A scalable multilevel algorithm for community structure detection

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A scalable multilevel algorithm for community
structure detection

• Melih Onus Hristo Djidjev
• Arizona State University Los Alamos National
  Laboratory

Models and Algorithms for the Web Graph (WAW
2006) November 29 December 2, 2006
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Community Structure Detection Problem

• The problem of identifying communities in a
  network is usually modeled as a graph clustering
  problem
• Vertices correspond to individual items
• Edges describe relationships
• The communities correspond to subgraphs
• Dense connections between vertices from the same
  subgraph
• Fewer connections between vertices in different
  subgraphs

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Motivation Why to detect communities?

• Analyze and understand the information contained
  in the huge amount of data available on the WWW
• Finding related commercial items
• Recommendation systems
• Important for
• Social networks
• Ad-hoc networks
• Protein interaction networks
• Genetic networks

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Motivation Why to detect communities?

• Predict how much someone going to love a
  movie based on their movie preferences

Grand Prize 1.000.000
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Outline of the talk

• Previous work
• Graph partitioning problem
• Our approach
• Modularity
• Reduction
• Multilevel graph partitioning
• Experimental results
• Conclusions

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Previous Work

• Two main classes
• Agglomerative Methods (addition of edges)
• Divisive Methods (removal of edges)
• Algorithms based on
• Laplacian Matrix
• Centrality measures
• Flow models
• Random walks
• Resistor networks
• Optimization
• Not fast enough or inaccurate

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Graph Partitioning Problem

• Given a graph G(V, E), find a partition such that
  
• The partition is balanced (i.e., the number of
  vertices of all subsets are roughly equal)
• Cut size is minimized (i.e., the number of the
  edges with endpoints in different subsets is
  minimized)
• Previous Work
• Kernighan-Lin algorithm
• Spectral partitioning
• Multilevel algorithms

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Kernighan - Lin Algorithm

• Find an initial random partition
• Improve by a greedy procedure that swaps pairs of
  vertices from different partitions
• Minimize the size of the cut set

u
v
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Graph Partitioning vs Graph Clustering

• Find Clusters
• Community sizes may differ
• Number of subsets varies

• Minimize cut size
• Equal number of vertices in each subset
• Number of subsets is an input

• Algorithms for graph partitioning can not be
  directly used to produce good quality clustering

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Our approach

• Convert original graph G into a complete graph G
• Find min-cut of G using modified graph
  partitioning method
• This will produce a good quality (high
  modularity) clustering for G

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Modularity

• A useful measure of clustering quality
• Introduced by Newman 6
• Modularity of a partitioning
• (number of edges within communities)
• (expected number of such edges)
• We are trying to find a division of graph with
  high modularity

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Reduction
Min-Cut Problem The problem of finding a
minimum cut in a complete edge-weighted graph G'
Graph Clustering Problem The problem of finding
a clustering of maximum modularity in G
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Reduction

• Maximize modularity of a partitioning
• (number of edges within communities)
• (expected number of such edges)

Graph Clustering Problem Maximize modularity
Minimize (- modularity) (cut size)
(expected cut size)
Min-Cut Problem Minimize cut size
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Random Graph Models
pij the probability that there is an edge
between vertices i and j in a random graph from a
given distribution

• Erdos - Renyi Model

Chung - Lu Model
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Multilevel graph partitioning

• Fast and an accurate method for producing
  high-quality partitions

• Consists of the three phases
• Coarsening phase
• Partitioning phase
• Uncoarsening and refinement phase

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Coarsening Phase

• Find a maximal matching and collapse edges to a
  vertex
• Recursive coarsening
• lt G G1, G2, , Gk gt

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Partitioning Phase

• Greedy graph growing partitioning
• Partition Gk

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Uncoarsening and Refinement Phase

• Project the partitioning Pi of Gi to Pi-1 of Gi-1
• More degrees of freedom at Gi than Gi-1
• Improve Pi using KL algorithm

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Implementation

• Our implementation is based on the graph
  partitioning package METIS 3 that employs a
  multilevel strategy
• Convert the graph partitioning algorithm into a
  clustering one
• The optimal clustering might not be balanced.
• We ignore the restrictions that control the
  sizes of the parts.
• The number of the parts in the optimal clustering
  is not known.
• We employ a recursive bisection procedure.
• The original graph G might be sparse, while the
  transformed one G' is complete. Our algorithm
  does not explicitly generate G.

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Modularity Erdos - Renyi Model

• (- Modularity) cut size n1n2p

(- Modularity) cut size (n11)(n2-1)p
n1
n2
Erdos - Renyi Model
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Modularity Chung - Lu Model

• (- Modularity) cut size w1w2/2m

(- Modularity) cut size
(w1 w(v))(w2 - w(v))/2m
w1
w2
wi Sum of degrees in partition i
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Analysis

• Time Complexity O(nm)
• Experiments
• Random Graphs
• k-community graphs
• nd.edu

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Experiment I Random Graphs

• We generated random graphs with 128 vertices and
  4 communities of size 32 each
• The expected degree of any vertex is 16
• Out degree varies

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Experiment II k-community graphs

• We generated graphs with k communities
• Size of each community is 100
• Expected number of edges in the community is
  equal to expected number of edges going outside
  from community.
• Probability of an edge in communities varies
  between 0.5 and 0.1.
• Results show that graphs are clustered especially
  99 correctly.

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Experiment III nd.edu

• Data consists of the complete map of the nd.edu
  domain, which contains 325,729 document and
  1090108 links
• Our algorithm clusters this graph into 280
  clusters with modularity 0.925579
• This high modularity indicates strong community
  structure in the graph
• We show the dendrogram generated by our
  algorithm.
• The size of rectangles are proportional to size
  of communities.

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Conclusions

• Community structure detection problem
• A scalable algorithm
• Based on multilevel graph partitioning
• Uses modularity as a quality measure