Ambedkar.Dukkipati.1..

67 Pages·2017·1.19 MB·English

by  Ambedkar Dukkipati

On statistical analysis of
spectral graph algorithms for
community detection in networks
Ambedkar Dukkipati
DepartmentofComputerScience&Automation
IndianInstituteofScience
17 April 2017
The Problem - Graph Clustering
(cid:73) Partition a graph G into k
‘clusters’.
(cid:73) Cluster Properties
(cid:73) Many edges within clusters
(cid:73) Few edges between clusters
(cid:73) Partitioning Objective
(cid:73) Cut across fewest edges possible
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Why is this hard?
(cid:73) Graph partitioning is NP-hard
(cid:73) Brute force?
(cid:73) For a small graph with 100 nodes, the number of different
partitions exceeds the number of atoms in the universe!
(cid:73) Heuristics?
(cid:73) Optimality, consistency, efficiency ...
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Spectral...Why and What?
Why?
(cid:73) Nice approximations that give rise to polynomial time
algorithms
(cid:73) with theoretical guarantees, provided by statistical analysis.
What?
(cid:73) Underlying objects in a problem can be represented as
matrices
(cid:73) Eigenvalues and eigenvectors of these matrices become a
clue to the solution.
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Spectral Clustering
(cid:73) Well studied in literature
(cid:73) Strong theoretical grounding
K-means
(cid:73) Spectral Graph Theory
(cid:73) Consistency results
(cid:73) Efficient linear algebraic
computations
Spectral Clustering
Ngetal. NIPS,2001
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Graph Coloring
Theorem (Brooks)
Apart from the following cases
1. G is complete
2. G has odd cycles
we have χ ≤ d
G max
Theorem (Gershgorin Disk)
Assume A is a nonnegative n×n real matrix. Then all
eigenvalues of A lie in the set
 
n
(cid:91) (cid:88) (cid:88)
Aii− Aij, Aii+ Aij
i=1 j(cid:54)=i j(cid:54)=i
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Graph Coloring
Lemma
Let A be the adjacency matrix of G = (V,E). Let
µ ≥ µ ≥ ... ≥ µ be the eigenvalues of A. Then µ ≤ d .
1 2 n 1 max
Proof: By Gershgorin theorem
 
(cid:88)
µ1 ≤ max Aii+ Aij
1≤i≤n
j(cid:54)=i
n
(cid:88)
= max A
ij
1≤i≤n
j=1
= max deg(i) = d
max
1≤i≤n
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Graph Coloring
The previous result can be proved using Rayleigh’s principle.
Theorem (Rayleigh’s Principle)
Let A be a nonnegative n×n real matrix and Let µ be the
1
largest eigenvalues of A then
vTAv
µ = max
1 v(cid:54)=0 vTv
Note: A is a adjacency matrix of graph G and let µ be the
1
largest eigenvalue of A. Then we already have the following:
(cid:73) χ ≤ d
n max
(cid:73) µ ≤ d
1 max
Theorem (Wilf, 1967)
χ ≤ (cid:98)µ (cid:99)+1
G 1
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Proof make use of Rayleigh’s principle.
Some matrices related to graphs
Let G = (V,E) be a graph. |V| = n and |E| = e.
(cid:73) Adjacency Matrix: A ∈ Rn×n such that

0 if i = j,





A = 1 if (i,j) ∈ E,
ij




 0 if (i,j) ∈/ E.
(cid:73) Degree Matrix: D ∈ Rn×n is diagonal matrix such that
D = deg(i)
ii
(cid:73) Incidence Matrix: B ∈ Rn×e, where rows indexed by
vertices and columns indexed by edges and B = 1 if vertex
ij
i lies on edge j.
(cid:73) Laplacian Matrix: L ∈ Rn×n is defined as L = D−A
(cid:73) Normalized Laplacian: L ∈ Rn×n is defined as
L = I −D−1/2AD−1/2
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Graph Laplacian
Let G = (V,E) be a graph. |V| = n and |E| = e. Laplacian:
L ∈ Rn×n such that

d if i = j,
 i




L = −1 if (i,j) ∈ E,
ij




 0 if (i,j) ∈/ E.
Theorem
Let λ ≤ λ ≤ ... ≤ λ be eigenvalues of L. Then
1 2 n
1. L is symmetric and positive semidefinite
2. λ = 0
1
3. λ > 0 iff G is connected
2
4. λ = 0 and λ > 0 iff G has exactly k-disjoint
k k+1
components
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