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Introduction of Network Science

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Introduction of Network Science

Prof. Cheng-Shang Chang (張正尚教授)

Institute of Communications Engineering National TsingHua University

Hsinchu Taiwan

- 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

- 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, …

- 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

- 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)

- 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)

- 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

- 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)

- 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

- 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

- 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

- 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

- 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

Self-edge

Multiedge

1

2

4

3

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

1

2

3

4

1

1

2

A=

2

4

3

4

3

- 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

4

1

1

2

A=

2

4

3

4

3

- 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)

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

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

- 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

- 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: 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.

- : the number of paths of length 2 from j to i
- : the number of paths of length 3 from j to i

- 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

- 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

- 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 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

- 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.

- is the degree of vertex i
- is the Kronecker delta, which is 1 if i=j and 0 otherwise.

- 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

- Suppose vi and λi are the ith eigenvector and eigenvalue.
- Guess the solution has the form

- 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.

- 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

- 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

- Sensor networks (measuring temperature)
- A flock of flying birds

- Constant edge weights
- In matrix form
- L=D-A is the Laplacian of the graph

- 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.

- 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

- Stephen Boyd, ArpitaGhosh, BalajiPrabhakar, 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

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- A general framework for distributed algorithm to achieve
- desynchronizationneeded 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

- The DESYNC-STALE algorithm

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.

Fire!

- 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.

Fire!

- The jumping function only uses the firing information of the node fires before it and the node fires after it.

Fire!

- 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

- 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.

- 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.

- 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.

Fire!

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

Random Graphs

- 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 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

- 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)

- 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.

- 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.

- 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).

- Given a specific degree sequence
- Give vertex iki “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.

- 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).

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

- 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

- 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.

Every vertex is connected to c nearest neighbors in a line (a) or in a ring (b). Here c=6.

- 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

- 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 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

- 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 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.

- Join work with Prof. Duan-Shin Lee
- Director of the Institute of Communications Engineering
- National TsingHua University

National Tsing-Hua University

Institute of Communications Engineering

- 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.

National Tsing-Hua University

Institute of Communications Engineering

t = 0

- when

National Tsing-Hua University

Institute of Communications Engineering

do nothing with probability 1-a

t = 1

- when

uniform attachment

triad formation with probability a

triad formation with probability a

National Tsing-Hua University

Institute of Communications Engineering

t = 2

- when

- INFOCOM 2011
- Cheng-Shang Chang, Chin-Yi Hsu, Jay Cheng, and Duan-Shin Lee
Institute of Communications Engineering

National TsingHuaUniversity

Taiwan, R.O.C.

- 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.

- Example :

- 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

Agglomerative Algorithms

- Example :

- 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.

- 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.

- 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).

- 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.

- Definition:For any two indicator random variables X and Y , is called a correlation measurein 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.

- Covariance: For two indicator random variables X and Y , we have

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

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

- 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.

- 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

- 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

- A path with length l is selected with probability

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

We then have the bivariate distribution

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

113

(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

(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 iand j.

115

(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

(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

Definition: A set of nodes S is a communityin 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

- 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.

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.

- 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

- 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.,

W. W. Zachary, J. Anthropol, Res. 33, 452, 1977.

- 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.

- 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?