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?

8. Resolution profile
10−3 10−2 10−1 100
103
104
105
106
γ
N E

9. Significance
How significant is a partition?

10. Significance
E = 14
E = 9
Fixed partition
E = 11
Better partition

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).

18. Significance
10−3 10−2 10−1 100
103
104
105
106
γ
N E

19. Significance
10−3 10−2 10−1 100
103
104
105
106
γ
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
µ
S∗
S
CPM+Sig
Significance
Modularity
Infomap
OSLOM

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