Modularity-Maximizing Graph Communities via Mathematical Programming

Transcription

1 Modularity-Maximizing Graph Communities via Mathematical Programming Ying Xuan February 24, 2009 Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

2 Table of contents 1 Problem Definition and Preliminaries Modularity Maximization Recent Efforts Main Contributions 2 Algorithms and Analysis Linear Programming Algorithm Vector Programming Algorithm Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

3 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]: is maximized. Here, Q(C) = 1 2m u,v (a u,v d ud v 2m ) δ(γ(u),γ(v)) 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. Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

4 Recent Efforts Problem Definition and Preliminaries 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. Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

5 Problem Definition and Preliminaries Main Contributions Main Contributions Two heuristics: LP relaxation and Distance-based Rounding Algorithm; Quadratic Programming and Randomized Rounding Algorithm. One potential method ratio analysis: Similarity with Min-Disagree problem (4-approx) in LP formulation. Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

6 Pitfalls Problem Definition and Preliminaries Main Contributions No ratio analysis over the LP rounding or randomized rounding for SDP; Significant huge resource requirement due to Θ(n 3 ) constraints in LP and Θ(n 2 ) variables in the vector programming; Huge time complexity and computation overhead; Performance of LP rounding relies on the selection of center vertex. Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

7 Algorithms and Analysis Linear Programming Algorithm Linear Programming Algorithm IP Formulation and LP relaxation; Distance-based Rounding; Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

8 IP formulation Algorithms and Analysis Linear Programming Algorithm Integer Program Maximize Subject to 1 2m (a u,v d ud v 2m ) (1 x u,v) u,v x u,w x u,v + x v,w x u,v {0,1} u,v,w V u,v V Let m u,v = a u,v 2m d ud v 4m 2 Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

9 Algorithms and Analysis Linear Programming Algorithm LP formulation Linear Program Maximize Subject to m u,v (1 x u,v ) u,v x u,w x u,v + x v,w x u,v 0 u,v,w V u,v V Use CPlex to solve it = Θ(n 3 )constraints Modularity-Maximizing Graph Communities via Mathematical February Programming 24, / 19

10 How good can this heuristic be? Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19 Algorithms and Analysis Distance-based rounding Linear Programming Algorithm Algorithm 1 dist(u,v) x u,v for LP solution X; S V ; while S do Select u S; randomly select T u {v dist(u,v) 1/2} if average dist(u,v) < 1/4 for all v {T u \ {u}} then Make C = T u a cluster; else Make C = {u} a singleton cluster S S \ C; Refine the result using local-search algorithm.

11 Algorithms and Analysis Linear Programming Algorithm Analysis Definition Min-Disagree problem formulated by [3]: Given a complete graph where all edges are labeled as respectively + or - to indicate the similarity or dis-similarity of vertex pair; partition the graph into clusters such that the number of errors ( - edges within clusters and + edges between clusters) are minimized. Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19

12 Analysis(Cont ) Algorithms and Analysis Linear Programming Algorithm Min-Disagree LP Minimize (u,v) E + x u,v + (u,v) E (1 x u,v ) E + (u,v) E µ u,v(1 x u,v ) Subject to x u,w x u,v + x v,w x u,v 0 u,v,w V u,v V where µ u,v = 1 if (u,v) is + edge, otherwise 0. Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19

13 Analysis(Cont ) Algorithms and Analysis Linear Programming Algorithm Further results: if define µ u,v = m u,v = a u,v 2m d ud v 4m 2 Mis-Degree formulation on complete graph is similar to the IP formulation of modularity maximization. Mis-Degree problem has a 4-approximation rounding algorithm; Due to existence of E +, it is hard to get a same approximation algorithm for modularity maximization. Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19

14 Algorithms and Analysis Vector Programming Algorithm Vector Programming Algorithm Kernel: Formulate the problem as Quadratic Program; Relax to vector program (Semi-definite programming); Use randomized cutting hyperplane to round the SDP solution. Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19

15 Algorithms and Analysis Vector Programming Algorithm Quadratic Programming Formulation Considering partitioning the graph into two communities (S, S) of maximum modularity, let y v = ±1 indicate that vertex v belongs (or not) to S. Therefore, the formulation is: Quadratic Program Maximize Subject to 1 4m u,v V m u,v (1 + y u y v ) y 2 v = 1 for all v Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19

16 Algorithms and Analysis Semi-definite Relaxation Vector Programming Algorithm Key idea: relax y = ±1 to n-dimensional vector y with y = 1. The product of y u y v is corresponding to inner product of y u y v = cos θ Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19

17 Algorithms and Analysis Randomized Rounding Vector Programming Algorithm randomly select a n-dimension vector s, where each component is following independent N(0,1) Gaussian. S = {v y v s 0} S = {v y v s < 0} Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19

18 Algorithms and Analysis Vector Programming Algorithm Algorithm Algorithm 2 Hierarchical Clustering: use Semi-definite programming find a near-optimal division of a larger cluster locally optimal; repeat the division until no further partition will increase the modularity; do local search post-processing to refine the solution. Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19

19 Bibliography Algorithms and Analysis Vector Programming Algorithm M. Newman and M. Girvan. Finding and evaluating community structure in networks. Physical Review E, 69, U. Brandes, D. Delling, M. Gaertler, R. Görke, M. Hoefer, Z. Nikoloski, and D. Wagner. On modularity clustering. IEEE Transactions on Knowledge and Data Engineering, 20(2): , M. Charikar, V. Guruswami, and A. Wirth. Clustering with qualitative information. Journal of Computer and System Sciences, pages , M. Newman. Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA, 103: , Modularity-Maximizing Graph Communities via Mathematical FebruaryProgramming 24, / 19