Data Mining for Complex Network

Introduction and Background

Welcome!

Instructor: Ruoming Jin Homepage: www.cs.kent.edu/~jin/ Office: 264 MCS Building Email: jin@cs.kent.edu

Course overview

The course goal First of all, this is a research course, or a

special topic course. There is no textbook and even no definition what topics are supposed to under the course name

Discussing the state-of-are techniques for mining complex networks

Review & Course Project

Simple Concepts (Review)

Erdős–Rényi model Random Graph Model

Markov Chain and Random Walk Maximal Likelihood/Model Selection

Requirement

Each of you will have one presentation Review Paper

Select a topic and review at least four papers Bonus: Develop a new idea on this topic

Course Project One or Two a group Collect data; preprocess data; analyzing the

data;

Final Grade: 30% presentations, 25% review, 35% project, and 10% class participation

What is a network?

Network: a collection of entities that are interconnected with links. people that are friends computers that are interconnected web pages that point to each other proteins that interact

Graphs

In mathematics, networks are called graphs, the entities are nodes, and the links are edges

Graph theory starts in the 18th century, with Leonhard Euler The problem of Königsberg bridges Since then graphs have been studied

extensively.

Networks in the past

Graphs have been used in the past to model existing networks (e.g., networks of highways, social networks) usually these networks were small network can be studied visual inspection

can reveal a lot of information

Networks now

More and larger networks appear Products of technological advancement

• e.g., Internet, Web

Result of our ability to collect more, better, and more complex data• e.g., gene regulatory networks

Networks of thousands, millions, or billions of nodes impossible to visualize

The internet map

Understanding large graphs

What are the statistics of real life networks?

Can we explain how the networks were generated?

What else? A still very young field! (What is the basic principles and

what those principle will mean?)

Measuring network properties

Around 1999 Watts and Strogatz, Dynamics and

small-world phenomenon Faloutsos3, On power-law relationships

of the Internet Topology Kleinberg et al., The Web as a graph Barabasi and Albert, The emergence of

scaling in real networks

Real network properties

Most nodes have only a small number of neighbors (degree), but there are some nodes with very high degree (power-law degree distribution) scale-free networks

If a node x is connected to y and z, then y and z are likely to be connected high clustering coefficient

Most nodes are just a few edges away on average. small world networks

Networks from very diverse areas (from internet to biological networks) have similar properties Is it possible that there is a unifying underlying generative

process?

Generating random graphs

Classic graph theory model (Erdös-Renyi) each edge is generated independently with

probability p

Very well studied model but: most vertices have about the same degree the probability of two nodes being linked is

independent of whether they share a neighbor the average paths are short

Modeling real networks

Real life networks are not “random” Can we define a model that

generates graphs with statistical properties similar to those in real life? a flurry of models for random graphs

Processes on networks

Why is it important to understand the structure of networks?

Epidemiology: Viruses propagate much faster in scale-free networks

Vaccination of random nodes does not work, but targeted vaccination is very effective

The future of networks

Networks seem to be here to stay More and more systems are modeled as

networks Scientists from various disciplines are

working on networks (physicists, computer scientists, mathematicians, biologists, sociologist, economists)

There are many questions to understand.

Basic Mathematical Tools

Graph theory Probability theory Linear Algebra

Graph Theory

Graph G=(V,E) V = set of vertices E = set of edges

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undirected graphE={(1,2),(1,3),(2,3),(3,4),(4,5)}

Graph Theory

Graph G=(V,E) V = set of vertices E = set of edges

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directed graphE={‹1,2›, ‹2,1› ‹1,3›, ‹3,2›, ‹3,4›, ‹4,5›}

Undirected graph

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degree d(i) of node i number of edges

incident on node i

degree sequence [d(1),d(2),d(3),d(4),d(5)] [2,2,2,1,1]

degree distribution [(1,2),(2,3)]

Directed Graph

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in-degree din(i) of node i number of edges pointing

to node i

out-degree dout(i) of node i number of edges leaving

node i in-degree sequence

[1,2,1,1,1]

out-degree sequence [2,1,2,1,0]

Paths

Path from node i to node j: a sequence of edges (directed or undirected from node i to node j) path length: number of edges on the path nodes i and j are connected cycle: a path that starts and ends at the same node

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Shortest Paths

Shortest Path from node i to node j also known as BFS path, or geodesic

path

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Diameter

The longest shortest path in the graph

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Undirected graph

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Connected graph: a graph where there every pair of nodes is connected

Disconnected graph: a graph that is not connected

Connected Components: subsets of vertices that are connected

Fully Connected Graph

Clique Kn

A graph that has all possible n(n-1)/2 edges

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Directed Graph

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Strongly connected graph: there exists a path from every i to every j

Weakly connected graph: If edges are made to be undirected the graph is connected

Subgraphs

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Subgraph: Given V’ V, and E’ E, the graph G’=(V’,E’) is a subgraph of G.

Induced subgraph: Given V’ V, let E’ E is the set of all edges between the nodes in V’. The graph G’=(V’,E’), is an induced subgraph of G

Trees

Connected Undirected graphs without cycles

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Bipartite graphs

Graphs where the set V can be partitioned into two sets L and R, such that all edges are between nodes in L and R, and there is no edge within L or R

Linear Algebra

Adjacency Matrix symmetric matrix for undirected graphs

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01000

10100

01011

00101

00110

A

Linear Algebra

Adjacency Matrix unsymmetric matrix for undirected

graphs

00000

10000

01010

00001

00110

A 1

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Random Walks

Start from a node, and follow links uniformly at random.

Stationary distribution: The fraction of times that you visit node i, as the number of steps of the random walk approaches infinity if the graph is strongly connected, the

stationary distribution converges to a unique vector.

Random Walks

stationary distribution: principal left eigenvector of the normalized adjacency matrix x = xP for undirected graphs, the degree distribution

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00001

10000

0210210

00001

0021210

P

Eigenvalues and Eigenvectors

The value λ is an eigenvalue of matrix A if there exists a non-zero vector x, such that Ax=λx. Vector x is an eigenvector of matrix A The largest eigenvalue is called the

principal eigenvalue The corresponding eigenvector is the

principal eigenvector Corresponds to the direction of

maximum change

Types of networks

Social networks Knowledge (Information) networks Technology networks Biological networks

Social Networks

Links denote a social interaction Networks of acquaintances actor networks co-authorship networks director networks phone-call networks e-mail networks IM networks

• Microsoft buddy network Bluetooth networks sexual networks home page networks

Knowledge (Information) Networks

Nodes store information, links associate information Citation network (directed acyclic) The Web (directed) Peer-to-Peer networks Word networks Networks of Trust Bluetooth networks

Technological networks

Networks built for distribution of commodity The Internet

• router level, AS level

Power Grids Airline networks Telephone networks Transportation Networks

• roads, railways, pedestrian traffic

Software graphs

Biological networks

Biological systems represented as networks Protein-Protein Interaction Networks Gene regulation networks Metabolic pathways The Food Web Neural Networks

Now what?

The world is full with networks. What do we do with them? understand their topology and measure

their properties study their evolution and dynamics create realistic models create algorithms that make use of the

network structure

Measuring Networks

Degree distributions Small world phenomena Clustering Coefficient Mixing patterns Degree correlations Communities and clusters

Degree distributions

Problem: find the probability distribution that best fits the observed data

degree

frequency

k

fk

fk = fraction of nodes with degree k = probability of a randomly selected node to have degree k

Power-law distributions

The degree distributions of most real-life networks follow a power law

Right-skewed/Heavy-tail distribution there is a non-negligible fraction of nodes that has very

high degree (hubs) scale-free: no characteristic scale, average is not

informative

In stark contrast with the random graph model! highly concentrated around the mean the probability of very high degree nodes is

exponentially small

p(k) = Ck-α

Power-law signature

Power-law distribution gives a line in the log-log plot

α : power-law exponent (typically 2 ≤ α ≤ 3)

degree

frequency

log degree

log frequency α

log p(k) = -α logk + logC

Examples

Taken from [Newman 2003]

A random graph example

Maximum degree

For random graphs, the maximum degree is highly concentrated around the average degree z

For power law graphs

Rough argument: solve nP[X≥k]=1

1)1/(αmax nk

Exponential distribution

Observed in some technological or collaboration networks

Identified by a line in the log-linear plot

p(k) = λe-λk

log p(k) = - λk + log λ

degree

log frequency λ

Collective Statistics (M. Newman 2003)

Clustering (Transitivity) coefficient

Measures the density of triangles (local clusters) in the graph

Two different ways to measure it:

The ratio of the means

i

i(1)

i nodeat centered triples

i nodeat centered trianglesC

Example

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583

6113

C(1)

Clustering (Transitivity) coefficient

Clustering coefficient for node i

The mean of the ratios

i nodeat centered triplesi nodeat centered triangles

Ci

i(2) C

n1

C

Example

The two clustering coefficients give different measures

C(2) increases with nodes with low degree

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3013

611151

C(2)

83

C(1)

Collective Statistics (M. Newman 2003)

Clustering coefficient for random graphs

The probability of two of your neighbors also being neighbors is p, independent of local structure clustering coefficient C = p when z is fixed C = z/n =O(1/n)

Small world phenomena

Small worlds: networks with short paths

Obedience to authority (1963)

Small world experiment (1967)

Stanley Milgram (1933-1984): “The man who shocked the world”

Small world experiment

Letters were handed out to people in Nebraska to be sent to a target in Boston

People were instructed to pass on the letters to someone they knew on first-name basis

The letters that reached the destination followed paths of length around 6

Six degrees of separation: (play of John Guare)

Also: The Kevin Bacon game The Erdös number

Small world project: http://smallworld.columbia.edu/index.html

Measuring the small world phenomenon

dij = shortest path between i and j Diameter:

Characteristic path length:

Harmonic mean

ijji,

dmaxd

ji

ijd1)/2-n(n

1

ji

1-ij

d1)/2-n(n

11

Collective Statistics (M. Newman 2003)

Mixing patterns

Assume that we have various types of nodes. What is the probability that two nodes of different type are linked? assortative mixing (homophily)

E : mixing matrix

p(i,j) = mixing probability

ji,

j)E(i,j)E(i,

j)p(i,

p(j | i) = conditional mixing probability

j

j)E(i,j)E(i,

i)|p(j

Mixing coefficient

Gupta, Anderson, May 1989

Advantages: Q=1 if the matrix is diagonal Q=0 if the matrix is uniform

Disadvantages sensitive to transposition does not weight the entries

1N

1-i)|p(iQ i

Mixing coefficient

Newman 2003

Advantages: r = 1 for diagonal matrix , r = 0 for uniform matrix not sensitive to transposition, accounts for

weighting

i

i i

a(i)b(i)1

a(i)b(i)-i)|p(ir

j

j)p(i,a(i)

j

i)p(j,b(i)

(row marginal)

(column marginal)

r=0.621 Q=0.528

Degree correlations

Do high degree nodes tend to link to high degree nodes?

Pastor Satoras et al. plot the mean degree of the neighbors

as a function of the degree Newman

compute the correlation coefficient of the degrees of the two endpoints of an edge

assortative/disassortative

Collective Statistics (M. Newman 2003)

Communities and Clusters

Use the graph structure to discover communities of nodes essentially clustering and classification

on graphs

Other measures

Frequent (or interesting) motifs bipartite cliques in the web graph patterns in biological and software

graphs

Use graphlets to compare models [Przulj,Corneil,Jurisica 2004]

Other measures

Network resilience against random or

targeted node deletions

Graph eigenvalues

Other measures

The giant component

Other?

References

M. E. J. Newman, The structure and function of complex networks, SIAM Reviews, 45(2): 167-256, 2003

M. E. J. Newman, Random graphs as models of networks in Handbook of Graphs and Networks, S. Bornholdt and H. G. Schuster (eds.), Wiley-VCH, Berlin (2003).

N. Alon J. Spencer, The Probabilistic Method