Research ArticleA Study of How the Watts-Strogatz Model Relates toan Economic Systemrsquos Utility

Lunhan Luo and Jianan Fang

School of Information Science and Technology Donghua University Shanghai 201620 China

Correspondence should be addressed to Jianan Fang jafangdhueducn

Received 7 February 2014 Revised 8 May 2014 Accepted 8 May 2014 Published 1 June 2014

Academic Editor He Huang

Copyright copy 2014 L Luo and J Fang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Watts-Strogatz model is a main mechanism to construct the small-world networks It is widely used in the simulations of small-world featured systems including economic system Formally the model contains a parameters set including three variablesrepresenting group size number of neighbors and rewiring probability This paper discusses how the parameters set relates tothe economic system performance which is utility growth rate In conclusion it is found that regardless of the group size andrewiring probability 2 to 18 neighbors can help the economic system reach the highest utility growth rate Furthermore given therange of neighbors and group size of aWatts-Strogatzmodel based system the range of its edges can be calculated too By examiningthe containment relationship between that range and the edge number of an actual equal-size economic system we could knowwhether the system structure has redundant edges or can achieve the highest utility growth ratio

1 Introduction

In the 1990s Watts and Strogatz [1] have shown that theconnection topology of biological technological and socialnetworks is neither completely regular [2] nor completelyrandom [3] but stays somehow in between these two extremecases [4] To get the small-world network they have proposedtheWatts-Strogatz model (WSmodel) to interpolate betweenregular and random networks Since then a rapid surge ofinterest for small-world networks throughout natural andsocial sciences has witnessed the diffusion of new concepts

Alexander-Bloch et al [5] systematically explored rela-tionships between functional connectivity small-world fea-tured complex network topology and anatomical (Euclidean)distance between connected brain regions Celine and Guy[6] developed new classification and clustering schemesbased on the relative local density of subgraphs on geographyand described how the notions and methods contribute on aconceptual level in terms of measures delineations explana-tory analyses and visualization of geographical phenomenaUsing the small-world approach Corso et al [7] suggesteda network model for economy Based on evolving networkmodel the wealth distribution of a society was constructedqualitatively Sparked by an increasing need in science and

technology for understanding complex interwoven systemsas diverse as the world wide web [8 9] cellular metabolismand human social interactions [10 11] there has been anexplosion of interest in network dynamics the computationalanalysis of the structure and function of large physicaland virtual networks Besides some researchers focused onresearching the mechanism of the WS model Kleinberg [12]explained why arbitrary pairs of strangers should be able tofind short chains of acquaintances that link them together

Generally the abovementioned researchers utilize theWSmodel as a modeling tool through which they constructthe systems with their expertise Their research looks at thesystemrsquos ldquosmall-worldrdquo characteristics such as the averageshortest path length [13] clustering coefficient [14] anddegree distribution [15] With the same route we havebuilt a small-world featured economic system and analyzedthe correlation between the system structure and utilitygrowth rate (UGR) A brief review of the works focusedon utility is listed as follows John et al [16] discussed theeconomic utility function in the supply chain managementcovering logistics and marketing scheme Ben and Mark[17] studied the relationship between net worth and eco-nomic performance measured by utility variation Due to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 693743 7 pageshttpdxdoiorg1011552014693743

2 Mathematical Problems in Engineering

the broad implication of ldquoutilityrdquo some researchers fromentities other than economic system are drawn into thetopic Torrance [18] provided a new utility measurement andcost-utility analysis method in healthcare Birati and Tziner[19] introduced the concept of economic utility into thetraining program usually representing major outlay for manycorporations

During our research we found out that the removalof edges (at least one in five) in the system would notraise any impact on the value of utility function It can bededuced that there are redundant edges in this small-worldfeatured economic system Many studies on the networkstability [20 21] also revealed that some edges missing causedby the intentionally attack would not reduce the networkconnectivity and reliability despite being considered theadvantage of small-world network However in an economicsystem the useless edges bringing costs but no profits arenot expected

The result inspires us to trace back the origins of theseredundant edges that is the WS model Since no modifica-tion to the group size and the connections among the nodesafter the modeling process has been taken we hypothesizethat the system structure is the product of WS modelEquivalently the WS model would surely exert influence onUGR because of the proven fact that the system structurehas impact on the economic systemrsquos performance [22 23]Normally the WS model is expressed as a function withthree parameters (group size number of neighbors andrewiring probability) controlling the modeling process Thefact is that during the modeling process ldquogroup sizerdquo isfixed and ldquorewiring probabilityrdquo has nothing to do with thetotal amount of edges and the only variable that may causeredundant edges is ldquonumber of neighborsrdquo In this paperour purpose is to find out how many neighbors and whatrewiring probability can help the given size economic systemreach the highest UGR and keep the redundant edges aslittle as possible Once found a benchmark network canbe generated in accordance with these parameters com-pared with which the amount of redundant edges can bedetected

The structure of the paper is designed as follows Section 2introduces the WS model and economy model Based onthe model and theory Section 3 designs the experimentplans Section 4 is the collection of experimental resultsIn Section 5 the main contributions of this research aresummarized

2 Literature Review

In this section we will conduct a literature review on relevantstudiesThe review includes the description ofWSmodel andeconomy model

21 WS Model TheWSmodel is a random graph generationmodel that produces graphs with small-world propertiesincluding short average path lengths and high clustering Itwas proposed by Watts and Strogatz in their joint Naturepaper [1]

The mechanism of the model is as follows(1) Create a ring over 119899 nodes(2) Each node in the ring is connected with its 119896 nearest

neighbors (119896-1 neighbors if 119896 is odd)(3) For each edge in the ldquo119899-ringwith 119896nearest neighborsrdquo

shortcuts are created by replacing the original edges119906-V with a new edge 119906-119908 with uniformly randomchoice of existing node 119908 at rewiring probability119901

The WS model is included in software package normallyexpressed as a function The model contains three formsincluding watts strogatz graph() newman watts strogatzgraph() [24] and connected watts strogatz graph() [25]These three models have the approximately samemechanismdescribed above but still there are differences The wattsstrogatz graph() is in strict accordance with the mecha-nismThe newman watts strogatz graph() differs in the thirdstep which is the rewiring probability substituted by theprobability of adding a new edge for each node Theconnected watts strogatz graph() will repeat the steps in themechanism till a connected watts-strogatz graph is con-structed In contrast with newman watts strogatz graph()the random rewiring does not increase the numberof edges and is not guaranteed to be connected as inconnected watts strogatz graph()

In NetworkX [26] a Python language software package[27] for the creationmanipulation and study of the structuredynamics and functions of complex network there is arandomWS small-world graph generator

119908119886119905119905119904 119904119905119903119900119892119886119905119911 119892119903119886119901ℎ (119899 119896 119901) (1)where 119899 is the number of nodes in the graph 119896 is the numberof neighbors each node connected to and 119901 is the probabilityof rewiring each edge in the graph

22 Economy Model This economy model derives from theone proposed by Wilhite [28] and is designed according tothe definition of utility [29] Utility is a means of accuratelymeasuring the desirability of various types of goods andservices and the degree of well-being those products providefor consumers This measure is normally presented in theform of a mathematical expression utility function [30] andcan be utilized with just about any type of goods or servicethat is secured and used by a consumer

In this model a certain amount of independent agents iscreated Two types of goods one of which must be traded inwhole units and the other is infinitely divisible are assignedto each agent as the existing wealth to circulate in themarket The portion of the goods is randomly assigned at thebeginning of the experiment There is no production and noimports thus the aggregate stock of goods at the beginning ofthe experiment is the stock at the end

Each agentrsquos objective is to improve its own Cobb-Douglas utility function [31] in each period by engaging involuntary trade Formally 119880119894 depends on the individualrsquosexisting wealth of 119892

1and 119892

2

119880119894= 119892119894

1119892119894

2 119894 isin 1 119899 (2)

where 119899 is the amount of agents

Mathematical Problems in Engineering 3

The entire economic society is composed of every trans-action A transaction is a process in which two nodesexchange the goods since one nodersquos questing for a negoti-ating price chance is being responded An opportunity formutually beneficial transaction exists if the marginal rate ofsubstitution (MRS) of two agents differs MRS reflects theagentrsquos willing to give up 119892

2for a unit of 119892

1 With the utility

function in (1) the MRS of agent 119894 is

MRS119894 =1198801015840(119892119894

1)

1198801015840(119892119894

2)

=

119892119894

2

119892119894

1

119894 isin 1 119899 (3)

where 1198801015840(sdot) is the first derivative of 119880Themodel assumes that each agent reveals itsMRS In the

experiment agents search for beneficial trade opportunitiesaccording to the MRS and then establish a price to initiate atransaction Any agent can either trade 119892

2for 1198921or trade 119892

1

for 1198922at the expense of priceThroughout these experiments

the trading price 119901119894119895

between agent 119894 and agent 119895 is setaccording to the following rule

119901119894119895=

119892119894

2+ 119892119895

2

119892119894

1+ 119892119895

1

119894 119895 isin 1 119899 (4)

The questing node would pay 119901119894119895by 1198922to exchange one unit

of 1198921 Meanwhile the responding node would add 119901

119894119895to

its stock of 1198922and sell one unit of 119892

1to the questing node

The transaction proceeds as long as the trade benefits eachnodersquos 119880119894 and stops when one of the nodes is in lack of 119892

1

or cannot afford the price 119901119894119895 Each transaction is atomic

in the experiments namely it would not suspend till thewhole process is fulfilled In the experiments every activetransaction will be considered as one time trade with twoportions of trade volumes since the income and outcomestring are both taken into account

Once a transaction stops the questing node will searchagain for a new opportunity according to the trade rulesuntil no node responds to it and so on Another node isselected as questing node to engage in a transaction Theeconomy society evolves like this and stops at the networkrsquosequilibriumThe utility growth rate (UGR) is

UGR =sum119899

119894=1119880119894

119890minus sum119899

119894=1119880119894

sum119899

119894=1119880119894 119894 isin 1 119899 (5)

where 119880119894119890is the posttrade economic utility of node 119894

Equilibrium is a point when agents cannot find tradingopportunities that benefit any individuals Feldman [32]studied the equilibrium characteristics of welfare-improvingbilateral trade and showed that as long as all agents possesssome nonzero amount of one of the commodities (all agentshave some 119892

1or all agents have some 119892

2) then the pairwise

optimal allocation is also a Pareto optimal allocation In thisexperiment all agents are initially endowed with a positiveamount of both goods thus the equilibrium is Pareto optimal[33]

3 Experimental Design

A series of experiments are designed to run in the environ-ment of Python program aiming at finding out how manyneighbors and what rewiring probability can help the givensize economic system reach the highest UGR

At the beginning of the experiments artificial economysociety is abstracted to a network composed of nodes andedges to conduct the analysis by the network theory Accord-ing to the economy model in Section 22 each society hasagents (represented by nodes) trading rules (represented byedges) and goods 119892

1and 119892

2(represented by the endowment

of each node)The design can discuss anymarket with nodes inmultiple

of 10 We list the economic system at the group size from10 to 200 in this paper After the generation of nodesa portion of the society wealth is assigned to each nodeand so does the same Cobb-Douglas utility it intends tomaximize Network structure defines the edges among thenodes constraining the extent of trade partners each nodecan reach Then the systems undertake the trading processkeeping to the one defined in the economy model Once theeconomic system reaches the equilibrium the UGR of thesystem can be computed and bonded to the parameters setas the correspondence

The network structures are generated by the randomWSmodel complying with the mechanism in Section 21 Thisis the key process of the experiments Different parametersets (PS) including the group size (gs) the number ofneighbors (nn) and the rewiring probability (rp) representedas PS = (gs nn rp) would be endowed to the WS model Theinitial parameter set is PS = (10 2 001) and each variablewould increase progressively To enhance the operability theparameters would increase linearly at different step size asfollows

(1) The group size starts at 10 and grows with a step sizeof 10

(2) The number of neighbors starts at 2 and grows witha step size of 2 The WS model constraints thatthe number of neighbor each node attach to cannotexceed half of the group size and must be integer

(3) The rewiring probability starts at 001 and grows withstep size of 001

The forming of a new parameter set would trigger themodeling and trading process

A hierarchical nested loop is designed for fulfilling theincrementing as follows

(1) Fixing gs and rp After the succeeding modelingand trading process increase nn by its step size Togenerate sufficient experimental data simulations aretaken repeatedly under the same parameter set

(2) Once nn reaches its upper bound increase rp by itsstep size and go to (1)

(3) Once rp reaches its upper bound the experimentsstop

Figure 1 describes the design of the experiment

4 Mathematical Problems in Engineering

No

No

No

No

TradeModeling Data recordingEquilibrium

Yes

Yes

Yes

Yes

Start Finish

rp assignment

gs assignment

nn assignment nn lt gs2

rp lt 1

gs le 200

Figure 1 Experimental design

0

1

2

3

45

6

7

8

9

10

11

12

13

1415

16

17

18

19

Figure 2 An instance of a small-world network

Figure 2 is an instance of a network assigned by theparameter set (20 6 002)

4 Experimental Result

In this section we collected the results of the experimentsFigure 3 illustrates under the same rp the probability whennn takes a defined value so that the economic systemgenerated by the WS model reaches the highest UGR The 119909axe represents the number of neighbors each node can haveand 119910 axe in unit of percent () shows the probability that thecorresponding nn achieves the highest UGR represented ashp Each subgraph is an average record of 100-time repeatedsimulations 6 different sized systems are showed as examplesThrough the observation it is found that regardless of

Table 1 A comparison of NE among different group size systems

Group size Uttermost neighbors the upper end of hp 119864hp

50 600 450 300100 2400 900 600150 5550 1350 900200 9800 1800 1200

the group size 2 to 18 neighbors can achieve the highest UGRof an economic system that is

hp isin [2 18] (6)

Moreover Figure 3 also reveals the rising tendency of hpwhen the group size grows

This conclusion is reconfirmed in Figure 4(a) whichdescribes the total sum of hp in the group size from 10to 200 Lining up the points in Figure 4(a) and gettingFigure 4(b) the pattern of the probability approximates thePoisson distribution [34] and the mathematical expectation119864hp = 12

Since the extent of hp has been found bonding withgs the extent of edges can be calculated also It can bededuced that once the value of nn goes beyond the extentthe WS model would surely generate redundant edges inthe economic system Table 1 compares the number of edges(NE) when each node has the uttermost neighbors upperend of hp and 119864hp neighbors Under the same group sizeuttermost neighbors generate the maximum edges the WSmodel can the upper end of hp generates the upper boundnumber of edges indexing that beyond this bound thesystem definitely has redundant edges 119864hp generates themathematical expectation number of edges

In Table 1 compared to the systems generated underthe upper end of hp and 119864hp the one under uttermostneighbors has multiple edges and the gap would exaggeratealong with the enlargement of group size Due to the guiding

Mathematical Problems in Engineering 5

0 2 4 6 8 10 12

0

10

20

30

40

50

(a) gs = 20 rp = 002

0 5 10 15 20 25

0

10

20

30

40

50

60

(b) gs = 50 rp = 002

0 10 20 30 40 50

0

10

20

30

40

50

60

(c) gs = 100 rp = 002

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

(d) gs = 150 rp = 002

0 20 40 60 80 100

0

10

20

30

40

50

(e) gs = 180 rp = 002

0 20 40 60 80 100

0

5

10

15

20

25

30

(f) gs = 200 rp = 002

Figure 3 The probability when nn takes a defined value so that the economic system reaches the highest UGR

0 20 40 60 80 100

0

5

10

15

20

25

30

(a)

0 20 40 60 80 100

0

5

10

15

20

25

30

(b)

Figure 4 The total sum of hp in the group size from 10 to 200

significance of the bound generated by the upper end of hpany system with NE beyond the bound has costly redundantedges However the redundant edges have advantages asshields when the economic system confronts the random ordeliberate attacks Thus the complete removal of redundantedges is unnecessary

In the random WS model variable rp cannot bringchanges in NE but can reform the network structure Figure 5combines a group of graphs describing the correlationbetween nn (119909 axe) and UGR (119910 axe) when gs = 80 It revealsthat under the same group size UGR differs if the network

structure and endowment changes are reflected in the patternof each subgraph varying However the pattern of each graphconforms to the range hp isin [2 18] and has similar curvature

Despite the observation on Figure 5 we suppose that thepattern of UGR would decrease gradually when nn is beyond18 which is the upper end of hp To verify the conjecturegs rp and nn in the parameter set would increase linearlyas defined in Section 3 and after repeated simulations theUGR of systems generated by WS model is collected inFigure 6 It is confirmed that independent of group size andrewiring probability if nn gt 18 the UGR would decrease

6 Mathematical Problems in Engineering

0 10 20 30 40

000

005

010

015

020

(a) gs = 80 rp = 001

0 10 20 30 40

000

005

010

015

020

(b) gs = 80 rp = 002

0 10 20 30 40

000

005

010

015

020

(c) gs = 80 rp = 003

0 10 20 30 40

000

005

010

015

020

(d) gs = 80 rp = 004

0 10 20 30 40

000

005

010

015

020

(e) gs = 80 rp = 005

0 10 20 30 40

000

005

010

015

020

(f) gs = 80 rp = 006

0 10 20 30 40

000

005

010

015

020

(g) gs = 80 rp = 007

0 10 20 30 40

000

005

010

015

020

(h) gs = 80 rp = 008

0 10 20 30 40

000

005

010

015

020

(i) gs = 80 rp = 009

0 10 20 30 40

000

005

010

015

020

(j) gs = 80 rp = 01

Figure 5 The pattern of UGR when gs = 80 and rp increases linearly

0 10 20 30000

005

010

015

020

(a) gs = 60 rp = 001

0 10 20 30 40 50

000

005

010

015

020

(b) gs = 120 rp = 005

0 10 20 30 40 50 60

000

005

010

015

020

(c) gs = 150 rp = 008

0 20 40 60 80

000

005

010

015

020

(d) gs = 170 rp = 01

000

005

010

015

020

0 20 40 60 80 100

(e) gs = 200 rp = 02

Figure 6 The pattern of UGR when gs rp and nn increase linearly

gradually Although it may not be a monotonic decreasingthe decreasing tendency would be kept

When constructing the systems in Figures 5 and 6 wealso consider that the possible effect on hp and UGR maybe exerted by the endowments to goods 119892

1and 119892

2 The fixed

group size systems in Figure 5 are endowed with randomlygenerated 119892

1and 119892

2and in Figure 5 different group size

systems are discussed which means that their endowmentscannot keep consistent with each otherThe result shows thatthe UGR varies on account of the endowments and networkstructure but the range of hp cannot be influenced

5 Conclusion

This research combines economic theory network theoryand computer simulations to examine the parameters setassigned to the WS model relating to an economic systemrsquosUGR We discovered that regardless of the group size andrewiring probability (1) 2 to 18 neighbors and the corre-sponding computable number of edges can help an economicsystem reach the highest UGR (2) if the node has morethan 18 neighbors the UGR would decrease gradually andthe pattern is not monotonic but would keep the decreasingtendency (3) different endowments to goods119892

1and1198922would

not impact the range of neighbors which can help the systemachieve the highest UGR In the practical application itcan be construed that (1) acting as a measurement judgeswhether a system can reach the highest UGR based on its

current structure (2) when the resource is in the hands of afraction of people the resource can derive the highest valuefor the owners and the rest of people (3) once keeping thesuperiority of minority owners the value of resource wouldnot decrease or increase according to its quantity

Conflict of Interests

The authors declare 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] W-K Chen Graph Theory and Its Engineering ApplicationsWorld Scientific Singapore 1997

[3] P Erdos and A Renyi ldquoOn random graphs Irdquo PublicationesMathematicae vol 6 pp 290ndash297 1959

[4] D J Watts ldquoNetworks dynamics and the small-world phe-nomenonrdquo The American Journal of Sociology vol 105 no 2pp 493ndash527 1999

[5] A F Alexander-Bloch P E Vertes R Stidd et al ldquoThe anatom-ical distance of functional connections predicts brain networktopology in health and schizophreniardquo Cerebral Cortex vol 23no 1 pp 127ndash138 2013

[6] R Celine and M Guy ldquoA small world perspective on urbansystemsrdquo inMethods for Multilevel Analysis and Visualisation ofGeographical Networks vol 11 pp 19ndash32 2013

Mathematical Problems in Engineering 7

[7] G Corso L S Lucena and Z D Thome ldquoThe small-worldof economy a speculative proposalrdquo Physica A StatisticalMechanics and Its Applications vol 324 no 1-2 pp 430ndash4362003

[8] K Musiał and P Kazienko ldquoSocial networks on the internetrdquoWorld Wide Web vol 16 no 1 pp 31ndash72 2013

[9] X Cheng J Liu and C Dale ldquoUnderstanding the char-acteristics of internet short video sharing a youtube-basedmeasurement studyrdquo IEEE Transactions on Multimedia vol 15no 5 pp 1184ndash1194 2013

[10] D Eppstein M T Goodrich M Loffler D Strash and LTrott ldquoCategory-based routing in social networks member-ship dimension and the small-world phenomenonrdquoTheoreticalComputer Science vol 514 pp 96ndash104 2013

[11] D Cesar and L Igor ldquoStructure and dynamics of transportationnetworks models methods and applicationsrdquo in The SAGEHandbook of Transport Studies pp 347ndash364 2013

[12] J Kleinberg ldquoThe small-world phenomenon an algorithm per-spectiverdquo in Proceedings of the 32nd Annual ACM SymposiumonTheory of Computing pp 163ndash170 2000

[13] S Milgram ldquoThe small-world problemrdquo Psychology Today vol2 pp 60ndash67 1967

[14] A Kemper Valuation of Network Effects in Software MarketsA Complex Networks Approach Springer Heidelberg Germany2010

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

[16] T John P Theodore and L Terry ldquoSupply chain managementand its relationship to logistics marketing production andoperations managementrdquo Journal of Business Logistics vol 29no 1 pp 31ndash46 2008

[17] B Ben and G Mark ldquoFinancial fragility and economic perfor-mancerdquo The Quarterly Journal of Economics vol 105 no 1 pp87ndash114 1990

[18] GW Torrance ldquoUtilitymeasurement in healthcare the things Inever got tordquo PharmacoEconomics vol 24 no 11 pp 1069ndash10782006

[19] A Birati andA Tziner ldquoEconomic utility of training programsrdquoJournal of Business and Psychology vol 14 no 1 pp 155ndash1641999

[20] C Li and G Chen ldquoStability of a neural network modelwith small-world connectionsrdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 68 no 5 Article ID052901 2003

[21] X Li and X Wang ldquoControlling the spreading in small-worldevolving networks stability oscillation and topologyrdquo IEEETransactions on Automatic Control vol 51 no 3 pp 534ndash5402006

[22] M O Jackson and AWolinsky ldquoA strategic model of social andeconomic networksrdquo Journal of Economic Theory vol 71 no 1pp 44ndash74 1996

[23] Y A Ioannides ldquoEvolution of trading structuresrdquo in TheEconomy as an Evolving Complex System II W B Arthur SDurlauf and D Lane Eds pp 129ndash168 1997

[24] M E J Newman and D J Watts ldquoRenormalization groupanalysis of the small-world network modelrdquo Physics Letters AGeneral Atomic and Solid State Physics vol 263 no 4ndash6 pp341ndash346 1999

[25] httpnetworkxgithubiodocumentationlatest

[26] A Hagberg and D Conway ldquoHacking social networks usingthe Python programming language (module IImdashwhy do SNAin NetworkX)rdquo in Proceedings of the International Network forSocial Network Analysis Sunbelt Conference 2010

[27] ldquoPython 340 beta 3rdquo Python Software Foundation 2014[28] A Wilhite ldquoBilateral trade and ldquosmall-worldrdquo networksrdquo Com-

putational Economics vol 18 no 1 pp 49ndash64 2001[29] A Marshall Principles of Economics An Introductory Volume

Macmillan London UK 8th edition[30] J E Ingersoll JrTheory of Financial Decision Making Rowman

and Littlefield Totowa NJ USA 1987[31] DW Caves and L R Christensen ldquoGlobal properties of flexible

functional formsrdquo The American Economic Review vol 70 no3 pp 422ndash432 1980

[32] A Feldman ldquoBilateral trading processes pairwise optimalityand Pareto optimalityrdquo Review of Economic Studies vol 40 no4 pp 463ndash473 1973

[33] A Sen ldquoMarkets and freedoms achievements and limitationsof the market mechanism in promoting individual freedomsrdquoOxford Economic Papers vol 45 no 4 pp 519ndash541 1993

[34] J H Ahrens and U Dieter ldquoComputer methods for samplingfrom gamma beta poisson and binomial distributionsrdquo Com-puting vol 12 no 3 pp 223ndash246 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the broad implication of ldquoutilityrdquo some researchers fromentities other than economic system are drawn into thetopic Torrance [18] provided a new utility measurement andcost-utility analysis method in healthcare Birati and Tziner[19] introduced the concept of economic utility into thetraining program usually representing major outlay for manycorporations

During our research we found out that the removalof edges (at least one in five) in the system would notraise any impact on the value of utility function It can bededuced that there are redundant edges in this small-worldfeatured economic system Many studies on the networkstability [20 21] also revealed that some edges missing causedby the intentionally attack would not reduce the networkconnectivity and reliability despite being considered theadvantage of small-world network However in an economicsystem the useless edges bringing costs but no profits arenot expected

The result inspires us to trace back the origins of theseredundant edges that is the WS model Since no modifica-tion to the group size and the connections among the nodesafter the modeling process has been taken we hypothesizethat the system structure is the product of WS modelEquivalently the WS model would surely exert influence onUGR because of the proven fact that the system structurehas impact on the economic systemrsquos performance [22 23]Normally the WS model is expressed as a function withthree parameters (group size number of neighbors andrewiring probability) controlling the modeling process Thefact is that during the modeling process ldquogroup sizerdquo isfixed and ldquorewiring probabilityrdquo has nothing to do with thetotal amount of edges and the only variable that may causeredundant edges is ldquonumber of neighborsrdquo In this paperour purpose is to find out how many neighbors and whatrewiring probability can help the given size economic systemreach the highest UGR and keep the redundant edges aslittle as possible Once found a benchmark network canbe generated in accordance with these parameters com-pared with which the amount of redundant edges can bedetected

The structure of the paper is designed as follows Section 2introduces the WS model and economy model Based onthe model and theory Section 3 designs the experimentplans Section 4 is the collection of experimental resultsIn Section 5 the main contributions of this research aresummarized

2 Literature Review

In this section we will conduct a literature review on relevantstudiesThe review includes the description ofWSmodel andeconomy model

21 WS Model TheWSmodel is a random graph generationmodel that produces graphs with small-world propertiesincluding short average path lengths and high clustering Itwas proposed by Watts and Strogatz in their joint Naturepaper [1]

The mechanism of the model is as follows(1) Create a ring over 119899 nodes(2) Each node in the ring is connected with its 119896 nearest

neighbors (119896-1 neighbors if 119896 is odd)(3) For each edge in the ldquo119899-ringwith 119896nearest neighborsrdquo

shortcuts are created by replacing the original edges119906-V with a new edge 119906-119908 with uniformly randomchoice of existing node 119908 at rewiring probability119901

The WS model is included in software package normallyexpressed as a function The model contains three formsincluding watts strogatz graph() newman watts strogatzgraph() [24] and connected watts strogatz graph() [25]These three models have the approximately samemechanismdescribed above but still there are differences The wattsstrogatz graph() is in strict accordance with the mecha-nismThe newman watts strogatz graph() differs in the thirdstep which is the rewiring probability substituted by theprobability of adding a new edge for each node Theconnected watts strogatz graph() will repeat the steps in themechanism till a connected watts-strogatz graph is con-structed In contrast with newman watts strogatz graph()the random rewiring does not increase the numberof edges and is not guaranteed to be connected as inconnected watts strogatz graph()

In NetworkX [26] a Python language software package[27] for the creationmanipulation and study of the structuredynamics and functions of complex network there is arandomWS small-world graph generator

119908119886119905119905119904 119904119905119903119900119892119886119905119911 119892119903119886119901ℎ (119899 119896 119901) (1)where 119899 is the number of nodes in the graph 119896 is the numberof neighbors each node connected to and 119901 is the probabilityof rewiring each edge in the graph

22 Economy Model This economy model derives from theone proposed by Wilhite [28] and is designed according tothe definition of utility [29] Utility is a means of accuratelymeasuring the desirability of various types of goods andservices and the degree of well-being those products providefor consumers This measure is normally presented in theform of a mathematical expression utility function [30] andcan be utilized with just about any type of goods or servicethat is secured and used by a consumer

In this model a certain amount of independent agents iscreated Two types of goods one of which must be traded inwhole units and the other is infinitely divisible are assignedto each agent as the existing wealth to circulate in themarket The portion of the goods is randomly assigned at thebeginning of the experiment There is no production and noimports thus the aggregate stock of goods at the beginning ofthe experiment is the stock at the end

Each agentrsquos objective is to improve its own Cobb-Douglas utility function [31] in each period by engaging involuntary trade Formally 119880119894 depends on the individualrsquosexisting wealth of 119892

1and 119892

2

119880119894= 119892119894

1119892119894

2 119894 isin 1 119899 (2)

where 119899 is the amount of agents

Mathematical Problems in Engineering 3

The entire economic society is composed of every trans-action A transaction is a process in which two nodesexchange the goods since one nodersquos questing for a negoti-ating price chance is being responded An opportunity formutually beneficial transaction exists if the marginal rate ofsubstitution (MRS) of two agents differs MRS reflects theagentrsquos willing to give up 119892

2for a unit of 119892

1 With the utility

function in (1) the MRS of agent 119894 is

MRS119894 =1198801015840(119892119894

1)

1198801015840(119892119894

2)

=

119892119894

2

119892119894

1

119894 isin 1 119899 (3)

where 1198801015840(sdot) is the first derivative of 119880Themodel assumes that each agent reveals itsMRS In the

experiment agents search for beneficial trade opportunitiesaccording to the MRS and then establish a price to initiate atransaction Any agent can either trade 119892

2for 1198921or trade 119892

1

for 1198922at the expense of priceThroughout these experiments

the trading price 119901119894119895

between agent 119894 and agent 119895 is setaccording to the following rule

119901119894119895=

119892119894

2+ 119892119895

2

119892119894

1+ 119892119895

1

119894 119895 isin 1 119899 (4)

The questing node would pay 119901119894119895by 1198922to exchange one unit

of 1198921 Meanwhile the responding node would add 119901

119894119895to

its stock of 1198922and sell one unit of 119892

1to the questing node

The transaction proceeds as long as the trade benefits eachnodersquos 119880119894 and stops when one of the nodes is in lack of 119892

1

or cannot afford the price 119901119894119895 Each transaction is atomic

in the experiments namely it would not suspend till thewhole process is fulfilled In the experiments every activetransaction will be considered as one time trade with twoportions of trade volumes since the income and outcomestring are both taken into account

Once a transaction stops the questing node will searchagain for a new opportunity according to the trade rulesuntil no node responds to it and so on Another node isselected as questing node to engage in a transaction Theeconomy society evolves like this and stops at the networkrsquosequilibriumThe utility growth rate (UGR) is

UGR =sum119899

119894=1119880119894

119890minus sum119899

119894=1119880119894

sum119899

119894=1119880119894 119894 isin 1 119899 (5)

where 119880119894119890is the posttrade economic utility of node 119894

Equilibrium is a point when agents cannot find tradingopportunities that benefit any individuals Feldman [32]studied the equilibrium characteristics of welfare-improvingbilateral trade and showed that as long as all agents possesssome nonzero amount of one of the commodities (all agentshave some 119892

1or all agents have some 119892

2) then the pairwise

optimal allocation is also a Pareto optimal allocation In thisexperiment all agents are initially endowed with a positiveamount of both goods thus the equilibrium is Pareto optimal[33]

3 Experimental Design

A series of experiments are designed to run in the environ-ment of Python program aiming at finding out how manyneighbors and what rewiring probability can help the givensize economic system reach the highest UGR

At the beginning of the experiments artificial economysociety is abstracted to a network composed of nodes andedges to conduct the analysis by the network theory Accord-ing to the economy model in Section 22 each society hasagents (represented by nodes) trading rules (represented byedges) and goods 119892

1and 119892

2(represented by the endowment

of each node)The design can discuss anymarket with nodes inmultiple

of 10 We list the economic system at the group size from10 to 200 in this paper After the generation of nodesa portion of the society wealth is assigned to each nodeand so does the same Cobb-Douglas utility it intends tomaximize Network structure defines the edges among thenodes constraining the extent of trade partners each nodecan reach Then the systems undertake the trading processkeeping to the one defined in the economy model Once theeconomic system reaches the equilibrium the UGR of thesystem can be computed and bonded to the parameters setas the correspondence

The network structures are generated by the randomWSmodel complying with the mechanism in Section 21 Thisis the key process of the experiments Different parametersets (PS) including the group size (gs) the number ofneighbors (nn) and the rewiring probability (rp) representedas PS = (gs nn rp) would be endowed to the WS model Theinitial parameter set is PS = (10 2 001) and each variablewould increase progressively To enhance the operability theparameters would increase linearly at different step size asfollows

(1) The group size starts at 10 and grows with a step sizeof 10

(2) The number of neighbors starts at 2 and grows witha step size of 2 The WS model constraints thatthe number of neighbor each node attach to cannotexceed half of the group size and must be integer

(3) The rewiring probability starts at 001 and grows withstep size of 001

The forming of a new parameter set would trigger themodeling and trading process

A hierarchical nested loop is designed for fulfilling theincrementing as follows

(1) Fixing gs and rp After the succeeding modelingand trading process increase nn by its step size Togenerate sufficient experimental data simulations aretaken repeatedly under the same parameter set

(2) Once nn reaches its upper bound increase rp by itsstep size and go to (1)

(3) Once rp reaches its upper bound the experimentsstop

Figure 1 describes the design of the experiment

4 Mathematical Problems in Engineering

No

No

No

No

TradeModeling Data recordingEquilibrium

Yes

Yes

Yes

Yes

Start Finish

rp assignment

gs assignment

nn assignment nn lt gs2

rp lt 1

gs le 200

Figure 1 Experimental design

0

1

2

3

45

6

7

8

9

10

11

12

13

1415

16

17

18

19

Figure 2 An instance of a small-world network

Figure 2 is an instance of a network assigned by theparameter set (20 6 002)

4 Experimental Result

In this section we collected the results of the experimentsFigure 3 illustrates under the same rp the probability whennn takes a defined value so that the economic systemgenerated by the WS model reaches the highest UGR The 119909axe represents the number of neighbors each node can haveand 119910 axe in unit of percent () shows the probability that thecorresponding nn achieves the highest UGR represented ashp Each subgraph is an average record of 100-time repeatedsimulations 6 different sized systems are showed as examplesThrough the observation it is found that regardless of

Table 1 A comparison of NE among different group size systems

Group size Uttermost neighbors the upper end of hp 119864hp

50 600 450 300100 2400 900 600150 5550 1350 900200 9800 1800 1200

the group size 2 to 18 neighbors can achieve the highest UGRof an economic system that is

hp isin [2 18] (6)

Moreover Figure 3 also reveals the rising tendency of hpwhen the group size grows

This conclusion is reconfirmed in Figure 4(a) whichdescribes the total sum of hp in the group size from 10to 200 Lining up the points in Figure 4(a) and gettingFigure 4(b) the pattern of the probability approximates thePoisson distribution [34] and the mathematical expectation119864hp = 12

Since the extent of hp has been found bonding withgs the extent of edges can be calculated also It can bededuced that once the value of nn goes beyond the extentthe WS model would surely generate redundant edges inthe economic system Table 1 compares the number of edges(NE) when each node has the uttermost neighbors upperend of hp and 119864hp neighbors Under the same group sizeuttermost neighbors generate the maximum edges the WSmodel can the upper end of hp generates the upper boundnumber of edges indexing that beyond this bound thesystem definitely has redundant edges 119864hp generates themathematical expectation number of edges

In Table 1 compared to the systems generated underthe upper end of hp and 119864hp the one under uttermostneighbors has multiple edges and the gap would exaggeratealong with the enlargement of group size Due to the guiding

Mathematical Problems in Engineering 5

0 2 4 6 8 10 12

0

10

20

30

40

50

(a) gs = 20 rp = 002

0 5 10 15 20 25

0

10

20

30

40

50

60

(b) gs = 50 rp = 002

0 10 20 30 40 50

0

10

20

30

40

50

60

(c) gs = 100 rp = 002

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

(d) gs = 150 rp = 002

0 20 40 60 80 100

0

10

20

30

40

50

(e) gs = 180 rp = 002

0 20 40 60 80 100

0

5

10

15

20

25

30

(f) gs = 200 rp = 002

Figure 3 The probability when nn takes a defined value so that the economic system reaches the highest UGR

0 20 40 60 80 100

0

5

10

15

20

25

30

(a)

0 20 40 60 80 100

0

5

10

15

20

25

30

(b)

Figure 4 The total sum of hp in the group size from 10 to 200

significance of the bound generated by the upper end of hpany system with NE beyond the bound has costly redundantedges However the redundant edges have advantages asshields when the economic system confronts the random ordeliberate attacks Thus the complete removal of redundantedges is unnecessary

In the random WS model variable rp cannot bringchanges in NE but can reform the network structure Figure 5combines a group of graphs describing the correlationbetween nn (119909 axe) and UGR (119910 axe) when gs = 80 It revealsthat under the same group size UGR differs if the network

structure and endowment changes are reflected in the patternof each subgraph varying However the pattern of each graphconforms to the range hp isin [2 18] and has similar curvature

Despite the observation on Figure 5 we suppose that thepattern of UGR would decrease gradually when nn is beyond18 which is the upper end of hp To verify the conjecturegs rp and nn in the parameter set would increase linearlyas defined in Section 3 and after repeated simulations theUGR of systems generated by WS model is collected inFigure 6 It is confirmed that independent of group size andrewiring probability if nn gt 18 the UGR would decrease

6 Mathematical Problems in Engineering

0 10 20 30 40

000

005

010

015

020

(a) gs = 80 rp = 001

0 10 20 30 40

000

005

010

015

020

(b) gs = 80 rp = 002

0 10 20 30 40

000

005

010

015

020

(c) gs = 80 rp = 003

0 10 20 30 40

000

005

010

015

020

(d) gs = 80 rp = 004

0 10 20 30 40

000

005

010

015

020

(e) gs = 80 rp = 005

0 10 20 30 40

000

005

010

015

020

(f) gs = 80 rp = 006

0 10 20 30 40

000

005

010

015

020

(g) gs = 80 rp = 007

0 10 20 30 40

000

005

010

015

020

(h) gs = 80 rp = 008

0 10 20 30 40

000

005

010

015

020

(i) gs = 80 rp = 009

0 10 20 30 40

000

005

010

015

020

(j) gs = 80 rp = 01

Figure 5 The pattern of UGR when gs = 80 and rp increases linearly

0 10 20 30000

005

010

015

020

(a) gs = 60 rp = 001

0 10 20 30 40 50

000

005

010

015

020

(b) gs = 120 rp = 005

0 10 20 30 40 50 60

000

005

010

015

020

(c) gs = 150 rp = 008

0 20 40 60 80

000

005

010

015

020

(d) gs = 170 rp = 01

000

005

010

015

020

0 20 40 60 80 100

(e) gs = 200 rp = 02

Figure 6 The pattern of UGR when gs rp and nn increase linearly

gradually Although it may not be a monotonic decreasingthe decreasing tendency would be kept

When constructing the systems in Figures 5 and 6 wealso consider that the possible effect on hp and UGR maybe exerted by the endowments to goods 119892

1and 119892

2 The fixed

group size systems in Figure 5 are endowed with randomlygenerated 119892

1and 119892

2and in Figure 5 different group size

systems are discussed which means that their endowmentscannot keep consistent with each otherThe result shows thatthe UGR varies on account of the endowments and networkstructure but the range of hp cannot be influenced

5 Conclusion

This research combines economic theory network theoryand computer simulations to examine the parameters setassigned to the WS model relating to an economic systemrsquosUGR We discovered that regardless of the group size andrewiring probability (1) 2 to 18 neighbors and the corre-sponding computable number of edges can help an economicsystem reach the highest UGR (2) if the node has morethan 18 neighbors the UGR would decrease gradually andthe pattern is not monotonic but would keep the decreasingtendency (3) different endowments to goods119892

1and1198922would

not impact the range of neighbors which can help the systemachieve the highest UGR In the practical application itcan be construed that (1) acting as a measurement judgeswhether a system can reach the highest UGR based on its

current structure (2) when the resource is in the hands of afraction of people the resource can derive the highest valuefor the owners and the rest of people (3) once keeping thesuperiority of minority owners the value of resource wouldnot decrease or increase according to its quantity

Conflict of Interests

The authors declare 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] W-K Chen Graph Theory and Its Engineering ApplicationsWorld Scientific Singapore 1997

[3] P Erdos and A Renyi ldquoOn random graphs Irdquo PublicationesMathematicae vol 6 pp 290ndash297 1959

[4] D J Watts ldquoNetworks dynamics and the small-world phe-nomenonrdquo The American Journal of Sociology vol 105 no 2pp 493ndash527 1999

[5] A F Alexander-Bloch P E Vertes R Stidd et al ldquoThe anatom-ical distance of functional connections predicts brain networktopology in health and schizophreniardquo Cerebral Cortex vol 23no 1 pp 127ndash138 2013

[6] R Celine and M Guy ldquoA small world perspective on urbansystemsrdquo inMethods for Multilevel Analysis and Visualisation ofGeographical Networks vol 11 pp 19ndash32 2013

Mathematical Problems in Engineering 7

[7] G Corso L S Lucena and Z D Thome ldquoThe small-worldof economy a speculative proposalrdquo Physica A StatisticalMechanics and Its Applications vol 324 no 1-2 pp 430ndash4362003

[8] K Musiał and P Kazienko ldquoSocial networks on the internetrdquoWorld Wide Web vol 16 no 1 pp 31ndash72 2013

[9] X Cheng J Liu and C Dale ldquoUnderstanding the char-acteristics of internet short video sharing a youtube-basedmeasurement studyrdquo IEEE Transactions on Multimedia vol 15no 5 pp 1184ndash1194 2013

[10] D Eppstein M T Goodrich M Loffler D Strash and LTrott ldquoCategory-based routing in social networks member-ship dimension and the small-world phenomenonrdquoTheoreticalComputer Science vol 514 pp 96ndash104 2013

[11] D Cesar and L Igor ldquoStructure and dynamics of transportationnetworks models methods and applicationsrdquo in The SAGEHandbook of Transport Studies pp 347ndash364 2013

[12] J Kleinberg ldquoThe small-world phenomenon an algorithm per-spectiverdquo in Proceedings of the 32nd Annual ACM SymposiumonTheory of Computing pp 163ndash170 2000

[13] S Milgram ldquoThe small-world problemrdquo Psychology Today vol2 pp 60ndash67 1967

[14] A Kemper Valuation of Network Effects in Software MarketsA Complex Networks Approach Springer Heidelberg Germany2010

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

[16] T John P Theodore and L Terry ldquoSupply chain managementand its relationship to logistics marketing production andoperations managementrdquo Journal of Business Logistics vol 29no 1 pp 31ndash46 2008

[17] B Ben and G Mark ldquoFinancial fragility and economic perfor-mancerdquo The Quarterly Journal of Economics vol 105 no 1 pp87ndash114 1990

[18] GW Torrance ldquoUtilitymeasurement in healthcare the things Inever got tordquo PharmacoEconomics vol 24 no 11 pp 1069ndash10782006

[19] A Birati andA Tziner ldquoEconomic utility of training programsrdquoJournal of Business and Psychology vol 14 no 1 pp 155ndash1641999

[20] C Li and G Chen ldquoStability of a neural network modelwith small-world connectionsrdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 68 no 5 Article ID052901 2003

[21] X Li and X Wang ldquoControlling the spreading in small-worldevolving networks stability oscillation and topologyrdquo IEEETransactions on Automatic Control vol 51 no 3 pp 534ndash5402006

[22] M O Jackson and AWolinsky ldquoA strategic model of social andeconomic networksrdquo Journal of Economic Theory vol 71 no 1pp 44ndash74 1996

[23] Y A Ioannides ldquoEvolution of trading structuresrdquo in TheEconomy as an Evolving Complex System II W B Arthur SDurlauf and D Lane Eds pp 129ndash168 1997

[24] M E J Newman and D J Watts ldquoRenormalization groupanalysis of the small-world network modelrdquo Physics Letters AGeneral Atomic and Solid State Physics vol 263 no 4ndash6 pp341ndash346 1999

[25] httpnetworkxgithubiodocumentationlatest

[26] A Hagberg and D Conway ldquoHacking social networks usingthe Python programming language (module IImdashwhy do SNAin NetworkX)rdquo in Proceedings of the International Network forSocial Network Analysis Sunbelt Conference 2010

[27] ldquoPython 340 beta 3rdquo Python Software Foundation 2014[28] A Wilhite ldquoBilateral trade and ldquosmall-worldrdquo networksrdquo Com-

putational Economics vol 18 no 1 pp 49ndash64 2001[29] A Marshall Principles of Economics An Introductory Volume

Macmillan London UK 8th edition[30] J E Ingersoll JrTheory of Financial Decision Making Rowman

and Littlefield Totowa NJ USA 1987[31] DW Caves and L R Christensen ldquoGlobal properties of flexible

functional formsrdquo The American Economic Review vol 70 no3 pp 422ndash432 1980

[32] A Feldman ldquoBilateral trading processes pairwise optimalityand Pareto optimalityrdquo Review of Economic Studies vol 40 no4 pp 463ndash473 1973

[33] A Sen ldquoMarkets and freedoms achievements and limitationsof the market mechanism in promoting individual freedomsrdquoOxford Economic Papers vol 45 no 4 pp 519ndash541 1993

[34] J H Ahrens and U Dieter ldquoComputer methods for samplingfrom gamma beta poisson and binomial distributionsrdquo Com-puting vol 12 no 3 pp 223ndash246 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

The entire economic society is composed of every trans-action A transaction is a process in which two nodesexchange the goods since one nodersquos questing for a negoti-ating price chance is being responded An opportunity formutually beneficial transaction exists if the marginal rate ofsubstitution (MRS) of two agents differs MRS reflects theagentrsquos willing to give up 119892

2for a unit of 119892

1 With the utility

function in (1) the MRS of agent 119894 is

MRS119894 =1198801015840(119892119894

1)

1198801015840(119892119894

2)

=

119892119894

2

119892119894

1

119894 isin 1 119899 (3)

where 1198801015840(sdot) is the first derivative of 119880Themodel assumes that each agent reveals itsMRS In the

experiment agents search for beneficial trade opportunitiesaccording to the MRS and then establish a price to initiate atransaction Any agent can either trade 119892

2for 1198921or trade 119892

1

for 1198922at the expense of priceThroughout these experiments

the trading price 119901119894119895

between agent 119894 and agent 119895 is setaccording to the following rule

119901119894119895=

119892119894

2+ 119892119895

2

119892119894

1+ 119892119895

1

119894 119895 isin 1 119899 (4)

The questing node would pay 119901119894119895by 1198922to exchange one unit

of 1198921 Meanwhile the responding node would add 119901

119894119895to

its stock of 1198922and sell one unit of 119892

1to the questing node

The transaction proceeds as long as the trade benefits eachnodersquos 119880119894 and stops when one of the nodes is in lack of 119892

1

or cannot afford the price 119901119894119895 Each transaction is atomic

in the experiments namely it would not suspend till thewhole process is fulfilled In the experiments every activetransaction will be considered as one time trade with twoportions of trade volumes since the income and outcomestring are both taken into account

Once a transaction stops the questing node will searchagain for a new opportunity according to the trade rulesuntil no node responds to it and so on Another node isselected as questing node to engage in a transaction Theeconomy society evolves like this and stops at the networkrsquosequilibriumThe utility growth rate (UGR) is

UGR =sum119899

119894=1119880119894

119890minus sum119899

119894=1119880119894

sum119899

119894=1119880119894 119894 isin 1 119899 (5)

where 119880119894119890is the posttrade economic utility of node 119894

Equilibrium is a point when agents cannot find tradingopportunities that benefit any individuals Feldman [32]studied the equilibrium characteristics of welfare-improvingbilateral trade and showed that as long as all agents possesssome nonzero amount of one of the commodities (all agentshave some 119892

1or all agents have some 119892

2) then the pairwise

optimal allocation is also a Pareto optimal allocation In thisexperiment all agents are initially endowed with a positiveamount of both goods thus the equilibrium is Pareto optimal[33]

3 Experimental Design

A series of experiments are designed to run in the environ-ment of Python program aiming at finding out how manyneighbors and what rewiring probability can help the givensize economic system reach the highest UGR

At the beginning of the experiments artificial economysociety is abstracted to a network composed of nodes andedges to conduct the analysis by the network theory Accord-ing to the economy model in Section 22 each society hasagents (represented by nodes) trading rules (represented byedges) and goods 119892

1and 119892

2(represented by the endowment

of each node)The design can discuss anymarket with nodes inmultiple

of 10 We list the economic system at the group size from10 to 200 in this paper After the generation of nodesa portion of the society wealth is assigned to each nodeand so does the same Cobb-Douglas utility it intends tomaximize Network structure defines the edges among thenodes constraining the extent of trade partners each nodecan reach Then the systems undertake the trading processkeeping to the one defined in the economy model Once theeconomic system reaches the equilibrium the UGR of thesystem can be computed and bonded to the parameters setas the correspondence

The network structures are generated by the randomWSmodel complying with the mechanism in Section 21 Thisis the key process of the experiments Different parametersets (PS) including the group size (gs) the number ofneighbors (nn) and the rewiring probability (rp) representedas PS = (gs nn rp) would be endowed to the WS model Theinitial parameter set is PS = (10 2 001) and each variablewould increase progressively To enhance the operability theparameters would increase linearly at different step size asfollows

(1) The group size starts at 10 and grows with a step sizeof 10

(2) The number of neighbors starts at 2 and grows witha step size of 2 The WS model constraints thatthe number of neighbor each node attach to cannotexceed half of the group size and must be integer

(3) The rewiring probability starts at 001 and grows withstep size of 001

The forming of a new parameter set would trigger themodeling and trading process

A hierarchical nested loop is designed for fulfilling theincrementing as follows

(1) Fixing gs and rp After the succeeding modelingand trading process increase nn by its step size Togenerate sufficient experimental data simulations aretaken repeatedly under the same parameter set

(2) Once nn reaches its upper bound increase rp by itsstep size and go to (1)

(3) Once rp reaches its upper bound the experimentsstop

Figure 1 describes the design of the experiment

4 Mathematical Problems in Engineering

No

No

No

No

TradeModeling Data recordingEquilibrium

Yes

Yes

Yes

Yes

Start Finish

rp assignment

gs assignment

nn assignment nn lt gs2

rp lt 1

gs le 200

Figure 1 Experimental design

0

1

2

3

45

6

7

8

9

10

11

12

13

1415

16

17

18

19

Figure 2 An instance of a small-world network

Figure 2 is an instance of a network assigned by theparameter set (20 6 002)

4 Experimental Result

In this section we collected the results of the experimentsFigure 3 illustrates under the same rp the probability whennn takes a defined value so that the economic systemgenerated by the WS model reaches the highest UGR The 119909axe represents the number of neighbors each node can haveand 119910 axe in unit of percent () shows the probability that thecorresponding nn achieves the highest UGR represented ashp Each subgraph is an average record of 100-time repeatedsimulations 6 different sized systems are showed as examplesThrough the observation it is found that regardless of

Table 1 A comparison of NE among different group size systems

Group size Uttermost neighbors the upper end of hp 119864hp

50 600 450 300100 2400 900 600150 5550 1350 900200 9800 1800 1200

the group size 2 to 18 neighbors can achieve the highest UGRof an economic system that is

hp isin [2 18] (6)

Moreover Figure 3 also reveals the rising tendency of hpwhen the group size grows

This conclusion is reconfirmed in Figure 4(a) whichdescribes the total sum of hp in the group size from 10to 200 Lining up the points in Figure 4(a) and gettingFigure 4(b) the pattern of the probability approximates thePoisson distribution [34] and the mathematical expectation119864hp = 12

Since the extent of hp has been found bonding withgs the extent of edges can be calculated also It can bededuced that once the value of nn goes beyond the extentthe WS model would surely generate redundant edges inthe economic system Table 1 compares the number of edges(NE) when each node has the uttermost neighbors upperend of hp and 119864hp neighbors Under the same group sizeuttermost neighbors generate the maximum edges the WSmodel can the upper end of hp generates the upper boundnumber of edges indexing that beyond this bound thesystem definitely has redundant edges 119864hp generates themathematical expectation number of edges

In Table 1 compared to the systems generated underthe upper end of hp and 119864hp the one under uttermostneighbors has multiple edges and the gap would exaggeratealong with the enlargement of group size Due to the guiding

Mathematical Problems in Engineering 5

0 2 4 6 8 10 12

0

10

20

30

40

50

(a) gs = 20 rp = 002

0 5 10 15 20 25

0

10

20

30

40

50

60

(b) gs = 50 rp = 002

0 10 20 30 40 50

0

10

20

30

40

50

60

(c) gs = 100 rp = 002

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

(d) gs = 150 rp = 002

0 20 40 60 80 100

0

10

20

30

40

50

(e) gs = 180 rp = 002

0 20 40 60 80 100

0

5

10

15

20

25

30

(f) gs = 200 rp = 002

Figure 3 The probability when nn takes a defined value so that the economic system reaches the highest UGR

0 20 40 60 80 100

0

5

10

15

20

25

30

(a)

0 20 40 60 80 100

0

5

10

15

20

25

30

(b)

Figure 4 The total sum of hp in the group size from 10 to 200

significance of the bound generated by the upper end of hpany system with NE beyond the bound has costly redundantedges However the redundant edges have advantages asshields when the economic system confronts the random ordeliberate attacks Thus the complete removal of redundantedges is unnecessary

In the random WS model variable rp cannot bringchanges in NE but can reform the network structure Figure 5combines a group of graphs describing the correlationbetween nn (119909 axe) and UGR (119910 axe) when gs = 80 It revealsthat under the same group size UGR differs if the network

structure and endowment changes are reflected in the patternof each subgraph varying However the pattern of each graphconforms to the range hp isin [2 18] and has similar curvature

Despite the observation on Figure 5 we suppose that thepattern of UGR would decrease gradually when nn is beyond18 which is the upper end of hp To verify the conjecturegs rp and nn in the parameter set would increase linearlyas defined in Section 3 and after repeated simulations theUGR of systems generated by WS model is collected inFigure 6 It is confirmed that independent of group size andrewiring probability if nn gt 18 the UGR would decrease

6 Mathematical Problems in Engineering

0 10 20 30 40

000

005

010

015

020

(a) gs = 80 rp = 001

0 10 20 30 40

000

005

010

015

020

(b) gs = 80 rp = 002

0 10 20 30 40

000

005

010

015

020

(c) gs = 80 rp = 003

0 10 20 30 40

000

005

010

015

020

(d) gs = 80 rp = 004

0 10 20 30 40

000

005

010

015

020

(e) gs = 80 rp = 005

0 10 20 30 40

000

005

010

015

020

(f) gs = 80 rp = 006

0 10 20 30 40

000

005

010

015

020

(g) gs = 80 rp = 007

0 10 20 30 40

000

005

010

015

020

(h) gs = 80 rp = 008

0 10 20 30 40

000

005

010

015

020

(i) gs = 80 rp = 009

0 10 20 30 40

000

005

010

015

020

(j) gs = 80 rp = 01

Figure 5 The pattern of UGR when gs = 80 and rp increases linearly

0 10 20 30000

005

010

015

020

(a) gs = 60 rp = 001

0 10 20 30 40 50

000

005

010

015

020

(b) gs = 120 rp = 005

0 10 20 30 40 50 60

000

005

010

015

020

(c) gs = 150 rp = 008

0 20 40 60 80

000

005

010

015

020

(d) gs = 170 rp = 01

000

005

010

015

020

0 20 40 60 80 100

(e) gs = 200 rp = 02

Figure 6 The pattern of UGR when gs rp and nn increase linearly

gradually Although it may not be a monotonic decreasingthe decreasing tendency would be kept

When constructing the systems in Figures 5 and 6 wealso consider that the possible effect on hp and UGR maybe exerted by the endowments to goods 119892

1and 119892

2 The fixed

group size systems in Figure 5 are endowed with randomlygenerated 119892

1and 119892

2and in Figure 5 different group size

systems are discussed which means that their endowmentscannot keep consistent with each otherThe result shows thatthe UGR varies on account of the endowments and networkstructure but the range of hp cannot be influenced

5 Conclusion

This research combines economic theory network theoryand computer simulations to examine the parameters setassigned to the WS model relating to an economic systemrsquosUGR We discovered that regardless of the group size andrewiring probability (1) 2 to 18 neighbors and the corre-sponding computable number of edges can help an economicsystem reach the highest UGR (2) if the node has morethan 18 neighbors the UGR would decrease gradually andthe pattern is not monotonic but would keep the decreasingtendency (3) different endowments to goods119892

1and1198922would

not impact the range of neighbors which can help the systemachieve the highest UGR In the practical application itcan be construed that (1) acting as a measurement judgeswhether a system can reach the highest UGR based on its

current structure (2) when the resource is in the hands of afraction of people the resource can derive the highest valuefor the owners and the rest of people (3) once keeping thesuperiority of minority owners the value of resource wouldnot decrease or increase according to its quantity

Conflict of Interests

The authors declare 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] W-K Chen Graph Theory and Its Engineering ApplicationsWorld Scientific Singapore 1997

[3] P Erdos and A Renyi ldquoOn random graphs Irdquo PublicationesMathematicae vol 6 pp 290ndash297 1959

[4] D J Watts ldquoNetworks dynamics and the small-world phe-nomenonrdquo The American Journal of Sociology vol 105 no 2pp 493ndash527 1999

[5] A F Alexander-Bloch P E Vertes R Stidd et al ldquoThe anatom-ical distance of functional connections predicts brain networktopology in health and schizophreniardquo Cerebral Cortex vol 23no 1 pp 127ndash138 2013

[6] R Celine and M Guy ldquoA small world perspective on urbansystemsrdquo inMethods for Multilevel Analysis and Visualisation ofGeographical Networks vol 11 pp 19ndash32 2013

Mathematical Problems in Engineering 7

[7] G Corso L S Lucena and Z D Thome ldquoThe small-worldof economy a speculative proposalrdquo Physica A StatisticalMechanics and Its Applications vol 324 no 1-2 pp 430ndash4362003

[8] K Musiał and P Kazienko ldquoSocial networks on the internetrdquoWorld Wide Web vol 16 no 1 pp 31ndash72 2013

[9] X Cheng J Liu and C Dale ldquoUnderstanding the char-acteristics of internet short video sharing a youtube-basedmeasurement studyrdquo IEEE Transactions on Multimedia vol 15no 5 pp 1184ndash1194 2013

[10] D Eppstein M T Goodrich M Loffler D Strash and LTrott ldquoCategory-based routing in social networks member-ship dimension and the small-world phenomenonrdquoTheoreticalComputer Science vol 514 pp 96ndash104 2013

[11] D Cesar and L Igor ldquoStructure and dynamics of transportationnetworks models methods and applicationsrdquo in The SAGEHandbook of Transport Studies pp 347ndash364 2013

[12] J Kleinberg ldquoThe small-world phenomenon an algorithm per-spectiverdquo in Proceedings of the 32nd Annual ACM SymposiumonTheory of Computing pp 163ndash170 2000

[13] S Milgram ldquoThe small-world problemrdquo Psychology Today vol2 pp 60ndash67 1967

[14] A Kemper Valuation of Network Effects in Software MarketsA Complex Networks Approach Springer Heidelberg Germany2010

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

[16] T John P Theodore and L Terry ldquoSupply chain managementand its relationship to logistics marketing production andoperations managementrdquo Journal of Business Logistics vol 29no 1 pp 31ndash46 2008

[17] B Ben and G Mark ldquoFinancial fragility and economic perfor-mancerdquo The Quarterly Journal of Economics vol 105 no 1 pp87ndash114 1990

[18] GW Torrance ldquoUtilitymeasurement in healthcare the things Inever got tordquo PharmacoEconomics vol 24 no 11 pp 1069ndash10782006

[19] A Birati andA Tziner ldquoEconomic utility of training programsrdquoJournal of Business and Psychology vol 14 no 1 pp 155ndash1641999

[20] C Li and G Chen ldquoStability of a neural network modelwith small-world connectionsrdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 68 no 5 Article ID052901 2003

[21] X Li and X Wang ldquoControlling the spreading in small-worldevolving networks stability oscillation and topologyrdquo IEEETransactions on Automatic Control vol 51 no 3 pp 534ndash5402006

[22] M O Jackson and AWolinsky ldquoA strategic model of social andeconomic networksrdquo Journal of Economic Theory vol 71 no 1pp 44ndash74 1996

[23] Y A Ioannides ldquoEvolution of trading structuresrdquo in TheEconomy as an Evolving Complex System II W B Arthur SDurlauf and D Lane Eds pp 129ndash168 1997

[24] M E J Newman and D J Watts ldquoRenormalization groupanalysis of the small-world network modelrdquo Physics Letters AGeneral Atomic and Solid State Physics vol 263 no 4ndash6 pp341ndash346 1999

[25] httpnetworkxgithubiodocumentationlatest

[26] A Hagberg and D Conway ldquoHacking social networks usingthe Python programming language (module IImdashwhy do SNAin NetworkX)rdquo in Proceedings of the International Network forSocial Network Analysis Sunbelt Conference 2010

[27] ldquoPython 340 beta 3rdquo Python Software Foundation 2014[28] A Wilhite ldquoBilateral trade and ldquosmall-worldrdquo networksrdquo Com-

putational Economics vol 18 no 1 pp 49ndash64 2001[29] A Marshall Principles of Economics An Introductory Volume

Macmillan London UK 8th edition[30] J E Ingersoll JrTheory of Financial Decision Making Rowman

and Littlefield Totowa NJ USA 1987[31] DW Caves and L R Christensen ldquoGlobal properties of flexible

functional formsrdquo The American Economic Review vol 70 no3 pp 422ndash432 1980

[32] A Feldman ldquoBilateral trading processes pairwise optimalityand Pareto optimalityrdquo Review of Economic Studies vol 40 no4 pp 463ndash473 1973

[33] A Sen ldquoMarkets and freedoms achievements and limitationsof the market mechanism in promoting individual freedomsrdquoOxford Economic Papers vol 45 no 4 pp 519ndash541 1993

[34] J H Ahrens and U Dieter ldquoComputer methods for samplingfrom gamma beta poisson and binomial distributionsrdquo Com-puting vol 12 no 3 pp 223ndash246 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

No

No

No

No

TradeModeling Data recordingEquilibrium

Yes

Yes

Yes

Yes

Start Finish

rp assignment

gs assignment

nn assignment nn lt gs2

rp lt 1

gs le 200

Figure 1 Experimental design

0

1

2

3

45

6

7

8

9

10

11

12

13

1415

16

17

18

19

Figure 2 An instance of a small-world network

Figure 2 is an instance of a network assigned by theparameter set (20 6 002)

4 Experimental Result

In this section we collected the results of the experimentsFigure 3 illustrates under the same rp the probability whennn takes a defined value so that the economic systemgenerated by the WS model reaches the highest UGR The 119909axe represents the number of neighbors each node can haveand 119910 axe in unit of percent () shows the probability that thecorresponding nn achieves the highest UGR represented ashp Each subgraph is an average record of 100-time repeatedsimulations 6 different sized systems are showed as examplesThrough the observation it is found that regardless of

Table 1 A comparison of NE among different group size systems

Group size Uttermost neighbors the upper end of hp 119864hp

50 600 450 300100 2400 900 600150 5550 1350 900200 9800 1800 1200

the group size 2 to 18 neighbors can achieve the highest UGRof an economic system that is

hp isin [2 18] (6)

Moreover Figure 3 also reveals the rising tendency of hpwhen the group size grows

This conclusion is reconfirmed in Figure 4(a) whichdescribes the total sum of hp in the group size from 10to 200 Lining up the points in Figure 4(a) and gettingFigure 4(b) the pattern of the probability approximates thePoisson distribution [34] and the mathematical expectation119864hp = 12

Since the extent of hp has been found bonding withgs the extent of edges can be calculated also It can bededuced that once the value of nn goes beyond the extentthe WS model would surely generate redundant edges inthe economic system Table 1 compares the number of edges(NE) when each node has the uttermost neighbors upperend of hp and 119864hp neighbors Under the same group sizeuttermost neighbors generate the maximum edges the WSmodel can the upper end of hp generates the upper boundnumber of edges indexing that beyond this bound thesystem definitely has redundant edges 119864hp generates themathematical expectation number of edges

In Table 1 compared to the systems generated underthe upper end of hp and 119864hp the one under uttermostneighbors has multiple edges and the gap would exaggeratealong with the enlargement of group size Due to the guiding

Mathematical Problems in Engineering 5

0 2 4 6 8 10 12

0

10

20

30

40

50

(a) gs = 20 rp = 002

0 5 10 15 20 25

0

10

20

30

40

50

60

(b) gs = 50 rp = 002

0 10 20 30 40 50

0

10

20

30

40

50

60

(c) gs = 100 rp = 002

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

(d) gs = 150 rp = 002

0 20 40 60 80 100

0

10

20

30

40

50

(e) gs = 180 rp = 002

0 20 40 60 80 100

0

5

10

15

20

25

30

(f) gs = 200 rp = 002

Figure 3 The probability when nn takes a defined value so that the economic system reaches the highest UGR

0 20 40 60 80 100

0

5

10

15

20

25

30

(a)

0 20 40 60 80 100

0

5

10

15

20

25

30

(b)

Figure 4 The total sum of hp in the group size from 10 to 200

significance of the bound generated by the upper end of hpany system with NE beyond the bound has costly redundantedges However the redundant edges have advantages asshields when the economic system confronts the random ordeliberate attacks Thus the complete removal of redundantedges is unnecessary

In the random WS model variable rp cannot bringchanges in NE but can reform the network structure Figure 5combines a group of graphs describing the correlationbetween nn (119909 axe) and UGR (119910 axe) when gs = 80 It revealsthat under the same group size UGR differs if the network

structure and endowment changes are reflected in the patternof each subgraph varying However the pattern of each graphconforms to the range hp isin [2 18] and has similar curvature

Despite the observation on Figure 5 we suppose that thepattern of UGR would decrease gradually when nn is beyond18 which is the upper end of hp To verify the conjecturegs rp and nn in the parameter set would increase linearlyas defined in Section 3 and after repeated simulations theUGR of systems generated by WS model is collected inFigure 6 It is confirmed that independent of group size andrewiring probability if nn gt 18 the UGR would decrease

6 Mathematical Problems in Engineering

0 10 20 30 40

000

005

010

015

020

(a) gs = 80 rp = 001

0 10 20 30 40

000

005

010

015

020

(b) gs = 80 rp = 002

0 10 20 30 40

000

005

010

015

020

(c) gs = 80 rp = 003

0 10 20 30 40

000

005

010

015

020

(d) gs = 80 rp = 004

0 10 20 30 40

000

005

010

015

020

(e) gs = 80 rp = 005

0 10 20 30 40

000

005

010

015

020

(f) gs = 80 rp = 006

0 10 20 30 40

000

005

010

015

020

(g) gs = 80 rp = 007

0 10 20 30 40

000

005

010

015

020

(h) gs = 80 rp = 008

0 10 20 30 40

000

005

010

015

020

(i) gs = 80 rp = 009

0 10 20 30 40

000

005

010

015

020

(j) gs = 80 rp = 01

Figure 5 The pattern of UGR when gs = 80 and rp increases linearly

0 10 20 30000

005

010

015

020

(a) gs = 60 rp = 001

0 10 20 30 40 50

000

005

010

015

020

(b) gs = 120 rp = 005

0 10 20 30 40 50 60

000

005

010

015

020

(c) gs = 150 rp = 008

0 20 40 60 80

000

005

010

015

020

(d) gs = 170 rp = 01

000

005

010

015

020

0 20 40 60 80 100

(e) gs = 200 rp = 02

Figure 6 The pattern of UGR when gs rp and nn increase linearly

gradually Although it may not be a monotonic decreasingthe decreasing tendency would be kept

When constructing the systems in Figures 5 and 6 wealso consider that the possible effect on hp and UGR maybe exerted by the endowments to goods 119892

1and 119892

2 The fixed

group size systems in Figure 5 are endowed with randomlygenerated 119892

1and 119892

2and in Figure 5 different group size

systems are discussed which means that their endowmentscannot keep consistent with each otherThe result shows thatthe UGR varies on account of the endowments and networkstructure but the range of hp cannot be influenced

5 Conclusion

This research combines economic theory network theoryand computer simulations to examine the parameters setassigned to the WS model relating to an economic systemrsquosUGR We discovered that regardless of the group size andrewiring probability (1) 2 to 18 neighbors and the corre-sponding computable number of edges can help an economicsystem reach the highest UGR (2) if the node has morethan 18 neighbors the UGR would decrease gradually andthe pattern is not monotonic but would keep the decreasingtendency (3) different endowments to goods119892

1and1198922would

not impact the range of neighbors which can help the systemachieve the highest UGR In the practical application itcan be construed that (1) acting as a measurement judgeswhether a system can reach the highest UGR based on its

current structure (2) when the resource is in the hands of afraction of people the resource can derive the highest valuefor the owners and the rest of people (3) once keeping thesuperiority of minority owners the value of resource wouldnot decrease or increase according to its quantity

Conflict of Interests

The authors declare 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] W-K Chen Graph Theory and Its Engineering ApplicationsWorld Scientific Singapore 1997

[3] P Erdos and A Renyi ldquoOn random graphs Irdquo PublicationesMathematicae vol 6 pp 290ndash297 1959

[4] D J Watts ldquoNetworks dynamics and the small-world phe-nomenonrdquo The American Journal of Sociology vol 105 no 2pp 493ndash527 1999

[5] A F Alexander-Bloch P E Vertes R Stidd et al ldquoThe anatom-ical distance of functional connections predicts brain networktopology in health and schizophreniardquo Cerebral Cortex vol 23no 1 pp 127ndash138 2013

[6] R Celine and M Guy ldquoA small world perspective on urbansystemsrdquo inMethods for Multilevel Analysis and Visualisation ofGeographical Networks vol 11 pp 19ndash32 2013

Mathematical Problems in Engineering 7

[7] G Corso L S Lucena and Z D Thome ldquoThe small-worldof economy a speculative proposalrdquo Physica A StatisticalMechanics and Its Applications vol 324 no 1-2 pp 430ndash4362003

[8] K Musiał and P Kazienko ldquoSocial networks on the internetrdquoWorld Wide Web vol 16 no 1 pp 31ndash72 2013

[9] X Cheng J Liu and C Dale ldquoUnderstanding the char-acteristics of internet short video sharing a youtube-basedmeasurement studyrdquo IEEE Transactions on Multimedia vol 15no 5 pp 1184ndash1194 2013

[10] D Eppstein M T Goodrich M Loffler D Strash and LTrott ldquoCategory-based routing in social networks member-ship dimension and the small-world phenomenonrdquoTheoreticalComputer Science vol 514 pp 96ndash104 2013

[11] D Cesar and L Igor ldquoStructure and dynamics of transportationnetworks models methods and applicationsrdquo in The SAGEHandbook of Transport Studies pp 347ndash364 2013

[12] J Kleinberg ldquoThe small-world phenomenon an algorithm per-spectiverdquo in Proceedings of the 32nd Annual ACM SymposiumonTheory of Computing pp 163ndash170 2000

[13] S Milgram ldquoThe small-world problemrdquo Psychology Today vol2 pp 60ndash67 1967

[14] A Kemper Valuation of Network Effects in Software MarketsA Complex Networks Approach Springer Heidelberg Germany2010

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

[16] T John P Theodore and L Terry ldquoSupply chain managementand its relationship to logistics marketing production andoperations managementrdquo Journal of Business Logistics vol 29no 1 pp 31ndash46 2008

[17] B Ben and G Mark ldquoFinancial fragility and economic perfor-mancerdquo The Quarterly Journal of Economics vol 105 no 1 pp87ndash114 1990

[18] GW Torrance ldquoUtilitymeasurement in healthcare the things Inever got tordquo PharmacoEconomics vol 24 no 11 pp 1069ndash10782006

[19] A Birati andA Tziner ldquoEconomic utility of training programsrdquoJournal of Business and Psychology vol 14 no 1 pp 155ndash1641999

[20] C Li and G Chen ldquoStability of a neural network modelwith small-world connectionsrdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 68 no 5 Article ID052901 2003

[21] X Li and X Wang ldquoControlling the spreading in small-worldevolving networks stability oscillation and topologyrdquo IEEETransactions on Automatic Control vol 51 no 3 pp 534ndash5402006

[22] M O Jackson and AWolinsky ldquoA strategic model of social andeconomic networksrdquo Journal of Economic Theory vol 71 no 1pp 44ndash74 1996

[23] Y A Ioannides ldquoEvolution of trading structuresrdquo in TheEconomy as an Evolving Complex System II W B Arthur SDurlauf and D Lane Eds pp 129ndash168 1997

[24] M E J Newman and D J Watts ldquoRenormalization groupanalysis of the small-world network modelrdquo Physics Letters AGeneral Atomic and Solid State Physics vol 263 no 4ndash6 pp341ndash346 1999

[25] httpnetworkxgithubiodocumentationlatest

[26] A Hagberg and D Conway ldquoHacking social networks usingthe Python programming language (module IImdashwhy do SNAin NetworkX)rdquo in Proceedings of the International Network forSocial Network Analysis Sunbelt Conference 2010

[27] ldquoPython 340 beta 3rdquo Python Software Foundation 2014[28] A Wilhite ldquoBilateral trade and ldquosmall-worldrdquo networksrdquo Com-

putational Economics vol 18 no 1 pp 49ndash64 2001[29] A Marshall Principles of Economics An Introductory Volume

Macmillan London UK 8th edition[30] J E Ingersoll JrTheory of Financial Decision Making Rowman

and Littlefield Totowa NJ USA 1987[31] DW Caves and L R Christensen ldquoGlobal properties of flexible

functional formsrdquo The American Economic Review vol 70 no3 pp 422ndash432 1980

[32] A Feldman ldquoBilateral trading processes pairwise optimalityand Pareto optimalityrdquo Review of Economic Studies vol 40 no4 pp 463ndash473 1973

[33] A Sen ldquoMarkets and freedoms achievements and limitationsof the market mechanism in promoting individual freedomsrdquoOxford Economic Papers vol 45 no 4 pp 519ndash541 1993

[34] J H Ahrens and U Dieter ldquoComputer methods for samplingfrom gamma beta poisson and binomial distributionsrdquo Com-puting vol 12 no 3 pp 223ndash246 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 2 4 6 8 10 12

0

10

20

30

40

50

(a) gs = 20 rp = 002

0 5 10 15 20 25

0

10

20

30

40

50

60

(b) gs = 50 rp = 002

0 10 20 30 40 50

0

10

20

30

40

50

60

(c) gs = 100 rp = 002

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

(d) gs = 150 rp = 002

0 20 40 60 80 100

0

10

20

30

40

50

(e) gs = 180 rp = 002

0 20 40 60 80 100

0

5

10

15

20

25

30

(f) gs = 200 rp = 002

Figure 3 The probability when nn takes a defined value so that the economic system reaches the highest UGR

0 20 40 60 80 100

0

5

10

15

20

25

30

(a)

0 20 40 60 80 100

0

5

10

15

20

25

30

(b)

Figure 4 The total sum of hp in the group size from 10 to 200

significance of the bound generated by the upper end of hpany system with NE beyond the bound has costly redundantedges However the redundant edges have advantages asshields when the economic system confronts the random ordeliberate attacks Thus the complete removal of redundantedges is unnecessary

In the random WS model variable rp cannot bringchanges in NE but can reform the network structure Figure 5combines a group of graphs describing the correlationbetween nn (119909 axe) and UGR (119910 axe) when gs = 80 It revealsthat under the same group size UGR differs if the network

structure and endowment changes are reflected in the patternof each subgraph varying However the pattern of each graphconforms to the range hp isin [2 18] and has similar curvature

Despite the observation on Figure 5 we suppose that thepattern of UGR would decrease gradually when nn is beyond18 which is the upper end of hp To verify the conjecturegs rp and nn in the parameter set would increase linearlyas defined in Section 3 and after repeated simulations theUGR of systems generated by WS model is collected inFigure 6 It is confirmed that independent of group size andrewiring probability if nn gt 18 the UGR would decrease

6 Mathematical Problems in Engineering

0 10 20 30 40

000

005

010

015

020

(a) gs = 80 rp = 001

0 10 20 30 40

000

005

010

015

020

(b) gs = 80 rp = 002

0 10 20 30 40

000

005

010

015

020

(c) gs = 80 rp = 003

0 10 20 30 40

000

005

010

015

020

(d) gs = 80 rp = 004

0 10 20 30 40

000

005

010

015

020

(e) gs = 80 rp = 005

0 10 20 30 40

000

005

010

015

020

(f) gs = 80 rp = 006

0 10 20 30 40

000

005

010

015

020

(g) gs = 80 rp = 007

0 10 20 30 40

000

005

010

015

020

(h) gs = 80 rp = 008

0 10 20 30 40

000

005

010

015

020

(i) gs = 80 rp = 009

0 10 20 30 40

000

005

010

015

020

(j) gs = 80 rp = 01

Figure 5 The pattern of UGR when gs = 80 and rp increases linearly

0 10 20 30000

005

010

015

020

(a) gs = 60 rp = 001

0 10 20 30 40 50

000

005

010

015

020

(b) gs = 120 rp = 005

0 10 20 30 40 50 60

000

005

010

015

020

(c) gs = 150 rp = 008

0 20 40 60 80

000

005

010

015

020

(d) gs = 170 rp = 01

000

005

010

015

020

0 20 40 60 80 100

(e) gs = 200 rp = 02

Figure 6 The pattern of UGR when gs rp and nn increase linearly

gradually Although it may not be a monotonic decreasingthe decreasing tendency would be kept

When constructing the systems in Figures 5 and 6 wealso consider that the possible effect on hp and UGR maybe exerted by the endowments to goods 119892

1and 119892

2 The fixed

group size systems in Figure 5 are endowed with randomlygenerated 119892

1and 119892

2and in Figure 5 different group size

systems are discussed which means that their endowmentscannot keep consistent with each otherThe result shows thatthe UGR varies on account of the endowments and networkstructure but the range of hp cannot be influenced

5 Conclusion

This research combines economic theory network theoryand computer simulations to examine the parameters setassigned to the WS model relating to an economic systemrsquosUGR We discovered that regardless of the group size andrewiring probability (1) 2 to 18 neighbors and the corre-sponding computable number of edges can help an economicsystem reach the highest UGR (2) if the node has morethan 18 neighbors the UGR would decrease gradually andthe pattern is not monotonic but would keep the decreasingtendency (3) different endowments to goods119892

1and1198922would

not impact the range of neighbors which can help the systemachieve the highest UGR In the practical application itcan be construed that (1) acting as a measurement judgeswhether a system can reach the highest UGR based on its

current structure (2) when the resource is in the hands of afraction of people the resource can derive the highest valuefor the owners and the rest of people (3) once keeping thesuperiority of minority owners the value of resource wouldnot decrease or increase according to its quantity

Conflict of Interests

The authors declare 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] W-K Chen Graph Theory and Its Engineering ApplicationsWorld Scientific Singapore 1997

[3] P Erdos and A Renyi ldquoOn random graphs Irdquo PublicationesMathematicae vol 6 pp 290ndash297 1959

[4] D J Watts ldquoNetworks dynamics and the small-world phe-nomenonrdquo The American Journal of Sociology vol 105 no 2pp 493ndash527 1999

[5] A F Alexander-Bloch P E Vertes R Stidd et al ldquoThe anatom-ical distance of functional connections predicts brain networktopology in health and schizophreniardquo Cerebral Cortex vol 23no 1 pp 127ndash138 2013

[6] R Celine and M Guy ldquoA small world perspective on urbansystemsrdquo inMethods for Multilevel Analysis and Visualisation ofGeographical Networks vol 11 pp 19ndash32 2013

Mathematical Problems in Engineering 7

[7] G Corso L S Lucena and Z D Thome ldquoThe small-worldof economy a speculative proposalrdquo Physica A StatisticalMechanics and Its Applications vol 324 no 1-2 pp 430ndash4362003

[8] K Musiał and P Kazienko ldquoSocial networks on the internetrdquoWorld Wide Web vol 16 no 1 pp 31ndash72 2013

[9] X Cheng J Liu and C Dale ldquoUnderstanding the char-acteristics of internet short video sharing a youtube-basedmeasurement studyrdquo IEEE Transactions on Multimedia vol 15no 5 pp 1184ndash1194 2013

[10] D Eppstein M T Goodrich M Loffler D Strash and LTrott ldquoCategory-based routing in social networks member-ship dimension and the small-world phenomenonrdquoTheoreticalComputer Science vol 514 pp 96ndash104 2013

[11] D Cesar and L Igor ldquoStructure and dynamics of transportationnetworks models methods and applicationsrdquo in The SAGEHandbook of Transport Studies pp 347ndash364 2013

[12] J Kleinberg ldquoThe small-world phenomenon an algorithm per-spectiverdquo in Proceedings of the 32nd Annual ACM SymposiumonTheory of Computing pp 163ndash170 2000

[13] S Milgram ldquoThe small-world problemrdquo Psychology Today vol2 pp 60ndash67 1967

[14] A Kemper Valuation of Network Effects in Software MarketsA Complex Networks Approach Springer Heidelberg Germany2010

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

[16] T John P Theodore and L Terry ldquoSupply chain managementand its relationship to logistics marketing production andoperations managementrdquo Journal of Business Logistics vol 29no 1 pp 31ndash46 2008

[17] B Ben and G Mark ldquoFinancial fragility and economic perfor-mancerdquo The Quarterly Journal of Economics vol 105 no 1 pp87ndash114 1990

[18] GW Torrance ldquoUtilitymeasurement in healthcare the things Inever got tordquo PharmacoEconomics vol 24 no 11 pp 1069ndash10782006

[19] A Birati andA Tziner ldquoEconomic utility of training programsrdquoJournal of Business and Psychology vol 14 no 1 pp 155ndash1641999

[20] C Li and G Chen ldquoStability of a neural network modelwith small-world connectionsrdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 68 no 5 Article ID052901 2003

[21] X Li and X Wang ldquoControlling the spreading in small-worldevolving networks stability oscillation and topologyrdquo IEEETransactions on Automatic Control vol 51 no 3 pp 534ndash5402006

[22] M O Jackson and AWolinsky ldquoA strategic model of social andeconomic networksrdquo Journal of Economic Theory vol 71 no 1pp 44ndash74 1996

[23] Y A Ioannides ldquoEvolution of trading structuresrdquo in TheEconomy as an Evolving Complex System II W B Arthur SDurlauf and D Lane Eds pp 129ndash168 1997

[24] M E J Newman and D J Watts ldquoRenormalization groupanalysis of the small-world network modelrdquo Physics Letters AGeneral Atomic and Solid State Physics vol 263 no 4ndash6 pp341ndash346 1999

[25] httpnetworkxgithubiodocumentationlatest

[26] A Hagberg and D Conway ldquoHacking social networks usingthe Python programming language (module IImdashwhy do SNAin NetworkX)rdquo in Proceedings of the International Network forSocial Network Analysis Sunbelt Conference 2010

[27] ldquoPython 340 beta 3rdquo Python Software Foundation 2014[28] A Wilhite ldquoBilateral trade and ldquosmall-worldrdquo networksrdquo Com-

putational Economics vol 18 no 1 pp 49ndash64 2001[29] A Marshall Principles of Economics An Introductory Volume

Macmillan London UK 8th edition[30] J E Ingersoll JrTheory of Financial Decision Making Rowman

and Littlefield Totowa NJ USA 1987[31] DW Caves and L R Christensen ldquoGlobal properties of flexible

functional formsrdquo The American Economic Review vol 70 no3 pp 422ndash432 1980

[32] A Feldman ldquoBilateral trading processes pairwise optimalityand Pareto optimalityrdquo Review of Economic Studies vol 40 no4 pp 463ndash473 1973

[33] A Sen ldquoMarkets and freedoms achievements and limitationsof the market mechanism in promoting individual freedomsrdquoOxford Economic Papers vol 45 no 4 pp 519ndash541 1993

[34] J H Ahrens and U Dieter ldquoComputer methods for samplingfrom gamma beta poisson and binomial distributionsrdquo Com-puting vol 12 no 3 pp 223ndash246 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 10 20 30 40

000

005

010

015

020

(a) gs = 80 rp = 001

0 10 20 30 40

000

005

010

015

020

(b) gs = 80 rp = 002

0 10 20 30 40

000

005

010

015

020

(c) gs = 80 rp = 003

0 10 20 30 40

000

005

010

015

020

(d) gs = 80 rp = 004

0 10 20 30 40

000

005

010

015

020

(e) gs = 80 rp = 005

0 10 20 30 40

000

005

010

015

020

(f) gs = 80 rp = 006

0 10 20 30 40

000

005

010

015

020

(g) gs = 80 rp = 007

0 10 20 30 40

000

005

010

015

020

(h) gs = 80 rp = 008

0 10 20 30 40

000

005

010

015

020

(i) gs = 80 rp = 009

0 10 20 30 40

000

005

010

015

020

(j) gs = 80 rp = 01

Figure 5 The pattern of UGR when gs = 80 and rp increases linearly

0 10 20 30000

005

010

015

020

(a) gs = 60 rp = 001

0 10 20 30 40 50

000

005

010

015

020

(b) gs = 120 rp = 005

0 10 20 30 40 50 60

000

005

010

015

020

(c) gs = 150 rp = 008

0 20 40 60 80

000

005

010

015

020

(d) gs = 170 rp = 01

000

005

010

015

020

0 20 40 60 80 100

(e) gs = 200 rp = 02

Figure 6 The pattern of UGR when gs rp and nn increase linearly

gradually Although it may not be a monotonic decreasingthe decreasing tendency would be kept

When constructing the systems in Figures 5 and 6 wealso consider that the possible effect on hp and UGR maybe exerted by the endowments to goods 119892

1and 119892

2 The fixed

group size systems in Figure 5 are endowed with randomlygenerated 119892

1and 119892

2and in Figure 5 different group size

systems are discussed which means that their endowmentscannot keep consistent with each otherThe result shows thatthe UGR varies on account of the endowments and networkstructure but the range of hp cannot be influenced

5 Conclusion

This research combines economic theory network theoryand computer simulations to examine the parameters setassigned to the WS model relating to an economic systemrsquosUGR We discovered that regardless of the group size andrewiring probability (1) 2 to 18 neighbors and the corre-sponding computable number of edges can help an economicsystem reach the highest UGR (2) if the node has morethan 18 neighbors the UGR would decrease gradually andthe pattern is not monotonic but would keep the decreasingtendency (3) different endowments to goods119892

1and1198922would

not impact the range of neighbors which can help the systemachieve the highest UGR In the practical application itcan be construed that (1) acting as a measurement judgeswhether a system can reach the highest UGR based on its

current structure (2) when the resource is in the hands of afraction of people the resource can derive the highest valuefor the owners and the rest of people (3) once keeping thesuperiority of minority owners the value of resource wouldnot decrease or increase according to its quantity

Conflict of Interests

The authors declare 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] W-K Chen Graph Theory and Its Engineering ApplicationsWorld Scientific Singapore 1997

[3] P Erdos and A Renyi ldquoOn random graphs Irdquo PublicationesMathematicae vol 6 pp 290ndash297 1959

[4] D J Watts ldquoNetworks dynamics and the small-world phe-nomenonrdquo The American Journal of Sociology vol 105 no 2pp 493ndash527 1999

[5] A F Alexander-Bloch P E Vertes R Stidd et al ldquoThe anatom-ical distance of functional connections predicts brain networktopology in health and schizophreniardquo Cerebral Cortex vol 23no 1 pp 127ndash138 2013

[6] R Celine and M Guy ldquoA small world perspective on urbansystemsrdquo inMethods for Multilevel Analysis and Visualisation ofGeographical Networks vol 11 pp 19ndash32 2013

Mathematical Problems in Engineering 7

[7] G Corso L S Lucena and Z D Thome ldquoThe small-worldof economy a speculative proposalrdquo Physica A StatisticalMechanics and Its Applications vol 324 no 1-2 pp 430ndash4362003

[8] K Musiał and P Kazienko ldquoSocial networks on the internetrdquoWorld Wide Web vol 16 no 1 pp 31ndash72 2013

[9] X Cheng J Liu and C Dale ldquoUnderstanding the char-acteristics of internet short video sharing a youtube-basedmeasurement studyrdquo IEEE Transactions on Multimedia vol 15no 5 pp 1184ndash1194 2013

[10] D Eppstein M T Goodrich M Loffler D Strash and LTrott ldquoCategory-based routing in social networks member-ship dimension and the small-world phenomenonrdquoTheoreticalComputer Science vol 514 pp 96ndash104 2013

[11] D Cesar and L Igor ldquoStructure and dynamics of transportationnetworks models methods and applicationsrdquo in The SAGEHandbook of Transport Studies pp 347ndash364 2013

[12] J Kleinberg ldquoThe small-world phenomenon an algorithm per-spectiverdquo in Proceedings of the 32nd Annual ACM SymposiumonTheory of Computing pp 163ndash170 2000

[13] S Milgram ldquoThe small-world problemrdquo Psychology Today vol2 pp 60ndash67 1967

[14] A Kemper Valuation of Network Effects in Software MarketsA Complex Networks Approach Springer Heidelberg Germany2010

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

[16] T John P Theodore and L Terry ldquoSupply chain managementand its relationship to logistics marketing production andoperations managementrdquo Journal of Business Logistics vol 29no 1 pp 31ndash46 2008

[17] B Ben and G Mark ldquoFinancial fragility and economic perfor-mancerdquo The Quarterly Journal of Economics vol 105 no 1 pp87ndash114 1990

[18] GW Torrance ldquoUtilitymeasurement in healthcare the things Inever got tordquo PharmacoEconomics vol 24 no 11 pp 1069ndash10782006

[19] A Birati andA Tziner ldquoEconomic utility of training programsrdquoJournal of Business and Psychology vol 14 no 1 pp 155ndash1641999

[20] C Li and G Chen ldquoStability of a neural network modelwith small-world connectionsrdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 68 no 5 Article ID052901 2003

[21] X Li and X Wang ldquoControlling the spreading in small-worldevolving networks stability oscillation and topologyrdquo IEEETransactions on Automatic Control vol 51 no 3 pp 534ndash5402006

[22] M O Jackson and AWolinsky ldquoA strategic model of social andeconomic networksrdquo Journal of Economic Theory vol 71 no 1pp 44ndash74 1996

[23] Y A Ioannides ldquoEvolution of trading structuresrdquo in TheEconomy as an Evolving Complex System II W B Arthur SDurlauf and D Lane Eds pp 129ndash168 1997

[24] M E J Newman and D J Watts ldquoRenormalization groupanalysis of the small-world network modelrdquo Physics Letters AGeneral Atomic and Solid State Physics vol 263 no 4ndash6 pp341ndash346 1999

[25] httpnetworkxgithubiodocumentationlatest

[26] A Hagberg and D Conway ldquoHacking social networks usingthe Python programming language (module IImdashwhy do SNAin NetworkX)rdquo in Proceedings of the International Network forSocial Network Analysis Sunbelt Conference 2010

[27] ldquoPython 340 beta 3rdquo Python Software Foundation 2014[28] A Wilhite ldquoBilateral trade and ldquosmall-worldrdquo networksrdquo Com-

putational Economics vol 18 no 1 pp 49ndash64 2001[29] A Marshall Principles of Economics An Introductory Volume

Macmillan London UK 8th edition[30] J E Ingersoll JrTheory of Financial Decision Making Rowman

and Littlefield Totowa NJ USA 1987[31] DW Caves and L R Christensen ldquoGlobal properties of flexible

functional formsrdquo The American Economic Review vol 70 no3 pp 422ndash432 1980

[32] A Feldman ldquoBilateral trading processes pairwise optimalityand Pareto optimalityrdquo Review of Economic Studies vol 40 no4 pp 463ndash473 1973

[33] A Sen ldquoMarkets and freedoms achievements and limitationsof the market mechanism in promoting individual freedomsrdquoOxford Economic Papers vol 45 no 4 pp 519ndash541 1993

[34] J H Ahrens and U Dieter ldquoComputer methods for samplingfrom gamma beta poisson and binomial distributionsrdquo Com-puting vol 12 no 3 pp 223ndash246 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

[7] G Corso L S Lucena and Z D Thome ldquoThe small-worldof economy a speculative proposalrdquo Physica A StatisticalMechanics and Its Applications vol 324 no 1-2 pp 430ndash4362003

[8] K Musiał and P Kazienko ldquoSocial networks on the internetrdquoWorld Wide Web vol 16 no 1 pp 31ndash72 2013

[9] X Cheng J Liu and C Dale ldquoUnderstanding the char-acteristics of internet short video sharing a youtube-basedmeasurement studyrdquo IEEE Transactions on Multimedia vol 15no 5 pp 1184ndash1194 2013

[10] D Eppstein M T Goodrich M Loffler D Strash and LTrott ldquoCategory-based routing in social networks member-ship dimension and the small-world phenomenonrdquoTheoreticalComputer Science vol 514 pp 96ndash104 2013

[11] D Cesar and L Igor ldquoStructure and dynamics of transportationnetworks models methods and applicationsrdquo in The SAGEHandbook of Transport Studies pp 347ndash364 2013

[12] J Kleinberg ldquoThe small-world phenomenon an algorithm per-spectiverdquo in Proceedings of the 32nd Annual ACM SymposiumonTheory of Computing pp 163ndash170 2000

[13] S Milgram ldquoThe small-world problemrdquo Psychology Today vol2 pp 60ndash67 1967

[14] A Kemper Valuation of Network Effects in Software MarketsA Complex Networks Approach Springer Heidelberg Germany2010

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

[16] T John P Theodore and L Terry ldquoSupply chain managementand its relationship to logistics marketing production andoperations managementrdquo Journal of Business Logistics vol 29no 1 pp 31ndash46 2008

[17] B Ben and G Mark ldquoFinancial fragility and economic perfor-mancerdquo The Quarterly Journal of Economics vol 105 no 1 pp87ndash114 1990

[18] GW Torrance ldquoUtilitymeasurement in healthcare the things Inever got tordquo PharmacoEconomics vol 24 no 11 pp 1069ndash10782006

[19] A Birati andA Tziner ldquoEconomic utility of training programsrdquoJournal of Business and Psychology vol 14 no 1 pp 155ndash1641999

[20] C Li and G Chen ldquoStability of a neural network modelwith small-world connectionsrdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 68 no 5 Article ID052901 2003

[21] X Li and X Wang ldquoControlling the spreading in small-worldevolving networks stability oscillation and topologyrdquo IEEETransactions on Automatic Control vol 51 no 3 pp 534ndash5402006

[22] M O Jackson and AWolinsky ldquoA strategic model of social andeconomic networksrdquo Journal of Economic Theory vol 71 no 1pp 44ndash74 1996

[23] Y A Ioannides ldquoEvolution of trading structuresrdquo in TheEconomy as an Evolving Complex System II W B Arthur SDurlauf and D Lane Eds pp 129ndash168 1997

[24] M E J Newman and D J Watts ldquoRenormalization groupanalysis of the small-world network modelrdquo Physics Letters AGeneral Atomic and Solid State Physics vol 263 no 4ndash6 pp341ndash346 1999

[25] httpnetworkxgithubiodocumentationlatest

[26] A Hagberg and D Conway ldquoHacking social networks usingthe Python programming language (module IImdashwhy do SNAin NetworkX)rdquo in Proceedings of the International Network forSocial Network Analysis Sunbelt Conference 2010

[27] ldquoPython 340 beta 3rdquo Python Software Foundation 2014[28] A Wilhite ldquoBilateral trade and ldquosmall-worldrdquo networksrdquo Com-

putational Economics vol 18 no 1 pp 49ndash64 2001[29] A Marshall Principles of Economics An Introductory Volume

Macmillan London UK 8th edition[30] J E Ingersoll JrTheory of Financial Decision Making Rowman

and Littlefield Totowa NJ USA 1987[31] DW Caves and L R Christensen ldquoGlobal properties of flexible

functional formsrdquo The American Economic Review vol 70 no3 pp 422ndash432 1980

[32] A Feldman ldquoBilateral trading processes pairwise optimalityand Pareto optimalityrdquo Review of Economic Studies vol 40 no4 pp 463ndash473 1973

[33] A Sen ldquoMarkets and freedoms achievements and limitationsof the market mechanism in promoting individual freedomsrdquoOxford Economic Papers vol 45 no 4 pp 519ndash541 1993

[34] J H Ahrens and U Dieter ldquoComputer methods for samplingfrom gamma beta poisson and binomial distributionsrdquo Com-puting vol 12 no 3 pp 223ndash246 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of