1 Modularity and Community Structure in Networks* Final project *Based on a paper by M.E.J Newman in PNAS 2006

2 Introduction

3 Networks A network: presented by a graph G(V,E): V = nodes, E = edges (link node pairs) Examples of real-life networks: –social networks (V = people) –World Wide Web (V= webpages) –protein-protein interaction networks (V = proteins)

4 Protein-protein Interaction Networks Nodes – proteins (6K), edges – interactions (15K). Reflect the cell’s machinery and signaling pathways.

5 Communities (clusters) in a network A community (cluster) is a densely connected group of vertices, with only sparser connections to other groups.

6 Searching for communities in a network There are numerous algorithms with different "target-functions": –"Homogenity" - dense connectivity clusters –"Separation"- graph partitioning, min-cut approach Clustering is important for understanding the structure of the network –Provides an overview of the network

7 Distilling Modules from Networks Motivation: identifying protein complexes responsible for certain functions in the cell

8 Newman's network division algorithm

9 Important features of Newman's clustering algorithm The number and size of the clusters are determined by the algorithm Attempts to find a division that maximizes a modularity score Q –heuristic algorithm Notifies when the network is non-modular

10 Overview of the algorithm

11 Spectral 2-division algorithm Input: adjacency matrix A (n vertices) Output: a (  1)-vector of size n representing the 2-division –"-1" cluster (vertices whose corresponding entry is -1) and "+1" cluster (vertices whose corresponding entry is +1) Build a modularity matrix B from A Compute the leading eigen-pair (u 1,  1 ) of B –u 1 is the eigen-vector (size n),  1 is the eigen-value. leading eigen-pair: Bu 1 =  1 u 1.  1 is the maximal eigen value If (  1 == 0) => the network is indivisible Else (heuristic...) –Transform u 1 into vector (  1)-vector s –Q = s T Bs –if (Q > 0) return s, else return (+1,....,+1)

12 Dividing into more than 2 How to compute into more than 2? Idea: apply the algorithm recursively* on every group.  The algorithm should be generalized for a 2- division of a group in the network

13 Newman's clustering algorithm P* = {{1,....,n}} (*singleton nodes should be removed) For each group g in P –Remove g from P –Perform a spectral 2-division on g if g is divisible - improve the 2-division by additional heuristic. Add* each subgroup in the 2-division to P * if the subgroup has more than one element, and is different from g.