Introduction of Network Science Prof. Cheng-Shang Chang ( 張正尚教授 ) Institute of...

Introduction of Network Science

Prof. Cheng-Shang Chang (張正尚教授 )Institute of Communications Engineering

National Tsing Hua UniversityHsinchu Taiwan

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

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, …

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

A social network (Newman, 2003)

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

Power grid maphttp://www.treehugger.com/files/2009/04/nprs-interactive-

power-grid-map-shows-whos-got-the-power.php

Citation networks http://www.public.asu.edu/~majansse/pubs/SupplementIHDP.htm

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

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)

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)

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)

1959-1960 Erdos and Renyi: random graph model

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)

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

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

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

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

Review of the mathematics of networks

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

Multiedge Self-edge

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 41234

A=

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.

1 2 3 41234

A=2

1

3

4

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)

Mean degree c: the mean degree of a vertex in an

undirected graph

The maximum possible number of edges is (n-1)n/2.

Density Density (connectance): the fraction

of the maximum number of edges that actually present

For large network (n is very large)

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

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

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.

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

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

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

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.

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

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.

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.

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

Solving the system of linear differential equations Suppose vi and λi are the ith eigenvector

and eigenvalue. Guess the solution has the form

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.

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

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

Motivation Sensor networks (measuring temperature)

A flock of flying birds

Distributed linear iterations Constant edge weights

In matrix form

L=D-A is the Laplacian of the graph

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.

Condition for convergence As

The condition for W to be a nonnegative matrix,

ki is the degree of vertex i Distributed linear iteration is guaranteed to

converge if

Randomized gossip algorithms Stephen Boyd, Arpita Ghosh, Balaji Prabhakar, and

Devavrat Shah, IEEE Transactions on Information Theory, VOL. 52, NO. 6, pp. 2508-2530, 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

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.

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.

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.

Influence network

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.

Command hierarchies

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

• Desynchronization has many applications

• Fair resource scheduling as Time Division Multiple Access.

• Resource scheduling in wireless sensor networks.

• Phenomenon of mutual synchronization

• The flashing of fireflies in south Asia.• Spreading identical oscillators into a round-robin

schedule.

Desynchronization algorithms

• A general framework for distributed algorithm to achieve desynchronization needed in TDMA.

• 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

Degesys, Rose, Patel, Nagpal (2007)

55


• The DESYNC-STALE algorithm

56

Fire!


• 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 from various computer simulations.

57


• When a node reaches the end of the cycle, it fires and resets its phase back to 0.

58

Fire! 𝜙=0

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

59

Fire!

Desynchronization algorithms• The jumping function only uses the firing information of

the node fires before it and the node fires after it.

60

Fire!


• An extension of the fair scheduling scheme to the GPS.

• 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.

• Generalized process sharing scheme (GPS)

Pagliari, Hong, Scaglione (2010)

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

61


• 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.

62


• 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.

63


• 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.

64

Dynamics in Anchored Desynchronization

65

Fire!

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

random triad connections

Random Graphs

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

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

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)

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.

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.

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).

A sample of an Erdos-Renyi graph http://igraph.sourceforge.net/screenshots2.html

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.

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).

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

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

The small-world model

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.

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.

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

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

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

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.

The small-world model

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.

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

Scaling function for the small-world model

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

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.

91Institute of Communications Engineering

National Tsing-Hua University

Uniform Attachment and Triad Formation

• when



t = 0


• when



t = 1

uniform attachment

triad formation with probability a

triad formation with probability a

do nothing with probability 1-a


• when



t = 2

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

Newman’s fast algorithm also belongs to this

class

Our algorithm belongs to this class

100

Agglomerative Algorithms Example :

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.

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

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).

103

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.

104

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.

( 1, 1) ( 1) ( 1)P X Y P X P Y

( , )X Y

( , )X Y

( , ) 0X Y ( , ) 0X Y

( , ) 0X Y

106

Covariance: For two indicator random variables X and Y , we have ( , ) ( 1, 1) ( 1) ( 1)Cov X Y P X Y P X P Y

Examples of Correlation Measures

107

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

,where

2( ) ( 1) ( ( 1))Var X P X P X 2( ) ( 1) ( ( 1))Var Y P Y P Y

[ , ]Correl X Y

[ , ][ , ]

( ) ( )

Cov X YCorrel X Y

Var X Var Y

( 1, 1) ( 1) ( 1)

( ) ( )

P X Y P X P Y

Var X Var Y


108

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

( , ) ( ( , )) ( ; )X Y Sgn Cov X Y I X Y

( ; )I X Y

,

,,

( , ) supp( )

( , )( ; ) ( , ) log

( ) ( )X Y

X YX Y

x y P X Y

P x yI X Y P x y

P x P y


109

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:

(( , ) ( , ))P V W v w

{ 1/2m, if vertices v and w are connected, 0, otherwise.

( )A

1( , ) ( , )

( ) vwP V v W w p v w AA

( ) vwv w

A A

Probabilistic Framework

110

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 1

( , ) ( )( ( )) vwP V v W w f Af A


111

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

0 1 2, , and are three nonnegative constants

20 1 2( ) If A A A

20 1 2( ( )) (I) ( ) ( )f A A A

/ ( ( ))l f A


112

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


,( )v wR R

, /v w vw vR A k

v / 2vk ml

1 1

1

,1

i i

l

l v v vi

R

1 2, , , lv v v v w

113

We then have the bivariate distribution

We can simply let l = 0 for all l > 2 and this leads to


1

2 1

1

,1 1

( , )i i

l

l

v l v vl v v i

p v w R

2 2

2 2

, ,1 2,

1

( , )2 2

nv v v w

v wv v

A Ap v w A

m m k

114

(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.

Distribution-based Clustering Algorithm

( , )p v w

( , )X Y

{ }iS iiS

115

(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.


iX

( 1) ( ) ( )i

i V Vv S

P X p v p i

( , )i jX Y

jY

( 1) ( ) ( )j

j W Ww S

P Y p w p j

,

( 1, 1) ( , ) ( , )i j

i jv S w S

P X Y p v w p i j

116

(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


( 1, 1) ( 1, 1)j i j jP X Y P X Y

( 1) ( 1) ( 1)k i jP X P X P X k i jS S S

( 1) ( 1) ( 1)k i jP Y P Y P Y

( 1, 1) ( 1, 1) ( 1, 1)k k i i i jP X Y P X Y P X Y

( 1, 1) ( 1, 1) ( 1, 1)k l i l j lP X Y P X Y P X Y ( 1, 1) ( 1, 1) ( 1, 1)l k l i l jP X Y P X Y P X Y

117

(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 .


i j

( , )l kX Y

( , ) 0i jX Y

l k ( , )k lX Y

118

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.

Definition of a Community

( , ) ( ) ( )P V S W S P V S P W S ( ) 0P W S

( | ) ( )P V S W S P V S

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.

( , )p v w2( , ) ( ( ))Vp v v p v

120

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.

Definition of the Modularity Index

( , )p v w

cS

'c cS S 'c c

1{1,2, , }

C

ccS n

cS

1

( ( , ) ( ) ( ))C

c c c cc

Q P V S W S P V S P W S

( , )p v w

121

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

Simulation Result

122

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.,

20 1 2( ) If A A A

( , )p v w

0 1 2( , , ) (0,1,0)

0 1 2( , , ) (1,1,0)

0 1 2( , , ) (1,0.5,0.25)

1( )f A A

2 ( ) If A A

23( ) I 0.5 0.25f A A A

Simulation Result

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

1( )f A A

Simulation Result

129

2 ( ) If A A

Simulation Result

130

23( ) I 0.5 0.25f A A A

Simulation Result

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.

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?

Introduction of Network Science Prof. Cheng-Shang Chang ( 張正尚教授 ) Institute of...

Documents

Transcript of Introduction of Network Science Prof. Cheng-Shang Chang ( 張正尚教授 ) Institute of...