Significant scales in community structure

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Presentation at ECCS 2013, Barcelona, September 17, 2013

Transcript of Significant scales in community structure

  • 1. Signicant scales in community structure V.A. Traag1,2, G. Krings3, P. Van Dooren4 1KITLV, Leiden, the Netherlands 2e-Humanities, KNAW, Amsterdam, the Netherlands 3Real Impact, Brussels, Belgium, 4UCL, Louvain-la-Neuve, Belgium September 17, 2013 eRoyal Netherlands Academy of Arts and Sciences Humanities

2. Community Detection Contant Potts Model (CPM) Minimize H() = ij (Aij )(i , j ) Resolution-limit-free Internal density pc > Density between pcd < 3. Community Detection Contant Potts Model (CPM) Minimize H() = ij (Aij )(i , j ) Resolution-limit-free Internal density pc > Density between pcd < 4. Community Detection Contant Potts Model (CPM) Minimize H() = ij (Aij )(i , j ) = c(ec n2 c) Resolution-limit-free Internal density pc > Density between pcd < 5. Community Detection Contant Potts Model (CPM) Minimize H() = ij (Aij )(i , j ) = c(ec n2 c) Resolution-limit-free Internal density pc > Density between pcd < 6. Community Detection Contant Potts Model (CPM) Minimize H() = ij (Aij )(i , j ) = c(ec n2 c) Resolution-limit-free Internal density pc > Density between pcd < 7. Community Detection Contant Potts Model (CPM) Minimize H() = ij (Aij )(i , j ) = c(ec n2 c) Resolution-limit-free Internal density pc > Density between pcd < How to choose ? 8. Resolution prole 103 102 101 100 103 104 105 106 N E 9. Signicance How signicant is a partition? 10. Signicance E = 14 E = 9 Fixed partition E = 11 Better partition 11. Signicance E = 14 E = 9 Fixed partition E = 11 Better partition Not: Probability to nd E edges in partition. But: Probability to nd partition with E edges. 12. Subgraph probability Decompose partition Probability to nd partition with E edges. Probability to nd communities with ec edges. Asymptotic estimate Probability for subgraph of nc nodes with density pc Pr(S(nc, pc) G(n, p)) exp n2 cD(pc p) Signicance Probability for all communities Pr() c exp n2 cD(pc p) . Signicance S() = log Pr() = c n2 cD(pc p). 13. Subgraph probability Decompose partition Probability to nd partition with E edges. Probability to nd communities with ec edges. Asymptotic estimate Probability for subgraph of nc nodes with density pc Pr(S(nc, pc) G(n, p)) exp n2 cD(pc p) Signicance Probability for all communities Pr() c exp n2 cD(pc p) . Signicance S() = log Pr() = c n2 cD(pc p). 14. Subgraph probability Decompose partition Probability to nd partition with E edges. Probability to nd communities with ec edges. Asymptotic estimate Probability for subgraph of nc nodes with density pc Pr(S(nc, pc) G(n, p)) exp n2 cD(pc p) Signicance Probability for all communities Pr() c exp n2 cD(pc p) . Signicance S() = log Pr() = c n2 cD(pc p). 15. Subgraph probability Decompose partition Probability to nd partition with E edges. Probability to nd communities with ec edges. Asymptotic estimate Probability for subgraph of nc nodes with density pc Pr(S(nc, pc) G(n, p)) exp n2 cD(pc p) Signicance Probability for all communities Pr() c exp n2 cD(pc p) . Signicance S() = log Pr() = c n2 cD(pc p). 16. Subgraph probability Decompose partition Probability to nd partition with E edges. Probability to nd communities with ec edges. Asymptotic estimate Probability for subgraph of nc nodes with density pc Pr(S(nc, pc) G(n, p)) exp n2 cD(pc p) Signicance Probability for all communities Pr() c exp n2 cD(pc p) . Signicance S() = log Pr() = c n2 cD(pc p). 17. Subgraph probability Decompose partition Probability to nd partition with E edges. Probability to nd communities with ec edges. Asymptotic estimate Probability for subgraph of nc nodes with density pc Pr(S(nc, pc) G(n, p)) exp n2 cD(pc p) Signicance Probability for all communities Pr() c exp n2 cD(pc p) . Signicance S() = log Pr() = c n2 cD(pc p). 18. Signicance 103 102 101 100 103 104 105 106 N E 19. Signicance 103 102 101 100 103 104 105 106 N E S 20. Benchmark 0.25 0.5 0.75 1 NMI n = 5000, Small 0 1 S S 0 0.2 0.4 0.6 0.8 1 0 1 S S CPM+Sig Signicance Modularity Infomap OSLOM 21. Conclusions Scan eciently. Signicance applicable in all methods. Correct comparison to random graph. Traag, Krings, Van Dooren Signicant scales in Community Structure arXiv:1306.3398 Thank you! Questions? e-mail: vincent@traag.net twitter: @vtraag