Significant scales in community structure
of 21
/21
-
Upload
vincent-traag -
Category
Science
-
view
98 -
download
3
Embed Size (px)
description
Presentation at ECCS 2013, Barcelona, September 17, 2013
Transcript of Significant scales in community structure
Significant scales in community structure
V.A. Traag1,2, G. Krings3, P. Van Dooren4
1KITLV, Leiden, the Netherlands2e-Humanities, KNAW, Amsterdam, the Netherlands
3Real Impact, Brussels, Belgium,4UCL, Louvain-la-Neuve, Belgium
September 17, 2013
eRoyal Netherlands Academy of Arts and SciencesHumanities
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
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
Community Detection
Contant Potts Model (CPM)
• Minimize H(γ) = −∑ij(Aij − γ)δ(σi , σj) = −∑
c(ec − γn2c)
• Resolution-limit-free
• Internal density pc > γ
• Density between pcd < γ
Community Detection
Contant Potts Model (CPM)
• Minimize H(γ) = −∑ij(Aij − γ)δ(σi , σj) = −∑
c(ec − γn2c)
• Resolution-limit-free
• Internal density pc > γ
• Density between pcd < γ
Community Detection
Contant Potts Model (CPM)
• Minimize H(γ) = −∑ij(Aij − γ)δ(σi , σj) = −∑
c(ec − γn2c)
• Resolution-limit-free
• Internal density pc > γ
• Density between pcd < γ
Community Detection
Contant Potts Model (CPM)
• Minimize H(γ) = −∑ij(Aij − γ)δ(σi , σj) = −∑
c(ec − γn2c)
• Resolution-limit-free
• Internal density pc > γ
• Density between pcd < γ
How to choose γ?
Resolution profile
10−3 10−2 10−1 100103
104
105
106
γ
N E
Significance
How significant is a partition?
Significance
E = 14
E = 9
Fixed partition
E = 11
Better partition
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.
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[−n2cD(pc ‖ p)
]
Significance
• Probability for all communities Pr(σ) ≈∏c
exp[−n2cD(pc ‖ p)
].
• Significance S(σ) = − log Pr(σ) =∑c
n2cD(pc ‖ p).
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[−n2cD(pc ‖ p)
]
Significance
• Probability for all communities Pr(σ) ≈∏c
exp[−n2cD(pc ‖ p)
].
• Significance S(σ) = − log Pr(σ) =∑c
n2cD(pc ‖ p).
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[−n2cD(pc ‖ p)
]
Significance
• Probability for all communities Pr(σ) ≈∏c
exp[−n2cD(pc ‖ p)
].
• Significance S(σ) = − log Pr(σ) =∑c
n2cD(pc ‖ p).
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[−n2cD(pc ‖ p)
]
Significance
• Probability for all communities Pr(σ) ≈∏c
exp[−n2cD(pc ‖ p)
].
• Significance S(σ) = − log Pr(σ) =∑c
n2cD(pc ‖ p).
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[−n2cD(pc ‖ p)
]
Significance
• Probability for all communities Pr(σ) ≈∏c
exp[−n2cD(pc ‖ p)
].
• Significance S(σ) = − log Pr(σ) =∑c
n2cD(pc ‖ p).
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[−n2cD(pc ‖ p)
]
Significance
• Probability for all communities Pr(σ) ≈∏c
exp[−n2cD(pc ‖ p)
].
• Significance S(σ) = − log Pr(σ) =∑c
n2cD(pc ‖ p).
Significance
10−3 10−2 10−1 100103
104
105
106
γ
N E
Significance
10−3 10−2 10−1 100103
104
105
106
γ
N E S
Benchmark
0.25
0.5
0.75
1
NM
In = 5000, Small
0
1S S∗
0 0.2 0.4 0.6 0.8 101
µ
S∗ 〈S〉
CPM+SigSignificanceModularity
InfomapOSLOM
Conclusions
• Scan γ efficiently.
• Significance applicable in all methods.
• Correct comparison to random graph.
Traag, Krings, Van Dooren Significant scales in Community StructurearXiv:1306.3398
Thank you!Questions?
e-mail: [email protected] twitter: @vtraag
