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Community Structure in Large Complex Networks

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Community Structure in Large Complex Networks. Liaoruo Wang and John E. Hopcroft Dept. of Computer Engineering & Computer Science, Cornell University In Proc. 7th Annual Conference on Theory and Applications of Models of Computation (TAMC) , June 2010 Presented by Nam Nguyen. Agenda. - PowerPoint PPT Presentation
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Community Structure in Large Complex Networks Liaoruo Wang and John E. Hopcroft Dept. of Computer Engineering & Computer Science, Cornell University In Proc. 7th Annual Conference on Theory and Applications of Models of Computation (TAMC), June 2010 Presented by Nam Nguyen
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Page 1: Community Structure in Large Complex Networks

Community Structure in Large Complex Networks

Liaoruo Wang and John E. HopcroftDept. of Computer Engineering & Computer Science, Cornell University

In Proc. 7th Annual Conference on Theory and Applications of Models of Computation (TAMC), June 2010

Presented by Nam Nguyen

Page 2: Community Structure in Large Complex Networks

Motivation Introduction Contributions of the paper Definitions WHISKER is NP-Complete. Algorithms.

Agenda

Page 3: Community Structure in Large Complex Networks

C.S is a classical but still-hot topic in complex networks.

Previous studies: Communities were assumed to be densely connected inside but sparsely connected outside.

A different point of view: We should disregard “whiskers” and elaborate “cores” in the networks.

Motivation

Page 4: Community Structure in Large Complex Networks

Roughly speaking◦ Whiskers: Subsets of vertices that are barely connected

to the rest of the network.◦ Cores: Connected subgraphs that are densely connected

inside and well-connected to the rest of the network, i.e., “real communities”

Why???◦ For real-world societies, communities are also well

connected to the rest of the network.◦ Imagine a close-nit community, CISE Dept., with only one

connection with the outer world. Definitions come right away.

Introduction

Page 5: Community Structure in Large Complex Networks

More concrete definitions of “whiskers” and “cores” in a networks.

WHISKER is NP-Complete Three heuristic algorithms for finding

approximate cores. Simulation results.

Contributions

Page 6: Community Structure in Large Complex Networks

Graph G = (V,E) undirected, A = (Ai,j). For S⊆V, let SC = V\S.

Conduction of S

where A suitable cut

Definition

Page 7: Community Structure in Large Complex Networks

A k-whisker

A maximal k-whisker

Definition(cont’d)

Page 8: Community Structure in Large Complex Networks

A whisker

A maximal whisker

Definition (cont’d)

Page 9: Community Structure in Large Complex Networks

A core

Definition (cont’d)

Page 10: Community Structure in Large Complex Networks

Lemmas

Proof

The only suitable cut of size = 26

|S ⋃ T| = 25

>

Page 11: Community Structure in Large Complex Networks

Lemmas (cont’d)

Proof

(1a) exr + exz + eyr + eyz ≤ vx + vy(1b) eyr + exy + ezr + exz ≤ vy + vz

(1c) exr + eyr + ezr > vx + vz

(1a) + (1b) and use (1c) givesexr+2eyr+ezr+exy+eyz+2exz ≤ vx+2vy+vz < exr+eyr+ezr+vy

eyr + exy + eyz < vy

Page 12: Community Structure in Large Complex Networks

NAE-3-SAT: The problem of determining whether there exists a truth assignment for a 3-CNF Boolean formula such that each clause has at least one true literal and at least one false literal.

Fact: NAE-3-SAT is NP-Complete [1]

WHISKER: Given an unweighted undirected graph, determine whether there exists a whisker or not.

WHISKER is NP-Complete(of course, from a reduction from NAE-3-SAT)

NP-Completeness

Page 13: Community Structure in Large Complex Networks

Road map◦ 1. Construct a special graph G of 2n

vertices and show that G admits 2n whiskers and no more.

◦ 2. Construct a G-like graph for the 3-SAT problem.

◦ 3. Make a reduction from NAE-3-SAT problem to WHISKER

WHISKER is NP-Complete

Page 14: Community Structure in Large Complex Networks

WHISKER is in NP Reduction from NAE-3-SAT to WHISKER

◦ Consider the following graph (constructed in poly time) At each row, pick only one vertex (i.e., either xi or ¬xi) The resulted graph G of n vertices is a whisker Total number of whiskers is 2n ………… And no more than that

NP-Completeness

Page 15: Community Structure in Large Complex Networks

2n whiskers and no more than that!!! Why???

Suppose there is a whisker W of 2k+j vertices

Cut size of W

By definition of suitable cut size, we have

which implies !!!!

NP-Complete

Page 16: Community Structure in Large Complex Networks

NAE-3-SAT ≤P WHISKER Consider an instance of NAE-3-SAT with n

variables and c clauses. Construct G1, G2, …, Gc as follow

NP-Complete

Page 17: Community Structure in Large Complex Networks

NAE-3-SAT ≤P WHISKER Now, combine all Gi’s and add up all edge weights to get G’.

Next

NP-Complete

G G

G’ G’G*3CNF has a satisfied

assignment contains a whisker

update

update

Page 18: Community Structure in Large Complex Networks

Update G ( )

Update G’◦ Amplify all edge weights of G’ by a small amount δ where cn2δ << 1

All whiskers in new G are the same as in old G.

NP-Complete

Page 19: Community Structure in Large Complex Networks

G* = G + G’

Goal: If the 3CNF instance has a satisfied truth assignment, then selecting true literal from each row of G* gives us a whisker of size n, and vice versa.

For any truth assignment of 3SAT, rearrange the literals in to TRUE and FALSE columns.

If there is a satisfied not-all-equal assignment for 3SAT◦ Each clause must have one TRUE and one FALSE literals.◦ Not all the literals in each clause can be in the same column.◦ For each ith clause, Gi contains n2-2 edges connecting its two columns◦ Total cut size is required to satisfied

NP-Complete

Page 20: Community Structure in Large Complex Networks

If there is NO satisfied not-all-equal assignment for 3SAT◦ At least one clause i has its literals located in the same column n2

edges between the two columns of Gi.◦ For the other (c-1) clauses, there are at most (n2-2) edges connecting the

their two columns. Total number of edges: (c-1)(n2-2)+n2 = cn2–2c+2.◦ Of course, we don’t want selecting the true literal in each row give us a

whisker, thus

Combining the two inequalities, if ℇ and δ is chosen such that

Then If the 3CNF instance has a satisfied truth assignment, then selecting true literal from each row of G* gives us a whisker of size n, and vice versa.

◦ Hence, NAE-3-CNF ≤P WHISKER □

NP-Complete

Page 21: Community Structure in Large Complex Networks

Heuristic Algorithms

Page 22: Community Structure in Large Complex Networks

On random graph

◦ Alg 2 can positively find an approximate core◦ Alg 3 fails to find approximate core◦ The size of core growing linearly with d = np (fixed n) and

logarithmically with n (fixed d)◦ ??? G(n,p) displays core structure with high probability when p > 1/n ???

Results

Page 23: Community Structure in Large Complex Networks

Textual graph◦ Vertices and Edges: Words and their semantic Correlations◦ Data is crawled from 10K scientific papers of KDD conf. (1992-2003)◦ Pointwise mutual information

◦ Total: 685 vertices and 6.432 edges

Results

Page 24: Community Structure in Large Complex Networks

Both alg 2 and 3 successfully find approximate cores. Higher values of λ indicate smaller core sizes. Fig (b), the best community of the textual graph has a large

conductance of .3 best community has as many internal edges as cut edges.

Alg 3 is believed to be more useful.

Results

Page 25: Community Structure in Large Complex Networks

Is a “whisker” make sense?

Comment

Page 26: Community Structure in Large Complex Networks

[1] Schaefer, T. J. The complexity of satisfiability problems. In Proc. 10th Ann. ACM Symp. on Theory of Computing (1978), Association for Computing Machinery, pp. 216-226.

Reference


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