FLOW OPTIMIZATIONIN COMPLEX NETWORKS

By

Rui Huang

A Thesis Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the Degree of

MASTER OF SCIENCE

Major Subject: COMPUTER SCIENCE

Approved:

Boleslaw K. Szymanski, Thesis Adviser

Gyorgy Korniss, Thesis Adviser

Rensselaer Polytechnic InstituteTroy, New York

May 2010(For Graduation May 2010)

c© Copyright

by

Rui Huang

All Rights Reserved

ii

CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. BACKGROUND AND RELATED PROBLEMS . . . . . . . . . . . . . . . 3

2.1 Complex networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Scale-free networks: The Barabasi-Albert model . . . . . . . . 5

2.1.3 Random Geometric Graphs (RGG) . . . . . . . . . . . . . . . 7

2.1.4 Small-world networks . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 EW Synchronization problem . . . . . . . . . . . . . . . . . . 10

2.2.2 Optimizing resistance in weighted resistor networks . . . . . . 12

2.2.3 Connection with random walks . . . . . . . . . . . . . . . . . 13

3. FLOW AND TRANSPORT IN COMPLEX NETWORKS . . . . . . . . . 16

3.1 Technical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Identical source/target rate case . . . . . . . . . . . . . . . . . 17

3.2.1.1 Barabasi-Albert Scale-free networks . . . . . . . . . . 18

3.2.1.2 Erdos and Renyi networks . . . . . . . . . . . . . . . 21

3.2.1.3 Random geometric graphs . . . . . . . . . . . . . . . 21

3.2.2 Heterogeneous source/target rate case . . . . . . . . . . . . . 25

3.2.2.1 Barabasi-Albert Scale-free networks . . . . . . . . . . 27

3.2.2.2 Random geometric graphs . . . . . . . . . . . . . . . 28

4. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

LITERATURE CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

iii

LIST OF FIGURES

2.1 Degree distribution of BA network with m = 5 and N = 1000. Theaverage degree here is 〈k〉 = 10. The solid line shows the power-lawbehavior in log-log scale with the power α = 0.3. From G. Korniss [17]. 6

2.2 Degree distribution of a random geometric network with 10, 000 nodesand the average degree 〈k〉 = 10. The solid line is a Poisson distribution.From Q. Lu [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Degree distribution of a random small-world network. The network isconstructed by adding pN/2 random links with p = 0.08 on top of a1d ring of size N = 100. The resulting network has the average degree〈k〉 = 10. The solid line is a Poisson distribution. From G. Korniss [17]. 10

3.1 Maximum current flow for different system size N and different β in theBA network with m = 5 and average degree 〈k〉 = 10. Inset shows thethroughput vs β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Average current flow for different system size N and different β in theBA network with m = 5 and average degree 〈k〉 = 10. . . . . . . . . . . 19

3.3 Current flow vs degree for β = −1 and β = 0 in the BA network withm = 5, average degree 〈k〉 = 10 and system size N = 100. . . . . . . . . 19

3.4 Normalized distribution of the current flow for β = −1 and β = 0 inthe BA network with m = 5, average degree 〈k〉 = 10 and system sizeN = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Maximum current flow for different system size N and different β inthe ER network (long dashed lines) and BA network (solid lines) withsame average degree 〈k〉 = 10. Inset shows the throughput vs β. . . . . 22

3.6 Average current flow for different system size N and different β in theER network (long dashed lines) and BA network (solid lines) with sameaverage degree 〈k〉 = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.7 Distribution of the current flow for β = −1 and β = 0 in the ER networkwith average degree 〈k〉 = 10 and N = 100. . . . . . . . . . . . . . . . . 23

3.8 Maximum current flow for different system size N and different β inthe RGG network with average degree 〈k〉 = 10 (solid line) and 〈k〉 = 8(long dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

iv

3.9 Maximum current flow for different system size N and different β inthe RGG SW network with average degree 〈k〉 = 10: average degree ofRGG is 10 (solid line), average degree of RGG is 9 with average degreeof added SW links is 1 (dashed line), average degree of RGG is 8 withaverage degree of added SW links is 2 (dashed line). Inset shows thethroughput vs β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.10 Average current flow for different system size N and different β in theRGG SW network with average degree 〈k〉 = 10: average degree ofRGG is 10 (solid line), average degree of RGG is 9 with average degreeof added SW links is 1 (dashed line), average degree of RGG is 8 withaverage degree of added SW links is 2 (dashed line). . . . . . . . . . . . 25

3.11 Maximum current flow for different system size N and different β inthe BA network with ρ = 1, m = 5 and average degree 〈k〉 = 10. Insetshows the throughput vs β. . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.12 Average current flow for different system size N and different β in theBA network with ρ = 1, m = 5 and average degree 〈k〉 = 10. . . . . . . 26

3.13 Current flow vs degree for β = −2, β = −1 and β = 0 in the BAnetwork with ρ = 1, m = 5, average degree 〈k〉 = 10 and system sizeN = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.14 Normalized distribution of the current flow for β = −2, β = −1 andβ = 0 in the BA network with ρ = 1, m = 5, average degree 〈k〉 = 10and system size N = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.15 Maximum current flow for different ρ in the BA network with β = 0,β = 1, m = 5, average degree 〈k〉 = 10 and system size N = 100. . . . . 29

3.16 Average current flow for different ρ in the BA network with β = 0,β = 1, m = 5, average degree 〈k〉 = 10 and system size N = 100. . . . . 29

3.17 Maximum current flow for different system size N and different β in theRGG network with ρ = 1 and average degree 〈k〉 = 10. . . . . . . . . . . 30

3.18 Average current flow for different system size N and different β in theRGG network with ρ = 1 and average degree 〈k〉 = 10. . . . . . . . . . . 30

v

ACKNOWLEDGMENT

I would like to thank Professor Boleslaw Szymanski for being helpful and support

to this thesis work. As a leading researcher in network science, Professor Szymanski

has assisted me in numerous ways. The discussions I had with him were invaluable.

Next, I thank Professor Gyorgy Korniss for many insightful conversations and help-

ful suggestions. It is my pleasure to work with him. I would also like to thank my

group members: Qiming Lu, David Hunt and Jierui Xie for your valuable advice

and friendship. It is a great time for me to work in this nice group.

My parents have been an inspiration throughout my life. They have always

supported my dreams and aspirations. I would like to thank them for all they are,

and all they have done for me. I also wish to thank my wife Zheng Xue for her

unlimited support during my research.

I acknowledge the financial support of the National Science Foundation Grant

No. DMR-0426488, DTRA Award No. HDTRA1-09-1-0049.

vi

ABSTRACT

Transport and flow on complex network have attracted lots of attention because

of their extensive applications to biological, transportation, communication and in-

frastructure networks. Recently, simple resistor networks were utilized to study

transport efficiency in scale-free and small-world networks.

In this Master Thesis, we investigate and characterize the statistics of the ex-

tremes in correlated load landscapes in the complex networks. Four network models:

scale-free network, Erdos-Renyi random graph, random geometric graph and small-

world network, are utilized here. Each of them could mimic certain properties of

real-world networks. We consider a specific form of the weights, where the strength

of a link is proportional to (kikj)β with ki and kj being the degrees of the nodes

connected by the link. We also add parameter ρ to control the probability for each

node to become either the source or target. Exact numerical diagonalization based

method and computational codes (for weighted network Laplacians) are employed

to extract flow-based load. Mainly, we focus on two important observables, the

maximal current flow and the average current flow in the network. Numerical re-

sults show that the optimal value of β for the maximum current flow is close to

−1 for homogenous source/target rate. Further, this optimal value can change for

different ρ depending on the network topology. Those results could help understand

the network vulnerability problem and thus further the future work on cascading

network failures.

vii

1. INTRODUCTION

Resistor networks have been widely studied since the 70’s as models for conductivity

problems and classical transport in disordered media [1]. With the recent surge of

research on complex networks [3], resistor-networks and related flow models have

been employed to study and explore community structures in social networks [4] and

to construct recommendation models for community networks [8]. Also, resistor

networks, as abstract models for network flows with a fundamental conservation

law [9], were utilized to study transport efficiency in complex networks [12, 13,

10, 14, 15, 16]. The disorder-averaged two-point resistance is obtained utilizing

analytic results for the propagator of the Edwards-Wilkinson (EW) process on small-

world networks [15]. The work by Lopez et al. [13] revealed that in scale-free (SF)

networks anomalous transport properties can emerge, displayed by the power-law

tail of distribution of the network conductance.

Previous works employed resistor networks and current flow models to charac-

terize betweenness in social networks, and in turn, to identify community structures

[4, 5, 6, 7]. Here, we utilize resistor networks and current flows to explore fundamen-

tal transport characteristics of distributed flows in networks [13, 15]. In particular,

we study flow optimization (minimizing the largest load, or equivalently, maximizing

throughput in the network) by employing weighted links in complex and spatially

embedded networks.

In this Master Thesis, we investigate and characterize the statistics of the ex-

tremes in correlated load landscapes in the complex networks. Here, we consider a

specific and symmetric form of the weights on different networks, being proportional

to (kikj)β where ki and kj are the degrees of the nodes connected by the link. The

above general form has been suggested by empirical studies of metabolic [33] and air-

line transportation networks [34]. Meanwhile, we add another parameter ρ to control

the probability for each node to become either the source or target. Source/target

rate will be proportional to (kikj)ρ. We ask what is the optimal allocation of the

weights for a given source/target rate (controlled by β and ρ, respectively), in order

1

2

to maximize throughput in the networks. Because both the EW process and resis-

tor networks are governed by the same network Laplacian, we could also related our

problems with the synchronization problems in noisy environments. Connections

between random walks and resistor networks have been discussed in several works

[35, 36]. The EW synchronization problems will be explained in detail in Chapter

2.

In this work, we employ four different complex structure models: scale-free

network, Erdos-Renyi random graph, random geometric graph and small-world net-

work, to study the properties of the current flow and transport in these networks.

We use Numerical Recipes [32] for finding the eigenvalues and eigenvectors of the

underlying network Laplacians.

2. BACKGROUND AND RELATED PROBLEMS

2.1 Complex networks

Complex networks describe a wide variety of the natural and social systems,

e.g. the World Wide Web, phone call networks, neural networks. Understanding

such systems has become significantly important but difficult. While traditionally

random graph theory, first modeled by Paul Erdos and Alfred Renyi [20, 21, 22],

was found useful in tackling these problems, it has been increasingly recognized that

real networks are “random”, with some organizing principles embedded. Motivated

by the growing interest in complex systems and dramatic advances in computing

technology, many new network models have been proposed and investigated in the

past few years. Among them, small-world network and scale-free network are two

widely used models for studying the properties of complex networks. In what follows,

we briefly describe some important aspects of those models which are of direct

relevance to complex networks [19].

2.1.1 Random Graphs

A random graph is generated by some random processes. Erdos and Renyi

[20] defines a random graph as N labeled nodes connected by n edges, which are

chosen randomly from the N(N − 1)/2 possible edges. The binomial model gives

an alternative way to define a random graph. Starting with N nodes, every pair

of nodes will be connected with probability p. Hence the total number of edges is

E(n) = p[N(N − 1)/2]. Random-graph theory studies the properties of probability

space associated with graphs with N nodes as N → ∞.

In the graph construction process, with larger and larger probabilities p, the

graph obtained will eventually become a fully connected graph (having the maximum

number of edges n = N(N − 1)/2) for p → 1. During this process, one of the most

significant discovery of Erdos and Renyi was that almost every graph generated

either has some property Q (e.g. every pair of nodes is connected by a path of

consecutive edges) or, conversely, almost no graph has it, at a given probability p

3

4

[3]. There is a critical probability pc(N) where the probability that the graph has

the property Q satisfies

limN→∞

PN,p(Q) =

⎧⎨⎩

0 if p(N)pc(N)

→ 0

1 if p(N)pc(N)

→ ∞. (2.1)

The degree distribution is an important property of graphs. In the random

graph with probability p, the degree ki of a node i follows a binomial distribution

with parameters N − 1 and p:

P (ki = k) =

(N − 1

k

)pk(1 − p)N−1−k , (2.2)

with the average degree 〈k〉 = (N − 1)p. Furthermore, in the limit of large network

size, the distribution approaches a Poisson distribution,

P (k) � e−〈k〉 〈k〉kk!

, (2.3)

where 〈k〉 � Np.

To characterize the spread of a graph, one way is to calculate the average

distance l between any pair of nodes (average path length). In random graph, it

varies logarithmically with the number of nodes:

l ∼ ln(N)

ln(pN)=

ln(N)

ln(〈k〉) . (2.4)

Eq. (2.4) gives a reasonable estimate of the average path length comparing with

real-world networks.

The clustering coefficient is another fundamental measure which assesses the

degree to which nodes tend to cluster together. The definition of the clustering

coefficient Ci for a vertex is the proportion of links between the vertices within

its neighborhood divided by the number of links that could possibly exist between

them,

Ci =Ei

ki(ki−1)2

, (2.5)

5

where ki is the degree of the vertex and Ei gives the actual edges among its neigh-

borhood. Most real-world networks exhibit a large degree of clustering. In the

random graph, the probability that two of the neighbors are connected is equal to

the probability that two randomly selected nodes are connected. Consequently, the

clustering coefficient is

C = 〈Ci〉 =〈k〉N

. (2.6)

The fraction C/〈k〉 decreases as N−1. However, many real-world networks do not

follow the prediction based on a random graph. The fraction is independent of N

[3, 19].

2.1.2 Scale-free networks: The Barabasi-Albert model

The degree distribution of real-world networks often follows a power law P (k) ∼k−γ. These are so-called “scale-free networks” [2]. In the following section, we will

introduce one widely used “scale-free” model: the Barabasi-Albert (BA) model.

Although it is possible to construct random graphs that have a power-law

degree distribution, Barabasi and Albert introduced a new model based on two

more realistic characteristics of real networks, growth and preferential attachment.

The algorithm for constructing the BA model is as follows [3]:

1. Growth: Starting with a small number (m0) of fully connected nodes, at every

time step, a new node will be added with m (≤ m0) edges that link the new

node to m different node already present in the system.

2. Preferential attachment: When choosing the nodes to which the new node

connects, assume that the probability Π that a new node will be connected to

node i depends on the degree ki of node i, such that Π(ki) = ki∑j kj

.

A network with N = t+m0 nodes and mt+ m0(m0−1)2

edges will be built after

t time steps above procedure. The dynamical properties of the scale-free model can

be analyzed using various approaches. The continuum theory proposed by Barabasi

and Albert proves that γ becomes asymptotically independent of m and t in the

large-t limit [2].

6

100

101

102

k

10−6

10−5

10−4

10−3

10−2

10−1

100

P(k

)

Figure 2.1: Degree distribution of BA network with m = 5 and N = 1000.The average degree here is 〈k〉 = 10. The solid line shows thepower-law behavior in log-log scale with the power α = 0.3.From G. Korniss [17].

While the BA model captures the power-law tail of the degree distribution

[Fig. 2.1], it has other properties needed to be addressed, e.g. average path length

and clustering coefficient.

Barabasi and Albert showed that the average path length in a BA network is

smaller than in a random graph for any system size N . It is also found that the

average path length of the BA network increases approximately logarithmically with

N , and the best fit following a generalized logarithmic form

l = A ln(N −B) + C . (2.7)

While there is no analytical prediction of the clustering coefficient for the BA

network, it is numerically studied, showing that the clustering coefficient of the

scale-free network follows approximately a power law C ∼ N−0.75 [3], a slower decay

than C = 〈k〉N−1 observed for random graphs [19].

7

2.1.3 Random Geometric Graphs (RGG)

Random geometric graphs have been used extensively in real networks mod-

eling and continuum percolation. For higher dimensions, they approach standard

random graphs. However, in some aspects, they are different from random graphs

[29, 30].

A random geometric graph is constructed by randomly distributing N nodes

in the box [0, 1]d. Two nodes are connected only if they fall within each other’s

range R. One key parameter in the graph is the average degree, or connectivity

α = 〈k〉 = 2K/N , where K is the total number of links. In random geometric

graphs, the critical connectivity αc should be reached so that the size of the largest

cluster becomes comparable to the total number of nodesN . In 2d random geometric

graphs, the critical connectivity is about αc � 4.52. Moreover, there exists a direct

relation between the degree α and the range R

α = πNR . (2.8)

Figure 2.2: Degree distribution of a random geometric network with10, 000 nodes and the average degree 〈k〉 = 10. The solid lineis a Poisson distribution. From Q. Lu [18].

8

Random geometric graphs also have a binomial degree distribution, which

approaches Poisson distribution for large N [Fig. 2.2] [30]

P (k) =

(N − 1

k

)pk(1 − p)N−1−k � e−αα

k

k!. (2.9)

The clustering coefficient C of random geometric graph could be analytically

derived in arbitrary dimensions d [30] as

C =

⎧⎨⎩

1 −Hd(1) for even d

32−Hd(

12) for odd d

, (2.10)

where

Hd(x) =1√π

d2∑

i=x

Γ(i)

Γ(i+ 12)(3

4)i+1/2 . (2.11)

When d = 2, Eq. (2.10) reduces to

C =1

π(3

4)3/2 . (2.12)

It is clear from Eq. (2.10) that the clustering coefficient is a function depending

only on the dimension d. Compared with C = 〈k〉/N = α/N in random graph, it is

inappropriate to believe the statement that random geometric graphs are identical

to random graphs [19].

2.1.4 Small-world networks

As mentioned before, pure random graphs exhibit a small average path length

along with a small clustering coefficient. Watts and Strogatz [23] found key mea-

surements that in fact many real-world networks have a small average path length,

and a clustering coefficient independent of the network size. Especially in ordered

lattices, clustering coefficient depends only on the coordination number. Watts and

Strogatz then proposed a one-parameter graph model that interpolates between an

ordered lattice and a random graph, now named the Watts-Strogatz (WS) model,

with high clustering coefficient and small average path length.

Several procedures are known to generate small-world networks. These proce-

9

dures start with an ordered lattice and randomly rewire existing links [23] (or add

new links [24]) between lattice nodes with probability p such that self-connections

and duplicate edges are excluded.

Next, let us focus on some main results regarding the properties of small-world

models. First, the average path length l changes as the fraction p of the rewired

edges is increased. For small p, l scales linearly with the system size, while for large

p the scaling is logarithmic. In general, l obeys the scaling form [25]

l(N, p) ∼ N1/d

Kf(pKN) , (2.13)

where f(u) is a universal scaling function that obeys

f(u) =

⎧⎨⎩

const if u� 1

ln(u)/u if u� 1. (2.14)

Here d is the dimension of the original lattice to which the random edges are added

and K is the number of neighbors in the ordered lattice.

In addition to a short average path length, small-world networks have a rela-

tively high clustering coefficient. In a regular lattice (p = 0), the clustering coeffi-

cient does not depend on the size of lattice but only on its topology. As the links

of the network are randomized, the clustering coefficient C(p) will be much greater

than C(1) (p = 1 corresponds to a random graph) over a broad interval of p [3].

In the WS model, for p = 0, each node has the same degree K. After intro-

ducing some randomness into the network, average degree still is maintained as K

while the degree distribution will be broaden. It has a pronounced peak at K and

decay exponentially [Fig. 2.3] [19].

10

0 20 40 60k

0

0.05

0.1

P(k

)

Figure 2.3: Degree distribution of a random small-world network. Thenetwork is constructed by adding pN/2 random links withp = 0.08 on top of a 1d ring of size N = 100. The resultingnetwork has the average degree 〈k〉 = 10. The solid line is aPoisson distribution. From G. Korniss [17].

2.2 Related problems

2.2.1 EW Synchronization problem

The EW process on a network is given by the Langevin equation

∂thi = −N∑

j=1

Cij(hi − hj) + ηi(t) , (2.15)

where hi(t) is the general stochastic field variable on a node (such as fluctuations in

the task-completion landscape in certain distributed parallel schemes on computer

networks [11, 31]) and ηi(t) is a delta-correlated noise with zero mean and variance

〈ηi(t)ηj(t′)〉 = 2δijδ(t− t′). Here, Cij = Cji > 0 is the symmetric coupling strength

between the nodes i and j (Cii ≡ 0). Defining the network Laplacian,

Γij = δij∑l �=i

Cil − Cij , (2.16)

11

one could have the steady-state equal-time two-point correlation function

Gij ≡ 〈(hi − h)(hj − h)〉 = Γ−1ij =

N−1∑k=1

1

λk

ψkiψkj , (2.17)

where h = (1/N)∑N

i=1 hi and 〈· · ·〉 denotes an ensemble average over the noise.

Here, Γ−1ij denotes the inverse of Γ in the space orthogonal to the zero mode. Also,

{ψki}Ni=1 and λk, k = 0, 1, ..., N − 1, denote the kth normalized eigenvectors and the

corresponding eigenvalues, respectively. The averaged steady-state spread or width

in the synchronization landscape can be written as [31]

〈ω2〉 ≡ 〈 1

N

N∑i=1

(hi − h)2〉 =1

N

N∑i=1

Gii =1

N

N−1∑i=1

1

λk

. (2.18)

Given the form of weights Wij ∝ (kikj)β and the fixed total cost constraint∑

i<j Cij = 12Ctot, the coupling strength between nodes i and j will be

Cij = NkAij(kikj)

β∑l,nAln(klkn)β

, (2.19)

where k =∑

i ki/N =∑

i,j Aij/N is the mean degree of the graph. The problem

then becomes to find the optimum among all networks with fixed cost, for which

the EW synchronization problem yields the minimum width.

For uncorrelated weighted random graphs, Eq. (2.19) can be approximated as

Ci ≈ 〈k〉 kβ+1i∫ ∞

mdk′k′β+1P (k′)

, (2.20)

where P (k) is the degree distribution. Here, SF degree distributions is employed,

P (k) = (γ − 1)mγ−1k−γ , (2.21)

where m is the minimum degree in the network and 2 ≤ γ ≤ 3. The average and

12

the minimum degree are related through 〈k〉 = m(γ − 1)/(γ − 2). Thus one finds

Ci ≈ γ − 2 − β

γ − 2

kβ+1i

mβ. (2.22)

The width of the synchronization landscape can be obtained,

〈ω2(β)〉 =1

N

N∑i=1

〈(hi − h)2〉 ≈ 1

N

∑ 1

Ci

=1

〈k〉(γ − 1)2

(γ − 2 − β)(γ + β).

(2.23)

From the above expression, one can get that 〈ω2(β)〉 reaches minimum at β = −1

with value 〈ω2〉min = 1/〈k〉. Numerical results, based on exact numerical diagonal-

ization of the corresponding network Laplacian, confirm the mean-field results, with

small corrections to the optimal value of β = −1 [10].

2.2.2 Optimizing resistance in weighted resistor networks

Considering an arbitrary (connected) network where Cij is the conductance of

the link between node i and j, one obtains

∑j

Cij(Vi − Vj) = I(δis − δit) , (2.24)

Nodes s and t are the nodes where a current I enters and leaves the network,

respectively. The above equation can be further rewritten as

∑j

LijVj = I(δis − δit) , (2.25)

where Lij is the same network Laplacian as introduced earlier. Then by introducing

the voltages measured from the mean at each node, Vi = Vi − V and employing the

inverse of L, one has

Vi = I(Gis −Git) . (2.26)

13

Here node s and t serve as source and target. If we change them to node i and j,

one immediately finds

V = Vi − Vj = Vi − Vj = I(Gii +Gjj − 2Gij) . (2.27)

For the equivalent two-point resistance, one finally obtains [15]

R ≡ V

I= (Gii +Gjj − 2Gij) =

N−1∑k=1

(ψ2ki + ψ2

kj − 2ψkiψkj) . (2.28)

Comparing Eqs. (2.17) and (2.28), the two-point resistance of a network is

the same as the steady-state height-difference correlation function of EW process

on the network,

〈(hi − hj)2〉 = Gii +Gjj − 2Gij = Rij . (2.29)

This relation could directly lead to R ≡ 1N(N−1)

∑i�=j Rij = 2〈w2〉 by summing up

over all i �= j pairs. This relationships are exact and valid for any graph. The

corresponding optimization problem for β-controlled weight resistor networks then

could be solved as follows: β = −1 and R = 2N/[(N − 1)〈k〉] � 2/〈k〉 in the

mean-filed approximation on uncorrelated random SF networks. Numerical results

for R(β) could also prove above statements [10].

2.2.3 Connection with random walks

Random walk problem has been studied for many years in probability theory.

A random walk (RW) can be defined with transition probabilities [36]

Pij ≡ Cij

Ci

, (2.30)

where Ci =∑

l Cil. Pij is the probability that the walker will hop from node i to

node j in the next step. Recall the weight Cij ∝ Aij(kikj)β, one can have

Pij =Aij(kikj)

β∑l Ail(kikl)β

=Aijk

βj∑

l Ailkβl

. (2.31)

14

Here any normalization prefactor associated with the conserved cost cancels out.

Betweenness is a fundamental measure in network problem to capture the

amount of traffic passing through a node or a link. For RW, the observable is the

node betweenness Bi: the expected number of visits to node i for a random walker

starting at source node s before reaching target node t, Es,ti , summed over all source-

target pairs, giving the weight form (kikj)β. For a general RW, Es,t

i can be obtained

using the framework of the equivalent resistor-network problem with a unit current

flowing from s to t [36]

Es,ti = Ci(Vi − Vt) . (2.32)

Utilizing the network propagator and Eq. (2.26), one gets

Es,ti = Ci(Vi − Vt) = Ci(Gis −Git −Gts +Gtt) . (2.33)

To calculate the node betweenness, one then has

Bi =∑s �=t

Es,ti =

1

2

∑s �=t

(Es,ti + Et,s

i )

=1

2

∑s �=t

Ci(Gss +Gtt − 2Gts) =Ci

2

∑s �=t

Rst

=Ci

2N(N − 1)R .

(2.34)

This expression is valid for any weighted RW defined by the transition probability

Eq. (2.30). The node betweenness is proportional to the product of a local topologi-

cal measure, the weighted degree Ci and a global flow measure, the average network

resistance R. Consider a special case of un-weighted RW (β = 0), Ci =∑

l Ail = ki.

So the node betweenness is exactly proportional to the degree of the node. Using

the results mentioned in the earlier chapter, Eq. (2.34) can be rewritten as

Bi(β) =Ci

2N(N − 1)R = CiN

2〈ω2〉 ≈ N2 γ − 1

γ + β

k1+βi

m1+β. (2.35)

15

The average “load” of the network is

B =1

N

∑i

Bi =

∑iCi

2(N − 1)R . (2.36)

For the β = 0 case, B = kN(N − 1)R/2. Utilizing the approximations for uncorre-

lated SF graphs, one has

B(β) =

∑iCi

2(N − 1)R

= (∑

i

Ci)N〈ω2〉

≈ N2 (γ − 1)2

(γ − 2 − β)(γ + β).

(2.37)

Thus, the average RW node betweenness is also minimal for β = −1 [10].

3. FLOW AND TRANSPORT IN COMPLEX

NETWORKS

3.1 Technical Approach

To study the flow and transport in complex networks, we start with Kirchhoff’s

and Ohm’s laws, ∑j

Cij(Vi − Vj) = I(δis − δit) , (3.1)

Cij is the conductance of the link between node i and j, as defined in Chapter 2.

Applying Eq. (2.26) to voltage difference between nodes i and j, we have

Vij = Vi − Vj = Vi − Vj = I(Gis −Git −Gjs +Gjt) , (3.2)

with Gij = (Γ−1)ij. The current through link (i,j) then becomes

Istij ≡ CijVij = Cij(Gis −Git −Gjs +Gjt) , (3.3)

considering unit current flows entering (leaving) the network of node s (t) (I = 1).

Note here that if we change the direction of the current I, then Istij = −I ts

ij and∑s,t I

stij = 0.

In the network, the sum of currents flowing into one node is equal to the sum

of currents flowing out of that node, Isti = 1

2

∑j |Ist

ij |. Following this line, we finally

derive the expression of our observable: current flow (or “current load”)

li =1

N − 1

∑s,t

|Isti |

=1

2

1

N − 1

∑j

∑s �=t

|Istij | .

(3.4)

Assuming uniform processing capabilities for each node, the network is congestion-

free as long as

φli < 1 , (3.5)

16

17

for every node i [26, 28]. As the φ (network throughput per node) increased to a

certain critical value, Eq. (3.5) will be violated. The throughput of the node with

the maximum current flow becomes the upper limit of the network throughput

φc =1

lmax

. (3.6)

As discussed before, we took the same specific form of weights, where Cij ∝Aij(kikj)

β with Aij, the adjacent matrix of the underlying network. Our goal is to

find the optimal β (allocate weights) to minimize the current flow (maximize the

network throughput).

3.2 Numerical Results

3.2.1 Identical source/target rate case

We first consider the situation where all nodes have equal probability of be-

coming a source or target is equal. This is similar to this simplest local “routing”

problem [26, 27]. We also assuming that the link processing capabilities are limited

and identical. Given the above conditions, link with highest “load” will experience

failure if its “load” reaches the pre-set processing capability. Thus, the maximum

current flow lmax is a standard measure to characterize the efficiency of the networks.

Generally, one can increase the processing capabilities of the links [28], change the

underlying network topology [26], or optimize routing scheme [27], to optimize the

network throughput. In our weight form, parameter β can be controlled to dis-

tribute traffic, in order to maximize global throughput. While our main focus is on

the network throughput, the average current flow of the network 〈l〉 is investigated

also.

We have performed exact numerical diagonalization and employed Eq. (3.4) to

extract current load for a given network. Generally, we carried out 1000 realizations

for smaller-size networks, and 500 realizations for networks with N = 1000.

18

3.2.1.1 Barabasi-Albert Scale-free networks

For comparison with the above theoretical results, we considered growing net-

works: BA SF networks. For these networks, degree-degree correlations are anomaly

(marginally) weak [13].

−4 −2 0 2 4β

0

100

200

300

400

500

600

l max

N=100N=200N=400N=1000

−4 −2 0 2 4β

0

0.05

0.1

0.15

0.2

throughput

Figure 3.1: Maximum current flow for different system size N and dif-ferent β in the BA network with m = 5 and average degree〈k〉 = 10. Inset shows the throughput vs β.

The result [Fig. 3.1] shows that the critical network throughput exhibits a

maximum at β ≈ −1, as predicted using the mean-field and uncorrelated approxi-

mation. Further, in the β > −1 region, where the long tail of degree distribution

dominates the network behavior, the maximum current flow lmax in the network rises

dramatically as we increase β. Systems with more nodes also experience higher lmax.

For β < −1, nodes with high betweenness are the nodes with a low degree, more

specially, nodes with a degree of order m. Thus, the actual distribution of the flow

on those links depends strongly on the “local” fluctuations of the network disorder

(randomness of the network structure).

Fig. 3.2 shows the result for average current flow in the BA network. Clearly,

it keeps decreasing as β increases. Here β ≈ −1 is the tradeoff between optimizing

19

−4 −2 0 2 4β

2

4

6

8

10

<l>

N=100N=200N=400N=1000

Figure 3.2: Average current flow for different system size N and differentβ in the BA network with m = 5 and average degree 〈k〉 = 10.

0 10 20 30 40 50 60k

0

0.05

0.1

0.15

0.2

l

β=0β=−1

Figure 3.3: Current flow vs degree for β = −1 and β = 0 in the BAnetwork with m = 5, average degree 〈k〉 = 10 and system sizeN = 100.

20

0 0.05 0.1 0.15 0.2l

0

20

40

60

P(l

)

β=0β=−1

Figure 3.4: Normalized distribution of the current flow for β = −1 andβ = 0 in the BA network with m = 5, average degree 〈k〉 = 10and system size N = 100.

the global throughput and keeping the average “load” low.

Next, we provide more details of distribution of current flow in these networks

[Fig. 3.3, 3.4]. We find that the current load is strongly correlated with the degree.

Current flow increases with larger degree for β = 0; while β = −1 case has more

balanced loads. Most links have small current flow with a tail distribution for β = 0;

while β = −1 case has Gaussian-like symmetrical shape. Further, for β > −1, the

tail of the degree distribution determines the tail of the distribution of current

flow. For β < −1, the large-l tail of the load distribution is governed by the small-k

behavior of the degree distribution with a cutoff m. Also, the tail of the betweenness

distribution is essentially independent of system size N [10].

To summarize the results, the above β-controlled weighted model on SF net-

works indicates that the current flow is optimal at the value β ≈ −1. At this point,

the load is balanced, the network throughput is maximum and the average current

flow is relatively low.

21

3.2.1.2 Erdos and Renyi networks

In this section, we will give some comparisons for ER networks. As mentioned

before, ER network follows a binomial degree distribution. Fig. 3.5 and Fig. 3.6 dis-

play the maximum current flow and average current flow in ER random graph with

system size varying from 100 to 1000. Compared with BA SF network, throughput

also exhibits a maximum at β ≈ −1. However, the maximum current flow is much

smaller in the ER networks, due to the existence of hubs in the SF networks. Most

of the nodes in the ER network have the same amount of links to other nodes.

So for β > −1, where the long tail of degree distribution dominates the network

behavior, the maximum current flow lmax in the ER network does not change too

much. On the other hand, for β < −1, nodes with high load are the nodes with a

low degree. In the ER network, no strict cutoff is defined. The actual distribution

of the betweenness for those nodes still depends on the pre-set average degree. This

could also explain the behavior of the average current flow. It is decreasing as β

increases. But it is larger in the region of β < −1 and smaller for β > −1 for BA

networks.

We also plot the distribution of the current flow in the ER network [Fig. 3.7].

As expected, the load is more balanced for β = −1, while in β = 0 case, long tail of

degree distribution generates larger link current flow li.

3.2.1.3 Random geometric graphs

We first consider pure random geometric graphs (RGG) with different average

degrees.

Fig. 3.8 shows histogram plot of maximum current flow for values of average

degree at 8 and 10. Also, β ≈ −1 is the optimal value for throughput. Further,

because nodes in the RGG with 〈k〉 = 10 have more connections with other nodes,

“load”s are more distributed. Thus, we reach a conclusion that RGG networks with

higher average degree always exhibit larger throughput.

By introducing additional links into the RGG network, a small world (SW)

network is created, where each node can be reached by shorter paths. Fig. 3.9, 3.10

shows the results of maximum current flow and average current flow in the RGG SW

22

−4 −2 0 2 4β

0

200

400

600

l max

N=100 BAN=200 BAN=400 BAN=1000 BAN=100 ERN=200 ERN=400 ER

−4 −2 0 2 4β

0

0.05

0.1

0.15

0.2throughput

Figure 3.5: Maximum current flow for different system size N and differ-ent β in the ER network (long dashed lines) and BA network(solid lines) with same average degree 〈k〉 = 10. Inset showsthe throughput vs β.

−4 −2 0 2 4β

2

4

6

8

10

<l>

N=100 BAN=200 BAN=400 BAN=1000 BAN=100 ERN=200 ERN=400 ER

Figure 3.6: Average current flow for different system size N and differentβ in the ER network (long dashed lines) and BA network(solid lines) with same average degree 〈k〉 = 10.

23

0 0.1 0.2 0.3 0.4 0.5l

0

0.05

0.1

0.15

0.2

P(l

)

β=0β=−1

Figure 3.7: Distribution of the current flow for β = −1 and β = 0 in theER network with average degree 〈k〉 = 10 and N = 100.

−4 −2 0 2 4β

0

50

100

150

200

250

300

l max

N=100 q=10N=200 q=10N=400 q=10N=100 q=8N=200 q=8N=400 q=8

−4 −2 0 2 4β

throughput

Figure 3.8: Maximum current flow for different system size N and differ-ent β in the RGG network with average degree 〈k〉 = 10 (solidline) and 〈k〉 = 8 (long dashed line).

24

−4 −2 0 2 4β

0

20

40

60

80

100

l max

N=100 q=10 q_SW=0N=100 q=9 q_SW=1N=100 q=8 q_SW=2N=200 q=10 q_SW=0N=200 q=9 q_SW=1N=200 q=8 q_SW=2

−4 −2 0 2 4β

00.020.040.060.080.1

0.120.14

N=100 q=10 q_SW=0N=100 q=9 q_SW=1N=100 q=8 q_SW=2N=200 q=10 q_SW=0N=200 q=9 q_SW=1N=200 q=8 q_SW=2

throughput

Figure 3.9: Maximum current flow for different system size N and differ-ent β in the RGG SW network with average degree 〈k〉 = 10:average degree of RGG is 10 (solid line), average degree ofRGG is 9 with average degree of added SW links is 1 (dashedline), average degree of RGG is 8 with average degree ofadded SW links is 2 (dashed line). Inset shows the through-put vs β.

networks with the same average degree 〈k〉 = 10. Additional SW links are added to

keep the average degree unchanged. In the previous Fig. 3.8, we’ve already seen the

effects of decreasing the average degree of the RGG network. However, adding SW

links makes the opposite changes. Both lmax and 〈l〉 drop as a result of the existence

of SW links.

In general, giving the specific form of the weights, where the strength of a

link is proportional to (kikj)β and the constraint that the total network cost is

fixed, we found that the throughput is optimal at β ≈ −1 based on exact numerical

diagonalization method. Loads are balanced at that point. This also confirmed the

mean-field results [10]. Besides, the average current flow keeps decreasing as β goes

up. But the result is acceptable at this critical point β = −1. In particular, by

comparing the betweennesses on different networks (BA SF, ER, RGG, RGG SW),

an optimally weighted ER and RGG (RGG SW) always outperform its BA scale-free

25

−4 −2 0 2 4β

0

2

4

6

8

10

12

14

16

18

20

<l>

N=100 q=10 q_SW=0N=100 q=9 q_SW=1N=100 q=8 q_SW=2N=200 q=10 q_SW=0N=200 q=9 q_SW=1N=200 q=8 q_SW=2

Figure 3.10: Average current flow for different system size N and differ-ent β in the RGG SW network with average degree 〈k〉 = 10:average degree of RGG is 10 (solid line), average degree ofRGG is 9 with average degree of added SW links is 1 (dashedline), average degree of RGG is 8 with average degree ofadded SW links is 2 (dashed line).

counterpart with the same average degree. Work by Danila et al. [27] have obtained

qualitatively similar results in actual traffic simulations.

3.2.2 Heterogeneous source/target rate case

One major simplification of the above prototype model of the transport prob-

lems was that the probability of becoming the source and target for all nodes was

identical, although the degree distribution is heterogeneous. This is the reason

why the optimal routing scheme is at the point of β ≈ −1. But considering a

realistic network-transport scenarios, where hubs not only has significantly higher

degree, but also have higher probability of generating current (e.g. packets), can

the “reweighted” flow be maximized? Next, we will give some preliminary results.

26

−4 −2 0 2 4β

0

100

200

300

400

500

l max

N=100N=200N=400N=1000

−4 −2 0 2 4β

0

0.05

0.1

0.15

0.2

throughput

Figure 3.11: Maximum current flow for different system size N and dif-ferent β in the BA network with ρ = 1, m = 5 and averagedegree 〈k〉 = 10. Inset shows the throughput vs β.

−4 −2 0 2 4β

2

4

6

8

10

12

<l>

N=100N=200N=400N=1000

Figure 3.12: Average current flow for different system size N and differ-ent β in the BA network with ρ = 1, m = 5 and averagedegree 〈k〉 = 10.

27

3.2.2.1 Barabasi-Albert Scale-free networks

We first study the BA SF networks with heterogeneous source/target rate.

This rate will be set proportional to (kikj)ρ, similar to the form of the weights. The

result [Fig. 3.11] shows that the critical network throughput still has a maximum.

But it occurs at β ≈ −2. Similarly, in the β > −2 and β < −2 region, the network

behavior is governed by the long tail of degree distribution and the nodes with a

degree of m, respectively. The network disorder still strongly affects the actual

distribution of the betweenness. Fig. 3.12 shows that the average current flows have

the same decreasing behaviors. In this case, it’s hard to decide whether β = −1 or

β = −2 provides better current flow scheme.

To further investigate the current flows in the network, Fig. 3.13 indicates that

current flow increases with larger degree for β = 0; while β = −1 case distributes

more loads on the links; and β = −2 generates the most balanced loads. Fig. 3.13

also provides evidence for above statement.

0 10 20 30 40 50 60k

0

0.05

0.1

0.15

0.2

l

β=−2β=−1β=0

Figure 3.13: Current flow vs degree for β = −2, β = −1 and β = 0 in theBA network with ρ = 1, m = 5, average degree 〈k〉 = 10 andsystem size N = 100.

To investigate the effects of adding heterogenous source/target rates, we ex-

amined current flows in the BA networks for different ρ with β = 0 and β = 1

28

0 0.05 0.1 0.15 0.2l

0

200

400

600

800

1000

P(l

)

β=−2β=−1β=0

Figure 3.14: Normalized distribution of the current flow for β = −2, β =−1 and β = 0 in the BA network with ρ = 1, m = 5, averagedegree 〈k〉 = 10 and system size N = 100.

[Fig. 3.15, 3.16]. Within the region ρ < 0, maximum current flow almost stays

the same; for ρ > 0, maximum current flow rises. This means by making hubs

send/receive more “goods” (ρ > 0), they will become the congested nodes and the

throughput will drop dramatically. Meanwhile, the average current flow also drops

for larger ρ. These two properties raise the problem of finding the best tradeoff to

balance for the flow inside the network.

3.2.2.2 Random geometric graphs

For comparison, we also studies the RGG SW networks with parameter ρ = 1.

Unlike the BA networks, The result [Fig. 3.17] shows that the critical network

throughput is still at β ≈ −1. Adding source/target rate hardly changes the be-

havior of the maximum current flow. Average current flow for ρ = 1 is also a little

lower than homogeneous case [Fig. 3.18]. Future work will focus on investigating

more on the effects of heterogeneous source/target rate.

29

−4 −2 0 2 4ρ

0

5

10

15

20

25

30

l max

β=0β=−1

Figure 3.15: Maximum current flow for different ρ in the BA networkwith β = 0, β = 1, m = 5, average degree 〈k〉 = 10 and systemsize N = 100.

−4 −2 0 2 4ρ

3

3.5

4

4.5

5

<l>

β=0β=−1

Figure 3.16: Average current flow for different ρ in the BA network withβ = 0, β = 1, m = 5, average degree 〈k〉 = 10 and system sizeN = 100.

30

−4 −2 0 2 4β

0

10

20

30

40

50

l max

N=100 ρ=1N=200 ρ=1N=100 ρ=0N=200 ρ=0

Figure 3.17: Maximum current flow for different system size N and dif-ferent β in the RGG network with ρ = 1 and average degree〈k〉 = 10.

−4 −2 0 2 4β

4

5

6

7

8

9

<l>

N=100 ρ=1N=200 ρ=1N=100 ρ=0N=100 ρ=0

Figure 3.18: Average current flow for different system size N and differ-ent β in the RGG network with ρ = 1 and average degree〈k〉 = 10.

4. SUMMARY

We studied the transport and flow problem on different random network models.

We found that using the specific form of the weights (kikj)β, the local routing is

optimal and the load is balanced, at the point of β ≈ −1. Numerical results also

showed that an optimally weighted RGG SW network always outperforms its BA

SF counterpart. By introducing another parameter ρ, the behavior of the current

load distribution changes for BA networks. But it only lowers the current flow for

RGG SW networks a little. More analyses of this part will be considered in future

works. We hope that using the results from above simplified model could help to

further our understanding on the network vulnerability problem as well as cascading

network failures.

31

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