Overlapping Communities

Transcription

1 Yangyang Hou, Mu Wang, Yongyang Yu Purdue Univiersity Department of Computer Science April 25, 2013

2 Overview Datasets Algorithm I Algorithm II Algorithm III Evaluation

3 Overview Graph models of many real world applications exhibit an overlapping community structure, which is hard to grasp with the classical graph clustering methods where each vertex of the graph is assigned to exactly one community. GOAL: Allow each vertex of the graph to belong to multiple communities at the same time.

4 Datasets Zachary s karate club: social network of friendships between 34 members of a karate club at a US university in 1970s karate dataset for result illustration American College football: network of American football games between Division IA colleges during regular season Fall 2000 (115 vertices, 615 edges)

5 Algorithm I: LA-IS2 (Baumes et al. 2005) Link-Aggregate and Improved Iterative Scan (LA-IS2) Optimize density metric: W ad (C) = 2 E(C) C E(C) set of edges with both endpoints in C Intuition: average degree, the bigger the better for local communities Step 1: generate community candidates (LA step) Step 2: refine community candidates and remove duplicates (IS2 step)

6 LA-LS2 algorithm framework 1 LA step Order the vertices based on PageRank values Add vi to C j if W ad (C j v i ) > W ad (C j ), else C k = {v i } 2 IS2 step for each candidate C j N Cj adj(c j ) for all v N, if v Cj, C C j \ {v}, else C C j {v} Cj C if W ad (C ) > W ad (C j ) until W ad (C j ) no longer increases

7 Algorithm II (Zhang et al. 2007) Spectal Mapping & Fuzzy C-means Minimize a modularity function: Q(U) = k c=1 [ A(V c, V c ) A(V, V ) ( ) ] A(Vc, V ) 2 A(V, V ) A(V c, V c ) = (u i,c + u j,c) w(i, j) 2 i V c,j V c A(V c, V ) = A(V c, V c ) + u i,c + (1 u j,c) w(i, j) 2 i V c,j / V c A(V, V ) = w(i, j) i V,j V

8 Algorithm II (Zhang et al. 2007) u(i, c) describes the possibility that vertex i belongs to cluster c u(i, c) is solved by fuzzy c-means (a method of clustering which allows one piece of data to belong to two or more clusters) Using a given threshold to classify J = N C u i,k x i c k i=1 k=1

9 Algorithm II (Zhang et al. 2007) 1 Spectral mapping: solve a generalized eigenvalue problem to get the top K eigenvectors Ax = λdx where A is the adjacency matrix, D is the degree diagonal 2 Fuzzy c-means: Normalize each row of the K 1 eigenvectors (the largest one is ignored) Cluster the row vectors using fuzzy c-means method to get U 3 Choose the largest result which maximizes Q(U)

10 Algorithm III (Nepusz et al., 2008 ) Partition Matrix U ( c V ): u ik [0, 1] for all 1 i c, 1 k N c i=1 u ik = 1 for all 1 k N 0 N k=1 u ik N for all 1 i c Derive the similarity matrix S ( V V ) from U: s ij = c k=1 u iku jk or S = U T U in the matrix form.

11 Algorithm III (Nepusz et al., 2008 ) Given the number of cluster c, the weight matrix W and the desired similarities S, find U that minimizes D G (U): Usually, we set S = A G. D G (U) = N N i=1 j=1 w ij ( s ij s ij ) 2

12 Algorithm III (Nepusz et al., 2008 ) Employ a gradient-based iterative optimization method. Initially, set c =2. Keep on increasing the number of communities until the newly introduced community does not improve the overall community structure of the network.

13 Algorthim I for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

14 Algorthim I for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

15 Algorthim I for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

16 Algorthim II for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

17 Algorthim II for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

18 Algorthim II for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

19 Algorthim II for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

20 Algorthim II for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

21 Algorthim III for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

22 Algorthim III for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

23 Algorthim III for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

24 Algorthim III for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

25 Algorthim III for karate dataset ""!(!# %!! '!' & " $! (!" "*!$ #! ) # #" "!!& ## #$!* ") "( "%!) "&!% "# #* "$ "'

26 Measurements Normalized Cut ( k ) N = #Cut Edges i=1 1 di This also shows the number of shared nodes. Average Degree W ad (C) = 2 E(C) C Hope this will be significantly larger than the average degree.

27 Result Karate Algorithm I Algorithm II Algorithm III #Community Normalized Cut Shared nodes Average Degree Football Algorithm I Algorithm II Algorithm III #Community Normalized Cut Shared nodes Average Degree

28 Currently working on.. Testing three algorithms on large datasets. e.g. partial facebook network.

29 References S. Fortunato. Community Detection in Graphs, arxiv: v2 [physics.soc-ph] J. Baumes, M. Goldberg, M. Krishnamoorhty, M. M. Ismail, and N. Preston. Finding Communities by Clustering a Graph into Overlapping Subgraphs, Proceedings of IADIS Applied Computing 2005, pages , February J. Baumes, M. Goldberg, and M. M. Ismail. Efficient Identification of, Proceedings of the 2005 IEEE International Conference on Intelligence and Security Informatics, pages 27 36, 2005 S. Zhang, RS. Wang, and XS. Zhang. Identification of Overlapping Community Structure in Complex Networks Using Fuzzy C-means Clustering, Physica A: Statistical Mechanics and its Applications, 374 (1), pages T. Nepusz, A. Petróczi, L. Négyessy, and F. Bazsó. Fuzzy Communities and the Concept of Bridgeness in Complex Networks, arxiv: v2 [physics.soc-ph]

30 QUESTION?