Mining Communities in Networks:
A Solution for Consistency and
Its Evaluation
Haewoon Kwak Yoonchan Choi* Young-Ho Eom
Hawoong Jeong Sue Moon
KAIST, Korea
*Samsung Advanced Institute of Technology, Korea
1

Outline
2

Outline
• Introduction to Community Identification
2

Outline
• Introduction to Community Identification
• Inconsistency problem in CI
2

Outline
• Introduction to Community Identification
• Inconsistency problem in CI
• Metrics for the inconsistency in CI
2

Outline
• Introduction to Community Identification
• Inconsistency problem in CI
• Metrics for the inconsistency in CI
• Empirical solution to remove inconsistency
2

Outline
• Introduction to Community Identification
• Inconsistency problem in CI
• Metrics for the inconsistency in CI
• Empirical solution to remove inconsistency
• The case study of AS network
2

“Sense of Community”
Introduction
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Definitions of community
• “Subsets of nodes characterized by having
more internal connections than external
connections between them”
• “Set of web pages dealing with similar
topics”
• “Functional units”
4

Community in ...
• Sociology
• Biology
• Epidemiology
• Information theory
• Social network analysis
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Community identification
• Graph partitioning based on betweenness
• Clique-based approach
• Link-pattern based approach
• Random walks on network
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How do we know whether
‘communities are well identified?’
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8

8

Which is better?
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Quantitative metric
• Modularity, Q [15]
• eii : ratio of the number of links between
nodes belonging to community i over all
links
‣ ai : ratio of ends of edges that are
attached to vertices in community i
10

<
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Problem transformation from
• Finding GOOD communities?
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Problem transformation to
• Achieving HIGH modularity!
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Greedy approach for high Q
• CNM algorithm
• Wakita algorithm
• Louvain algorithm
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CNM: Initialization
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Calculate ∆Q
‣
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Calculate ∆Q
‣
0.051 0.041 0.026 0.041
0.041
0.041
0.051
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Select ‘global’ max ∆Q
‣
0.026 0.041
0.041
0.051
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Update ∆Q
‣
0.026 0.041
0.082 0.041
0.051
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Keep going...
‣
-0.005
0.041
0.041
0.051
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CNM - Final
‣
-0.005
Q=2 x ((6/14 - (7/14)^2)) = 0.357
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Wakita algorithm
• CNM + heuristic
‣ More scalable, but not high Q
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Louvain algorithm
• Iterative 2-phase algorithm
‣ P(I) : Finding ‘local’ max ∆Q
‣ P(II): Rebuild the weighted network
• Node ← community
• Weight ← ∑ (weights btwn. nodes)
• Until Q becomes larger
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Louvain: 1st phase
‣
0.051 0.041
0.041
0.051
Q=2 x ((6/14 - (7/14)^2)) = 0.357
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Louvain: 2nd phase
‣
Q’ = (2/2 - (2/2)^2) = 0 < Q
Louvain stops here
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“Similar, but not the same”
Inconsistency problem
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Multiple max ∆Q
• When 2 or more max ∆Q exist, how do
we pick two communities to merge?
• Inconsistent communities are produced!!
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Inconsistent results
13 0
21 19
2 9
22 12
23 29 30 29
17 4
5 6
27 12 25 31
7 11 22 3
6 0 15 10
28 18 17 20
25 24 27 28
30 31 11 7
19 26
14 15 9 5 24
2
4 32
1 20 16 8
10 18
26 21
32 13
16 3 1 23
33 8 14 33
(a) Q=0.273176 (b) Q=0.380671
FIG. 1: [Color Online] Visualization of inconsistent community identification in the K
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belong to the same community, and node ordering is depicted as the number in the nod

“New metrics”
Evaluate inconsistency
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Datasets
• 12 network: 34 to 11M nodes
‣ Online social network
‣ Biological network
‣ Internet AS network
‣ Wikipedia link network
‣ WWW network
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Overview of datasets
10
9
Orkut Cyworld
# of edges (log)
8
Flickr Wikipedia
7 YouTube
Facebook WWW
6
5
BBS AS Graph
4
3 C. Elegans Protein
2 Karate
1
0
0 1 2 3 4 5 6 7 8 9 10
# of nodes (log)
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Measurement methodology
• Choosing one of max ∆Q is related to
input order of nodes
• For each network,
‣ Generating N sets with different order
‣ Finding communities in N sets
‣ Comparing identified communities
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Variance of modularity
b (b) C.Elegans (c) Prot
(e) AS graph (f)
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We learned...
(a) Karate club (b) C.Elegans (c) Protein Interaction
(d) BBS (e) AS graph (f) Facebook
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We learned...
Louvain algorithm produces highest Q
CNM shows the smallest variance
(a) Karate club (b) C.Elegans (c) Protein Interaction
*Only Louvain works in a huge network
(d) BBS (e) AS graph (f) Facebook
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Figure 6: Consistency (no data available
Pairwise membershiprandomly orde
runs of an algorithm, each over a prob.
Over runs of an algorithm, each over a ra
uantify set, we an algorithm, eachpaira of nodespair of
the likelihood of alikelihood of aordered inp
Over• The likelihood of athe of nodes resulting
runs of quantify pair over randomly
resulti
munity as: community as: aover of runs resulting in t
set, we quantify same community pair N nodes
same the likelihood of
in the
same community as:
(
where where
1 if = in the th dataset
1 1 the th dataset
0if otherwise in
= if = in the
0 otherwise 0 otherwise
and and are nodes and and represent communities that
and belong to, respectively. 34We call this metric pairwise mem
and and are nodes and and represe

Distribution of p.m.p.
(b) C.Elegans
(e) AS graph
35

Distribution of p.m.p.
(a) Karate club (b) C.Elegans (c) Protein Interaction
(d) BBS (e) AS graph (f) Facebook
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Distribution of p.m.p.
There are many edges whose
pairwise membership prob. is not (c) Protein Interaction
(a) Karate club (b) C.Elegans 0 or 1
(d) BBS (e) AS graph (f) Facebook
36

ms produce pairwise membership probabilities of
’s. For the remaining nine networks, Louvain p
Consistency, C
t consistent outcome and, for (g) to (h), the only ou
der to quantify network-wide community members
, we define a metric of consistency for the entire
• To quantify network-wide consistency,
Normalization
sistency Weighing p.m.p. pairwise 0.5
weighs the away from membership prob
om . The second term in (4) normalizes from
e of communities detected by CNM algorithm in th
37

C in 12 networks
Figure 6: Consistency (no data available by CNM and Wakita for Wikipedia and Cyworld)
ver runs of an algorithm, each over a randomly ordered input
we quantify the likelihood of a pair of nodes resulting in the 38

C in 12 networks
No one outperforms the other two
Figure 6: Consistency (no data available by CNM and Wakita for Wikipedia and Cyworld)
ver runs of an algorithm, each over a randomly ordered input
we quantify the likelihood of a pair of nodes resulting in the 38

“Totally intuitive”
Our approach
39

Intuitions behind our approach
• Every edge has pairwise membership prob.
• High pairwise membership probability
indicates that two nodes are likely to be in
the same community
• All 3 algorithms in weighted network place
edge of high weight within the community
40

Reinforcing p.m.p.
41

Reinforcing p.m.p.
• After a cycle of N runs,
‣ Calculate pairwise membership prob.
‣ Assign p.m.p. as edge weight
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Reinforcing p.m.p.
• After a cycle of N runs,
‣ Calculate pairwise membership prob.
‣ Assign p.m.p. as edge weight
• Return to another cycle of N runs
41

Reinforcing p.m.p.
• After a cycle of N runs,
‣ Calculate pairwise membership prob.
‣ Assign p.m.p. as edge weight
• Return to another cycle of N runs
• Continue until C gains no improvement
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Convergence of C
Figure 8: Convergence of consistency
erforms the other two in all networks and no consistent correla-
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between the consistency and the topological characteristics of

Convergence of C
Except Orkut & Cyworld,
C converges to 1 within 5 cycles
Figure 8: Convergence of consistency
erforms the other two in all networks and no consistent correla-
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between the consistency and the topological characteristics of

Agreement btwn. trials
nvergence of consistency
la-
of
us-
FI-
the
all
all
hip
is, Figure 10: Comparison of community size distribution in 4 tri-
43

Agreement btwn. trials
nvergence of consistency
la-
of
us-
Communities of independent trials
FI- are almost identical
the
all
all
hip
is, Figure 10: Comparison of community size distribution in 4 tri-
43

For non-converging case
• Is not enough N = 100 ?
• Resolution limit in community detection ?
44

For non-converging case
• Is not enough N = 100 ?
• Resolution limit in community detection ?
We are building an analytical framework
to explain inconsistency problems
44

“Internet Jellyfish”
Preliminary analysis of
AS graph
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Communities in AS graph
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Communities in AS graph
The largest community,
The geographically concentrated comm.,
The star-shaped community
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The largest community, L
• 32.3% of all ASes
• MCI, Level3, AT&T WorldNet, Sprint, ...
• 9 of top 10 listed in AS ranking of CAIDA
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Reapplying our approach to L
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Reapplying our approach to L
3 of 9 falls in the same community,
remaining 6 fall into different community
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Geographically
concentrated community
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Geographically
concentrated community
97.4% of ASes in Korea
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Star-shaped community
All relations are provider-customer
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Summary
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Summary
• We identify inconsistency
in community identification
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Summary
• We identify inconsistency
in community identification
• We define new metrics
for measuring inconsistency
51

Summary
• We identify inconsistency
in community identification
• We define new metrics
for measuring inconsistency
• We propose empirical solutions
reinforcing pairwise membership probability
51

Summary
• We identify inconsistency
in community identification
• We define new metrics
for measuring inconsistency
• We propose empirical solutions
reinforcing pairwise membership probability
• We present preliminary analysis
of communities in AS graph
51

Supplementary material
• http://an.kaist.ac.kr/traces/IMC2009-kwak.html
52

Supplementary material
• http://arxiv.org/abs/0910.1508
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Thank you
54

Backup slides
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Change of p.m.p.
(a) Facebook (a) Facebook
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