1. Discovering Community Structure
by Optimizing Community Quality Metrics
Mingming Chen
Department of Computer Science
Rensselaer Polytechnic Institute
09/28/2015

2. Community Structure Community is the basic structure in many networks
Community is a group of people that are similar to each other
Community is a group of nodes that are more densely connected with each other than to the rest of the network
Disjoint community structure
Each node belongs to one and only one community
Overlapping community structure
Each node belongs to one or more communities
Disjoint
Overlapping

3. Community Structure
Strong community: each node has more connections inside its community than with the rest of the network
Weak community: the total internal degree of c exceeds its total external degree
Radicchi et al., Proc. Natl. Acad. Sci. 101, 2658–2663 (2004)

4. Zachary’s Karate Club Network
Nodes: Club members
Edges: Interactions

5. American College Football Network
NCAA conferences
Nodes: Football teams
Edges: Games played

6. Collaboration Network between Scientists
at Santa Fe Institute
Research divisions
Nodes: Scientists
Edges: Collaboration

7. Facebook Network High school Summer internship Stanford (Basketball)
Social communities
High school
Summer internship
Stanford (Basketball)
Stanford (Squash)
Nodes: Facebook users
Edges: Friendships

8. Protein-Protein Interactions
Functional modules
Nodes: Proteins
Edges: Physical interactions

9. A New Community Quality Metric
Modularity Density:
A New Community Quality Metric
Mingming Chen, Tommy Nguyen, and Boleslaw K. Szymanski, “A New Metric for Quality of Network Community Structure”, ASE Human Journal, vol. 2, no. 4, Sep. 2013, pp
Mingming Chen, Tommy Nguyen, and Boleslaw K. Szymanski, “On Measuring the Quality of a Network Community Structure”, The ASE/IEEE International Conference on Social Computing (SocialCom), Washington D.C., Sep. 2013, pp

10. What Makes a Good Community Structure?
Part I
What Makes a Good Community Structure?

11. Community Quality Metrics
Part I
Community Quality Metrics
How to measure the quality of the community structure found with community detection algorithms
Community quality metrics: modularity

12. Part I
Modularity
Modularity (Q): the fraction of edges inside the communities minus the expected value in an equivalent network with edges placed at random
Newman and Girvan, Phys. Rev. E 69, (2004)
Newman, Proc. Natl. Acad. Sci. 103, 8577–8582 (2006)

13. Two Problems of Modularity Maximization
Part I
Two Problems of Modularity Maximization
In some cases, it splits large communities by favoring small communities
In other cases, it favors large communities by failing to discover communities smaller than a certain size even when such communities are well defined
This size depends on the total number of edges in the network and the degree of interconnectedness of communities
Also known as the resolution limit problem of modularity
Fortunato et al., 2008; Li et al., 2008; Arenas et al., 2008; Berry et al., 2009; Good et al., 2010; Ronhovde et al., 2010; Fortunato, 2010; Lancichinetti et al., 2011; Traag et al., 2011; Darst et al., 2013.

14. Multi-resolution Modularity
Part I
Multi-resolution Modularity
Multi-resolution modularity (Qλ): introduce the resolution parameter λ into modularity
High values of λ lead to smaller communities
Low values of λ lead to larger communities
Lancichinetti and Fortunato, Phys. Rev. E 84, (2011)

15. Multi-resolution Modularity
Qλ still suffers from the two opposite yet coexisting issues
Favoring small communities: e.g. split random graph
Resolution limit problem: e.g. merge loosely connected cliques
Often very difficult and impossible to tune the resolution parameter so as to avoid both problems simultaneously
Heterogonous distribution of community sizes
Lancichinetti and Fortunato, Phys. Rev. E 84, (2011)
Schematic network with a random subgraph and two cliques

16. Modularity with Split Penalty
Part I
Modularity with Split Penalty
Split penalty (SP): the fraction of edges that connect nodes of different communities
Qs = Q – SP: solving the problem of favoring small communities of modularity

17. Qs with Community Density: Modularity Density
Part I
Qs with Community Density: Modularity Density
Supplement both modularity and split penalty with edge densities to arrive at Modularity Density
Modularity Density solves both problems of modularity: the resolution limit problem and favoring small communities problem

18. Two Very Well-Separated Communities
Part I
Two Very Well-Separated Communities
Modularity (Q)
Split Penalty (SP)
Qs = Q – SP
Qds
2 communities
0.5
1 community
0.245

19. Two Well-Separated Communities
Part I
Two Well-Separated Communities
Modularity (Q)
Split Penalty (SP)
Qs = Q – SP
Qds
2 communities
0.357
0.143
0.214
0.339
1 community
0.25

20. Two Weakly Connected Communities
Part I
Two Weakly Connected Communities
Modularity (Q)
Split Penalty (SP)
Qs = Q – SP
Qds
2 communities
0.3
0.2
0.1
0.263
1 community
0.249

21. Ambiguity between One and Two Communities
Part I
Ambiguity between One and Two Communities
Modularity (Q)
Split Penalty (SP)
Qs = Q – SP
Qds
2 communities
0.25
0.188
1 community
0.245

22. One Well Connected Community
Part I
One Well Connected Community
Modularity (Q)
Split Penalty (SP)
Qs = Q – SP
Qds
2 communities
0.167
0.333
-0.167
0.0417
1 community
0.23

23. One Very Well Connected Community
Part I
One Very Well Connected Community
Modularity (Q)
Split Penalty (SP)
Qs = Q – SP
Qds
2 communities
0.0455
0.455
-0.409
-0.239
1 community
0.168

24. Community quality on a complete graph with 8 nodes
Part I
One Complete Graph
Community quality on a complete graph with 8 nodes
Modularity (Q)
Split Penalty (SP)
Qs = Q – SP
Qds
2 communities
0.571
-0.643
1 community

25. Modularity Has Nothing to Do with #Nodes
Part I
Modularity Has Nothing to Do with #Nodes

26. Example of Resolution Limit Problem
Part I
Example of Resolution Limit Problem
Modularity (Q)
Split Penalty (SP)
Qs = Q – SP
Qds
30 communities
0.8758
0.7848
0.8721
15 communities
0.8879
0.8424
0.4305
∆Qs=( )= > ∆Q=( )=0.0121

27. Proof of Solving Resolution Limit Problem
Part I
Proof of Solving Resolution Limit Problem
(a) Modularity density Qds does not merge two or more consecutive clqiues.
(b) Qds does not merge two small communities.

28. Proof of Solving the Two Problems
Part I
Proof of Solving the Two Problems
Modularity density does not split the random subgraph
Modularity density does not merge the two cliques
Schematic network with a random subgraph and two cliques

29. Other Community Quality Metrics
Part I
Other Community Quality Metrics
The number of Intra-edges:
Contraction: , average number of edges per node inside community c
The number of Inter-edges:
Expansion: , average number of edges per node that point outside community c
Conductance: , the fraction of the total number of edges that point outside community c

30. Evaluation and Analysis
Part I
Evaluation and Analysis
Senate dataset
Totally 111 snapshots over 220 years
Nodes are senators; weight on the edge between two senators is the fraction of times they voted similarly
Reality mining Bluetooth scan data
Each week is a snapshot, totally 43 snapshots
Nodes are subjects; weight on the edge is the number of Bluetooth scans between two subjects
Best value of parameter q (0.05≤q≤0.95) of LabelRankT compared with Estrangement Q Qs
Qds
#Intra-edges
Contraction
#Inter-edges
Expansion
Conductance
Senate
0.2
0.6
0.7, 0.8
Reality mining
0.5
0.7

31. Summary
Modularity density solves the two issues simultaneously without the trouble of specifying any particular parameter
We demonstrated with proofs and experiments on real dynamic datasets that modularity density is an effective alternative to modularity.
Modularity density for overlapping community structure
Mingming Chen, Konstantin Kuzmin, and Boleslaw Szymanski, “Extension of Modularity Density for Overlapping Community Structure”, IEEE/ACM ASONAM Workshop on Social Network Analysis in Applications (SNAA), Beijing, China, Aug. 2014, pp
Mingming Chen and Boleslaw K. Szymanski, “Fuzzy Overlapping Community Quality Metrics”, Social Network Analysis and Mining 5:40, Jul

32. Thanks!
Q & A

33. Fine-tuned Disjoint Community
Detection Algorithm
Mingming Chen, Konstantin Kuzmin, and Boleslaw Szymanski, ``Community Detection via Maximization of Modularity and Its Variants'', IEEE transactions on Computational Social Systems 1(1) Mar. 2014, pp

34. Introduction Optimize community quality metrics to detect communities
Part II
Introduction
Optimize community quality metrics to detect communities
Community quality metrics: modularity and modularity density
Modularity optimization methods
Greedy algorithms
Spectral algorithms
Fine-tuned disjoint community detection algorithm
Iteratively tries to improve the quality metrics by splitting and merging the given community structure
Combines both greedy and spectral methods, but a little more sophisticated

35. How to Find Communities: Splitting and Merging
Part II
How to Find Communities: Splitting and Merging
Spectral algorithm (top down): split the network (as a whole community) until each node is a community of itself
Greedy algorithm (bottom up): merge two communities until there is only a single community left

36. Spectral Partitioning: Laplacian Matrix
Part II
Spectral Partitioning: Laplacian Matrix 1 2 3 4 5 6 -1
Laplacian matrix (L)
|V|  |V| symmetric matrix
What is trivial eigenpair?
𝒙=(𝟏,…,𝟏) then 𝑳⋅𝒙=𝟎 and so 𝝀=𝝀 𝟏 =𝟎
Important properties:
Eigenvalues are non-negative real numbers
Eigenvectors are real and orthogonal 1 3 2 5 4 6
𝑳 = 𝑫 − 𝑨

37. How to Split a Community?
Part II
How to Split a Community? C1 C
Approximation C2
Fiedler vector
Fiedler vector: the eigenvector of the Laplacian matrix corresponding to the second smallest eigenvalue
Split: put the nodes corresponding to the positive values of the Fiedler vector into one group and the other nodes into the other group

38. Example: Spectral Partitioning
Part II
Example: Spectral Partitioning
Components of x2
Value of x2
Rank in x2

39. G Fine-tuned Algorithm
Part II
Fine-tuned Algorithm
Iteratively split and merge the community structure until doing so cannot improve the community quality metrics
Split stage
Merging stage
COMMUN I T Y S R U C E G

40. Split Stage Clique Based on Fiedler vector Fiedler vector
Part II
Split Stage
Clique
Clique
Based on Fiedler vector
Sort its elements in decreasing (or increasing) order, then cut them into two groups in each of the |c| - 1 possible ways
Choose the one that improves the metric the largest
X2 = [ ]
Fiedler vector
0.8
0.03

41. Evaluation and Analysis
Part II
Evaluation and Analysis
Three community detection algorithms
Greedy Q*, greedy algorithm of modularity maximization
Fine-tuned Q, fine-tuned algorithm to maximize modularity
Fine-tuned Qds, fine-tuned algorithm to optimize modularity density
Metrics with ground truth community structure
Information theory based metrics
Variation of Information (VI), Normalized Mutual Information (NMI)
Cluster matching based metrics
F-measure, Normalized Van Dongen metric (NVD)
Pair counting based metrics
Rand Index (RI), Adjusted Rand Index (ARI), Jaccard Index (JI)
Network datasets
Zachary's karate club network
American college football network
Clique network for resolution limit problem
LFR benchmark networks (0.1 ≤μ≤0.5)
*Clauset et al., Phys. Rev. E 70, (2004)

42. Zachary’s Karate Club Network
Part II
Zachary’s Karate Club Network

43. American College Football Network
Part II
American College Football Network
American college football network: the schedule of games between American college football teams in a single season
115 nodes and 613 edges
12 ground truth communities

44. Part II
12 communities
7 communities
9 communities
12 communities

45. Part II
Clique Network

46. LFR Benchmark Networks
Part II
LFR Benchmark Networks
μ is the mixing parameter. Low values of μ indicate strong community structure.

47. LFR Benchmark Networks
Part II
LFR Benchmark Networks

48. Part II
Summary
Fine-tuned Qds performs the best among all the three algorithms, followed by fine-tuned Q, and both are much more effective than Greedy Q
Fine-tuned Qds can be used to significantly improve the community detection results of other algorithms
All the seven quality metrics based on ground truth community structure are consistent with Qds, but not consistent with Q
Superiority of Qds over Q as a community quality metric

49. Resources Papers and book chapters worth to read
S. Fortunato, “Community detection in graphs,” Physics Reports, vol. 486, pp. 75–174, 2010.
J. Xie, S. Kelley, and B. K. Szymanski, “Overlapping community detection in networks: The state-of-the-art and comparative study,” ACM Comput. Surv., vol. 45, no. 4, pp. 43:1–43:35, Aug
M. Chen, K. Kuzmin, and B. K. Szymanski, “Community detection via maximization of modularity and its variants,” IEEE Trans. Comput. Soc. Syst., vol. 1, no. 1, pp. 46–65, 2014.
Community Detection Algorithms
Greedy Q or Fast Modularity:
Fine-tuned Algorithm:
Louvain Algorithm:
GANXiS:
CFinder:
Datasets:
Visualization tools
Gephi:
Cytoscape:

50. Thanks!
Q & A