson correlation coefficient 119903 of vertices connected by anedge [12 13]

119903 =|119864|sum119894

119895119894

119896119894

minus (sum119894

(12) (119895119894

+ 119896119894

))2

|119864| sum119894

(12) (1198952

119894

+ 1198962

119894

) minus (sum119894

(12) (119895119894

+ 119896119894

))2

(13)

where 119864 is the edge set of the graph in question and 119895119894

and 119896119894

are the degrees of the vertices at the ends of the 119894th edge with119894 = 1 2 |119864| It lies in the range minus1 le 119903 le 1This coefficientis zero for uncorrelated graph and positive or negative forassortative or disassortative mixing respectively Let 119903(119905) bethe degree-degree Pearson correlation coefficient of 119866(119905) Wehave the following result

The Scientific World Journal 5

Proposition 4 The Pearson correlation coefficient of 119866(119905) is

119903 (119905) = (1 + 119900 (1))

((((2119898)2

+ 1) (2119898 minus 1)) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

(((3(2119898)2

+ 1) (2 (2119898 minus 1))) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

997888rarr2(2119898)

2

+ 2 minus (9 (2119898 minus 1))

3(2119898)2

+ 1 minus (9 (2119898 minus 1))

(14)

as 119905 rarr infin

It is direct to check that (10) is positive for all 119898 ge 1 Itis an increasing function with 119898 and has upper bound 23Therefore for large 119905 the growth model 119866(119898 119905) is assortativefor all 119898 ge 1 and the level of assortativity increases with 119898This also justifies the above discussion of assortativity basedon local quantity 119896

119899119899

(120575)

Proof Following the notation in [14] we denote by ⟨119895119894

119896119894

⟩ the119894th edge in119866(119905) connecting two vertices with degree 119895

119894

and 119896119894

respectively By (5) the edge in119866(0) is thus ⟨1+119898+(119905minus1)119898

21+119898+(119905minus1)119898

2

⟩ (2119898)119905

119894 new edges are added to the networkat iteration step 119905

119894

ge 1 These edges will connect new verticesto every vertex in119866(119905

119894

minus1) whose degree distribution at 119905119894

minus1

is 120575(119897 119905119894

minus 1) = 2(119898(119905119894

minus 1 minus 119897) + 1) for 1 le 119897 le 119905119894

minus 1 and120575(0 119905119894

minus1) = 1+119898+(119905119894

minus2)1198982 Here the 120575 notation is defined

in the proof of Proposition 3At each of the subsequent steps of 119905

119894

minus 1 the degrees of allthese vertices will gain 2119898 except V

0

and V1

whose degreeswill gain119898 Consequently at iteration step 119905 ge 119905

119894

the numberof edges ⟨2119898(119905minus119905

119894

)+2 2(119898(119905119894

minus119897)+1)⟩ for 1 le 119897 le 119905119894

minus1 is (2119898)119897

and the number of edges ⟨2119898(119905 minus 119905119894

) + 2 1 +119898+ (119905 minus 1)1198982

⟩ is2119898

We now can evaluate these sums in (13) for large 119905

|119864(119905)|

sum

119894=1

119895119894

119896119894

= (1 + 119898 + (119905 minus 1)1198982

)2

+ 2119898 (1 + 119898 + (119905 minus 1)1198982

)

times

119905

sum

119905

119894=1

(2119898 (119905 minus 119905119894

) + 2)

+ 2

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2119898 (119905 minus 119905119894

) + 2)

times (119898 (119905119894

minus 1) + 1) (2119898)119897

= ((2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(15)

Likewise we have

|119864(119905)|

sum

119894=1

(119895119894

+ 119896119894

)

= 2(1 + 119898 + (119905 minus 1)1198982

)2

+

119905

sum

119905

119894=1

(2119898 (1 + 119898 + (119905 minus 1)1198982

) + 2119898 (119905 minus 119905119894

) + 2)

+

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2 (119898 (119905119894

minus 119897) + 1) + 2119898 (119905 minus 119905119894

) + 2) (2119898)119897

=3

2119898 minus 1(2119898)3+119905

+ 119900 ((2119898)119905

)

|119864(119905)|

sum

119894=1

(1198952

119894

+ 1198962

119894

) = (3(2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(16)Feeding these quantities into the definition (13) we thenarrive at the desired result

34 Clustering Coefficient The clustering coefficient [1] is agood indicator of local clustering namely the local densityof triangles and thus often used to characterize small-worldnetworks In a network 119866 = (119881 119864) the clustering coefficient119888(V) of a vertex V isin 119881 is the ratio of the total number 119890V ofedges that actually exist between all its 120575V nearest neighborsand the number 120575V(120575V minus 1)2 of all possible edges betweenthem More precisely

119888 (V) =2119890V

120575V (120575V minus 1) (17)

The clustering coefficient 119888(119866) of the whole network 119866 is theaverage of all individual 119888(V)rsquos

119888 (119866) =sumVisin119881 119888 (V)

|119881| (18)

In what follows we compute the clustering coefficient for thegrowth model 119866(119905)

Proposition 5 The clustering coefficient of 119866(119905) is119888 (119866 (119905)) = 119888 (119905)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(19)

6 The Scientific World Journal

where the function Φ represents the Lerch transcendent (see[29 Section 111])

Proof When a new vertex V is added to the graph it is easyto see 120575V = 2 and 119890V = 1 Furthermore every subsequentaddition of an edge attached to this vertex will increase bothparameters by one unit Therefore we have 119890V = 120575V minus 1 forevery vertex at every step Thus

119888 (V) =2119890V

120575V (120575V minus 1)=

2

120575V (20)

Drawing on this relationship the degree distributionobtained in Proposition 2 can be useful for calculation of theclustering coefficient of 119866(119905)

Indeed the number of vertices with clustering coefficient1 1(119898+1) 1(2119898+1) 1(119898(119905minus1)+1) 2(1+119898+(119905minus1)119898

2

)equals respectively119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 2Consequently we obtain

119888 (119866 (119905)) =1

|119881 (119905)|

times (

119905

sum

119894=1

1

(119894 minus 1)119898 + 1sdot 119898(2119898)

119905minus119894

+4

1 + 119898 + (119905 minus 1)1198982)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(21)

as desired

For large graphs (ie 119905 rarr infin) the right-hand side of (19)approaches

(2119898 minus 1) (2119898)1119898

21198982ln (2119898) (22)

which is a decreasing function with respect to 119898 Hencefor larger 119898 the level of local clustering becomes lowereventually This is not quite surprising since a large bunch ofvertices will be added to the network at each iteration when119898 and 119905 become large which mitigate the coefficient

35 Transitivity Coefficient Transitivity is an importantproperty especially in the analysis of social networks see forexample [21 30 31] Let 119879(119866) be the number of triangles and119876(119866) be the number of paths of length two in a graph119866Thenthe transitivity coefficient 1198881015840(119866) of 119866 can be defined as

1198881015840

(119866) =3119879 (119866)

119876 (119866) (23)

A brief discussion of the relationship between clustering andtransitivity coefficients can be found for example in [14]

Proposition 6 The transitivity coefficient of 119866(119905) is

1198881015840

(119866 (119905)) = 1198881015840

(119905)

= (1 + 119900 (1))31198983

((2119898)119905

minus 1)

(2119898 minus 1) ((2119898)2

+ 1) (2119898)119905+1

997888rarr31198982

2 (2119898 minus 1) ((2119898)2

+ 1)

(24)

as 119905 rarr infin

Proof We first calculate 119879(119866(119905)) Note that if the number ofgenerating edges after iteration 119905 minus 1 is 119886 the number of newtriangles introduced to the graph after iteration 119905 is 3119886 Since119886 = 119871

119864

(119905 minus 1) we obtain

119879 (119866 (119905)) = 119879 (119866 (119905 minus 1)) + 119898119871119864

(119905 minus 1)

= 119879 (119866 (119905 minus 1)) + 119898(2119898)119905minus1

(25)

which together with the initial value 119879(119866(1)) = 119898 gives

119879 (119866 (119905)) =119898 ((2119898)

119905

minus 1)

2119898 minus 1 (26)

for 119905 ge 1The number of paths of length two 119876(119866(119905)) can be

derived as follows by using the degree distribution again

119876 (119866 (119905)) = 119898(2119898)119905minus1

+ 119898

119905minus1

sum

119896=1

(2 (119896119898 + 1)

2) (2119898)

119905minus119896minus1

+ 2(1 + 119898 + (119905 minus 1)119898

2

2)

=(2119898)2

+ 1

1198982(2119898)119905+1

+ 119900 ((2119898)119905

)

(27)

which along with (26) leads to the stated result

Clearly the left-hand side of (24) is a decreasing functionof119898 Recalling the comments after Proposition 5 we see thatthe difference between clustering and transitivity coefficientsof 119866(119905) is by and large quantitative This is because theymeasure a quite similar property of networks

36 Diameter Network diameter namely the largest lengthof the shortest paths between all pairs of vertices is a mea-sure of the transmission performance and communicationefficiency We show analytically the diameter of our growthmodel and find a quantitative difference between 119898 = 1 and119898 gt 1

Proposition 7 The diameter diam(119866(119898 119905)) = diam(119905) of119866(119898 119905) equals 119905 for119898 = 1 and 119905 + 1 for119898 ge 2

The Scientific World Journal 7

Proof The case of 119898 = 1 was shown in [14] In what followswe take over their method to study119898 ge 2

Clearly diam(119866(119898 0)) = 1 and diam(119866(119898 1)) = 2At each step 119905 ge 2 the longest distance between twovertices is for some vertices added at this step correspondingto different generating edges at the last step Consider twovertices introduced at step 119905 ge 2 corresponding to differentgenerating edges say 119906

119905

and V119905

The vertex 119906119905

is adjacent totwo vertices and one of them must have been added to thegraph at step 119905 minus 2 or earlier

If 119905 = 2119896 is even 119906119905

can reach some vertex in 119866(119898 0) by119896 jumps and the same thing is true for vertex V

119905

Thereforediam(119866(119898 2119896)) le 2119896 + 1 If 119905 = 2119896 + 1 is odd 119906

119905

can reachsome vertex in119866(119898 1) by 119896 jumps and the same thing is truefor vertex V

119905

Therefore diam(119866(119898 2119896 + 1)) le 2119896 + 2 Thesebounds are attained by pairs of vertices 119906

119905

and V119905

created atiteration 119905 which correspond to different generating edgesand have the property of being connected to two verticesintroduced at steps 119905minus1 and 119905minus2 respectively Consequentlywe have diam(119866(119898 119905)) = 119905 + 1 for all119898 ge 2 and 119905 ge 1

From Proposition 1 we have for 119905 large

ln |119881 (119905)| = ln(119898(2119898)119905

+ 3119898 minus 2

2119898 minus 1) sim 119905 ln (2119898) (28)

Hence we obtain the logarithmic scale

diam (119905) simln |119881 (119905)|

ln (2119898) (29)

which together with high clustering (Propositions 5 and 6)justifies the small-world characteristics [1] of our growthmodel

4 Conclusion

We have studied a geometric assortative growth model119866(119898 119905) for small-world networks in a deterministic wayWe obtain analytical solutions of main properties of themodel such as the degree distribution and correlationsclustering and transitivity coefficients and graph diameter inthe full spectrum of parameter 119898 The 119866(119898 119905) model holdsboth tunable small-world and tunable assortative mixingbehaviors This should be useful to guide the research anddevelopment of varied social networks On the other handthe deterministic character of this graph family shouldfacilitate the exact calculation of other network-orientedquantities including average path length hyperbolicity [32]modular structure and motifs [33]

The introduction of tunable parameter 119898 also brings arange of open questions for future research In addition tothose mentioned before here are more examples how canwemake a trade-off between local clustering and assortativityby tuning119898 since they have opposite monotonicity What if119898 = 119898(119905) is a function of time

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[2] L A N Amara A Scala M Barthelemy and H E StanleyldquoClasses of small-world networksrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 97 no21 pp 11149ndash11152 2000

[3] S Jespersen and A Blumen ldquoSmall-world networks links withlong-tailed distributionsrdquo Physical Review E vol 62 no 5 pp6270ndash6274 2000

[4] M Kuperman and G Abramson ldquoComplex structures ingeneralized small worldsrdquo Physical Review E vol 64 no 4Article ID 047103 4 pages 2001

[5] M E J Newman ldquoModels of the small worldrdquo Journal ofStatistical Physics vol 101 no 3-4 pp 819ndash841 2000

[6] Y Shang ldquoFast distributed consensus seeking in large-scalesensor networks via shortcutsrdquo International Journal of Compu-tational Science and Engineering vol 7 no 2 pp 121ndash124 2012

[7] Y Shang ldquoA sharp threshold for rainbow connection in small-world networksrdquoMiskolc Mathematical Notes vol 13 no 2 pp493ndash497 2012

[8] F Comellas and M Sampels ldquoDeterministic small-world net-worksrdquo Physica A vol 309 no 1-2 pp 231ndash235 2002

[9] W Xiao and B Parhami ldquoCayley graphs as models of determin-istic small-world networksrdquo Information Processing Letters vol97 no 3 pp 115ndash117 2006

[10] S Boettcher B Goncalves and H Guclu ldquoHierarchical regularsmall-world networksrdquo Journal of Physics A vol 41 no 25Article ID 252001 2008

[11] Y Zhang Z Zhang S Zhou and J Guan ldquoDeterministicweighted scale-free small-world networksrdquo Physica A vol 389no 16 pp 3316ndash3324 2010

[12] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview Letters vol 89 no 20 Article ID 208701 4 pages 2002

[13] M E J Newman ldquoMixing patterns in networksrdquo PhysicalReview E vol 67 no 2 Article ID 026126 13 pages 2003

[14] Z Zhang and F Comellas ldquoFarey graphs as models for complexnetworksrdquo Theoretical Computer Science vol 412 no 8ndash10 pp865ndash875 2011

[15] C J Colbourn ldquoFarey series and maximal outerplanar graphsrdquoSIAM Journal on Algebraic Discrete Methods vol 3 no 2 pp187ndash189 1982

[16] Z Zhang S Zhou Z Wang and Z Shen ldquoA geometricgrowth model interpolating between regular and small-worldnetworksrdquo Journal of Physics A vol 40 no 39 pp 11863ndash118762007

[17] Y Shang ldquoDistinct clusterings and characteristic path lengthsin dynamic smallworld networks with identical limit degreedistributionrdquo Journal of Statistical Physics vol 149 no 3 pp505ndash518 2012

[18] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoPseud-ofractal scale-free webrdquo Physical Review E vol 65 no 6 ArticleID 066122 4 pages 2002

[19] Y Shang ldquoMean commute time for randomwalks on hierarchi-cal scale-free networksrdquo Internet Mathematics vol 8 no 4 pp321ndash337 2012

[20] R Albert and A L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

8 The Scientific World Journal

[21] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[22] S Jung S Kim and B Kahng ldquoGeometric fractal growthmodelfor scale-free networksrdquo Physical Review E vol 65 no 5 ArticleID 056101 6 pages 2002

[23] J S Andrade Jr H J Herrmann R F S Andrade and LR Da Silva ldquoApollonian networks simultaneously scale-freesmall world euclidean space filling andwithmatching graphsrdquoPhysical Review Letters vol 94 no 1 Article ID 018702 4 pages2009

[24] A-L Barabasi E Ravasz and T Vicsek ldquoDeterministic scale-free networksrdquo Physica A vol 299 no 3-4 pp 559ndash564 2001

[25] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[26] R Pastor-Satorras A Vazquez and A Vespignani ldquoDynamicaland correlation properties of the internetrdquo Physical ReviewLetters vol 87 no 25 Article ID 258701 4 pages 2001

[27] W F de la Vega and Z Tuza ldquoGroupies in random graphsrdquoInformation Processing Letters vol 109 no 7 pp 339ndash340 2009

[28] Y Shang ldquoGroupies in random bipartite graphsrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 278ndash2832010

[29] H Bateman and A Erdelyi Higher Transcendental Functionsvol 1 McGraw-Hill New York NY USA 1953

[30] M E J NewmanD JWatts and SH Strogatz ldquoRandomgraphmodels of social networksrdquo Proceedings of theNational Academyof Sciences of the United States of America vol 99 no 1 pp2566ndash2572 2002

[31] H Ebel L-I Mielsch and S Bornholdt ldquoScale-free topology ofe-mail networksrdquo Physical Review E vol 66 no 3 Article ID035103 4 pages 2002

[32] Y Shang ldquoLack of Gromov-hyperbolicity in small-world net-worksrdquo Central European Journal of Mathematics vol 10 no 3pp 1152ndash1158 2012

[33] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 The Scientific World Journal

Table 1 Properties of model 119866(119898 119905)

119898 = 1a

119898 ge 2

Cumulative degree distribution 119875cum(120575) (2119898)minus1205752119898 for large 119905

Average neighbor degree 119896nn(120575)1198982

120575

2(2119898 minus 1)for large 119905

Pearson correlation coefficient 119903(119905) Increase with119898 as 2(2119898)2

+ 2 minus (9 (2119898 minus 1))

3(2119898)2

+ 1 minus (9 (2119898 minus 1))for large 119905

Clustering coefficient 119888(119905) Decrease with119898 as (2119898 minus 1) (2119898)1119898

21198982ln(2119898) for large 119905

Transitivity coefficient 1198881015840(119905) Decrease with119898 as 31198982

2 (2119898 minus 1) ((2119898)2

+ 1)

for large 119905

Diameter diam(119905) 119905 119905 + 1

a the properties for 119866(1 119905) were obtained in [14]

Here we should mention prior work that is conceptuallyor spiritually relevant The 119898 = 1 case of our model wasproposed in [14] as an alternative construction for Fareygraphs Farey graphs have many interesting properties suchas minimally 3-colorable uniquely Hamiltonian maximallyouterplanar and perfect see for example [15] Randomconstructions of Farey graph were explored in [16 17] wherean edge is removed with some probability 119902 and 119902(119905) at eachstep respectively Also for a different purpose Dorogovtsevet al [18] used a similar deterministic iteration process togenerate pseudofractal scale-free networks (see also [19])They have relevant but distinct properties with respect to ourmodel

The rest of the paper is organized as follows In Section 2we present our growth model for small-world networks Wereport the structure properties of the model in Section 3 Weconclude the paper in Section 4 with open problems

2 The Network Model 119866(119898119905)

In this section we introduce the geometric assortativegrowth model for small-world networks in a deterministicmanner and we denote the network graph by 119866(119898 119905) =

(119881(119898 119905) 119864(119898 119905)) with vertex set119881(119898 119905) and edge set 119864(119898 119905)

after 119905 iteration steps The construction algorithm of themodel is the following (i) for 119905 = 0 119866(119898 0) contains twoinitial vertices and an edge joining them namely 119870

2

(ii) for119905 ge 1 119866(119898 119905) is obtained from 119866(119898 119905 minus 1) by adding 119898 newvertices for each edge introduced at step 119905 minus 1 and attachingthem to two end vertices of this edge As such we will call anedge a generating edge if it is used to introduce new verticesin the next iteration step The first three steps of generationprocess of the growth model are shown in Figure 1

In what follows wewill oftenwrite119866(119905)119881(119905)119864(119905) and soforth suppressing the variable 119898 if we do not emphasize thespecific value of119898 We denote the two initial vertices in 119866(0)

by V0

and V1

and the number of new vertices and edges addedat step 119905 by 119871

119881

(119905) and 119871119864

(119905) respectively Therefore we have

t = 0

t = 1 t = 2

Figure 1 A depiction of graphs 119866(119898 119905) produced at iterations 119905 =

0 1 2 with119898 = 2

119871119881

(0) = 2 and 119871119864

(0) = 1 From the above construction it iseasy to see that 119871

119881

(119905) = 119898119871119864

(119905 minus 1) and 119871119864

(119905) = 2119898119871119864

(119905 minus 1)which give rise to 119871

119864

(119905) = (2119898)119905 and 119871

119881

(119905) = 119898(2119898)119905minus1 for

any 119905 ge 1 We have the following result

Proposition 1 The order and size of the graph 119866(119905) are

|119881 (119905)| =119898(2119898)

119905

+ 3119898 minus 2

2119898 minus 1 |119864 (119905)| =

(2119898)119905+1

minus 1

2119898 minus 1

(1)

respectively Moreover the average degree of 119866(119905) is

120575 (119905) = 4 minus12119898 minus 6

119898(2119898)119905

+ 3119898 minus 2 (2)

Proof They can be directly checked by |119881(119905)| = sum119905

119894=0

119871119881

(119894)|119864(119905)| = sum

119905

119894=0

119871119864

(119894) and 120575(119905) = 2|119864(119905)||119881(119905)|

Note that the average degree tends to 4 as 119905 rarr infin

irrespective of 119898 This kind of sparse networks are commonin both humanly constructed and natural networks [20 21]Some more sophisticated properties will be addressed inthe following We will for example improve the one-pointaverage-degree characterization of a network by consideringassortativity a two-point correlation quantity

The Scientific World Journal 3

3 Topological Properties of 119866(119898119905)

Thanks to the deterministic nature of the graphs 119866(119898 119905)in this section we will derive analytically some main topo-logical properties namely the degree distribution degreecorrelations clustering coefficient transitivity coefficientand diameter

31 Degree Distribution A fundamental quantity character-izing the structure and driving the behavior of a large networkis the probability distribution function 119875(120575) of vertex degree120575 It is the probability that a randomly chosen vertex has120575 direct neighbors It is often convenient to consider thecumulative degree distribution [17 21 22]

119875cum (120575) =

infin

sum

120575

1015840=120575

119875 (1205751015840

) (3)

which indicates the proportion of the vertices whose degreeis greater than or equal to 120575 An appealing property of thecumulative distribution is Networks with exponential degreedistribution namely 119875(120575) sim 119890

minus120572120575 also have exponentialcumulative distribution with the same exponent Indeed

119875cum (120575) =

infin

sum

120575

1015840=120575

119875 (1205751015840

) asymp

infin

sum

120575

1015840=120575

119890minus120572120575

1015840

= (119890120572

119890120572 minus 1) 119890minus120572120575

(4)

The Watts-Strogatz small-world model [1] also has an expo-nential degree distribution as we will study hereWementionthat there are some other geometric growthmodels proposedin the literature which follow another ubiquitous degreedistribution scale-free distributions see for example [18 2324]

Proposition 2 The cumulative degree distribution of 119866(119905)follows an exponential distribution 119875

119888119906119898

(120575) sim (2119898)minus(1205752119898) for

large 119905

Proof Let 120575V(119905) denote the degree of vertex V in 119866(119905) Let119905119894V be the step at which a vertex V is added to the graphFrom the construction all the vertices in the graph (excepttwo initial vertices V

0

and V1

) are always connected to twogenerating edges and will increase their degrees by 2119898 at thenext iteration

At 119905 = 0 the graph has two initial vertices V0

and V1

with degree 1 that is 120575V0

(0) = 120575V1

(0) = 1 For 119905 ge 1 byconstruction we have

120575V0

(119905) = 120575V1

(119905) = 1 + 119898 + (119905 minus 1)1198982

(5)

For other vertices we have 120575V(119905119894V) = 2 and 120575V(119905 + 1) = 120575V(119905) +

2119898 Thus

120575V (119905) = 2 (119898 (119905 minus 119905119894V) + 1) (6)

for 119905 ge 119905119894V Hence the degree distribution of the graph

119866(119905) is as follows The number of vertices of degree 2 sdot

1 2 sdot (119898 + 1) 2 sdot (2119898 + 1) 2 sdot (119898(119905 minus 1) + 1) equals119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 respectively and thedegrees of two initial vertices are given by (5)

Using (6) we have 119875cum(120575) = 119875(1199051015840

le 120591 = 119905 minus (120575 minus 2)2119898)Thus by exploiting Proposition 1 we obtain

119875cum (120575) asymp

120591

sum

119905

1015840=0

119871119881

(1199051015840

)

|119881 (119905)|

=2 (2119898 minus 1)

119898(2119898)119905

+ 3119898 minus 2

+

120591

sum

119905

1015840=1

119898(2119898)119905

1015840minus1

(2119898 minus 1)

119898(2119898)119905

+ 3119898 minus 2

=119898(2119898)

120591

+ 3119898 minus 2

119898(2119898)119905

+ 3119898 minus 2sim (2119898)

minus(1205752119898)

(7)

for large graphs (ie 119905 rarr infin)

We will make use of the exact degree distribution of 119866(119905)obtained in the above proof to study the clustering coefficientin the sequel

32 Degree Correlations (Average Neighbor Degree) Touncover correlations between the degrees of connected ver-tices the average neighbor degree 119896

119899119899

(120575) for vertices ofdegree 120575 is defined as the average degree of nearest neighborsof vertices with degree 120575 as a function of this degree value[25 26] If 119896

119899119899

(120575) is an increasing function of 120575 verticeswith high-degree have a larger probability to be connectedwith large degree vertices In this case the graph is saidto be assortative and this property is referred to in socialsciences as assortative mixing [12] Generally assortativity isthe tendency of entities to seek out and group with thoseother entities that exhibit similar characteristics In contrasta decreasing behavior of 119896

119899119899

(120575) defines a disassortative graphin the sense that high-degree vertices have a majority ofneighbors with low-degree whereas the opposite holds forlow-degree vertices In the absence of degree correlations119896119899119899

(120575) is a constantWe remark here that the concept of 119896119899119899

(120575)

is related to the groupie in graphs (see eg [27 28])

Proposition 3 The average neighbor degree for 119866(119905) isrespectively

(i)

119896119899119899

(1205750

) =1199051205750

+ 119898119905 + 21198982

119905 + 2119898 minus 119905

1 + 119898 + (119905 minus 1)1198982 (8)

where 1205750

= 1+119898+(119905minus1)1198982 (cf (5)) is the degree of two initial

vertices V0

and V1

(ii)

119896119899119899

(120575) =1198982

120575

2 (2119898 minus 1)+

119898(2119898)3

(2119898 minus 1)2

120575+

1198982

2119898 minus 1

minus119898 (119905 + 3) (2119898)

(1+(1205752))119898

(2119898)119905

120575

(9)

where 120575 = 2(119898(119905minus119905119894

)+1) (cf (6)) is the degree of other verticesadded to the network at step 119905

119894

ge 1

4 The Scientific World Journal

Proof We first show (9) It is clear that all vertices introducedat the same iteration step have the same degree No vertices(except V

0

and V1

) added to the network at the same step willbe connected to each other When a new vertex is added tothe network it connects vertices with larger degrees and itwill connect vertices with smaller degrees in the subsequentsteps From (6) for vertices introduced to the network at step119905119894

ge 1 they have the same degree 120575 = 2(119898(119905 minus 119905119894

) + 1)Let 120575(119905

119894

119905) represent the degree at step 119905 of a vertex thatwas generated at step 119905

119894

Thus 120575(119905119894

119905) = 2(119898(119905 minus 119905119894

) + 1) Wehave

119896119899119899

(120575) =1

119871119881

(119905119894

) 120575 (119905119894

119905)

times (2119898

119905

119894minus1

sum

119905

1015840

119894=1

119871119881

(1199051015840

119894

) 120575 (1199051015840

119894

119905)

+ 2119898

119905

sum

119905

1015840

119894=119905

119894+1

119871119881

(1199051015840

119894

) 120575 (1199051015840

119894

119905)

+119871119881

(0) 120575 (0 119905))

(10)

The first sum on the left-hand side of (10) accounts for theadjacencies made to vertices with larger degree namely 1 le

1199051015840

119894

lt 119905119894

and the second sum represents the edges introducedto vertices with a smaller degree at each step 119905

1015840

119894

gt 119905119894

The lastterm in (10) accounts for the adjacencies made to the initialvertices V

0

and V1

From (10) we derive that

119896119899119899

(120575) =1

(2119898)119905

119894 (119898 (119905 minus 119905119894

) + 1)

times (

2119898 (2119898 + 21198982

119905) ((2119898)119905

119894minus1

minus 1)

2119898 minus 1

minus

41198983

((2119898)119905

119894minus1

minus 1 + 2119898 (119905119894

minus 1))

2119898 minus 1

+

41198983

((2119898)119905

119894minus1

minus 1)

(2119898 minus 1)2

+

(2119898)119905

119894+1

(2119898 + 21198982

119905) ((2119898)119905minus119905

119894 minus 1)

2119898 minus 1

minus ((41198983

((119905119894

+ 1) (2119898)119905

119894 ((2119898)119905minus119905

119894 minus 1)

+ (2119898)119905

(119905 minus 119905119894

)))

times (2119898 minus 1)minus1

)

+41198983

(2119898)119905

119894+1

((2119898)119905minus119905

119894 minus 1)

(2119898 minus 1)2

+2 (1 + 119898 + (119905 minus 1)1198982

))

(11)

Feed 120575 = 2(119898(119905 minus 119905119894

) + 1) into the above expression eliminate119905119894

and simplify the consequential expression giving rise to (9)finally

Next for the two initial vertices with degree 1205750

= 1 +119898+

(119905 minus 1)1198982 we obtain

119896119899119899

(1205750

) =119898

1205750

119905

sum

119905

1015840

119894=0

120575 (1199051015840

119894

119905) =2119898 (119905 + 1) + 119898

2

119905 (119905 + 1)

1 + 119898 + (119905 minus 1)1198982 (12)

which yields to (8) as desired

Note that as 119905 tends to infinity (8) is tantamount to(1205750

+ 119898 + 21198982

minus 1)1198982 and the last term on the right-

hand side of (9) is vanishing Therefore we conclude that119896119899119899

(120575) is approximately a linear function of 120575 for large 119905which implies that our model 119866(119905) undergoes assortativegrowth

To find the impact of parameter 119898 we note that (8)decreases with119898 while (9) increases with119898 for large 119905 Sincethe contribution to the degree correlation of the two initialvertices of 119866(119905) is small we can safely think of 119896

119899119899

(120575) asan increasing function with respect to 119898 for large graphsmeaning that 119866(119898 119905) shows more significant assortativemixing for larger119898 This fact will be even clearer drawing onthe correlation coefficient (see below)

33 Degree Correlations (Pearson Correlation Coefficient)Another quantity often used to probe the assortativity is thePearson correlation coefficient 119903 of vertices connected by anedge [12 13]

119903 =|119864|sum119894

119895119894

119896119894

minus (sum119894

(12) (119895119894

+ 119896119894

))2

|119864| sum119894

(12) (1198952

119894

+ 1198962

119894

) minus (sum119894

(12) (119895119894

+ 119896119894

))2

(13)

where 119864 is the edge set of the graph in question and 119895119894

and 119896119894

are the degrees of the vertices at the ends of the 119894th edge with119894 = 1 2 |119864| It lies in the range minus1 le 119903 le 1This coefficientis zero for uncorrelated graph and positive or negative forassortative or disassortative mixing respectively Let 119903(119905) bethe degree-degree Pearson correlation coefficient of 119866(119905) Wehave the following result

The Scientific World Journal 5

Proposition 4 The Pearson correlation coefficient of 119866(119905) is

119903 (119905) = (1 + 119900 (1))

((((2119898)2

+ 1) (2119898 minus 1)) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

(((3(2119898)2

+ 1) (2 (2119898 minus 1))) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

997888rarr2(2119898)

2

+ 2 minus (9 (2119898 minus 1))

3(2119898)2

+ 1 minus (9 (2119898 minus 1))

(14)

as 119905 rarr infin

It is direct to check that (10) is positive for all 119898 ge 1 Itis an increasing function with 119898 and has upper bound 23Therefore for large 119905 the growth model 119866(119898 119905) is assortativefor all 119898 ge 1 and the level of assortativity increases with 119898This also justifies the above discussion of assortativity basedon local quantity 119896

119899119899

(120575)

Proof Following the notation in [14] we denote by ⟨119895119894

119896119894

⟩ the119894th edge in119866(119905) connecting two vertices with degree 119895

119894

and 119896119894

respectively By (5) the edge in119866(0) is thus ⟨1+119898+(119905minus1)119898

21+119898+(119905minus1)119898

2

⟩ (2119898)119905

119894 new edges are added to the networkat iteration step 119905

119894

ge 1 These edges will connect new verticesto every vertex in119866(119905

119894

minus1) whose degree distribution at 119905119894

minus1

is 120575(119897 119905119894

minus 1) = 2(119898(119905119894

minus 1 minus 119897) + 1) for 1 le 119897 le 119905119894

minus 1 and120575(0 119905119894

minus1) = 1+119898+(119905119894

minus2)1198982 Here the 120575 notation is defined

in the proof of Proposition 3At each of the subsequent steps of 119905

119894

minus 1 the degrees of allthese vertices will gain 2119898 except V

0

and V1

whose degreeswill gain119898 Consequently at iteration step 119905 ge 119905

119894

the numberof edges ⟨2119898(119905minus119905

119894

)+2 2(119898(119905119894

minus119897)+1)⟩ for 1 le 119897 le 119905119894

minus1 is (2119898)119897

and the number of edges ⟨2119898(119905 minus 119905119894

) + 2 1 +119898+ (119905 minus 1)1198982

⟩ is2119898

We now can evaluate these sums in (13) for large 119905

|119864(119905)|

sum

119894=1

119895119894

119896119894

= (1 + 119898 + (119905 minus 1)1198982

)2

+ 2119898 (1 + 119898 + (119905 minus 1)1198982

)

times

119905

sum

119905

119894=1

(2119898 (119905 minus 119905119894

) + 2)

+ 2

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2119898 (119905 minus 119905119894

) + 2)

times (119898 (119905119894

minus 1) + 1) (2119898)119897

= ((2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(15)

Likewise we have

|119864(119905)|

sum

119894=1

(119895119894

+ 119896119894

)

= 2(1 + 119898 + (119905 minus 1)1198982

)2

+

119905

sum

119905

119894=1

(2119898 (1 + 119898 + (119905 minus 1)1198982

) + 2119898 (119905 minus 119905119894

) + 2)

+

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2 (119898 (119905119894

minus 119897) + 1) + 2119898 (119905 minus 119905119894

) + 2) (2119898)119897

=3

2119898 minus 1(2119898)3+119905

+ 119900 ((2119898)119905

)

|119864(119905)|

sum

119894=1

(1198952

119894

+ 1198962

119894

) = (3(2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(16)Feeding these quantities into the definition (13) we thenarrive at the desired result

34 Clustering Coefficient The clustering coefficient [1] is agood indicator of local clustering namely the local densityof triangles and thus often used to characterize small-worldnetworks In a network 119866 = (119881 119864) the clustering coefficient119888(V) of a vertex V isin 119881 is the ratio of the total number 119890V ofedges that actually exist between all its 120575V nearest neighborsand the number 120575V(120575V minus 1)2 of all possible edges betweenthem More precisely

119888 (V) =2119890V

120575V (120575V minus 1) (17)

The clustering coefficient 119888(119866) of the whole network 119866 is theaverage of all individual 119888(V)rsquos

119888 (119866) =sumVisin119881 119888 (V)

|119881| (18)

In what follows we compute the clustering coefficient for thegrowth model 119866(119905)

Proposition 5 The clustering coefficient of 119866(119905) is119888 (119866 (119905)) = 119888 (119905)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(19)

6 The Scientific World Journal

where the function Φ represents the Lerch transcendent (see[29 Section 111])

Proof When a new vertex V is added to the graph it is easyto see 120575V = 2 and 119890V = 1 Furthermore every subsequentaddition of an edge attached to this vertex will increase bothparameters by one unit Therefore we have 119890V = 120575V minus 1 forevery vertex at every step Thus

119888 (V) =2119890V

120575V (120575V minus 1)=

2

120575V (20)

Drawing on this relationship the degree distributionobtained in Proposition 2 can be useful for calculation of theclustering coefficient of 119866(119905)

Indeed the number of vertices with clustering coefficient1 1(119898+1) 1(2119898+1) 1(119898(119905minus1)+1) 2(1+119898+(119905minus1)119898

2

)equals respectively119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 2Consequently we obtain

119888 (119866 (119905)) =1

|119881 (119905)|

times (

119905

sum

119894=1

1

(119894 minus 1)119898 + 1sdot 119898(2119898)

119905minus119894

+4

1 + 119898 + (119905 minus 1)1198982)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(21)

as desired

For large graphs (ie 119905 rarr infin) the right-hand side of (19)approaches

(2119898 minus 1) (2119898)1119898

21198982ln (2119898) (22)

which is a decreasing function with respect to 119898 Hencefor larger 119898 the level of local clustering becomes lowereventually This is not quite surprising since a large bunch ofvertices will be added to the network at each iteration when119898 and 119905 become large which mitigate the coefficient

35 Transitivity Coefficient Transitivity is an importantproperty especially in the analysis of social networks see forexample [21 30 31] Let 119879(119866) be the number of triangles and119876(119866) be the number of paths of length two in a graph119866Thenthe transitivity coefficient 1198881015840(119866) of 119866 can be defined as

1198881015840

(119866) =3119879 (119866)

119876 (119866) (23)

A brief discussion of the relationship between clustering andtransitivity coefficients can be found for example in [14]

Proposition 6 The transitivity coefficient of 119866(119905) is

1198881015840

(119866 (119905)) = 1198881015840

(119905)

= (1 + 119900 (1))31198983

((2119898)119905

minus 1)

(2119898 minus 1) ((2119898)2

+ 1) (2119898)119905+1

997888rarr31198982

2 (2119898 minus 1) ((2119898)2

+ 1)

(24)

as 119905 rarr infin

Proof We first calculate 119879(119866(119905)) Note that if the number ofgenerating edges after iteration 119905 minus 1 is 119886 the number of newtriangles introduced to the graph after iteration 119905 is 3119886 Since119886 = 119871

119864

(119905 minus 1) we obtain

119879 (119866 (119905)) = 119879 (119866 (119905 minus 1)) + 119898119871119864

(119905 minus 1)

= 119879 (119866 (119905 minus 1)) + 119898(2119898)119905minus1

(25)

which together with the initial value 119879(119866(1)) = 119898 gives

119879 (119866 (119905)) =119898 ((2119898)

119905

minus 1)

2119898 minus 1 (26)

for 119905 ge 1The number of paths of length two 119876(119866(119905)) can be

derived as follows by using the degree distribution again

119876 (119866 (119905)) = 119898(2119898)119905minus1

+ 119898

119905minus1

sum

119896=1

(2 (119896119898 + 1)

2) (2119898)

119905minus119896minus1

+ 2(1 + 119898 + (119905 minus 1)119898

2

2)

=(2119898)2

+ 1

1198982(2119898)119905+1

+ 119900 ((2119898)119905

)

(27)

which along with (26) leads to the stated result

Clearly the left-hand side of (24) is a decreasing functionof119898 Recalling the comments after Proposition 5 we see thatthe difference between clustering and transitivity coefficientsof 119866(119905) is by and large quantitative This is because theymeasure a quite similar property of networks

36 Diameter Network diameter namely the largest lengthof the shortest paths between all pairs of vertices is a mea-sure of the transmission performance and communicationefficiency We show analytically the diameter of our growthmodel and find a quantitative difference between 119898 = 1 and119898 gt 1

Proposition 7 The diameter diam(119866(119898 119905)) = diam(119905) of119866(119898 119905) equals 119905 for119898 = 1 and 119905 + 1 for119898 ge 2

The Scientific World Journal 7

Proof The case of 119898 = 1 was shown in [14] In what followswe take over their method to study119898 ge 2

Clearly diam(119866(119898 0)) = 1 and diam(119866(119898 1)) = 2At each step 119905 ge 2 the longest distance between twovertices is for some vertices added at this step correspondingto different generating edges at the last step Consider twovertices introduced at step 119905 ge 2 corresponding to differentgenerating edges say 119906

119905

and V119905

The vertex 119906119905

is adjacent totwo vertices and one of them must have been added to thegraph at step 119905 minus 2 or earlier

If 119905 = 2119896 is even 119906119905

can reach some vertex in 119866(119898 0) by119896 jumps and the same thing is true for vertex V

119905

Thereforediam(119866(119898 2119896)) le 2119896 + 1 If 119905 = 2119896 + 1 is odd 119906

119905

can reachsome vertex in119866(119898 1) by 119896 jumps and the same thing is truefor vertex V

119905

Therefore diam(119866(119898 2119896 + 1)) le 2119896 + 2 Thesebounds are attained by pairs of vertices 119906

119905

and V119905

created atiteration 119905 which correspond to different generating edgesand have the property of being connected to two verticesintroduced at steps 119905minus1 and 119905minus2 respectively Consequentlywe have diam(119866(119898 119905)) = 119905 + 1 for all119898 ge 2 and 119905 ge 1

From Proposition 1 we have for 119905 large

ln |119881 (119905)| = ln(119898(2119898)119905

+ 3119898 minus 2

2119898 minus 1) sim 119905 ln (2119898) (28)

Hence we obtain the logarithmic scale

diam (119905) simln |119881 (119905)|

ln (2119898) (29)

which together with high clustering (Propositions 5 and 6)justifies the small-world characteristics [1] of our growthmodel

4 Conclusion

We have studied a geometric assortative growth model119866(119898 119905) for small-world networks in a deterministic wayWe obtain analytical solutions of main properties of themodel such as the degree distribution and correlationsclustering and transitivity coefficients and graph diameter inthe full spectrum of parameter 119898 The 119866(119898 119905) model holdsboth tunable small-world and tunable assortative mixingbehaviors This should be useful to guide the research anddevelopment of varied social networks On the other handthe deterministic character of this graph family shouldfacilitate the exact calculation of other network-orientedquantities including average path length hyperbolicity [32]modular structure and motifs [33]

The introduction of tunable parameter 119898 also brings arange of open questions for future research In addition tothose mentioned before here are more examples how canwemake a trade-off between local clustering and assortativityby tuning119898 since they have opposite monotonicity What if119898 = 119898(119905) is a function of time

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[2] L A N Amara A Scala M Barthelemy and H E StanleyldquoClasses of small-world networksrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 97 no21 pp 11149ndash11152 2000

[3] S Jespersen and A Blumen ldquoSmall-world networks links withlong-tailed distributionsrdquo Physical Review E vol 62 no 5 pp6270ndash6274 2000

[4] M Kuperman and G Abramson ldquoComplex structures ingeneralized small worldsrdquo Physical Review E vol 64 no 4Article ID 047103 4 pages 2001

[5] M E J Newman ldquoModels of the small worldrdquo Journal ofStatistical Physics vol 101 no 3-4 pp 819ndash841 2000

[6] Y Shang ldquoFast distributed consensus seeking in large-scalesensor networks via shortcutsrdquo International Journal of Compu-tational Science and Engineering vol 7 no 2 pp 121ndash124 2012

[7] Y Shang ldquoA sharp threshold for rainbow connection in small-world networksrdquoMiskolc Mathematical Notes vol 13 no 2 pp493ndash497 2012

[8] F Comellas and M Sampels ldquoDeterministic small-world net-worksrdquo Physica A vol 309 no 1-2 pp 231ndash235 2002

[9] W Xiao and B Parhami ldquoCayley graphs as models of determin-istic small-world networksrdquo Information Processing Letters vol97 no 3 pp 115ndash117 2006

[10] S Boettcher B Goncalves and H Guclu ldquoHierarchical regularsmall-world networksrdquo Journal of Physics A vol 41 no 25Article ID 252001 2008

[11] Y Zhang Z Zhang S Zhou and J Guan ldquoDeterministicweighted scale-free small-world networksrdquo Physica A vol 389no 16 pp 3316ndash3324 2010

[12] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview Letters vol 89 no 20 Article ID 208701 4 pages 2002

[13] M E J Newman ldquoMixing patterns in networksrdquo PhysicalReview E vol 67 no 2 Article ID 026126 13 pages 2003

[14] Z Zhang and F Comellas ldquoFarey graphs as models for complexnetworksrdquo Theoretical Computer Science vol 412 no 8ndash10 pp865ndash875 2011

[15] C J Colbourn ldquoFarey series and maximal outerplanar graphsrdquoSIAM Journal on Algebraic Discrete Methods vol 3 no 2 pp187ndash189 1982

[16] Z Zhang S Zhou Z Wang and Z Shen ldquoA geometricgrowth model interpolating between regular and small-worldnetworksrdquo Journal of Physics A vol 40 no 39 pp 11863ndash118762007

[17] Y Shang ldquoDistinct clusterings and characteristic path lengthsin dynamic smallworld networks with identical limit degreedistributionrdquo Journal of Statistical Physics vol 149 no 3 pp505ndash518 2012

[18] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoPseud-ofractal scale-free webrdquo Physical Review E vol 65 no 6 ArticleID 066122 4 pages 2002

[19] Y Shang ldquoMean commute time for randomwalks on hierarchi-cal scale-free networksrdquo Internet Mathematics vol 8 no 4 pp321ndash337 2012

[20] R Albert and A L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

8 The Scientific World Journal

[21] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[22] S Jung S Kim and B Kahng ldquoGeometric fractal growthmodelfor scale-free networksrdquo Physical Review E vol 65 no 5 ArticleID 056101 6 pages 2002

[23] J S Andrade Jr H J Herrmann R F S Andrade and LR Da Silva ldquoApollonian networks simultaneously scale-freesmall world euclidean space filling andwithmatching graphsrdquoPhysical Review Letters vol 94 no 1 Article ID 018702 4 pages2009

[24] A-L Barabasi E Ravasz and T Vicsek ldquoDeterministic scale-free networksrdquo Physica A vol 299 no 3-4 pp 559ndash564 2001

[25] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[26] R Pastor-Satorras A Vazquez and A Vespignani ldquoDynamicaland correlation properties of the internetrdquo Physical ReviewLetters vol 87 no 25 Article ID 258701 4 pages 2001

[27] W F de la Vega and Z Tuza ldquoGroupies in random graphsrdquoInformation Processing Letters vol 109 no 7 pp 339ndash340 2009

[28] Y Shang ldquoGroupies in random bipartite graphsrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 278ndash2832010

[29] H Bateman and A Erdelyi Higher Transcendental Functionsvol 1 McGraw-Hill New York NY USA 1953

[30] M E J NewmanD JWatts and SH Strogatz ldquoRandomgraphmodels of social networksrdquo Proceedings of theNational Academyof Sciences of the United States of America vol 99 no 1 pp2566ndash2572 2002

[31] H Ebel L-I Mielsch and S Bornholdt ldquoScale-free topology ofe-mail networksrdquo Physical Review E vol 66 no 3 Article ID035103 4 pages 2002

[32] Y Shang ldquoLack of Gromov-hyperbolicity in small-world net-worksrdquo Central European Journal of Mathematics vol 10 no 3pp 1152ndash1158 2012

[33] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

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Electrical and Computer Engineering

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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httpwwwhindawicom Volume 2014

Advances in

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International Journal of

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 3

3 Topological Properties of 119866(119898119905)

Thanks to the deterministic nature of the graphs 119866(119898 119905)in this section we will derive analytically some main topo-logical properties namely the degree distribution degreecorrelations clustering coefficient transitivity coefficientand diameter

31 Degree Distribution A fundamental quantity character-izing the structure and driving the behavior of a large networkis the probability distribution function 119875(120575) of vertex degree120575 It is the probability that a randomly chosen vertex has120575 direct neighbors It is often convenient to consider thecumulative degree distribution [17 21 22]

119875cum (120575) =

infin

sum

120575

1015840=120575

119875 (1205751015840

) (3)

which indicates the proportion of the vertices whose degreeis greater than or equal to 120575 An appealing property of thecumulative distribution is Networks with exponential degreedistribution namely 119875(120575) sim 119890

minus120572120575 also have exponentialcumulative distribution with the same exponent Indeed

119875cum (120575) =

infin

sum

120575

1015840=120575

119875 (1205751015840

) asymp

infin

sum

120575

1015840=120575

119890minus120572120575

1015840

= (119890120572

119890120572 minus 1) 119890minus120572120575

(4)

The Watts-Strogatz small-world model [1] also has an expo-nential degree distribution as we will study hereWementionthat there are some other geometric growthmodels proposedin the literature which follow another ubiquitous degreedistribution scale-free distributions see for example [18 2324]

Proposition 2 The cumulative degree distribution of 119866(119905)follows an exponential distribution 119875

119888119906119898

(120575) sim (2119898)minus(1205752119898) for

large 119905

Proof Let 120575V(119905) denote the degree of vertex V in 119866(119905) Let119905119894V be the step at which a vertex V is added to the graphFrom the construction all the vertices in the graph (excepttwo initial vertices V

0

and V1

) are always connected to twogenerating edges and will increase their degrees by 2119898 at thenext iteration

At 119905 = 0 the graph has two initial vertices V0

and V1

with degree 1 that is 120575V0

(0) = 120575V1

(0) = 1 For 119905 ge 1 byconstruction we have

120575V0

(119905) = 120575V1

(119905) = 1 + 119898 + (119905 minus 1)1198982

(5)

For other vertices we have 120575V(119905119894V) = 2 and 120575V(119905 + 1) = 120575V(119905) +

2119898 Thus

120575V (119905) = 2 (119898 (119905 minus 119905119894V) + 1) (6)

for 119905 ge 119905119894V Hence the degree distribution of the graph

119866(119905) is as follows The number of vertices of degree 2 sdot

1 2 sdot (119898 + 1) 2 sdot (2119898 + 1) 2 sdot (119898(119905 minus 1) + 1) equals119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 respectively and thedegrees of two initial vertices are given by (5)

Using (6) we have 119875cum(120575) = 119875(1199051015840

le 120591 = 119905 minus (120575 minus 2)2119898)Thus by exploiting Proposition 1 we obtain

119875cum (120575) asymp

120591

sum

119905

1015840=0

119871119881

(1199051015840

)

|119881 (119905)|

=2 (2119898 minus 1)

119898(2119898)119905

+ 3119898 minus 2

+

120591

sum

119905

1015840=1

119898(2119898)119905

1015840minus1

(2119898 minus 1)

119898(2119898)119905

+ 3119898 minus 2

=119898(2119898)

120591

+ 3119898 minus 2

119898(2119898)119905

+ 3119898 minus 2sim (2119898)

minus(1205752119898)

(7)

for large graphs (ie 119905 rarr infin)

We will make use of the exact degree distribution of 119866(119905)obtained in the above proof to study the clustering coefficientin the sequel

32 Degree Correlations (Average Neighbor Degree) Touncover correlations between the degrees of connected ver-tices the average neighbor degree 119896

119899119899

(120575) for vertices ofdegree 120575 is defined as the average degree of nearest neighborsof vertices with degree 120575 as a function of this degree value[25 26] If 119896

119899119899

(120575) is an increasing function of 120575 verticeswith high-degree have a larger probability to be connectedwith large degree vertices In this case the graph is saidto be assortative and this property is referred to in socialsciences as assortative mixing [12] Generally assortativity isthe tendency of entities to seek out and group with thoseother entities that exhibit similar characteristics In contrasta decreasing behavior of 119896

119899119899

(120575) defines a disassortative graphin the sense that high-degree vertices have a majority ofneighbors with low-degree whereas the opposite holds forlow-degree vertices In the absence of degree correlations119896119899119899

(120575) is a constantWe remark here that the concept of 119896119899119899

(120575)

is related to the groupie in graphs (see eg [27 28])

Proposition 3 The average neighbor degree for 119866(119905) isrespectively

(i)

119896119899119899

(1205750

) =1199051205750

+ 119898119905 + 21198982

119905 + 2119898 minus 119905

1 + 119898 + (119905 minus 1)1198982 (8)

where 1205750

= 1+119898+(119905minus1)1198982 (cf (5)) is the degree of two initial

vertices V0

and V1

(ii)

119896119899119899

(120575) =1198982

120575

2 (2119898 minus 1)+

119898(2119898)3

(2119898 minus 1)2

120575+

1198982

2119898 minus 1

minus119898 (119905 + 3) (2119898)

(1+(1205752))119898

(2119898)119905

120575

(9)

where 120575 = 2(119898(119905minus119905119894

)+1) (cf (6)) is the degree of other verticesadded to the network at step 119905

119894

ge 1

4 The Scientific World Journal

Proof We first show (9) It is clear that all vertices introducedat the same iteration step have the same degree No vertices(except V

0

and V1

) added to the network at the same step willbe connected to each other When a new vertex is added tothe network it connects vertices with larger degrees and itwill connect vertices with smaller degrees in the subsequentsteps From (6) for vertices introduced to the network at step119905119894

ge 1 they have the same degree 120575 = 2(119898(119905 minus 119905119894

) + 1)Let 120575(119905

119894

119905) represent the degree at step 119905 of a vertex thatwas generated at step 119905

119894

Thus 120575(119905119894

119905) = 2(119898(119905 minus 119905119894

) + 1) Wehave

119896119899119899

(120575) =1

119871119881

(119905119894

) 120575 (119905119894

119905)

times (2119898

119905

119894minus1

sum

119905

1015840

119894=1

119871119881

(1199051015840

119894

) 120575 (1199051015840

119894

119905)

+ 2119898

119905

sum

119905

1015840

119894=119905

119894+1

119871119881

(1199051015840

119894

) 120575 (1199051015840

119894

119905)

+119871119881

(0) 120575 (0 119905))

(10)

The first sum on the left-hand side of (10) accounts for theadjacencies made to vertices with larger degree namely 1 le

1199051015840

119894

lt 119905119894

and the second sum represents the edges introducedto vertices with a smaller degree at each step 119905

1015840

119894

gt 119905119894

The lastterm in (10) accounts for the adjacencies made to the initialvertices V

0

and V1

From (10) we derive that

119896119899119899

(120575) =1

(2119898)119905

119894 (119898 (119905 minus 119905119894

) + 1)

times (

2119898 (2119898 + 21198982

119905) ((2119898)119905

119894minus1

minus 1)

2119898 minus 1

minus

41198983

((2119898)119905

119894minus1

minus 1 + 2119898 (119905119894

minus 1))

2119898 minus 1

+

41198983

((2119898)119905

119894minus1

minus 1)

(2119898 minus 1)2

+

(2119898)119905

119894+1

(2119898 + 21198982

119905) ((2119898)119905minus119905

119894 minus 1)

2119898 minus 1

minus ((41198983

((119905119894

+ 1) (2119898)119905

119894 ((2119898)119905minus119905

119894 minus 1)

+ (2119898)119905

(119905 minus 119905119894

)))

times (2119898 minus 1)minus1

)

+41198983

(2119898)119905

119894+1

((2119898)119905minus119905

119894 minus 1)

(2119898 minus 1)2

+2 (1 + 119898 + (119905 minus 1)1198982

))

(11)

Feed 120575 = 2(119898(119905 minus 119905119894

) + 1) into the above expression eliminate119905119894

and simplify the consequential expression giving rise to (9)finally

Next for the two initial vertices with degree 1205750

= 1 +119898+

(119905 minus 1)1198982 we obtain

119896119899119899

(1205750

) =119898

1205750

119905

sum

119905

1015840

119894=0

120575 (1199051015840

119894

119905) =2119898 (119905 + 1) + 119898

2

119905 (119905 + 1)

1 + 119898 + (119905 minus 1)1198982 (12)

which yields to (8) as desired

Note that as 119905 tends to infinity (8) is tantamount to(1205750

+ 119898 + 21198982

minus 1)1198982 and the last term on the right-

hand side of (9) is vanishing Therefore we conclude that119896119899119899

(120575) is approximately a linear function of 120575 for large 119905which implies that our model 119866(119905) undergoes assortativegrowth

To find the impact of parameter 119898 we note that (8)decreases with119898 while (9) increases with119898 for large 119905 Sincethe contribution to the degree correlation of the two initialvertices of 119866(119905) is small we can safely think of 119896

119899119899

(120575) asan increasing function with respect to 119898 for large graphsmeaning that 119866(119898 119905) shows more significant assortativemixing for larger119898 This fact will be even clearer drawing onthe correlation coefficient (see below)

33 Degree Correlations (Pearson Correlation Coefficient)Another quantity often used to probe the assortativity is thePearson correlation coefficient 119903 of vertices connected by anedge [12 13]

119903 =|119864|sum119894

119895119894

119896119894

minus (sum119894

(12) (119895119894

+ 119896119894

))2

|119864| sum119894

(12) (1198952

119894

+ 1198962

119894

) minus (sum119894

(12) (119895119894

+ 119896119894

))2

(13)

where 119864 is the edge set of the graph in question and 119895119894

and 119896119894

are the degrees of the vertices at the ends of the 119894th edge with119894 = 1 2 |119864| It lies in the range minus1 le 119903 le 1This coefficientis zero for uncorrelated graph and positive or negative forassortative or disassortative mixing respectively Let 119903(119905) bethe degree-degree Pearson correlation coefficient of 119866(119905) Wehave the following result

The Scientific World Journal 5

Proposition 4 The Pearson correlation coefficient of 119866(119905) is

119903 (119905) = (1 + 119900 (1))

((((2119898)2

+ 1) (2119898 minus 1)) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

(((3(2119898)2

+ 1) (2 (2119898 minus 1))) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

997888rarr2(2119898)

2

+ 2 minus (9 (2119898 minus 1))

3(2119898)2

+ 1 minus (9 (2119898 minus 1))

(14)

as 119905 rarr infin

It is direct to check that (10) is positive for all 119898 ge 1 Itis an increasing function with 119898 and has upper bound 23Therefore for large 119905 the growth model 119866(119898 119905) is assortativefor all 119898 ge 1 and the level of assortativity increases with 119898This also justifies the above discussion of assortativity basedon local quantity 119896

119899119899

(120575)

Proof Following the notation in [14] we denote by ⟨119895119894

119896119894

⟩ the119894th edge in119866(119905) connecting two vertices with degree 119895

119894

and 119896119894

respectively By (5) the edge in119866(0) is thus ⟨1+119898+(119905minus1)119898

21+119898+(119905minus1)119898

2

⟩ (2119898)119905

119894 new edges are added to the networkat iteration step 119905

119894

ge 1 These edges will connect new verticesto every vertex in119866(119905

119894

minus1) whose degree distribution at 119905119894

minus1

is 120575(119897 119905119894

minus 1) = 2(119898(119905119894

minus 1 minus 119897) + 1) for 1 le 119897 le 119905119894

minus 1 and120575(0 119905119894

minus1) = 1+119898+(119905119894

minus2)1198982 Here the 120575 notation is defined

in the proof of Proposition 3At each of the subsequent steps of 119905

119894

minus 1 the degrees of allthese vertices will gain 2119898 except V

0

and V1

whose degreeswill gain119898 Consequently at iteration step 119905 ge 119905

119894

the numberof edges ⟨2119898(119905minus119905

119894

)+2 2(119898(119905119894

minus119897)+1)⟩ for 1 le 119897 le 119905119894

minus1 is (2119898)119897

and the number of edges ⟨2119898(119905 minus 119905119894

) + 2 1 +119898+ (119905 minus 1)1198982

⟩ is2119898

We now can evaluate these sums in (13) for large 119905

|119864(119905)|

sum

119894=1

119895119894

119896119894

= (1 + 119898 + (119905 minus 1)1198982

)2

+ 2119898 (1 + 119898 + (119905 minus 1)1198982

)

times

119905

sum

119905

119894=1

(2119898 (119905 minus 119905119894

) + 2)

+ 2

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2119898 (119905 minus 119905119894

) + 2)

times (119898 (119905119894

minus 1) + 1) (2119898)119897

= ((2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(15)

Likewise we have

|119864(119905)|

sum

119894=1

(119895119894

+ 119896119894

)

= 2(1 + 119898 + (119905 minus 1)1198982

)2

+

119905

sum

119905

119894=1

(2119898 (1 + 119898 + (119905 minus 1)1198982

) + 2119898 (119905 minus 119905119894

) + 2)

+

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2 (119898 (119905119894

minus 119897) + 1) + 2119898 (119905 minus 119905119894

) + 2) (2119898)119897

=3

2119898 minus 1(2119898)3+119905

+ 119900 ((2119898)119905

)

|119864(119905)|

sum

119894=1

(1198952

119894

+ 1198962

119894

) = (3(2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(16)Feeding these quantities into the definition (13) we thenarrive at the desired result

34 Clustering Coefficient The clustering coefficient [1] is agood indicator of local clustering namely the local densityof triangles and thus often used to characterize small-worldnetworks In a network 119866 = (119881 119864) the clustering coefficient119888(V) of a vertex V isin 119881 is the ratio of the total number 119890V ofedges that actually exist between all its 120575V nearest neighborsand the number 120575V(120575V minus 1)2 of all possible edges betweenthem More precisely

119888 (V) =2119890V

120575V (120575V minus 1) (17)

The clustering coefficient 119888(119866) of the whole network 119866 is theaverage of all individual 119888(V)rsquos

119888 (119866) =sumVisin119881 119888 (V)

|119881| (18)

In what follows we compute the clustering coefficient for thegrowth model 119866(119905)

Proposition 5 The clustering coefficient of 119866(119905) is119888 (119866 (119905)) = 119888 (119905)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(19)

6 The Scientific World Journal

where the function Φ represents the Lerch transcendent (see[29 Section 111])

Proof When a new vertex V is added to the graph it is easyto see 120575V = 2 and 119890V = 1 Furthermore every subsequentaddition of an edge attached to this vertex will increase bothparameters by one unit Therefore we have 119890V = 120575V minus 1 forevery vertex at every step Thus

119888 (V) =2119890V

120575V (120575V minus 1)=

2

120575V (20)

Drawing on this relationship the degree distributionobtained in Proposition 2 can be useful for calculation of theclustering coefficient of 119866(119905)

Indeed the number of vertices with clustering coefficient1 1(119898+1) 1(2119898+1) 1(119898(119905minus1)+1) 2(1+119898+(119905minus1)119898

2

)equals respectively119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 2Consequently we obtain

119888 (119866 (119905)) =1

|119881 (119905)|

times (

119905

sum

119894=1

1

(119894 minus 1)119898 + 1sdot 119898(2119898)

119905minus119894

+4

1 + 119898 + (119905 minus 1)1198982)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(21)

as desired

For large graphs (ie 119905 rarr infin) the right-hand side of (19)approaches

(2119898 minus 1) (2119898)1119898

21198982ln (2119898) (22)

which is a decreasing function with respect to 119898 Hencefor larger 119898 the level of local clustering becomes lowereventually This is not quite surprising since a large bunch ofvertices will be added to the network at each iteration when119898 and 119905 become large which mitigate the coefficient

35 Transitivity Coefficient Transitivity is an importantproperty especially in the analysis of social networks see forexample [21 30 31] Let 119879(119866) be the number of triangles and119876(119866) be the number of paths of length two in a graph119866Thenthe transitivity coefficient 1198881015840(119866) of 119866 can be defined as

1198881015840

(119866) =3119879 (119866)

119876 (119866) (23)

A brief discussion of the relationship between clustering andtransitivity coefficients can be found for example in [14]

Proposition 6 The transitivity coefficient of 119866(119905) is

1198881015840

(119866 (119905)) = 1198881015840

(119905)

= (1 + 119900 (1))31198983

((2119898)119905

minus 1)

(2119898 minus 1) ((2119898)2

+ 1) (2119898)119905+1

997888rarr31198982

2 (2119898 minus 1) ((2119898)2

+ 1)

(24)

as 119905 rarr infin

Proof We first calculate 119879(119866(119905)) Note that if the number ofgenerating edges after iteration 119905 minus 1 is 119886 the number of newtriangles introduced to the graph after iteration 119905 is 3119886 Since119886 = 119871

119864

(119905 minus 1) we obtain

119879 (119866 (119905)) = 119879 (119866 (119905 minus 1)) + 119898119871119864

(119905 minus 1)

= 119879 (119866 (119905 minus 1)) + 119898(2119898)119905minus1

(25)

which together with the initial value 119879(119866(1)) = 119898 gives

119879 (119866 (119905)) =119898 ((2119898)

119905

minus 1)

2119898 minus 1 (26)

for 119905 ge 1The number of paths of length two 119876(119866(119905)) can be

derived as follows by using the degree distribution again

119876 (119866 (119905)) = 119898(2119898)119905minus1

+ 119898

119905minus1

sum

119896=1

(2 (119896119898 + 1)

2) (2119898)

119905minus119896minus1

+ 2(1 + 119898 + (119905 minus 1)119898

2

2)

=(2119898)2

+ 1

1198982(2119898)119905+1

+ 119900 ((2119898)119905

)

(27)

which along with (26) leads to the stated result

Clearly the left-hand side of (24) is a decreasing functionof119898 Recalling the comments after Proposition 5 we see thatthe difference between clustering and transitivity coefficientsof 119866(119905) is by and large quantitative This is because theymeasure a quite similar property of networks

36 Diameter Network diameter namely the largest lengthof the shortest paths between all pairs of vertices is a mea-sure of the transmission performance and communicationefficiency We show analytically the diameter of our growthmodel and find a quantitative difference between 119898 = 1 and119898 gt 1

Proposition 7 The diameter diam(119866(119898 119905)) = diam(119905) of119866(119898 119905) equals 119905 for119898 = 1 and 119905 + 1 for119898 ge 2

The Scientific World Journal 7

Proof The case of 119898 = 1 was shown in [14] In what followswe take over their method to study119898 ge 2

Clearly diam(119866(119898 0)) = 1 and diam(119866(119898 1)) = 2At each step 119905 ge 2 the longest distance between twovertices is for some vertices added at this step correspondingto different generating edges at the last step Consider twovertices introduced at step 119905 ge 2 corresponding to differentgenerating edges say 119906

119905

and V119905

The vertex 119906119905

is adjacent totwo vertices and one of them must have been added to thegraph at step 119905 minus 2 or earlier

If 119905 = 2119896 is even 119906119905

can reach some vertex in 119866(119898 0) by119896 jumps and the same thing is true for vertex V

119905

Thereforediam(119866(119898 2119896)) le 2119896 + 1 If 119905 = 2119896 + 1 is odd 119906

119905

can reachsome vertex in119866(119898 1) by 119896 jumps and the same thing is truefor vertex V

119905

Therefore diam(119866(119898 2119896 + 1)) le 2119896 + 2 Thesebounds are attained by pairs of vertices 119906

119905

and V119905

created atiteration 119905 which correspond to different generating edgesand have the property of being connected to two verticesintroduced at steps 119905minus1 and 119905minus2 respectively Consequentlywe have diam(119866(119898 119905)) = 119905 + 1 for all119898 ge 2 and 119905 ge 1

From Proposition 1 we have for 119905 large

ln |119881 (119905)| = ln(119898(2119898)119905

+ 3119898 minus 2

2119898 minus 1) sim 119905 ln (2119898) (28)

Hence we obtain the logarithmic scale

diam (119905) simln |119881 (119905)|

ln (2119898) (29)

which together with high clustering (Propositions 5 and 6)justifies the small-world characteristics [1] of our growthmodel

4 Conclusion

We have studied a geometric assortative growth model119866(119898 119905) for small-world networks in a deterministic wayWe obtain analytical solutions of main properties of themodel such as the degree distribution and correlationsclustering and transitivity coefficients and graph diameter inthe full spectrum of parameter 119898 The 119866(119898 119905) model holdsboth tunable small-world and tunable assortative mixingbehaviors This should be useful to guide the research anddevelopment of varied social networks On the other handthe deterministic character of this graph family shouldfacilitate the exact calculation of other network-orientedquantities including average path length hyperbolicity [32]modular structure and motifs [33]

The introduction of tunable parameter 119898 also brings arange of open questions for future research In addition tothose mentioned before here are more examples how canwemake a trade-off between local clustering and assortativityby tuning119898 since they have opposite monotonicity What if119898 = 119898(119905) is a function of time

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[2] L A N Amara A Scala M Barthelemy and H E StanleyldquoClasses of small-world networksrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 97 no21 pp 11149ndash11152 2000

[3] S Jespersen and A Blumen ldquoSmall-world networks links withlong-tailed distributionsrdquo Physical Review E vol 62 no 5 pp6270ndash6274 2000

[4] M Kuperman and G Abramson ldquoComplex structures ingeneralized small worldsrdquo Physical Review E vol 64 no 4Article ID 047103 4 pages 2001

[5] M E J Newman ldquoModels of the small worldrdquo Journal ofStatistical Physics vol 101 no 3-4 pp 819ndash841 2000

[6] Y Shang ldquoFast distributed consensus seeking in large-scalesensor networks via shortcutsrdquo International Journal of Compu-tational Science and Engineering vol 7 no 2 pp 121ndash124 2012

[7] Y Shang ldquoA sharp threshold for rainbow connection in small-world networksrdquoMiskolc Mathematical Notes vol 13 no 2 pp493ndash497 2012

[8] F Comellas and M Sampels ldquoDeterministic small-world net-worksrdquo Physica A vol 309 no 1-2 pp 231ndash235 2002

[9] W Xiao and B Parhami ldquoCayley graphs as models of determin-istic small-world networksrdquo Information Processing Letters vol97 no 3 pp 115ndash117 2006

[10] S Boettcher B Goncalves and H Guclu ldquoHierarchical regularsmall-world networksrdquo Journal of Physics A vol 41 no 25Article ID 252001 2008

[11] Y Zhang Z Zhang S Zhou and J Guan ldquoDeterministicweighted scale-free small-world networksrdquo Physica A vol 389no 16 pp 3316ndash3324 2010

[12] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview Letters vol 89 no 20 Article ID 208701 4 pages 2002

[13] M E J Newman ldquoMixing patterns in networksrdquo PhysicalReview E vol 67 no 2 Article ID 026126 13 pages 2003

[14] Z Zhang and F Comellas ldquoFarey graphs as models for complexnetworksrdquo Theoretical Computer Science vol 412 no 8ndash10 pp865ndash875 2011

[15] C J Colbourn ldquoFarey series and maximal outerplanar graphsrdquoSIAM Journal on Algebraic Discrete Methods vol 3 no 2 pp187ndash189 1982

[16] Z Zhang S Zhou Z Wang and Z Shen ldquoA geometricgrowth model interpolating between regular and small-worldnetworksrdquo Journal of Physics A vol 40 no 39 pp 11863ndash118762007

[17] Y Shang ldquoDistinct clusterings and characteristic path lengthsin dynamic smallworld networks with identical limit degreedistributionrdquo Journal of Statistical Physics vol 149 no 3 pp505ndash518 2012

[18] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoPseud-ofractal scale-free webrdquo Physical Review E vol 65 no 6 ArticleID 066122 4 pages 2002

[19] Y Shang ldquoMean commute time for randomwalks on hierarchi-cal scale-free networksrdquo Internet Mathematics vol 8 no 4 pp321ndash337 2012

[20] R Albert and A L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

8 The Scientific World Journal

[21] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[22] S Jung S Kim and B Kahng ldquoGeometric fractal growthmodelfor scale-free networksrdquo Physical Review E vol 65 no 5 ArticleID 056101 6 pages 2002

[23] J S Andrade Jr H J Herrmann R F S Andrade and LR Da Silva ldquoApollonian networks simultaneously scale-freesmall world euclidean space filling andwithmatching graphsrdquoPhysical Review Letters vol 94 no 1 Article ID 018702 4 pages2009

[24] A-L Barabasi E Ravasz and T Vicsek ldquoDeterministic scale-free networksrdquo Physica A vol 299 no 3-4 pp 559ndash564 2001

[25] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[26] R Pastor-Satorras A Vazquez and A Vespignani ldquoDynamicaland correlation properties of the internetrdquo Physical ReviewLetters vol 87 no 25 Article ID 258701 4 pages 2001

[27] W F de la Vega and Z Tuza ldquoGroupies in random graphsrdquoInformation Processing Letters vol 109 no 7 pp 339ndash340 2009

[28] Y Shang ldquoGroupies in random bipartite graphsrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 278ndash2832010

[29] H Bateman and A Erdelyi Higher Transcendental Functionsvol 1 McGraw-Hill New York NY USA 1953

[30] M E J NewmanD JWatts and SH Strogatz ldquoRandomgraphmodels of social networksrdquo Proceedings of theNational Academyof Sciences of the United States of America vol 99 no 1 pp2566ndash2572 2002

[31] H Ebel L-I Mielsch and S Bornholdt ldquoScale-free topology ofe-mail networksrdquo Physical Review E vol 66 no 3 Article ID035103 4 pages 2002

[32] Y Shang ldquoLack of Gromov-hyperbolicity in small-world net-worksrdquo Central European Journal of Mathematics vol 10 no 3pp 1152ndash1158 2012

[33] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

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Advances in

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International Journal of

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 The Scientific World Journal

Proof We first show (9) It is clear that all vertices introducedat the same iteration step have the same degree No vertices(except V

0

and V1

) added to the network at the same step willbe connected to each other When a new vertex is added tothe network it connects vertices with larger degrees and itwill connect vertices with smaller degrees in the subsequentsteps From (6) for vertices introduced to the network at step119905119894

ge 1 they have the same degree 120575 = 2(119898(119905 minus 119905119894

) + 1)Let 120575(119905

119894

119905) represent the degree at step 119905 of a vertex thatwas generated at step 119905

119894

Thus 120575(119905119894

119905) = 2(119898(119905 minus 119905119894

) + 1) Wehave

119896119899119899

(120575) =1

119871119881

(119905119894

) 120575 (119905119894

119905)

times (2119898

119905

119894minus1

sum

119905

1015840

119894=1

119871119881

(1199051015840

119894

) 120575 (1199051015840

119894

119905)

+ 2119898

119905

sum

119905

1015840

119894=119905

119894+1

119871119881

(1199051015840

119894

) 120575 (1199051015840

119894

119905)

+119871119881

(0) 120575 (0 119905))

(10)

The first sum on the left-hand side of (10) accounts for theadjacencies made to vertices with larger degree namely 1 le

1199051015840

119894

lt 119905119894

and the second sum represents the edges introducedto vertices with a smaller degree at each step 119905

1015840

119894

gt 119905119894

The lastterm in (10) accounts for the adjacencies made to the initialvertices V

0

and V1

From (10) we derive that

119896119899119899

(120575) =1

(2119898)119905

119894 (119898 (119905 minus 119905119894

) + 1)

times (

2119898 (2119898 + 21198982

119905) ((2119898)119905

119894minus1

minus 1)

2119898 minus 1

minus

41198983

((2119898)119905

119894minus1

minus 1 + 2119898 (119905119894

minus 1))

2119898 minus 1

+

41198983

((2119898)119905

119894minus1

minus 1)

(2119898 minus 1)2

+

(2119898)119905

119894+1

(2119898 + 21198982

119905) ((2119898)119905minus119905

119894 minus 1)

2119898 minus 1

minus ((41198983

((119905119894

+ 1) (2119898)119905

119894 ((2119898)119905minus119905

119894 minus 1)

+ (2119898)119905

(119905 minus 119905119894

)))

times (2119898 minus 1)minus1

)

+41198983

(2119898)119905

119894+1

((2119898)119905minus119905

119894 minus 1)

(2119898 minus 1)2

+2 (1 + 119898 + (119905 minus 1)1198982

))

(11)

Feed 120575 = 2(119898(119905 minus 119905119894

) + 1) into the above expression eliminate119905119894

and simplify the consequential expression giving rise to (9)finally

Next for the two initial vertices with degree 1205750

= 1 +119898+

(119905 minus 1)1198982 we obtain

119896119899119899

(1205750

) =119898

1205750

119905

sum

119905

1015840

119894=0

120575 (1199051015840

119894

119905) =2119898 (119905 + 1) + 119898

2

119905 (119905 + 1)

1 + 119898 + (119905 minus 1)1198982 (12)

which yields to (8) as desired

Note that as 119905 tends to infinity (8) is tantamount to(1205750

+ 119898 + 21198982

minus 1)1198982 and the last term on the right-

hand side of (9) is vanishing Therefore we conclude that119896119899119899

(120575) is approximately a linear function of 120575 for large 119905which implies that our model 119866(119905) undergoes assortativegrowth

To find the impact of parameter 119898 we note that (8)decreases with119898 while (9) increases with119898 for large 119905 Sincethe contribution to the degree correlation of the two initialvertices of 119866(119905) is small we can safely think of 119896

119899119899

(120575) asan increasing function with respect to 119898 for large graphsmeaning that 119866(119898 119905) shows more significant assortativemixing for larger119898 This fact will be even clearer drawing onthe correlation coefficient (see below)

33 Degree Correlations (Pearson Correlation Coefficient)Another quantity often used to probe the assortativity is thePearson correlation coefficient 119903 of vertices connected by anedge [12 13]

119903 =|119864|sum119894

119895119894

119896119894

minus (sum119894

(12) (119895119894

+ 119896119894

))2

|119864| sum119894

(12) (1198952

119894

+ 1198962

119894

) minus (sum119894

(12) (119895119894

+ 119896119894

))2

(13)

where 119864 is the edge set of the graph in question and 119895119894

and 119896119894

are the degrees of the vertices at the ends of the 119894th edge with119894 = 1 2 |119864| It lies in the range minus1 le 119903 le 1This coefficientis zero for uncorrelated graph and positive or negative forassortative or disassortative mixing respectively Let 119903(119905) bethe degree-degree Pearson correlation coefficient of 119866(119905) Wehave the following result

The Scientific World Journal 5

Proposition 4 The Pearson correlation coefficient of 119866(119905) is

119903 (119905) = (1 + 119900 (1))

((((2119898)2

+ 1) (2119898 minus 1)) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

(((3(2119898)2

+ 1) (2 (2119898 minus 1))) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

997888rarr2(2119898)

2

+ 2 minus (9 (2119898 minus 1))

3(2119898)2

+ 1 minus (9 (2119898 minus 1))

(14)

as 119905 rarr infin

It is direct to check that (10) is positive for all 119898 ge 1 Itis an increasing function with 119898 and has upper bound 23Therefore for large 119905 the growth model 119866(119898 119905) is assortativefor all 119898 ge 1 and the level of assortativity increases with 119898This also justifies the above discussion of assortativity basedon local quantity 119896

119899119899

(120575)

Proof Following the notation in [14] we denote by ⟨119895119894

119896119894

⟩ the119894th edge in119866(119905) connecting two vertices with degree 119895

119894

and 119896119894

respectively By (5) the edge in119866(0) is thus ⟨1+119898+(119905minus1)119898

21+119898+(119905minus1)119898

2

⟩ (2119898)119905

119894 new edges are added to the networkat iteration step 119905

119894

ge 1 These edges will connect new verticesto every vertex in119866(119905

119894

minus1) whose degree distribution at 119905119894

minus1

is 120575(119897 119905119894

minus 1) = 2(119898(119905119894

minus 1 minus 119897) + 1) for 1 le 119897 le 119905119894

minus 1 and120575(0 119905119894

minus1) = 1+119898+(119905119894

minus2)1198982 Here the 120575 notation is defined

in the proof of Proposition 3At each of the subsequent steps of 119905

119894

minus 1 the degrees of allthese vertices will gain 2119898 except V

0

and V1

whose degreeswill gain119898 Consequently at iteration step 119905 ge 119905

119894

the numberof edges ⟨2119898(119905minus119905

119894

)+2 2(119898(119905119894

minus119897)+1)⟩ for 1 le 119897 le 119905119894

minus1 is (2119898)119897

and the number of edges ⟨2119898(119905 minus 119905119894

) + 2 1 +119898+ (119905 minus 1)1198982

⟩ is2119898

We now can evaluate these sums in (13) for large 119905

|119864(119905)|

sum

119894=1

119895119894

119896119894

= (1 + 119898 + (119905 minus 1)1198982

)2

+ 2119898 (1 + 119898 + (119905 minus 1)1198982

)

times

119905

sum

119905

119894=1

(2119898 (119905 minus 119905119894

) + 2)

+ 2

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2119898 (119905 minus 119905119894

) + 2)

times (119898 (119905119894

minus 1) + 1) (2119898)119897

= ((2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(15)

Likewise we have

|119864(119905)|

sum

119894=1

(119895119894

+ 119896119894

)

= 2(1 + 119898 + (119905 minus 1)1198982

)2

+

119905

sum

119905

119894=1

(2119898 (1 + 119898 + (119905 minus 1)1198982

) + 2119898 (119905 minus 119905119894

) + 2)

+

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2 (119898 (119905119894

minus 119897) + 1) + 2119898 (119905 minus 119905119894

) + 2) (2119898)119897

=3

2119898 minus 1(2119898)3+119905

+ 119900 ((2119898)119905

)

|119864(119905)|

sum

119894=1

(1198952

119894

+ 1198962

119894

) = (3(2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(16)Feeding these quantities into the definition (13) we thenarrive at the desired result

34 Clustering Coefficient The clustering coefficient [1] is agood indicator of local clustering namely the local densityof triangles and thus often used to characterize small-worldnetworks In a network 119866 = (119881 119864) the clustering coefficient119888(V) of a vertex V isin 119881 is the ratio of the total number 119890V ofedges that actually exist between all its 120575V nearest neighborsand the number 120575V(120575V minus 1)2 of all possible edges betweenthem More precisely

119888 (V) =2119890V

120575V (120575V minus 1) (17)

The clustering coefficient 119888(119866) of the whole network 119866 is theaverage of all individual 119888(V)rsquos

119888 (119866) =sumVisin119881 119888 (V)

|119881| (18)

In what follows we compute the clustering coefficient for thegrowth model 119866(119905)

Proposition 5 The clustering coefficient of 119866(119905) is119888 (119866 (119905)) = 119888 (119905)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(19)

6 The Scientific World Journal

where the function Φ represents the Lerch transcendent (see[29 Section 111])

Proof When a new vertex V is added to the graph it is easyto see 120575V = 2 and 119890V = 1 Furthermore every subsequentaddition of an edge attached to this vertex will increase bothparameters by one unit Therefore we have 119890V = 120575V minus 1 forevery vertex at every step Thus

119888 (V) =2119890V

120575V (120575V minus 1)=

2

120575V (20)

Drawing on this relationship the degree distributionobtained in Proposition 2 can be useful for calculation of theclustering coefficient of 119866(119905)

Indeed the number of vertices with clustering coefficient1 1(119898+1) 1(2119898+1) 1(119898(119905minus1)+1) 2(1+119898+(119905minus1)119898

2

)equals respectively119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 2Consequently we obtain

119888 (119866 (119905)) =1

|119881 (119905)|

times (

119905

sum

119894=1

1

(119894 minus 1)119898 + 1sdot 119898(2119898)

119905minus119894

+4

1 + 119898 + (119905 minus 1)1198982)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(21)

as desired

For large graphs (ie 119905 rarr infin) the right-hand side of (19)approaches

(2119898 minus 1) (2119898)1119898

21198982ln (2119898) (22)

which is a decreasing function with respect to 119898 Hencefor larger 119898 the level of local clustering becomes lowereventually This is not quite surprising since a large bunch ofvertices will be added to the network at each iteration when119898 and 119905 become large which mitigate the coefficient

35 Transitivity Coefficient Transitivity is an importantproperty especially in the analysis of social networks see forexample [21 30 31] Let 119879(119866) be the number of triangles and119876(119866) be the number of paths of length two in a graph119866Thenthe transitivity coefficient 1198881015840(119866) of 119866 can be defined as

1198881015840

(119866) =3119879 (119866)

119876 (119866) (23)

A brief discussion of the relationship between clustering andtransitivity coefficients can be found for example in [14]

Proposition 6 The transitivity coefficient of 119866(119905) is

1198881015840

(119866 (119905)) = 1198881015840

(119905)

= (1 + 119900 (1))31198983

((2119898)119905

minus 1)

(2119898 minus 1) ((2119898)2

+ 1) (2119898)119905+1

997888rarr31198982

2 (2119898 minus 1) ((2119898)2

+ 1)

(24)

as 119905 rarr infin

Proof We first calculate 119879(119866(119905)) Note that if the number ofgenerating edges after iteration 119905 minus 1 is 119886 the number of newtriangles introduced to the graph after iteration 119905 is 3119886 Since119886 = 119871

119864

(119905 minus 1) we obtain

119879 (119866 (119905)) = 119879 (119866 (119905 minus 1)) + 119898119871119864

(119905 minus 1)

= 119879 (119866 (119905 minus 1)) + 119898(2119898)119905minus1

(25)

which together with the initial value 119879(119866(1)) = 119898 gives

119879 (119866 (119905)) =119898 ((2119898)

119905

minus 1)

2119898 minus 1 (26)

for 119905 ge 1The number of paths of length two 119876(119866(119905)) can be

derived as follows by using the degree distribution again

119876 (119866 (119905)) = 119898(2119898)119905minus1

+ 119898

119905minus1

sum

119896=1

(2 (119896119898 + 1)

2) (2119898)

119905minus119896minus1

+ 2(1 + 119898 + (119905 minus 1)119898

2

2)

=(2119898)2

+ 1

1198982(2119898)119905+1

+ 119900 ((2119898)119905

)

(27)

which along with (26) leads to the stated result

Clearly the left-hand side of (24) is a decreasing functionof119898 Recalling the comments after Proposition 5 we see thatthe difference between clustering and transitivity coefficientsof 119866(119905) is by and large quantitative This is because theymeasure a quite similar property of networks

36 Diameter Network diameter namely the largest lengthof the shortest paths between all pairs of vertices is a mea-sure of the transmission performance and communicationefficiency We show analytically the diameter of our growthmodel and find a quantitative difference between 119898 = 1 and119898 gt 1

Proposition 7 The diameter diam(119866(119898 119905)) = diam(119905) of119866(119898 119905) equals 119905 for119898 = 1 and 119905 + 1 for119898 ge 2

The Scientific World Journal 7

Proof The case of 119898 = 1 was shown in [14] In what followswe take over their method to study119898 ge 2

Clearly diam(119866(119898 0)) = 1 and diam(119866(119898 1)) = 2At each step 119905 ge 2 the longest distance between twovertices is for some vertices added at this step correspondingto different generating edges at the last step Consider twovertices introduced at step 119905 ge 2 corresponding to differentgenerating edges say 119906

119905

and V119905

The vertex 119906119905

is adjacent totwo vertices and one of them must have been added to thegraph at step 119905 minus 2 or earlier

If 119905 = 2119896 is even 119906119905

can reach some vertex in 119866(119898 0) by119896 jumps and the same thing is true for vertex V

119905

Thereforediam(119866(119898 2119896)) le 2119896 + 1 If 119905 = 2119896 + 1 is odd 119906

119905

can reachsome vertex in119866(119898 1) by 119896 jumps and the same thing is truefor vertex V

119905

Therefore diam(119866(119898 2119896 + 1)) le 2119896 + 2 Thesebounds are attained by pairs of vertices 119906

119905

and V119905

created atiteration 119905 which correspond to different generating edgesand have the property of being connected to two verticesintroduced at steps 119905minus1 and 119905minus2 respectively Consequentlywe have diam(119866(119898 119905)) = 119905 + 1 for all119898 ge 2 and 119905 ge 1

From Proposition 1 we have for 119905 large

ln |119881 (119905)| = ln(119898(2119898)119905

+ 3119898 minus 2

2119898 minus 1) sim 119905 ln (2119898) (28)

Hence we obtain the logarithmic scale

diam (119905) simln |119881 (119905)|

ln (2119898) (29)

which together with high clustering (Propositions 5 and 6)justifies the small-world characteristics [1] of our growthmodel

4 Conclusion

We have studied a geometric assortative growth model119866(119898 119905) for small-world networks in a deterministic wayWe obtain analytical solutions of main properties of themodel such as the degree distribution and correlationsclustering and transitivity coefficients and graph diameter inthe full spectrum of parameter 119898 The 119866(119898 119905) model holdsboth tunable small-world and tunable assortative mixingbehaviors This should be useful to guide the research anddevelopment of varied social networks On the other handthe deterministic character of this graph family shouldfacilitate the exact calculation of other network-orientedquantities including average path length hyperbolicity [32]modular structure and motifs [33]

The introduction of tunable parameter 119898 also brings arange of open questions for future research In addition tothose mentioned before here are more examples how canwemake a trade-off between local clustering and assortativityby tuning119898 since they have opposite monotonicity What if119898 = 119898(119905) is a function of time

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[2] L A N Amara A Scala M Barthelemy and H E StanleyldquoClasses of small-world networksrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 97 no21 pp 11149ndash11152 2000

[3] S Jespersen and A Blumen ldquoSmall-world networks links withlong-tailed distributionsrdquo Physical Review E vol 62 no 5 pp6270ndash6274 2000

[4] M Kuperman and G Abramson ldquoComplex structures ingeneralized small worldsrdquo Physical Review E vol 64 no 4Article ID 047103 4 pages 2001

[5] M E J Newman ldquoModels of the small worldrdquo Journal ofStatistical Physics vol 101 no 3-4 pp 819ndash841 2000

[6] Y Shang ldquoFast distributed consensus seeking in large-scalesensor networks via shortcutsrdquo International Journal of Compu-tational Science and Engineering vol 7 no 2 pp 121ndash124 2012

[7] Y Shang ldquoA sharp threshold for rainbow connection in small-world networksrdquoMiskolc Mathematical Notes vol 13 no 2 pp493ndash497 2012

[8] F Comellas and M Sampels ldquoDeterministic small-world net-worksrdquo Physica A vol 309 no 1-2 pp 231ndash235 2002

[9] W Xiao and B Parhami ldquoCayley graphs as models of determin-istic small-world networksrdquo Information Processing Letters vol97 no 3 pp 115ndash117 2006

[10] S Boettcher B Goncalves and H Guclu ldquoHierarchical regularsmall-world networksrdquo Journal of Physics A vol 41 no 25Article ID 252001 2008

[11] Y Zhang Z Zhang S Zhou and J Guan ldquoDeterministicweighted scale-free small-world networksrdquo Physica A vol 389no 16 pp 3316ndash3324 2010

[12] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview Letters vol 89 no 20 Article ID 208701 4 pages 2002

[13] M E J Newman ldquoMixing patterns in networksrdquo PhysicalReview E vol 67 no 2 Article ID 026126 13 pages 2003

[14] Z Zhang and F Comellas ldquoFarey graphs as models for complexnetworksrdquo Theoretical Computer Science vol 412 no 8ndash10 pp865ndash875 2011

[15] C J Colbourn ldquoFarey series and maximal outerplanar graphsrdquoSIAM Journal on Algebraic Discrete Methods vol 3 no 2 pp187ndash189 1982

[16] Z Zhang S Zhou Z Wang and Z Shen ldquoA geometricgrowth model interpolating between regular and small-worldnetworksrdquo Journal of Physics A vol 40 no 39 pp 11863ndash118762007

[17] Y Shang ldquoDistinct clusterings and characteristic path lengthsin dynamic smallworld networks with identical limit degreedistributionrdquo Journal of Statistical Physics vol 149 no 3 pp505ndash518 2012

[18] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoPseud-ofractal scale-free webrdquo Physical Review E vol 65 no 6 ArticleID 066122 4 pages 2002

[19] Y Shang ldquoMean commute time for randomwalks on hierarchi-cal scale-free networksrdquo Internet Mathematics vol 8 no 4 pp321ndash337 2012

[20] R Albert and A L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

8 The Scientific World Journal

[21] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[22] S Jung S Kim and B Kahng ldquoGeometric fractal growthmodelfor scale-free networksrdquo Physical Review E vol 65 no 5 ArticleID 056101 6 pages 2002

[23] J S Andrade Jr H J Herrmann R F S Andrade and LR Da Silva ldquoApollonian networks simultaneously scale-freesmall world euclidean space filling andwithmatching graphsrdquoPhysical Review Letters vol 94 no 1 Article ID 018702 4 pages2009

[24] A-L Barabasi E Ravasz and T Vicsek ldquoDeterministic scale-free networksrdquo Physica A vol 299 no 3-4 pp 559ndash564 2001

[25] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[26] R Pastor-Satorras A Vazquez and A Vespignani ldquoDynamicaland correlation properties of the internetrdquo Physical ReviewLetters vol 87 no 25 Article ID 258701 4 pages 2001

[27] W F de la Vega and Z Tuza ldquoGroupies in random graphsrdquoInformation Processing Letters vol 109 no 7 pp 339ndash340 2009

[28] Y Shang ldquoGroupies in random bipartite graphsrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 278ndash2832010

[29] H Bateman and A Erdelyi Higher Transcendental Functionsvol 1 McGraw-Hill New York NY USA 1953

[30] M E J NewmanD JWatts and SH Strogatz ldquoRandomgraphmodels of social networksrdquo Proceedings of theNational Academyof Sciences of the United States of America vol 99 no 1 pp2566ndash2572 2002

[31] H Ebel L-I Mielsch and S Bornholdt ldquoScale-free topology ofe-mail networksrdquo Physical Review E vol 66 no 3 Article ID035103 4 pages 2002

[32] Y Shang ldquoLack of Gromov-hyperbolicity in small-world net-worksrdquo Central European Journal of Mathematics vol 10 no 3pp 1152ndash1158 2012

[33] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

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Distributed Sensor Networks

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Volume 2014

International Journal of

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 5

Proposition 4 The Pearson correlation coefficient of 119866(119905) is

119903 (119905) = (1 + 119900 (1))

((((2119898)2

+ 1) (2119898 minus 1)) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

(((3(2119898)2

+ 1) (2 (2119898 minus 1))) (2119898)5+2119905

) minus ((9 (4(2119898 minus 1)2

)) (2119898)6+2119905

)

997888rarr2(2119898)

2

+ 2 minus (9 (2119898 minus 1))

3(2119898)2

+ 1 minus (9 (2119898 minus 1))

(14)

as 119905 rarr infin

It is direct to check that (10) is positive for all 119898 ge 1 Itis an increasing function with 119898 and has upper bound 23Therefore for large 119905 the growth model 119866(119898 119905) is assortativefor all 119898 ge 1 and the level of assortativity increases with 119898This also justifies the above discussion of assortativity basedon local quantity 119896

119899119899

(120575)

Proof Following the notation in [14] we denote by ⟨119895119894

119896119894

⟩ the119894th edge in119866(119905) connecting two vertices with degree 119895

119894

and 119896119894

respectively By (5) the edge in119866(0) is thus ⟨1+119898+(119905minus1)119898

21+119898+(119905minus1)119898

2

⟩ (2119898)119905

119894 new edges are added to the networkat iteration step 119905

119894

ge 1 These edges will connect new verticesto every vertex in119866(119905

119894

minus1) whose degree distribution at 119905119894

minus1

is 120575(119897 119905119894

minus 1) = 2(119898(119905119894

minus 1 minus 119897) + 1) for 1 le 119897 le 119905119894

minus 1 and120575(0 119905119894

minus1) = 1+119898+(119905119894

minus2)1198982 Here the 120575 notation is defined

in the proof of Proposition 3At each of the subsequent steps of 119905

119894

minus 1 the degrees of allthese vertices will gain 2119898 except V

0

and V1

whose degreeswill gain119898 Consequently at iteration step 119905 ge 119905

119894

the numberof edges ⟨2119898(119905minus119905

119894

)+2 2(119898(119905119894

minus119897)+1)⟩ for 1 le 119897 le 119905119894

minus1 is (2119898)119897

and the number of edges ⟨2119898(119905 minus 119905119894

) + 2 1 +119898+ (119905 minus 1)1198982

⟩ is2119898

We now can evaluate these sums in (13) for large 119905

|119864(119905)|

sum

119894=1

119895119894

119896119894

= (1 + 119898 + (119905 minus 1)1198982

)2

+ 2119898 (1 + 119898 + (119905 minus 1)1198982

)

times

119905

sum

119905

119894=1

(2119898 (119905 minus 119905119894

) + 2)

+ 2

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2119898 (119905 minus 119905119894

) + 2)

times (119898 (119905119894

minus 1) + 1) (2119898)119897

= ((2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(15)

Likewise we have

|119864(119905)|

sum

119894=1

(119895119894

+ 119896119894

)

= 2(1 + 119898 + (119905 minus 1)1198982

)2

+

119905

sum

119905

119894=1

(2119898 (1 + 119898 + (119905 minus 1)1198982

) + 2119898 (119905 minus 119905119894

) + 2)

+

119905

sum

119905

119894=2

119905

119894minus1

sum

119897=1

(2 (119898 (119905119894

minus 119897) + 1) + 2119898 (119905 minus 119905119894

) + 2) (2119898)119897

=3

2119898 minus 1(2119898)3+119905

+ 119900 ((2119898)119905

)

|119864(119905)|

sum

119894=1

(1198952

119894

+ 1198962

119894

) = (3(2119898)2

+ 1) (2119898)4+119905

+ 119900 ((2119898)119905

)

(16)Feeding these quantities into the definition (13) we thenarrive at the desired result

34 Clustering Coefficient The clustering coefficient [1] is agood indicator of local clustering namely the local densityof triangles and thus often used to characterize small-worldnetworks In a network 119866 = (119881 119864) the clustering coefficient119888(V) of a vertex V isin 119881 is the ratio of the total number 119890V ofedges that actually exist between all its 120575V nearest neighborsand the number 120575V(120575V minus 1)2 of all possible edges betweenthem More precisely

119888 (V) =2119890V

120575V (120575V minus 1) (17)

The clustering coefficient 119888(119866) of the whole network 119866 is theaverage of all individual 119888(V)rsquos

119888 (119866) =sumVisin119881 119888 (V)

|119881| (18)

In what follows we compute the clustering coefficient for thegrowth model 119866(119905)

Proposition 5 The clustering coefficient of 119866(119905) is119888 (119866 (119905)) = 119888 (119905)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(19)

6 The Scientific World Journal

where the function Φ represents the Lerch transcendent (see[29 Section 111])

Proof When a new vertex V is added to the graph it is easyto see 120575V = 2 and 119890V = 1 Furthermore every subsequentaddition of an edge attached to this vertex will increase bothparameters by one unit Therefore we have 119890V = 120575V minus 1 forevery vertex at every step Thus

119888 (V) =2119890V

120575V (120575V minus 1)=

2

120575V (20)

Drawing on this relationship the degree distributionobtained in Proposition 2 can be useful for calculation of theclustering coefficient of 119866(119905)

Indeed the number of vertices with clustering coefficient1 1(119898+1) 1(2119898+1) 1(119898(119905minus1)+1) 2(1+119898+(119905minus1)119898

2

)equals respectively119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 2Consequently we obtain

119888 (119866 (119905)) =1

|119881 (119905)|

times (

119905

sum

119894=1

1

(119894 minus 1)119898 + 1sdot 119898(2119898)

119905minus119894

+4

1 + 119898 + (119905 minus 1)1198982)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(21)

as desired

For large graphs (ie 119905 rarr infin) the right-hand side of (19)approaches

(2119898 minus 1) (2119898)1119898

21198982ln (2119898) (22)

which is a decreasing function with respect to 119898 Hencefor larger 119898 the level of local clustering becomes lowereventually This is not quite surprising since a large bunch ofvertices will be added to the network at each iteration when119898 and 119905 become large which mitigate the coefficient

35 Transitivity Coefficient Transitivity is an importantproperty especially in the analysis of social networks see forexample [21 30 31] Let 119879(119866) be the number of triangles and119876(119866) be the number of paths of length two in a graph119866Thenthe transitivity coefficient 1198881015840(119866) of 119866 can be defined as

1198881015840

(119866) =3119879 (119866)

119876 (119866) (23)

A brief discussion of the relationship between clustering andtransitivity coefficients can be found for example in [14]

Proposition 6 The transitivity coefficient of 119866(119905) is

1198881015840

(119866 (119905)) = 1198881015840

(119905)

= (1 + 119900 (1))31198983

((2119898)119905

minus 1)

(2119898 minus 1) ((2119898)2

+ 1) (2119898)119905+1

997888rarr31198982

2 (2119898 minus 1) ((2119898)2

+ 1)

(24)

as 119905 rarr infin

Proof We first calculate 119879(119866(119905)) Note that if the number ofgenerating edges after iteration 119905 minus 1 is 119886 the number of newtriangles introduced to the graph after iteration 119905 is 3119886 Since119886 = 119871

119864

(119905 minus 1) we obtain

119879 (119866 (119905)) = 119879 (119866 (119905 minus 1)) + 119898119871119864

(119905 minus 1)

= 119879 (119866 (119905 minus 1)) + 119898(2119898)119905minus1

(25)

which together with the initial value 119879(119866(1)) = 119898 gives

119879 (119866 (119905)) =119898 ((2119898)

119905

minus 1)

2119898 minus 1 (26)

for 119905 ge 1The number of paths of length two 119876(119866(119905)) can be

derived as follows by using the degree distribution again

119876 (119866 (119905)) = 119898(2119898)119905minus1

+ 119898

119905minus1

sum

119896=1

(2 (119896119898 + 1)

2) (2119898)

119905minus119896minus1

+ 2(1 + 119898 + (119905 minus 1)119898

2

2)

=(2119898)2

+ 1

1198982(2119898)119905+1

+ 119900 ((2119898)119905

)

(27)

which along with (26) leads to the stated result

Clearly the left-hand side of (24) is a decreasing functionof119898 Recalling the comments after Proposition 5 we see thatthe difference between clustering and transitivity coefficientsof 119866(119905) is by and large quantitative This is because theymeasure a quite similar property of networks

36 Diameter Network diameter namely the largest lengthof the shortest paths between all pairs of vertices is a mea-sure of the transmission performance and communicationefficiency We show analytically the diameter of our growthmodel and find a quantitative difference between 119898 = 1 and119898 gt 1

Proposition 7 The diameter diam(119866(119898 119905)) = diam(119905) of119866(119898 119905) equals 119905 for119898 = 1 and 119905 + 1 for119898 ge 2

The Scientific World Journal 7

Proof The case of 119898 = 1 was shown in [14] In what followswe take over their method to study119898 ge 2

Clearly diam(119866(119898 0)) = 1 and diam(119866(119898 1)) = 2At each step 119905 ge 2 the longest distance between twovertices is for some vertices added at this step correspondingto different generating edges at the last step Consider twovertices introduced at step 119905 ge 2 corresponding to differentgenerating edges say 119906

119905

and V119905

The vertex 119906119905

is adjacent totwo vertices and one of them must have been added to thegraph at step 119905 minus 2 or earlier

If 119905 = 2119896 is even 119906119905

can reach some vertex in 119866(119898 0) by119896 jumps and the same thing is true for vertex V

119905

Thereforediam(119866(119898 2119896)) le 2119896 + 1 If 119905 = 2119896 + 1 is odd 119906

119905

can reachsome vertex in119866(119898 1) by 119896 jumps and the same thing is truefor vertex V

119905

Therefore diam(119866(119898 2119896 + 1)) le 2119896 + 2 Thesebounds are attained by pairs of vertices 119906

119905

and V119905

created atiteration 119905 which correspond to different generating edgesand have the property of being connected to two verticesintroduced at steps 119905minus1 and 119905minus2 respectively Consequentlywe have diam(119866(119898 119905)) = 119905 + 1 for all119898 ge 2 and 119905 ge 1

From Proposition 1 we have for 119905 large

ln |119881 (119905)| = ln(119898(2119898)119905

+ 3119898 minus 2

2119898 minus 1) sim 119905 ln (2119898) (28)

Hence we obtain the logarithmic scale

diam (119905) simln |119881 (119905)|

ln (2119898) (29)

which together with high clustering (Propositions 5 and 6)justifies the small-world characteristics [1] of our growthmodel

4 Conclusion

We have studied a geometric assortative growth model119866(119898 119905) for small-world networks in a deterministic wayWe obtain analytical solutions of main properties of themodel such as the degree distribution and correlationsclustering and transitivity coefficients and graph diameter inthe full spectrum of parameter 119898 The 119866(119898 119905) model holdsboth tunable small-world and tunable assortative mixingbehaviors This should be useful to guide the research anddevelopment of varied social networks On the other handthe deterministic character of this graph family shouldfacilitate the exact calculation of other network-orientedquantities including average path length hyperbolicity [32]modular structure and motifs [33]

The introduction of tunable parameter 119898 also brings arange of open questions for future research In addition tothose mentioned before here are more examples how canwemake a trade-off between local clustering and assortativityby tuning119898 since they have opposite monotonicity What if119898 = 119898(119905) is a function of time

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[2] L A N Amara A Scala M Barthelemy and H E StanleyldquoClasses of small-world networksrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 97 no21 pp 11149ndash11152 2000

[3] S Jespersen and A Blumen ldquoSmall-world networks links withlong-tailed distributionsrdquo Physical Review E vol 62 no 5 pp6270ndash6274 2000

[4] M Kuperman and G Abramson ldquoComplex structures ingeneralized small worldsrdquo Physical Review E vol 64 no 4Article ID 047103 4 pages 2001

[5] M E J Newman ldquoModels of the small worldrdquo Journal ofStatistical Physics vol 101 no 3-4 pp 819ndash841 2000

[6] Y Shang ldquoFast distributed consensus seeking in large-scalesensor networks via shortcutsrdquo International Journal of Compu-tational Science and Engineering vol 7 no 2 pp 121ndash124 2012

[7] Y Shang ldquoA sharp threshold for rainbow connection in small-world networksrdquoMiskolc Mathematical Notes vol 13 no 2 pp493ndash497 2012

[8] F Comellas and M Sampels ldquoDeterministic small-world net-worksrdquo Physica A vol 309 no 1-2 pp 231ndash235 2002

[9] W Xiao and B Parhami ldquoCayley graphs as models of determin-istic small-world networksrdquo Information Processing Letters vol97 no 3 pp 115ndash117 2006

[10] S Boettcher B Goncalves and H Guclu ldquoHierarchical regularsmall-world networksrdquo Journal of Physics A vol 41 no 25Article ID 252001 2008

[11] Y Zhang Z Zhang S Zhou and J Guan ldquoDeterministicweighted scale-free small-world networksrdquo Physica A vol 389no 16 pp 3316ndash3324 2010

[12] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview Letters vol 89 no 20 Article ID 208701 4 pages 2002

[13] M E J Newman ldquoMixing patterns in networksrdquo PhysicalReview E vol 67 no 2 Article ID 026126 13 pages 2003

[14] Z Zhang and F Comellas ldquoFarey graphs as models for complexnetworksrdquo Theoretical Computer Science vol 412 no 8ndash10 pp865ndash875 2011

[15] C J Colbourn ldquoFarey series and maximal outerplanar graphsrdquoSIAM Journal on Algebraic Discrete Methods vol 3 no 2 pp187ndash189 1982

[16] Z Zhang S Zhou Z Wang and Z Shen ldquoA geometricgrowth model interpolating between regular and small-worldnetworksrdquo Journal of Physics A vol 40 no 39 pp 11863ndash118762007

[17] Y Shang ldquoDistinct clusterings and characteristic path lengthsin dynamic smallworld networks with identical limit degreedistributionrdquo Journal of Statistical Physics vol 149 no 3 pp505ndash518 2012

[18] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoPseud-ofractal scale-free webrdquo Physical Review E vol 65 no 6 ArticleID 066122 4 pages 2002

[19] Y Shang ldquoMean commute time for randomwalks on hierarchi-cal scale-free networksrdquo Internet Mathematics vol 8 no 4 pp321ndash337 2012

[20] R Albert and A L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

8 The Scientific World Journal

[21] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[22] S Jung S Kim and B Kahng ldquoGeometric fractal growthmodelfor scale-free networksrdquo Physical Review E vol 65 no 5 ArticleID 056101 6 pages 2002

[23] J S Andrade Jr H J Herrmann R F S Andrade and LR Da Silva ldquoApollonian networks simultaneously scale-freesmall world euclidean space filling andwithmatching graphsrdquoPhysical Review Letters vol 94 no 1 Article ID 018702 4 pages2009

[24] A-L Barabasi E Ravasz and T Vicsek ldquoDeterministic scale-free networksrdquo Physica A vol 299 no 3-4 pp 559ndash564 2001

[25] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[26] R Pastor-Satorras A Vazquez and A Vespignani ldquoDynamicaland correlation properties of the internetrdquo Physical ReviewLetters vol 87 no 25 Article ID 258701 4 pages 2001

[27] W F de la Vega and Z Tuza ldquoGroupies in random graphsrdquoInformation Processing Letters vol 109 no 7 pp 339ndash340 2009

[28] Y Shang ldquoGroupies in random bipartite graphsrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 278ndash2832010

[29] H Bateman and A Erdelyi Higher Transcendental Functionsvol 1 McGraw-Hill New York NY USA 1953

[30] M E J NewmanD JWatts and SH Strogatz ldquoRandomgraphmodels of social networksrdquo Proceedings of theNational Academyof Sciences of the United States of America vol 99 no 1 pp2566ndash2572 2002

[31] H Ebel L-I Mielsch and S Bornholdt ldquoScale-free topology ofe-mail networksrdquo Physical Review E vol 66 no 3 Article ID035103 4 pages 2002

[32] Y Shang ldquoLack of Gromov-hyperbolicity in small-world net-worksrdquo Central European Journal of Mathematics vol 10 no 3pp 1152ndash1158 2012

[33] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 The Scientific World Journal

where the function Φ represents the Lerch transcendent (see[29 Section 111])

Proof When a new vertex V is added to the graph it is easyto see 120575V = 2 and 119890V = 1 Furthermore every subsequentaddition of an edge attached to this vertex will increase bothparameters by one unit Therefore we have 119890V = 120575V minus 1 forevery vertex at every step Thus

119888 (V) =2119890V

120575V (120575V minus 1)=

2

120575V (20)

Drawing on this relationship the degree distributionobtained in Proposition 2 can be useful for calculation of theclustering coefficient of 119866(119905)

Indeed the number of vertices with clustering coefficient1 1(119898+1) 1(2119898+1) 1(119898(119905minus1)+1) 2(1+119898+(119905minus1)119898

2

)equals respectively119898(2119898)

119905minus1

119898(2119898)119905minus2

119898(2119898)119905minus3

119898 2Consequently we obtain

119888 (119866 (119905)) =1

|119881 (119905)|

times (

119905

sum

119894=1

1

(119894 minus 1)119898 + 1sdot 119898(2119898)

119905minus119894

+4

1 + 119898 + (119905 minus 1)1198982)

=2119898 minus 1

119898(2119898)119905

+ 3119898 minus 2

times ((2119898)119905minus1+(1119898) ln (2119898)

minus1

2119898Φ(2119898 1 119905 +

1

119898)

+4

1 + 119898 + (119905 minus 1)1198982)

(21)

as desired

For large graphs (ie 119905 rarr infin) the right-hand side of (19)approaches

(2119898 minus 1) (2119898)1119898

21198982ln (2119898) (22)

which is a decreasing function with respect to 119898 Hencefor larger 119898 the level of local clustering becomes lowereventually This is not quite surprising since a large bunch ofvertices will be added to the network at each iteration when119898 and 119905 become large which mitigate the coefficient

35 Transitivity Coefficient Transitivity is an importantproperty especially in the analysis of social networks see forexample [21 30 31] Let 119879(119866) be the number of triangles and119876(119866) be the number of paths of length two in a graph119866Thenthe transitivity coefficient 1198881015840(119866) of 119866 can be defined as

1198881015840

(119866) =3119879 (119866)

119876 (119866) (23)

A brief discussion of the relationship between clustering andtransitivity coefficients can be found for example in [14]

Proposition 6 The transitivity coefficient of 119866(119905) is

1198881015840

(119866 (119905)) = 1198881015840

(119905)

= (1 + 119900 (1))31198983

((2119898)119905

minus 1)

(2119898 minus 1) ((2119898)2

+ 1) (2119898)119905+1

997888rarr31198982

2 (2119898 minus 1) ((2119898)2

+ 1)

(24)

as 119905 rarr infin

Proof We first calculate 119879(119866(119905)) Note that if the number ofgenerating edges after iteration 119905 minus 1 is 119886 the number of newtriangles introduced to the graph after iteration 119905 is 3119886 Since119886 = 119871

119864

(119905 minus 1) we obtain

119879 (119866 (119905)) = 119879 (119866 (119905 minus 1)) + 119898119871119864

(119905 minus 1)

= 119879 (119866 (119905 minus 1)) + 119898(2119898)119905minus1

(25)

which together with the initial value 119879(119866(1)) = 119898 gives

119879 (119866 (119905)) =119898 ((2119898)

119905

minus 1)

2119898 minus 1 (26)

for 119905 ge 1The number of paths of length two 119876(119866(119905)) can be

derived as follows by using the degree distribution again

119876 (119866 (119905)) = 119898(2119898)119905minus1

+ 119898

119905minus1

sum

119896=1

(2 (119896119898 + 1)

2) (2119898)

119905minus119896minus1

+ 2(1 + 119898 + (119905 minus 1)119898

2

2)

=(2119898)2

+ 1

1198982(2119898)119905+1

+ 119900 ((2119898)119905

)

(27)

which along with (26) leads to the stated result

Clearly the left-hand side of (24) is a decreasing functionof119898 Recalling the comments after Proposition 5 we see thatthe difference between clustering and transitivity coefficientsof 119866(119905) is by and large quantitative This is because theymeasure a quite similar property of networks

36 Diameter Network diameter namely the largest lengthof the shortest paths between all pairs of vertices is a mea-sure of the transmission performance and communicationefficiency We show analytically the diameter of our growthmodel and find a quantitative difference between 119898 = 1 and119898 gt 1

Proposition 7 The diameter diam(119866(119898 119905)) = diam(119905) of119866(119898 119905) equals 119905 for119898 = 1 and 119905 + 1 for119898 ge 2

The Scientific World Journal 7

Proof The case of 119898 = 1 was shown in [14] In what followswe take over their method to study119898 ge 2

Clearly diam(119866(119898 0)) = 1 and diam(119866(119898 1)) = 2At each step 119905 ge 2 the longest distance between twovertices is for some vertices added at this step correspondingto different generating edges at the last step Consider twovertices introduced at step 119905 ge 2 corresponding to differentgenerating edges say 119906

119905

and V119905

The vertex 119906119905

is adjacent totwo vertices and one of them must have been added to thegraph at step 119905 minus 2 or earlier

If 119905 = 2119896 is even 119906119905

can reach some vertex in 119866(119898 0) by119896 jumps and the same thing is true for vertex V

119905

Thereforediam(119866(119898 2119896)) le 2119896 + 1 If 119905 = 2119896 + 1 is odd 119906

119905

can reachsome vertex in119866(119898 1) by 119896 jumps and the same thing is truefor vertex V

119905

Therefore diam(119866(119898 2119896 + 1)) le 2119896 + 2 Thesebounds are attained by pairs of vertices 119906

119905

and V119905

created atiteration 119905 which correspond to different generating edgesand have the property of being connected to two verticesintroduced at steps 119905minus1 and 119905minus2 respectively Consequentlywe have diam(119866(119898 119905)) = 119905 + 1 for all119898 ge 2 and 119905 ge 1

From Proposition 1 we have for 119905 large

ln |119881 (119905)| = ln(119898(2119898)119905

+ 3119898 minus 2

2119898 minus 1) sim 119905 ln (2119898) (28)

Hence we obtain the logarithmic scale

diam (119905) simln |119881 (119905)|

ln (2119898) (29)

which together with high clustering (Propositions 5 and 6)justifies the small-world characteristics [1] of our growthmodel

4 Conclusion

We have studied a geometric assortative growth model119866(119898 119905) for small-world networks in a deterministic wayWe obtain analytical solutions of main properties of themodel such as the degree distribution and correlationsclustering and transitivity coefficients and graph diameter inthe full spectrum of parameter 119898 The 119866(119898 119905) model holdsboth tunable small-world and tunable assortative mixingbehaviors This should be useful to guide the research anddevelopment of varied social networks On the other handthe deterministic character of this graph family shouldfacilitate the exact calculation of other network-orientedquantities including average path length hyperbolicity [32]modular structure and motifs [33]

The introduction of tunable parameter 119898 also brings arange of open questions for future research In addition tothose mentioned before here are more examples how canwemake a trade-off between local clustering and assortativityby tuning119898 since they have opposite monotonicity What if119898 = 119898(119905) is a function of time

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[2] L A N Amara A Scala M Barthelemy and H E StanleyldquoClasses of small-world networksrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 97 no21 pp 11149ndash11152 2000

[3] S Jespersen and A Blumen ldquoSmall-world networks links withlong-tailed distributionsrdquo Physical Review E vol 62 no 5 pp6270ndash6274 2000

[4] M Kuperman and G Abramson ldquoComplex structures ingeneralized small worldsrdquo Physical Review E vol 64 no 4Article ID 047103 4 pages 2001

[5] M E J Newman ldquoModels of the small worldrdquo Journal ofStatistical Physics vol 101 no 3-4 pp 819ndash841 2000

[6] Y Shang ldquoFast distributed consensus seeking in large-scalesensor networks via shortcutsrdquo International Journal of Compu-tational Science and Engineering vol 7 no 2 pp 121ndash124 2012

[7] Y Shang ldquoA sharp threshold for rainbow connection in small-world networksrdquoMiskolc Mathematical Notes vol 13 no 2 pp493ndash497 2012

[8] F Comellas and M Sampels ldquoDeterministic small-world net-worksrdquo Physica A vol 309 no 1-2 pp 231ndash235 2002

[9] W Xiao and B Parhami ldquoCayley graphs as models of determin-istic small-world networksrdquo Information Processing Letters vol97 no 3 pp 115ndash117 2006

[10] S Boettcher B Goncalves and H Guclu ldquoHierarchical regularsmall-world networksrdquo Journal of Physics A vol 41 no 25Article ID 252001 2008

[11] Y Zhang Z Zhang S Zhou and J Guan ldquoDeterministicweighted scale-free small-world networksrdquo Physica A vol 389no 16 pp 3316ndash3324 2010

[12] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview Letters vol 89 no 20 Article ID 208701 4 pages 2002

[13] M E J Newman ldquoMixing patterns in networksrdquo PhysicalReview E vol 67 no 2 Article ID 026126 13 pages 2003

[14] Z Zhang and F Comellas ldquoFarey graphs as models for complexnetworksrdquo Theoretical Computer Science vol 412 no 8ndash10 pp865ndash875 2011

[15] C J Colbourn ldquoFarey series and maximal outerplanar graphsrdquoSIAM Journal on Algebraic Discrete Methods vol 3 no 2 pp187ndash189 1982

[16] Z Zhang S Zhou Z Wang and Z Shen ldquoA geometricgrowth model interpolating between regular and small-worldnetworksrdquo Journal of Physics A vol 40 no 39 pp 11863ndash118762007

[17] Y Shang ldquoDistinct clusterings and characteristic path lengthsin dynamic smallworld networks with identical limit degreedistributionrdquo Journal of Statistical Physics vol 149 no 3 pp505ndash518 2012

[18] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoPseud-ofractal scale-free webrdquo Physical Review E vol 65 no 6 ArticleID 066122 4 pages 2002

[19] Y Shang ldquoMean commute time for randomwalks on hierarchi-cal scale-free networksrdquo Internet Mathematics vol 8 no 4 pp321ndash337 2012

[20] R Albert and A L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

8 The Scientific World Journal

[21] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[22] S Jung S Kim and B Kahng ldquoGeometric fractal growthmodelfor scale-free networksrdquo Physical Review E vol 65 no 5 ArticleID 056101 6 pages 2002

[23] J S Andrade Jr H J Herrmann R F S Andrade and LR Da Silva ldquoApollonian networks simultaneously scale-freesmall world euclidean space filling andwithmatching graphsrdquoPhysical Review Letters vol 94 no 1 Article ID 018702 4 pages2009

[24] A-L Barabasi E Ravasz and T Vicsek ldquoDeterministic scale-free networksrdquo Physica A vol 299 no 3-4 pp 559ndash564 2001

[25] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[26] R Pastor-Satorras A Vazquez and A Vespignani ldquoDynamicaland correlation properties of the internetrdquo Physical ReviewLetters vol 87 no 25 Article ID 258701 4 pages 2001

[27] W F de la Vega and Z Tuza ldquoGroupies in random graphsrdquoInformation Processing Letters vol 109 no 7 pp 339ndash340 2009

[28] Y Shang ldquoGroupies in random bipartite graphsrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 278ndash2832010

[29] H Bateman and A Erdelyi Higher Transcendental Functionsvol 1 McGraw-Hill New York NY USA 1953

[30] M E J NewmanD JWatts and SH Strogatz ldquoRandomgraphmodels of social networksrdquo Proceedings of theNational Academyof Sciences of the United States of America vol 99 no 1 pp2566ndash2572 2002

[31] H Ebel L-I Mielsch and S Bornholdt ldquoScale-free topology ofe-mail networksrdquo Physical Review E vol 66 no 3 Article ID035103 4 pages 2002

[32] Y Shang ldquoLack of Gromov-hyperbolicity in small-world net-worksrdquo Central European Journal of Mathematics vol 10 no 3pp 1152ndash1158 2012

[33] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 7

Proof The case of 119898 = 1 was shown in [14] In what followswe take over their method to study119898 ge 2

Clearly diam(119866(119898 0)) = 1 and diam(119866(119898 1)) = 2At each step 119905 ge 2 the longest distance between twovertices is for some vertices added at this step correspondingto different generating edges at the last step Consider twovertices introduced at step 119905 ge 2 corresponding to differentgenerating edges say 119906

119905

and V119905

The vertex 119906119905

is adjacent totwo vertices and one of them must have been added to thegraph at step 119905 minus 2 or earlier

If 119905 = 2119896 is even 119906119905

can reach some vertex in 119866(119898 0) by119896 jumps and the same thing is true for vertex V

119905

Thereforediam(119866(119898 2119896)) le 2119896 + 1 If 119905 = 2119896 + 1 is odd 119906

119905

can reachsome vertex in119866(119898 1) by 119896 jumps and the same thing is truefor vertex V

119905

Therefore diam(119866(119898 2119896 + 1)) le 2119896 + 2 Thesebounds are attained by pairs of vertices 119906

119905

and V119905

created atiteration 119905 which correspond to different generating edgesand have the property of being connected to two verticesintroduced at steps 119905minus1 and 119905minus2 respectively Consequentlywe have diam(119866(119898 119905)) = 119905 + 1 for all119898 ge 2 and 119905 ge 1

From Proposition 1 we have for 119905 large

ln |119881 (119905)| = ln(119898(2119898)119905

+ 3119898 minus 2

2119898 minus 1) sim 119905 ln (2119898) (28)

Hence we obtain the logarithmic scale

diam (119905) simln |119881 (119905)|

ln (2119898) (29)

which together with high clustering (Propositions 5 and 6)justifies the small-world characteristics [1] of our growthmodel

4 Conclusion

We have studied a geometric assortative growth model119866(119898 119905) for small-world networks in a deterministic wayWe obtain analytical solutions of main properties of themodel such as the degree distribution and correlationsclustering and transitivity coefficients and graph diameter inthe full spectrum of parameter 119898 The 119866(119898 119905) model holdsboth tunable small-world and tunable assortative mixingbehaviors This should be useful to guide the research anddevelopment of varied social networks On the other handthe deterministic character of this graph family shouldfacilitate the exact calculation of other network-orientedquantities including average path length hyperbolicity [32]modular structure and motifs [33]

The introduction of tunable parameter 119898 also brings arange of open questions for future research In addition tothose mentioned before here are more examples how canwemake a trade-off between local clustering and assortativityby tuning119898 since they have opposite monotonicity What if119898 = 119898(119905) is a function of time

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[2] L A N Amara A Scala M Barthelemy and H E StanleyldquoClasses of small-world networksrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 97 no21 pp 11149ndash11152 2000

[3] S Jespersen and A Blumen ldquoSmall-world networks links withlong-tailed distributionsrdquo Physical Review E vol 62 no 5 pp6270ndash6274 2000

[4] M Kuperman and G Abramson ldquoComplex structures ingeneralized small worldsrdquo Physical Review E vol 64 no 4Article ID 047103 4 pages 2001

[5] M E J Newman ldquoModels of the small worldrdquo Journal ofStatistical Physics vol 101 no 3-4 pp 819ndash841 2000

[6] Y Shang ldquoFast distributed consensus seeking in large-scalesensor networks via shortcutsrdquo International Journal of Compu-tational Science and Engineering vol 7 no 2 pp 121ndash124 2012

[7] Y Shang ldquoA sharp threshold for rainbow connection in small-world networksrdquoMiskolc Mathematical Notes vol 13 no 2 pp493ndash497 2012

[8] F Comellas and M Sampels ldquoDeterministic small-world net-worksrdquo Physica A vol 309 no 1-2 pp 231ndash235 2002

[9] W Xiao and B Parhami ldquoCayley graphs as models of determin-istic small-world networksrdquo Information Processing Letters vol97 no 3 pp 115ndash117 2006

[10] S Boettcher B Goncalves and H Guclu ldquoHierarchical regularsmall-world networksrdquo Journal of Physics A vol 41 no 25Article ID 252001 2008

[11] Y Zhang Z Zhang S Zhou and J Guan ldquoDeterministicweighted scale-free small-world networksrdquo Physica A vol 389no 16 pp 3316ndash3324 2010

[12] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview Letters vol 89 no 20 Article ID 208701 4 pages 2002

[13] M E J Newman ldquoMixing patterns in networksrdquo PhysicalReview E vol 67 no 2 Article ID 026126 13 pages 2003

[14] Z Zhang and F Comellas ldquoFarey graphs as models for complexnetworksrdquo Theoretical Computer Science vol 412 no 8ndash10 pp865ndash875 2011

[15] C J Colbourn ldquoFarey series and maximal outerplanar graphsrdquoSIAM Journal on Algebraic Discrete Methods vol 3 no 2 pp187ndash189 1982

[16] Z Zhang S Zhou Z Wang and Z Shen ldquoA geometricgrowth model interpolating between regular and small-worldnetworksrdquo Journal of Physics A vol 40 no 39 pp 11863ndash118762007

[17] Y Shang ldquoDistinct clusterings and characteristic path lengthsin dynamic smallworld networks with identical limit degreedistributionrdquo Journal of Statistical Physics vol 149 no 3 pp505ndash518 2012

[18] S N Dorogovtsev A V Goltsev and J F F Mendes ldquoPseud-ofractal scale-free webrdquo Physical Review E vol 65 no 6 ArticleID 066122 4 pages 2002

[19] Y Shang ldquoMean commute time for randomwalks on hierarchi-cal scale-free networksrdquo Internet Mathematics vol 8 no 4 pp321ndash337 2012

[20] R Albert and A L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

8 The Scientific World Journal

[21] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[22] S Jung S Kim and B Kahng ldquoGeometric fractal growthmodelfor scale-free networksrdquo Physical Review E vol 65 no 5 ArticleID 056101 6 pages 2002

[23] J S Andrade Jr H J Herrmann R F S Andrade and LR Da Silva ldquoApollonian networks simultaneously scale-freesmall world euclidean space filling andwithmatching graphsrdquoPhysical Review Letters vol 94 no 1 Article ID 018702 4 pages2009

[24] A-L Barabasi E Ravasz and T Vicsek ldquoDeterministic scale-free networksrdquo Physica A vol 299 no 3-4 pp 559ndash564 2001

[25] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[26] R Pastor-Satorras A Vazquez and A Vespignani ldquoDynamicaland correlation properties of the internetrdquo Physical ReviewLetters vol 87 no 25 Article ID 258701 4 pages 2001

[27] W F de la Vega and Z Tuza ldquoGroupies in random graphsrdquoInformation Processing Letters vol 109 no 7 pp 339ndash340 2009

[28] Y Shang ldquoGroupies in random bipartite graphsrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 278ndash2832010

[29] H Bateman and A Erdelyi Higher Transcendental Functionsvol 1 McGraw-Hill New York NY USA 1953

[30] M E J NewmanD JWatts and SH Strogatz ldquoRandomgraphmodels of social networksrdquo Proceedings of theNational Academyof Sciences of the United States of America vol 99 no 1 pp2566ndash2572 2002

[31] H Ebel L-I Mielsch and S Bornholdt ldquoScale-free topology ofe-mail networksrdquo Physical Review E vol 66 no 3 Article ID035103 4 pages 2002

[32] Y Shang ldquoLack of Gromov-hyperbolicity in small-world net-worksrdquo Central European Journal of Mathematics vol 10 no 3pp 1152ndash1158 2012

[33] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 The Scientific World Journal

[21] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[22] S Jung S Kim and B Kahng ldquoGeometric fractal growthmodelfor scale-free networksrdquo Physical Review E vol 65 no 5 ArticleID 056101 6 pages 2002

[23] J S Andrade Jr H J Herrmann R F S Andrade and LR Da Silva ldquoApollonian networks simultaneously scale-freesmall world euclidean space filling andwithmatching graphsrdquoPhysical Review Letters vol 94 no 1 Article ID 018702 4 pages2009

[24] A-L Barabasi E Ravasz and T Vicsek ldquoDeterministic scale-free networksrdquo Physica A vol 299 no 3-4 pp 559ndash564 2001

[25] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[26] R Pastor-Satorras A Vazquez and A Vespignani ldquoDynamicaland correlation properties of the internetrdquo Physical ReviewLetters vol 87 no 25 Article ID 258701 4 pages 2001

[27] W F de la Vega and Z Tuza ldquoGroupies in random graphsrdquoInformation Processing Letters vol 109 no 7 pp 339ndash340 2009

[28] Y Shang ldquoGroupies in random bipartite graphsrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 278ndash2832010

[29] H Bateman and A Erdelyi Higher Transcendental Functionsvol 1 McGraw-Hill New York NY USA 1953

[30] M E J NewmanD JWatts and SH Strogatz ldquoRandomgraphmodels of social networksrdquo Proceedings of theNational Academyof Sciences of the United States of America vol 99 no 1 pp2566ndash2572 2002

[31] H Ebel L-I Mielsch and S Bornholdt ldquoScale-free topology ofe-mail networksrdquo Physical Review E vol 66 no 3 Article ID035103 4 pages 2002

[32] Y Shang ldquoLack of Gromov-hyperbolicity in small-world net-worksrdquo Central European Journal of Mathematics vol 10 no 3pp 1152ndash1158 2012

[33] M Babaei H Ghassemieh and M Jalili ldquoCascading failuretolerance of modular small-world networksrdquo IEEE Transactionson Circuits and Systems II vol 58 no 8 pp 527ndash531 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014