1. Introduction of Network Science
Prof. Cheng-Shang Chang (張正尚教授)
Institute of Communications Engineering National Tsing Hua University
Hsinchu Taiwan

2. Outline What is network science? A brief history of network science
Review of the mathematics of networks
Diffusion, distributed averaging, random gossip, synchronization
Network formation
Structure of networks (Community detection)
Conclusion

3. What is network science?
2005 National Research Council of the National Academies
“Organized knowledge of networks based on their study using the scientific method”
Social networks, biological networks, communication networks, power grids, …

4. A visualization of the network structure of the Internet at the level of “autonomous systems” (Newman, 2003)

5. A social network (Newman, 2003)

6. A food web of predator-prey interactions between species in a freshwater lake (Newman, 2003)

7. Power grid map http://www. treehugger

8. Citation networks http://www. public. asu

10. Two key ingredients
The study of a collections of nodes and links (graphs) that represent something real
The study of dynamic behavior of the aggregation of nodes and links

11. Definition of Network G(t)={V(t), E(t), f(t): J(t)} t: time
V: node (vertex, actor)
E: link (edge)
f: NxN topology (adjacency matrix)
J: algorithm for the evolution of the network (microrule)

12. Definition of Network Science (by Ted G. Lewis)
The study of the theoretical foundation of network structure/dynamic behaviors and the application of network to many subfields
Social network analysis (SNA)
Collaboration networks (citations, online social networks)
Emergent systems (power grids, the Internet)
Physical science systems (phase transition, percolation theory)
Life science systems (epidemics, metabolic processes)

13. A brief history: The pre-network period (1736-1966)
1736 Leonhard Euler: seven bridge of Konigsberg problem
1925 Yule: preferential attachment
An explanation for the evolution of the Internet and WWW
1927 Kermack and McKendrick: epidemic model (diffusion of innovation, the spread of information)
Erdos and Renyi: random graph model

14. The meso-network period (1967-1998)
1967 Stanley Milgram
“Six degree of separation”
Communication project
If you do not know the target person, forward the request to a personal acquaintance
Small-world effect: the diameter of a network increases as ln(n)

15. The meso-network period (1967-1998)
1972 Bonacich :influence network
Distributed consensus
Kirchhoff’s network: the value of a node is equal to the difference between the sum of values from input and output links
States and differential equations
Fixed point (steady state)
1984 Kuramoto: synchronization in coupled linear systems

16. The modern period (1998-present)
1998 Holland: emergence as the final state (of the fixed point problem)
1998 Watts and Strogatz: a generative procedure of rewiring the links in a regular graph
The small-world model
Crossover point and phase transition

17. The modern period (1998-present)
1999 M. Faloutsos, P. Faloutsos and C. Faloutsos: observed a power law in their graph of the Internet
1999 Barabasi: math model for scale-free networks
2000 Dorogovtsev: power law in many biological systems
1999 Kleinberg: power law in webgraph
2002 Girvan and Newman: community structure

18. The modern period (1998-present)
Atay network (a generalization of the Kirchhoff network)
Emergence and synchronization:
Heart beating
The chirping of crickets
Distributed consensus
Propagation of influence

19. Review of the mathematics of networks

20. Networks and their representations
A networks is a graph
Vertices (nodes, sites, actors)
Edges (links, bonds, ties)
n: number of nodes
m: number of edges
Multiedges
Self-edges (self-loops)
Simple network (simple graph): a network that has neither self-edges nor multiedges
Multigraph: a network with multiedges 2 1 3 4
Self-edge
Multiedge

21. Adjacency matrix A: an n×n matrix
Aij=1 if there is an edge between vertices i and j.
Aij=0 otherwise.
For a network with no self-edges, the diagonal elements are all zero.
It is symmetric. 2 1 3 4 1 2 3 4 A=

22. Directed networks
Adjacency matrix: Aij =1 if there is an edge from j to i.
With self-edges: Aii =1 for a single edge from vertex i to itself in a directed network. 2 1 3 4 1 2 3 4 A=

23. Degree The degree of a vertex is the number of edges connected to it.
ki: the degree of vertex i
m: number of edges
2m ends of edges (every edge has two ends)

24. Mean degree c: the mean degree of a vertex in an undirected graph
The maximum possible number of edges is (n-1)n/2.

25. Density
Density (connectance): the fraction of the maximum number of edges that actually present
For large network (n is very large)

26. Density
A (large) network is said to be dense if the density ρ tends to a constant as
On the other hand, it is said to be sparse if ρ tends to 0 as

27. Regular graphs
A regular graph is a graph in which all the vertices have the same degree.
k-regular graph: every vertex has degree k
2-regular: ring
4-regular: square lattice

28. Path Path: a sequence of connected vertices
Self-avoiding path: a path that does not intersect itself
Length of a path: the number of edges in the path
If there is a path of length 2 from j to i via k, then AikAkj=1.

29. Paths and adjacency matrix
: the number of paths of length 2 from j to i
: the number of paths of length 3 from j to i

30. Geodesic paths
A geodesic path (shortest path) is a path between two vertices that no shorter path exists
Geodesic distance (shortest distance): the length of a geodesic path
The smallest value r such that
Geodesic paths are self-avoiding (Why?)
Geodesic paths are not necessarily unique

31. Diameter
The diameter d of a graph is the length of the longest geodesic paths between any pairs of vertices in a network.
Suppose that is the geodesic distance between vertices i and j

32. Components
A network is connected if there is a path from every vertex to any other vertex.
Disconnected networks can be separated into several components.
Components:
There is a path from every vertex in the subnetwork to any other vertex in the same subnetwork.
No other vextex can be added while preserving this property.

33. Diffusion
Diffusion is the process by which gas moves from regions of high density to regions of low, driven by the relative pressure of the different regions.
Diffusion in a network (Influence network):
The spread of an idea
The spread of a disease

34. Diffusion in a network
Suppose that we have some commodity on the vertices.
Let be the amount of the commodity at vertex i at time t
Suppose that community moves from vertex j to an adjacent vertex i at rate
C is called the diffusion constant.

35. Governing equation for diffusion in a network
is the degree of vertex i
is the Kronecker delta, which is 1 if i=j and 0 otherwise.

36. Governing equation for diffusion in a network
Let D be the diagonal matrix with vertex degrees along its diagonal.
Graph Laplacian: L=D-A
In matrix form,
A system of linear differential equations

37. Solving the system of linear differential equations
Suppose vi and λi are the ith eigenvector and eigenvalue.
Guess the solution has the form

38. Eigenvalues of the graph Laplacian
The Laplacian is symmetric.
It has real eigenvalues.
The Laplacian is positive-semidefinite.
All its eigenvalues are nonnegative.
The vector (1,1,…,1) is an eigenvector with eigenvalue 0.

39. Algebraic connectivity
The number of zero eigenvalues of the Laplacian is the number of components.
The Laplacian can be written in a block form.
The network is connected if and only if the second smallest eigenvalue of the Laplacian is nonzero.
Algebraic connectivity: the second smallest eigenvalue of the Laplacian

40. Distributed averaging consensus
Lin Xiao and Stephen Boyd, “Systems & Control Letters,” 53 (2004) 65 – 78.
Consider a network (connected graph) G=(V,E)
Each vertex i holds an initial scalar value xi(0) in R, and x(0)=(x1(0),…, xn(0))
Two vertices can communicate with each other, if and only if they are neighbors.
The problem is to compute the average of the initial values, ,via a distributed algorithm

41. Motivation Sensor networks (measuring temperature)
A flock of flying birds

42. Distributed linear iterations
Constant edge weights
In matrix form
L=D-A is the Laplacian of the graph

43. Distributed linear iterations
W=I- L
The vector (1,1,…,1) is an eigenvector with eigenvalue 0 of the Laplacain L.
L is symmetric for an undirected graph
W is a doubly stochastic matrix, i.e., all the row sums and column sums are all equal to 1.
If W is a nonnegative matrix, then W can be viewed as the probability transition matrix of a Markov chain and
where is a matrix with all its elements being 1.

44. Condition for convergenceAs
The condition for W to be a nonnegative matrix,
ki is the degree of vertex i
Distributed linear iteration is guaranteed to converge if

45. Randomized gossip algorithms
Stephen Boyd, Arpita Ghosh, Balaji Prabhakar, and Devavrat Shah, IEEE Transactions on Information Theory, VOL. 52, NO. 6, pp , JUNE 2006.
Gossip algorithm: an algorithm in which each node can communicate with no more than one neighbor in each time slot.
Consider a network (connected graph) G=(V,E)
Each vertex i holds an initial scalar value xi(0) in R, and x(0)=(x1(0),…, xn(0))
The problem is to compute the average of the initial values, ,via a gossip algorithm

46. Asynchronous time model
Each vertex has a clock which ticks at the times of a rate 1 Poisson process.
Superposition of independent Poisson processes is also a Poisson process with the rate equal to the sum of the rates of the original Poisson processes.
Uniformization: consider a Poisson process with rate n for clock ticks (as there are n vertices).
With probability 1/n, a clock tick is chosen for vertex i.

47. Asynchronous time model
In the kth time slot, let node i’s clock tick and let it contact some neighboring node j with probability Pij.
Both vertices set their values equal to the average of their current values.
With probability , the random matrix W(k) is
where Q is the permutation matrix that interchange the ith and jth coordinates.

48. Spread of information Other objective functions, e.g., max, min.
How fast is information distributed over a network via a randomized gossip algorithm?
Start from the initial state x(0)=(1,0,…,0), i.e., only the first vertex has the information.
If xi(t)>0, then vertex i must have been “visited” (at least once) by time t via the randomized gossip algorithm.
can be used to bound the probability that all the vertices received the information.

49. Influence network :the influence from j to i
xi(t): the degree of influence (power) of vertex i at time t
:the influence from j to i
We still have
But the weight matrix W is much more complicated. It may not be nonnegative, or doubly stochastic.
Convergence might be a problem.

50. Command hierarchies

51. Emergent power
Define power as the degree of influence in a social network
How does one increase his/her power?
Acquisition of weight influence: increase the influence to others and reduce the influence from others
Acquisition of link influence: rewiring links (by knowing more important people) is less effective.

52. Synchronization and desynchronization
Phenomenon of mutual synchronization
The flashing of fireflies in south Asia.
Spreading identical oscillators into a round-robin schedule.
Desynchronization has many applications
Resource scheduling in wireless sensor networks.
Fair resource scheduling as Time Division Multiple Access.

55. Desynchronization algorithms
A general framework for distributed algorithm to achieve
desynchronization needed in TDMA.
Degesys, Rose, Patel, Nagpal (2007)
All the nodes can communicate with each other.
Each node is modelled by an oscillator with the same
fundamental frequency.
There is no clock drift in every oscillator.
J.Degesys, I. Rose, A. Patel, R. Nagpal, “Desync: Self-Organizing desynchronization and TDMA on wireless sensor networks,” IPSN, 2007

56. Desynchronization algorithms
The DESYNC-STALE algorithm
Fire!

57. Desynchronization algorithms
The DESYNC-STALE algorithm
When a node reaches the end of the cycle, it fires and
resets its phase back to 0.
It waits for the next node to fire and jump to a new phase
according to a certain function.
The jumping function only uses the firing information of the node fires before it and the node fires after it.
The rate of convergence is only conjectured to be 𝑂(𝑛 2 ) from various computer simulations.

58. Desynchronization algorithms
Fire!
𝜙=0
When a node reaches the end of the cycle, it fires and
resets its phase back to 0.

59. Desynchronization algorithms
It waits for the next node to fire and jump to a new phase
according to a certain function.
Fire!

60. Desynchronization algorithms
The jumping function only uses the firing information of the node fires before it and the node fires after it.
Fire!

61. Desynchronization algorithms
Generalized process sharing scheme (GPS)
An extension of the fair scheduling scheme to the GPS.
Pagliari, Hong, Scaglione (2010)
Every node is assigned a weight and the amount of bandwidth received by a node is proportional to its weight.
They proposed an algorithm with two oscillators in each node and showed the convergence in the ideal case.
R. Pagliari, Y.-W. Hong, and A Scaglione “Bio-inspired algorithms for decentralized round-robin and proportional fair scheduling,” IEEE Journal on Selected Areas in Communications: Special Issue on Bio-Inspired Networking, 2010

62. Desynchronization algorithms
Both the DESYNC-STALE and the extension of the GPS scheme are shown to work properly by various computer simulations.
Lack of rigorous theoretical proofs in many aspects
The rate of convergence of the DESYNC-STALE algorithm.
The convergence of the stale GPS scheme.

63. Desynchronization algorithms
All the node are not likely to be identical
A particular node need to interact with the “outside” world, and might not have the freedom to adjust its local clock.
The master node in Bluetooth.
The collector node in a wireless sensor network.
The master clock in parallel analog-to-digital converters.

64. Desynchronization algorithms
Consider the desynchronization problem with an anchored node
The anchored node never adjusts its phase.
Except the anchored node, all the other nodes are identical and they do not know which node the anchored node is.

65. Dynamics in Anchored Desynchronization
Fire!

66. Network formation Erdos-Renyi random graph Configuration model
Preferential attachment
Small world
Formation of social networks by random triad connections

67. Random Graphs

68. The G(n,p) model There are n vertices.
With probability p, we place an edge independently between each distinct pair.
First studied by Solomonoff and Rapoport in 1951.
Erdos and Renyi published a series of papers on this model.
For a sample G in G(n,p) with m edges,
The probability of drawing a graph with m edges is then

69. The mean degree in the G(n,p) model
The mean value of m is
This is a direct result of the binomial distribution as there are n(n-1)/2 independent Bernoulli random variables with parameter p.
The mean degree in a graph with m edges is 2m/n. Thus the mean degree in G(n,p) is

70. Degree distribution
A given vertex in G(n,p) is connected with probability p to each of the n-1 other vertices.
The degree of of a given vertex is thus a random variable with the binominal distribution B(n-1,p), i.e.,
Let the mean degree c=(n-1)p be fixed as n goes to
Then the binomial distribution B(n-1,p) converges to the Poisson distribution with mean c, i.e.,
This is called Poisson random graph (as the limit of the Erdos and Renyi random graph)

71. Clustering coefficient
The clustering coefficient C is a measure of transitivity.
It is defined as the probability that two network neighbors of a vertex are also neighbors of each other.
As each edge is connected independently with probability p,
This is one of several aspects that the random graph differs sharply most from real-world networks.

72. Giant component Consider the Poisson random graph
For the case p=0, there are no edges in the network at all. Each vertex is completely isolated.
For the case p=1, every possible edge in the network is present and the network is an n-clique.
Phase transition: an interesting question is how the transition between the two extremes occurs if we increase p from 0 to 1.
Giant component: a network component whose size grows in proportion to n.

73. The size of the giant component
Let u be the average fraction of the vertices in the random graph that do not belong to the giant component.
Suppose that a randomly chosen vertex i does not belong to the giant component (with probability u).
For any other vertex j
Either i is not connected to j (with probability 1-p),
or i is connected to j but j itself is not a member of the giant component (with probability pu).

74. A sample of an Erdos-Renyi graph http://igraph. sourceforge

75. The configuration model
Given a specific degree sequence
Give vertex i ki “stubs”
Choose two of the stubs uniformly at random and create an edge between the two vertices of the two chosen stubs. (they might be the same vertices (self edges))
Repeat the process until all the stubs are chosen.

76. From degree sequence to degree distribution
Specify the degree distribution pk
Draw a degree sequence k1,k2, … with probability
Then use the degree sequence to generate a random graph via using the configuration model.
Scale free networks: the degree distribution obeys the power law (Pareto distribution).

77. Preferential attachment
Richer-get-richer effect
Cumulative advantage
Experience of shopping (品牌效應)

78. Price’s model
Every newly appearing paper cites previous ones chosen at random proportional to the number of citations these previous papers already have.
Degree distributions obey the power law.
A special case is the Barabasi and Albert model

79. The small-world model

80. Regular graphs vs. random graphs
Random graphs have low transitivity (clustering coefficient).
Random graphs have small diameter.
On the other hand, regular graphs have large diameter and some of them have large transitivity.

81. A simple one-dimensional network model (ring)
Every vertex is connected to c nearest neighbors in a line (a) or in a ring (b). Here c=6.

82. Clustering coefficient
Count the number of triangles
Two steps forward and one step back
The length of the last step can be chosen between 1,2,…c/2.
The number of ways to choose the first two steps is the number of positive integer solutions to , which is

83. Clustering coefficient
The total number of triangles is
Each vertex has degree c and the number of connected triples centered at a vertex is
This clustering coefficient varies from 0 for c=2 up to a maximum of ¾ when c

84. The mean shortest path
The farthest one can move around the ring in a single step is c/2.
So two vertices m lattice spacing apart are connected by a shortest path of 2m/c steps.
Averaging over the complete range of m from 0 to n/2 gives a mean shortest path of n/2c.
By contrast, the mean shortest path in the ER random graph is

85. The small-world model Watts and Strogatz 1998
Interpolate between the ring model and the random graph by moving or rewiring edges from the ring to random positions.
Start with a ring model with n vertices in which every vertex has degree c.
Go through each edge in turn and with probability p we remove that edge and replace it with one (shortcut) that joins two vertices chosen uniformly at random.

86. The small-world model

87. The small-world model
The crucial point about the Watts-Strogatz small-world model is that as p is increased from 0 the clustering coefficient is maintained up to quite large values of p while the small-world behavior, meaning short average path lengths, already appears for quite modest values of p.
As a result, there is a substantial range of intermediate values for which the model shows both effects simultaneously, i.e., large clustering coefficient and short average path length.

88. Clustering coefficient (solid line) and average path length (dashed line)

89. Scaling function for the small-world model

90. Formation of Social Networks by Random Triad Connections
Join work with Prof. Duan-Shin Lee
Director of the Institute of Communications Engineering
National Tsing Hua University

91. A Network Formation Model for Social Networks
Institute of Communications Engineering
National Tsing-Hua University
A Network Formation Model for Social Networks
At time zero, the network consists of a clique with m0 vertices.
At time t, which is a non-negative integer, a new vertex is attached to one of the existing vertices in the network.
The attached existing vertex is selected with equal probability.
This step is called the uniform attachment step.
Each neighbor of the attached existing vertex is attached to the new vertex with probability a and not attached with probability 1-a.
This step is called the triad formation step.
Friends’ friends are more likely to be friends.

92. Uniform Attachment and Triad Formation
Institute of Communications Engineering
National Tsing-Hua University
Uniform Attachment and Triad Formation
when
t = 0

93. Uniform Attachment and Triad Formation
Institute of Communications Engineering
National Tsing-Hua University
Uniform Attachment and Triad Formation
do nothing with probability 1-a
when
t = 1
uniform attachment
triad formation with probability a
triad formation with probability a

94. Uniform Attachment and Triad Formation
Institute of Communications Engineering
National Tsing-Hua University
Uniform Attachment and Triad Formation
when
t = 2

95. Community detection INFOCOM 2011
Cheng-Shang Chang, Chin-Yi Hsu, Jay Cheng, and Duan-Shin Lee
Institute of Communications Engineering
National Tsing Hua University
Taiwan, R.O.C.

96. Detecting Community Community :
It is the appearance of densely connected groups of vertices, with only sparser connections between groups.
Modularity (Girman and Newman 2002) :
It is a property of a network and a specifically proposed division of that network into communities.
It measures when the division is a good one, in the sense that there are fewer than expected edges between communities.

97. Detecting Community
Example :

98. Algorithms for Detecting Community
Many algorithms have been proposed in the literature. Basically, they can be classified into four categories:
(1) divisive algorithms
(2) agglomerative algorithms
(3) graph partitioning and clustering algorithms
(4) data compression algorithms
Our algorithm belongs to this class
Newman’s fast algorithm also belongs to this class

99. Agglomerative algorithms
Features of agglomerative algorithms
Clusters are iteratively merged if their similarity is sufficiently high.
Our distribution based clustering algorithms starts by viewing each vertex as a community with only one member.
The algorithm repeatedly merges the two most positively correlated communities into a new community until all the remaining communities are negatively correlated.

100. Agglomerative Algorithms
Example :

101. Our Contributions
In spite of all the efforts in developing community detection algorithms, there are still many questions that we do not have satisfactory answers.
What is a community in a network?
Even with a definition of a community, what would be the right index for measuring the performance of a graph partition?
We will provide a general probabilistic framework for these questions.

102. Our Contributions
In spite of all the efforts in developing community detection algorithms, there are still many questions that we do not have satisfactory answers.
What is a community in a network?
Even with a definition of a community, what would be the right index for measuring the performance of a graph partition?
We will provide a general probabilistic framework for these questions.

103. Our Contributions
Characterization of a graph: the key idea of our framework is to characterize a graph by a bivariate distribution that specifies the probability of the two vertices appearing at both ends of a “randomly” selected path in the graph.
Definition of a community: With such a bivariate distribution, we can then define a community as a set of vertices with the property that it is more likely to find the other end in the same community given one of the two ends in a randomly selected path is already in the community.
Correlation measures: To detect communities, we define a class of correlation measures that can be used for measuring how two vertices (and two communities) are related. Two communities are positively (resp. negatively) correlated if the value of a correlation measure for these two communities is positive (resp. negative).

104. Our Contributions
A class of distribution-based clustering algorithms : as a generalization of Newman’s fast algorithm, we propose a class of distribution-based clustering algorithms for community detection.
Two theoretic results that can be proved for a distribution-based clustering algorithm:
(i) it guarantees that every community detected by the algorithm satisfies the definition of a community under certain technical conditions for the bivariate distribution,
(ii) the algorithm increases the “modularity” index in each merge of two positively correlated communities.

105. Correlation Measures
Definition: For any two indicator random variables X and Y , is called a correlation measure in this paper if
(1) is solely determined by the bivariate distribution of X and Y ,
(2) if and only if X and Y are independent,
(3) if and only if X and Y are positively correlated, i.e.,
From (2) and (3), we also know that if and only if X and Y are negatively correlated.

106. Examples of Correlation Measures
Covariance: For two indicator random variables X and Y , we have

107. Examples of Correlation Measures
Correlation: Note that the correlation of two random variables X and Y , denoted by , can be computed as follows:
,where

108. Examples of Correlation Measures
Mutual information: The mutual information of two random variables X and Y, denoted by , can be computed as follows:

109. Probabilistic Framework
Instead of characterizing a network by a graph, we characterize a network by a bivariate distribution.
As mentioned before,
Now, let be the sum of all the elements in a matrix A, i.e.,
Then, we can rewrite the bivariate distribution: {
1/2m, if vertices v and w are connected,
0, otherwise.

110. Probabilistic Framework
Recall that the bivariate distribution above is the probability for the two ends of a randomly selected edge in a graph.
Now, our idea is to generate the needed bivariate distribution by randomly selecting the two ends of a path.
We first consider a function f that maps an adjacency matrix A to another matrix f(A).
Then we define a bivariate distribution from f(A) by

111. Probabilistic Framework
A random selection of a path with length not greater than 2: Consider a graph with an n n adjacency matrix A and
A path with length l is selected with probability
for l = 0, 1, and 2

112. Probabilistic Framework
A random walk on a graph: It can be characterized by a Markov chain with the n n transition probability matrix , where
is the transition probability from vertex v to vertex w.
The stationary probability that the Markov chain is in vertex v, denoted by , is
is the probability that we select a path with length l, l = 1, 2, … .
The probability of selecting a random walk (path) with vertices is

113. Probabilistic Framework
We then have the bivariate distribution
We can simply let l = 0 for all l > 2 and this leads to
113

114. Distribution-based Clustering Algorithm
(1) Input a bivariate distribution , v,w = 1, 2, … , n that characterizes the two randomly selected nodes V and W, and a correlation measure for two indicator random variables.
(2) Initially, there are n communities, indexed from 1 to n, with each community containing exactly one node. Specifically, let be the set of nodes in community i. Then , i = 1, 2, … , n.
114

115. Distribution-based Clustering Algorithm
(3) Let (resp. ) be the indicator random variable for the event that V is in community i (resp. W is in community j). Then
Compute for all i and j.
115

116. Distribution-based Clustering Algorithm
(4) Find the two (distinct) communities that have the largest correlation measure. Group these two communities into a new community. Suppose that community i and community j are grouped into a new community k. Then and update
116

117. Distribution-based Clustering Algorithm
(5) For all , compute and
(6) Repeat (4) until either there is only one community left or all the remaining pairs of communities have negative measures, i.e., for all
117

118. Definition of a Community
Definition: A set of nodes S is a community in a probabilistic sense if
If , then this is equivalent to
It is more likely to find the other node in the same community given that one of a randomly selected pair of two nodes is already in the community.

119. Definition of a Community
Theorem 1: Suppose that is symmetric and
, for all v = 1, 2, … , n. Then every community detected by any distribution-based clustering algorithm is a community in the probabilistic sense.

120. Definition of the Modularity Index
Definition: Consider a bivariate distribution with v, w = 1, 2, … , n. Let , c = 1, 2, … ,C, be a partition of {1, 2, … , n}, i.e., is an empty set for and .
The modularity index Q with respect to the partition where c = 1, 2, … ,C, is
Theorem 2: Suppose that is symmetric. Then for any distribution-based clustering algorithm, the modularity index is non-decreasing in every iteration.

121. Simulation Result
Each point in these figures is an average over 100 random graphs. In these figures, we also show 95% confidence intervals for all data points.
In our simulation results, we will consider three distribution-based clustering algorithms:
(1) covariance algorithm
(2) correlation algorithm
(3) mutual information algorithm

122. Simulation Result
To map a graph with an adjacency matrix A to a bivariate distribution Recall that,
, we will consider the following three types of functions:
(1) , i.e.,
(2) , i.e.,
(3) , i.e.,

123. Simulation Result

124. Simulation Result

125. Simulation Result

126. Simulation Result

127. Simulation Result
W. W. Zachary, J. Anthropol, Res. 33, 452, 1977.

128. Simulation Result

129. Simulation Result

130. Simulation Result

131. Conclusion
In 2005, the National Science Foundation in the U.S. realized that there is a need for “organized knowledge of networks” based on the scientific method.
This will require the integration of the knowledge in various fields, including the Internet, power grids, social networks, physical networks, and biological networks. The main mathematical tool for network science is the study of the dynamics of graphs.

132. Research problems
How is life formed? Is the emergence of life through random rewiring of DNAs according a certain microrule?
How powerful is a person in a community? How much is he/she worth? Can these be evaluated by the people he/she knows?
How can one bring down the Internet? What is the best strategy to defend one’s network from malicious attacks? How are these related to the topology of a network?
Why is there a phase change from water to ice? Can this be explained by using the percolation theory? Does the large deviation theory play a role here?