1. A scalable multilevel algorithm for community structure detection
My name is Melih Onus. I am a PhD student in ASU.
I will present my work on community structure detection
that I did at Los Alamos this summer.
This is a joint work with my mentor Hristo Dijidjev.
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

2. 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

3. 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

4. Motivation: Why to detect communities?
Predict how much someone going to love a movie based on their movie preferences
Grand Prize $

5. Outline of the talk Previous work Graph partitioning problem
Our approach
Modularity
Reduction
Multilevel graph partitioning
Experimental results
Conclusions

6. Previous Work Two main classes Algorithms based on
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

7. 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
, an initial random partition isimproved by a greedy procedure that swaps pairs ofvertices from different partitions so that the size of thecutset is reduced by a maximum amount, until a localoptimum is reached. We will discuss this algorithm ingreater detail in the following sections.
, is based on the Laplacian matrix of a graph and is usually more precise, but relatively slow compared to the KL method.

8. 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 u v

9. Graph Partitioning vs Graph Clustering
Minimize cut size
Equal number of vertices in each subset
Number of subsets is an input
Find Clusters
Community sizes may differ
Number of subsets varies
Algorithms for graph partitioning can not be directly used to produce good quality clustering

10. 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

11. 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

12. 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

13. 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

14. Random Graph Models Erdos - Renyi Model: Chung - Lu Model:
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:

15. 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
Graph is coarsened recursively until we get a graph of sufficiently small size.

16. Coarsening Phase
Find a maximal matching and collapse edges to a vertex
Recursive coarsening:
< G = G1, G2, …, Gk >
Graph is coarsened recursively until we get a graph of sufficiently small size.

17. Partitioning Phase Greedy graph growing partitioning Partition Gk
Graph is coarsened recursively until we get a graph of sufficiently small size.

18. 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
Graph is coarsened recursively until we get a graph of sufficiently small size.

19. 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’.

20. Modularity: Erdos - Renyi Model
(- Modularity) = cut size – n1n2p
(- Modularity)’ = cut size’ – (n1+1)(n2-1)p n1 n2
Erdos - Renyi Model:

21. 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

22. Analysis Time Complexity: O(n+m) Experiments Random Graphs
k-community graphs
nd.edu

23. 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

24. 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.

25. Experiment III: nd.edu
Data consists of the complete map of the nd.edu domain, which contains 325,729 document and links
Our algorithm clusters this graph into 280 clusters with modularity
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.

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