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Graph Analysis: Community Detection

Giri Iyengar

Cornell University

[email protected]

March 19, 2018

Giri Iyengar (Cornell Tech) Graph Analysis March 19, 2018 1 / 25

Overview

1 Overview

2 Spectral Clustering

3 Markov Cluster Algorithm


Overview

1 Overview




Graph Partitioning

Partitioning a GraphIn mathematics, the graph partition problem is defined on data representedin the form of a graph G = (V,E), with V vertices and E edges, such thatit is possible to partition G into smaller components with specificproperties. For instance, a k-way partition divides the vertex set into ksmaller components. A good partition is defined as one in which thenumber of edges running between separated components is small. Uniformgraph partition is a type of graph partitioning problem that consists ofdividing a graph into components, such that the components are of aboutthe same size and there are few connections between the components.

Source: Wikipedia


Graph Partitioning

Partitioning a GraphImportant applications of graph partitioning include scientific computing,partitioning various stages of a VLSI design circuit and task scheduling inmulti-processor systems. Recently, the graph partition problem has gainedimportance due to its application for clustering and detection of cliques insocial, pathological and biological networks. For a survey on recent trendsin computational methods and applications see Buluc et al. (2013).

Source: Wikipedia


Graph Partitioning

Natural clusters have nodes withmany edges between themAnd few edges across clustersA random walk on the graphwill infrequently go from onecluster to anotherAdjacency matrix will haveblock-like structureEigenvalues of Adjacency /Laplacian matrix will berevealing

Figure: Carlos Castillo, SapienzaUniversity


Overview

1 Overview




Spectral Clustering

Spectral ClusteringIn multivariate statistics and the clustering of data, spectral clusteringtechniques make use of the spectrum (eigenvalues) of the similarity matrixof the data to perform dimensionality reduction before clustering in fewerdimensions. The similarity matrix is provided as an input and consists of aquantitative assessment of the relative similarity of each pair of points inthe dataset.

Source: Wikipedia


Spectral Clustering

Initially used for Image Segmentation

Generally recognized as a technique to cluster dataAn alternative to K-MeansWidely used in Graph Partitioning / Community Detection


Spectral Clustering

Initially used for Image SegmentationGenerally recognized as a technique to cluster data

An alternative to K-MeansWidely used in Graph Partitioning / Community Detection


Spectral Clustering

Initially used for Image SegmentationGenerally recognized as a technique to cluster dataAn alternative to K-Means

Widely used in Graph Partitioning / Community Detection


Spectral Clustering

Initially used for Image SegmentationGenerally recognized as a technique to cluster dataAn alternative to K-MeansWidely used in Graph Partitioning / Community Detection


Spectral Clustering: Definitions

Graph: G(V, E)Weighted Graph. That is between vi and vj , there is a weight wijUndirected. wij == wjiDefine degree of vertex di =

∑nj=1 wij


Spectral Clustering: Definitions

W(A,B) =∑

i∈A,j∈B wij . A, B ⊂ V|A|: Number of vertices in Avol(A) =

∑i∈A di

A ∩ Ā = ∅


Spectral Clustering: Properties of Laplacian

fT Lf = 12∑n

i,j=1 wij(fi − fj)2

fT Lf = fT Df − fT Af =∑n

i=1 dif2i −

∑ni,j=1 wijfifj

fT Lf = 12(∑n

i=1 dif2i − 2

∑ni,j=1 wijfifj +

∑nj=1 djf

2j )

L is symmetric (follows from symmetry of D and A)L is positive semi-definiteSmallest eigenvalue of L is 0 and has 1n as the eigenvector


Spectral Clustering: Properties of Laplacian

Number of connected componentsNumber of connected components in the Graph is given by multiplicity ofeigenvalue 0. Trivial to see from previous slide.


Spectral Clustering Algorithm

Compute the un-normalized Laplacian LCompute the first k eigenvectors of L. These are the smallest keigenvectorsU matrix of the first k eigenvectors is an n× k matrixTake the Rk row-vectors of UPerform k-way K-Means on these vectors


Variations of Spectral Clustering

Instead of the un-normalized Laplacian, start with normalizedLaplacian matricesInstead of solving the eigenvalue problem, solve the generalizedeigenvalue problem instead


Spectral Clustering for Image Segmentation

Figure: Source - Scikit-learn Figure: Source - Scikit-learn


Overview

1 Overview




Markov Cluster Algorithm

MCL Simulation


https://micans.org/mcl/ani/mcl-animation.html

MCL Algorithm

MCLThe MCL algorithm is short for the Markov Cluster Algorithm, a fast andscalable unsupervised cluster algorithm for graphs (also known asnetworks) based on simulation of (stochastic) flow in graphs.

Heavily used in clustering Proteins based on their Amino acid sequences


MCL Algorithm

Finds clusters by simulating random walksAssumption: Random walks in a graph with tend to spend more timein Natural clustersRandom walks will infrequently go between clustersWithin a cluster, much more likely to find longer-length random walkbetween any pair of nodes


MCL Algorithm

Start with a Column Stochastic matrix of the GraphExpansion Step: Power iteration of the MatrixInflation Step: Hadamard power followed by normalizationRepeat Expansion and Inflation till convergence


MCL Algorithm

M1: Column Stochastic MatrixExpansion: M2 = M1 ×M1Hadamard: M3 = [mr2ij ]. Typically r = 2Normalization: M1 = [ m3ij∑

jm3ij

]


Hadamard power operation on Matrix

[0.8 0.6 0.90.2 0.4 0.1

] [0.64 0.36 0.810.04 0.16 0.01

]→

[0.94 0.69 0.990.06 0.31 0.01

]


MCL Algorithm

Figure: Source - Stijn van Dongen



OverviewSpectral ClusteringMarkov Cluster Algorithm

graph analysis: community detection - github pages · giri iyengar (cornell tech) graph analysis...

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