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modularity

modularity - Modularity and Community Structure M.E.J...

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Click to edit Master subtitle style Modularity and Community Structure M.E.J Newman in PNAS 2006 11

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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) 22
Protein-protein 33 Nodes – proteins (6K), edges – interactions (15K). Reflect the cell’s machinery and signaling pathways.

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Communities (clusters) in a network A community (cluster) is a densely connected group of vertices, with only sparser connections to other groups . 44
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 55

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Distilling Modules from Networks 66 Motivation: identifying protein complexes responsible for certain functions in the cell
Click to edit Master subtitle style Modularity (Newman) 77

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Modularity of a division (Q) 88 Q = #( edges within groups ) - E(#( edges within groups in a RANDOM graph with same node degrees )) Trivial division : all vertices in one group ==> Q(trivial division) = 0 Edges within groups ki = degree of node i M = l ki = 2|E| Aij otherwise Eij = expected number of edges between i and j in a random graph with same node degrees. Lemma : Eij ki*kj / M Q = &(Aij - ki*kj/M | i,j in the same
Modularity 99 Are two definitions of modularity equivalent ?

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Methods to Optimize Q 1010 Fast modularity Greedily iterative agglomeration of small communities Choosing at each step the join that results in the greatest increase (or smallest decrease) in Q Can be generalized to weighted networks Extreme methods: Simulated Annealing, GA Heuristic algorithm Spectral Partitioning
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 1111

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Algorithm 1: Division into two groups Suppose we have n vertices {1,. ..,n} s - { 1} vector of size n. Represent a 2-division:
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modularity - Modularity and Community Structure M.E.J...

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