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Significant scales in community structure

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Presentation at ECCS 2013, Barcelona, September 17, 2013

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Significant scales in community structure

  1. 1. Significant 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. 2. Community Detection Contant Potts Model (CPM) • Minimize H(γ) = − ij (Aij − γ)δ(σi , σj ) • Resolution-limit-free • Internal density pc > γ • Density between pcd < γ
  3. 3. Community Detection Contant Potts Model (CPM) • Minimize H(γ) = − ij (Aij − γ)δ(σi , σj ) • Resolution-limit-free • Internal density pc > γ • Density between pcd < γ
  4. 4. Community Detection Contant Potts Model (CPM) • Minimize H(γ) = − ij (Aij − γ)δ(σi , σj ) = − c(ec − γn2 c) • Resolution-limit-free • Internal density pc > γ • Density between pcd < γ
  5. 5. Community Detection Contant Potts Model (CPM) • Minimize H(γ) = − ij (Aij − γ)δ(σi , σj ) = − c(ec − γn2 c) • Resolution-limit-free • Internal density pc > γ • Density between pcd < γ
  6. 6. Community Detection Contant Potts Model (CPM) • Minimize H(γ) = − ij (Aij − γ)δ(σi , σj ) = − c(ec − γn2 c) • Resolution-limit-free • Internal density pc > γ • Density between pcd < γ
  7. 7. Community Detection Contant Potts Model (CPM) • Minimize H(γ) = − ij (Aij − γ)δ(σi , σj ) = − c(ec − γn2 c) • Resolution-limit-free • Internal density pc > γ • Density between pcd < γ How to choose γ?
  8. 8. Resolution profile 10−3 10−2 10−1 100 103 104 105 106 γ N E
  9. 9. Significance How significant is a partition?
  10. 10. Significance E = 14 E = 9 Fixed partition E = 11 Better partition
  11. 11. Significance E = 14 E = 9 Fixed partition E = 11 Better partition • Not: Probability to find E edges in partition. • But: Probability to find partition with E edges.
  12. 12. Subgraph probability Decompose partition • Probability to find partition with E edges. • Probability to find 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) Significance • Probability for all communities Pr(σ) ≈ c exp −n2 cD(pc p) . • Significance S(σ) = − log Pr(σ) = c n2 cD(pc p).
  13. 13. Subgraph probability Decompose partition • Probability to find partition with E edges. • Probability to find 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) Significance • Probability for all communities Pr(σ) ≈ c exp −n2 cD(pc p) . • Significance S(σ) = − log Pr(σ) = c n2 cD(pc p).
  14. 14. Subgraph probability Decompose partition • Probability to find partition with E edges. • Probability to find 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) Significance • Probability for all communities Pr(σ) ≈ c exp −n2 cD(pc p) . • Significance S(σ) = − log Pr(σ) = c n2 cD(pc p).
  15. 15. Subgraph probability Decompose partition • Probability to find partition with E edges. • Probability to find 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) Significance • Probability for all communities Pr(σ) ≈ c exp −n2 cD(pc p) . • Significance S(σ) = − log Pr(σ) = c n2 cD(pc p).
  16. 16. Subgraph probability Decompose partition • Probability to find partition with E edges. • Probability to find 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) Significance • Probability for all communities Pr(σ) ≈ c exp −n2 cD(pc p) . • Significance S(σ) = − log Pr(σ) = c n2 cD(pc p).
  17. 17. Subgraph probability Decompose partition • Probability to find partition with E edges. • Probability to find 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) Significance • Probability for all communities Pr(σ) ≈ c exp −n2 cD(pc p) . • Significance S(σ) = − log Pr(σ) = c n2 cD(pc p).
  18. 18. Significance 10−3 10−2 10−1 100 103 104 105 106 γ N E
  19. 19. Significance 10−3 10−2 10−1 100 103 104 105 106 γ N E S
  20. 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 µ S∗ S CPM+Sig Significance Modularity Infomap OSLOM
  21. 21. Conclusions • Scan γ efficiently. • Significance applicable in all methods. • Correct comparison to random graph. Traag, Krings, Van Dooren Significant scales in Community Structure arXiv:1306.3398 Thank you! Questions? e-mail: vincent@traag.net twitter: @vtraag

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