Community Detection and Labeling Nodes

Transcription

1 and Labeling Nodes Hao Chen Department of Statistics, Stanford Jan. 25, 2011 (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

2 Community Detection - Network: n nodes, m edges - detecting clusters - to find groups of nodes that have more connections within groups than than between groups and the remainder of the network - to choose an objective function that captures this property (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

3 Community Detection - Network: n nodes, m edges - detecting clusters - to find groups of nodes that have more connections within groups than than between groups and the remainder of the network - to choose an objective function that captures this property Notations: - A ij : the number of edges between vertices i and j - A = {A ij }: adjacency matrix - k i : degree of vertex i (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

4 An Review of Existing Methods 1 Minimum-cut method: number of parts predetermined, number of edges between groups minimized. 2 Hierarchical clustering: define a similarity measure on node pairs, eg. cosine similarity between rows of the adjacency matrix, and group similar nodes into communities in two approaches - bottom up and top down. 3 Girvan-Newman algorithm: assigns a number to each edge, which is large if the edge lies between many pairs of nodes, identifies these edges and removes them. 4 Modularity maximization (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

5 Modularity Community Detection Q = 1 4m ijr ( A ij k ) ik j S ir S jr 2m S ir = I vertex i belongs to group r (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

6 Modularity Q = 1 4m ijr ( A ij k ) ik j S ir S jr 2m S ir = I vertex i belongs to group r Approximate algorithms 1 Greedy algorithms - The Louvain method: first looks for small communities by optimizing modularity in a local way, then aggregates nodes of the same community and builds a new network whose nodes are the communities. 2 Simulated annealing: computational costly for large networks 3 Spectral optimization (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

7 Spectral Optimization [Newman, 2006] Two-group case Q = 1 2m s i = ij ( A ij k ) ik j s i s j 2m { 1 if i is in group 1 1 if i is in group 2 Let B ij = A ij k i k j 2m, s = (s 1, s 2..., s n ) T β 1 β 2 β n be eigenvalues of B and u 1, u 2,..., u n be corresponding eigenvectors. (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

8 Q = 1 4m st Bs = 1 4m (u T i s) 2 β i Q is maximized when s is proportional to u 1, however, impossible because s only takes values in 1 and -1. To maximize the term in β 1 - divide the vertices according to the signs of the elements in u 1. i (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

9 Dividing the network to more than two communities Repeat division into two. For subgroup g, B (g) ij = A ij k [ ] ik j 2m δ ij k (g) d g i k i 2m - k (g) i : the degree of vertex i within subgraph g - d g : the sum of the total degree k i of the vertices in the subgraph Stops when no division of a subgraph will increase the modularity of the network. (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

10 Labeling Nodes Labeling Nodes Mixed membership stochastic blockmodels [Airoldi et al. 2008] For each node u, draw a K dimensional mixed membership vector π u from Dirichlet(α). For each pair of nodes u, v, the number of link between then is sampled from Bernoulli(z T uvbz vu ) - z uv Multinomial(π u ) - z vu Multinomial(π v ) - B: a K K matrix, with B ij be the probability of having a link between a node from group i and a node from group j. For various K, estimate the hyper-parameters α and B, determine the number of groups through the BIC criterion. (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9

11 Interesting Problems Interesting Problems 1 To determine number of communities. (Department of Statistics, Stanford) Community Detection and Labeling Nodes Jan. 25, / 9