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. 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. 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. 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. 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. 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. 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?
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. 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. 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. 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. 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. 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. 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).
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