1. Affiliation Network Models of Clusters in Networks
Jure Leskovec
Stanford University
Joint work with Jaewon Yang

2. Behind each of these complex systems there is a network, that defines the interactions between the components
And thus basically everything around us is a network of interconnected parts.
And we have been dealing and living with networks all our lives
Networks

3. Network Clusters Networks are not uniform/homogeneous
They exhibit clusters!
Why clusters? What do they correspond to?
Blogosphere [Adamic&Glance]
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

4. From Clusters to Communities
Clusters form communities
Cluster: A set of appropriately connected nodes
Community: Nodes with a shared latent property
Many reasons for why communities form:
World Wide Web
Citation networks
Social networks
Metabolic networks
Blogosphere [Adamic&Glance]
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

5. Basis for Community Formation
How and why do communities form?
Granovetter’s Strength of weak ties suggest and the models of small-world suggest:
Strong ties are well embedded in the network
Weak ties span long-ranges
Given a network, how to find communities?
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

6. SFI collaboration network [Newman]
Finding Communities
Q: Given a network, how to find communities?
A: Find weak ties and identify communities
Girvan-Newman’s betweenness centrality, Modularity, and Graph partitioning methods are all based on this idea
SFI collaboration network [Newman]
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

7. Overlapping Communities
Non-overlapping vs. overlapping communities
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

8. Overlaps of Social Circles
[Palla et al., ‘05]
Overlaps of Social Circles
A node belongs to many social circles
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

9. Clique Percolation Method (CPM)
[Palla et al., ‘05]
Clique Percolation Method (CPM)
Two nodes belong to the same community if they can be connected through adjacent k-cliques:
k-clique:
Fully connected graph on k nodes
Adjacent k-cliques:
overlap in k-1 nodes
k-clique community
Set of nodes that can be reached through a sequence of adjacent k-cliques
k-clique
(k=3)
adjacent
cliques (k=3) A A C D B B D C
k=3
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

10. Mixed-Membership Block Models
Mixed-Membership Stochastic Block Models [Airoldi et al.]
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

11. Step Back: Community Detection
(1) Take a dataset
(2) Represent it as a graph
(3) Identify communities (really, clusters)
(4) Interpret clusters as “real” communities
Nodes that have “something” in common C A B D E H F G C A B D E H F G
work in the same area
publish in same journals
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

12. Ground-Truth Networks with a an explicit notion of Ground-Truth:
Collaborations: Conferences & Journals as proxies for scientific areas
Social Networks: People join to groups, create lists
Information Networks: Users create topic based groups C A B D E H F G C A B D E H F G
work in the same area
publish in same journals
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

13. Examples of Ground-Truth
LiveJournal social network:
Users create and join to groups created around culture, entertainment, expression, fandom, life/style, life/support, gaming, sports, student life and technology
TuDiabetes network
Groups form around specific types of diabetes, different age groups, emotional and social support, arts and crafts groups, different geo regions
A node can be a member of 0 or more groups
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

14. Networks with Ground-Truth
N … # of nodes
E … # of edges
C … # of ground-truth communities
S … average community size
A … memberships per node
For example:
… fans of Real Madrid
... subscribe to Lady Gaga videos
… follow Volvo Ocean Race
Youtube social network
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

15. Ground-Truth: Consequences≈
Ground-truth groups
Inferred communities
How groups map on the network?
 Insights for better models
How to evaluate and interpret?
 “Accuracy” of methods
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

16. Consequence: Overlaps are DENSER!
Edge Probability
Nodes u and v share k groups
What is edge prob. P(edge | k) as a func. of k?
P(edge | k)
Consequence: Overlaps are DENSER! k
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

17. Communities in Networks
Does it matter at all?
Ganovetter and all non-overlapping methods
Palla et al., MMSB and overlapping methods as well
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

18. Detecting Dense Overlaps?
Can present community detection methods detect dense overlaps? No!
(Mixed-Membership) Stochastic block model
𝐸[𝑃 𝑖,𝑗 ]= 𝑐 𝑃 𝑖,𝑐 𝑃 𝑗,𝑐 𝑃 𝑒 (𝑐)
Clique percolation
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

19. Dense Overlaps: Natural Model
Communities, C
Memberships, M
Nodes , V
Community-Affiliation Graph Model
B(V,C,M,{pc})
Provably generates power-law degree distributions and other patterns real-world networks exhibit [Lattanzi, Sivakumar, STOC ‘09]
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

20. Community-Affiliation Graph Model
AGM is flexible and can express variety of network structures: Non-overlapping, Nested, Overlapping
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

21. Model-based Community DetectionH F G
Given a Graph, find the Model
Affiliation graph B
Number of communities
Parameters pc
Yes, we can!
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

22. MAG Model Fitting Task: Optimization problem (MLE) How to solve?
Given network G(V,E). Find B(V, C, M, {pc})
Optimization problem (MLE)
How to solve?
Approach: Coordinate ascent
(1) Stochastic search over B, while keeping {pc} fixed
(2) Optimize {pc}, while keeping B fixed (convex!)
Works well in practice! C A B D E H F G
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

23. Experimental Setup
Communities!
Fit C A B D E H F G
Evaluate
Evaluation: How well do inferred communities correspond to ground-truth?
F1 score, Ω-index, NMI, …
Algorithms for comparison:
MMSB [Airoldi et al., JMLR]
Link Clustering [Ahn et al., Nature]
Clique Percolation [Palla et al., Nature]
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

24. Example: Facebook Ego-Net
High-school
Workplace
Stanford, Squash
89% accuracy
Stanford, Basketball
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

25. Experiments: Ground-truth
Overall (only overlaps) AGM improves (F1≈0.6)
57% (21%) over Link clustering
48% (22%) over CPM
10% (26%) over MMSB
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

26. Experiments: Meta-data Based
Evaluation based on node metadata [Ahn et al. ‘10]
Similar level of improvement
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

27. Conclusion & Reflections
Ground-Truth Communities
 Overlaps are denser
Present methods can’t detect such overlaps
Community-Affiliation Graph Model
 Model-based Community Detection
Outperforms state-of-the-art
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

28. Connections: Nested Core-Periphery
Denser and denser core of the network
Core contains 60% node and 80% edges
Whiskers are responsible for good communities
Nested Core-Periphery (jellyfish, octopus)

29. Communities & Core-Periphery
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Jure Leskovec

30. Explanation
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Jure Leskovec: Affiliation Network Models of Clusters in Networks

31. THANKS! http://snap.stanford.edu 5/27/2019
Jure Leskovec: Affiliation Network Models of Clusters in Networks

32. References
J. Yang, J. Leskovec. Structure and Overlaps of Communities in Networks.
J. Yang, J. Leskovec. Defining and Evaluating Network Communities based on Ground-truth.
J. Leskovec, K. Lang, A. Dasgupta, M. Mahoney. Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters. In Internet Mathematics.
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Jure Leskovec: Affiliation Network Models of Clusters in Networks