# Com_SDP - Modularity-Maximizing Graph Communities via...

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Unformatted text preview: Modularity-Maximizing Graph Communities via Mathematical Programming Ying Xuan February 24, 2009 Ying Xuan () Modularity-Maximizing Graph Communities via Mathematical Programming February 24, 2009 1 / 19 Table of contents 1 Problem Definition and Preliminaries Modularity Maximization Recent Efforts Main Contributions 2 Algorithms and Analysis Linear Programming Algorithm Vector Programming Algorithm Ying Xuan () Modularity-Maximizing Graph Communities via Mathematical Programming February 24, 2009 2 / 19 Problem Definition and Preliminaries Modularity Maximization Modularity Maximization Given undirected graph G = ( V , E ), find a clustering { C 1 , . . . , C k } which is a disjoint partition of V such that the modularity of the clustering C [1]: Q ( C ) = 1 2 m summationdisplay u , v ( a u , v − d u d v 2 m ) · δ ( γ ( u ) , γ ( v )) is maximized . Here, a u , v = a v , u = 1 if ( u , v ) ∈ E , otherwise 0; d u denotes the degree of any vertex u ; m is the number of edges in G ; γ ( v ) denotes the (unique) index of the cluster to which v belongs; δ ( x , y ) is the Kronecker Delta, which equals to 1 if x = y , otherwise 0. Ying Xuan () Modularity-Maximizing Graph Communities via Mathematical Programming February 24, 2009 3 / 19 Problem Definition and Preliminaries Recent Efforts Recent Efforts This maximization problem is NP-complete[2]; Correlation Clustering[3] interprets “partial membership of the same cluster” as a distance metric, group nearby ones together; Spectral Clustering[4] repeatedly divides clusters based on the largest eigenvalue and corresponding eigenvector of the modularity matrix. Ying Xuan () Modularity-Maximizing Graph Communities via Mathematical Programming February 24, 2009 4 / 19 Problem Definition and Preliminaries Main Contributions Main Contributions Two heuristics: LP relaxation and Distance-based Rounding Algorithm; Quadratic Programming and Randomized Rounding Algorithm....
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Com_SDP - Modularity-Maximizing Graph Communities via...

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