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I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion Community Detection in Multiplex Networks: A survey Rushed Kanawati A 3 , LIPN, CNRS UMR 7030 USPC - University Paris Nord http://lipn.fr/ kanawati

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INTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion

Community Detection in Multiplex Networks: A survey

source: muxviz

Rushed Kanawati

A3, LIPN, CNRS UMR 7030 USPC - University Paris Nord

http://lipn.fr/∼kanawati rushed.kanawati@lipn.univ-paris13.fr

AAFD’14, Villetaneuse, 29 April 2014

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INTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion

PLAN

1

INTRODUCTION

2

Community Detection in Multiplex Networks Applying monoplex community detection algorithms Adaptation of monoplex community detection algorithms Community evaluation approaches

3

Challenges

4

Conclusion

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BACKGROUND

Complex network ? Graph modeling direct or indirect interactions among a set of actors. Basic topological features

◮ Low Density ◮ Small Diameter ◮ Heterogeneous degree distribution. ◮ High Clustering coefficient ◮ Community structure

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COMPLEX NETWORKS: EXAMPLES

DBLP: co-authorship network Mubarak’s resignation on twitter (Gephi.org) Co-rating of films: MovieLens Temporal access similarity of places: Bourget 4 / 44

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MULTIPLEX NETWORK

Definition A set of actors related by different types of relations

Source: muxviz

Motivation ◮ Real networks are dynamic. ◮ Real networks are heterogeneous. ◮ Nodes are usually qualified by a set

  • f attributes.

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MULTIPLEX NETWORKS: RELATED TERMS

Recommended readings ✏ S. Mikko Kivel¨ a et. al.. Multilayer Networks. arXiv:1309.7233, March 2014

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MULTIPLEX NETWORKS: EXAMPLES

Airlines Europe Network Source: muxviz DBLP co-auorship network Source: muxviz D4D dataset (cˆ

  • te d’ivoire)

Source: muxviz 7 / 44

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MULTIPLEX NETWORK: NOTATIONS

G =< V, E1, . . . , Eα : Ek ⊆ V × V ∀k ∈ {1, . . . , α} > ◮ V: set of nodes (a.k.a. vertices, actors, sites) ◮ Ek: set of edges of type k (a.k.a. ties, links, bonds) Notations

◮ A[k] Adjacency Matrix of slice k : a[k]

ij

= 0 si les nœuds (vi, vj) ∈ Ek, 0 otherwise. ◮ n = |V| ◮ mk = |Ek|. We have often m ∼ n ◮ Neighbor’s of v in slice k: Γ(v)[k] = {x ∈ V : (x, v) ∈ Ek}. ◮ Node degree in slice k: dk

v = Γ(v)[k] 8 / 44

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COMMUNITY ?

Some definitions

◮ A dense subgraph loosely coupled to

  • ther modules in the network

◮ A community is a set of nodes seen as

  • ne by nodes outside the community

◮ A subgraph where almost all nodes are linked to other nodes in the community.

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COMMUNITY ?

Some definitions

◮ A dense subgraph loosely coupled to

  • ther modules in the network

◮ A community is a set of nodes seen as

  • ne by nodes outside the community

◮ A subgraph where almost all nodes are linked to other nodes in the community.

Applications

◮ Gaining insights into complex interaction patterns

Identifying useful groups of actors: functional units, recommender systems, . . . , etc

◮ Network Visualization ◮ Parallel computation ◮ Network compression

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COMMUNITY DETECTION IN MULTIPLEX NETWORKS

Definition ? What is a dense subgraph in a multiplex network ?

[BCG11] 11 / 44

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COMMUNITY DETECTION IN MULTIPLEX NETWORKS

Approaches 1 Transformation into a monoplex community detection problem

◮ Layer aggregation approaches. ◮ Hypergraph transformation based approaches ◮ Ensemble clustering approaches

2 Generalization of monoplex oriented algorithms to multiplex networks.

◮ Group-based approaches. ◮ Network-based approaches. ◮ Propagation-based approaches. ◮ Seed-centric approaches.

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LAYER AGGREGATION

Principle ◮ A ← F(A[1], . . . , A[α]) ◮ Apply a monoplex community detection algorithm on graph having A as adjacency matrix Aggregation functions

Aij =

  • 1

∃1 ≤ l ≤ α : A[l]

ij = 0

  • therwise

Aij = {d : A[d]

ij

= 0} Aij = 1 α

α

  • k=1

wkA[k]

ij

Aij = sim(vi, vj)

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K-UNIFORM HYPERGRAPH TRANSFORMATION

Principle ◮ A k-uniform hypergraph is a hypergraph in which the cardinality

  • f each hyperedge is exactly k

◮ Mapping a multiplex to a 3-uniform hypergraph H = (V, E) such that : V = V ∪ {1, . . . , α} (u, v, i) ∈ E if ∃l : A[l]

uv = 0, u, v ∈ V, i ∈ {1, . . . , α}

◮ Apply community detection approaches in Hypergraphs (Ex. tensor factorization approaches)

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ENSEMBLE CLUSTERING APPROACHES

Principle ◮ Apply a monoplex algorithm on each layer of the multiplex ◮ Apply clustering ensemble on obtained α clusterings.

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ENSEMBLE CLUSTERING: APPROACHES

CSPA: Cluster-based Similarity Partitioning Algorithm ◮ Let K be the number of basic models, Ci(x) be the cluster in model i to which x belongs. ◮ Define a similarity graph on objects : sim(v, u) =

K

  • i=1

δ(Ci(v),Ci(u)) K

◮ Cluster the obtained graph : Isolate connected components after prunning edges Apply community detection approach ◮ Complexity : O(n2kr) : n # objects, k # of clusters, r# of clustering solutions

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CSPA : EXEMPLE

from Seifi, M. Cœurs stables de communaut´ es dans les graphes de terrain. Th` ese de l’universit´ e Paris 6, 2012 17 / 44

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ENSEMBLE CLUSTERING: APPROACHES

HGPA: HyperGraph-Partitioning Algorithm ◮ Construct a hypergraph where nodes are objects and hyperedges are clusters. ◮ Partition the hypergraph by minimizing the number of cut hyperedges ◮ Each component forms a meta cluster ◮ Complexity : O(nkr)

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ENSEMBLE CLUSTERING: APPROACHES

MCLA: Meta-Clustering Algorithm ◮ Each cluster from a base model is an item ◮ Similarity is defined as the percentage of shared common objects ◮ Conduct meta-clustering on these clusters ◮ Assign an object to its most associated meta-cluster ◮ Complexity : O(nk2r2)

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GROUP-BASED APPROCHES

Principle Search for special (dense) subgraphs: ◮ k-clique ◮ n-clique ◮ γ-dense clique ◮ K-core

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GROUP-BASED APPROCHES

from Symeon Papadopoulos, Community Detection in Social Media, CERTH-ITI, 22 June 2011 21 / 44

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EXAMPLE: CLIQUE PERCOLATION

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EXAMPLE: CLIQUE PERCOLATION

◮ Suits fairly dense graph. ◮ Cohesive group concept are not yet generalized to multiplex networks

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NETWORK-BASED APPROCHES

Clustering approaches ◮ Apply classical clustering approaches using graph-based diastase function ◮ Different types of Graph-based distances: neighborhood-based, path-based (Random-walk) ◮ Usually requires the number of clusters to discover

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MODULARITY OPTIMIZATION APPROACHES

Modularity: a partition quality criteria Q(P) = 1 2m

  • c∈P
  • i,j∈c

(Aij − λ × didj 2m )

Figure: For λ = 1, Q = (15+6)−(11.25+2.56)

25

= 0.275

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MODULARITY OPTIMIZATION APPROACHES

◮ Applying classical optimization algorithms (ex. Genetic algorithms [Piz12]). ◮ Applying hierarchical clustering and select the level with Qmax (ex. Walktrap [PL06]) ◮ Divisive approach : Girvan-Newman algorithm [GN02] ◮ Greedy optimization : Louvain algorithm [BGL08] ◮ . . .

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MULTIPLEX MODULARITY

Generalized modularity [MRM+10] ◮ Qmultiplex(P) = 1 2µ

  • c∈P
  • i,j∈c

k,l:1→α

   A[s]

ij − λk

d[k]

i d[k] j

2m[k]   δkl + δijCkl

ij

  ◮ µ =

  • j∈V

k,l:1→α

m[k] + Cl

jk

◮ Ckl

ij Inter slice coupling = 0 ∀i = j

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MODULARITY OPTIMIZATION LIMITATIONS

Hypothesis ◮ The best partition of a graph is the one that maximize the modularity. ◮ If a network has a community structure, then it is possible to find a precise partition with maximal modularity. ◮ If a network has a community structure, then partitions having high modularity values are structurally similar.

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MODULARITY OPTIMIZATION LIMITATIONS

Hypothesis ◮ The best partition of a graph is the one that maximize the modularity. ◮ If a network has a community structure, then it is possible to find a precise partition with maximal modularity. ◮ If a network has a community structure, then partitions having high modularity values are structurally similar. All three hypothesis do not hold [GdMC10, LF11].

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PROPAGATION-BASED APPROACHES

Algorithm 1 Label propagation Require: G =< V, E > a connected graph,

1: Initialize nodes with unique labels 2: while Labels are not stable do 3:

for v ∈ V do adopt dominant label in Γ(v)

4:

end for

5: end while 6: return communities from labels

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LABEL PROPAGATION ON MULTIPLEX NETWORK

Features ◮ Complexity: O(m) (on monoplex) ◮ Highly parallel ◮ Unstable solutions → Usage of ensemble clustering approaches. Neighborhood in a multiplex ◮ Γ(v, σ) = set of nodes that are neighboring v in at least σ layer. ◮ Similarity-driven neighborhood [HK14]: Γ(v, δ) = {x ∈ ∪kΓ(v)[k] : sim(v, x) > δ}

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SEED-CENTRIC ALGORITHMS [KAN14]

Algorithm 2 General seed-centric community detection algorithm Require: G =< V, E > a connected graph,

1: C ← ∅ 2: S ← compute seeds(G) 3: for s ∈ S do 4:

Cs ← compute local com(s,G)

5:

C ← C + Cs

6: end for 7: return compute community(C)

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EXEMPLE: LICOD [YK14] I

Licod : the idea ! 1 Compute a set of seeds that are likely to be leaders in their communities

Heuristic : nodes having higher centralities than their neighbors

2 Each node in the graph ranks seeds in function of its own preference 3 Each node modify its preference vector in function of neighbor’s preferences 4 iterate max times or till convergence.

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EXEMPLE: LICOD [YK14] II

Adaptation to Multiplex [HK14] ◮ Multiplex degree centrality : dmultiplex

i

= −

α

  • k=1

d[k]

i

d[tot]

i

log

  • d[k]

i

d[tot]

i

  • ◮ Multiplex shortest path [PK13]

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EVALUATION METHODS

◮ Topological criteria : is it useful for applications ? ◮ Ground-truth comparaison : hard to find on large-scale ◮ Task-oriented approaches : Clustering , Recommendation, link prediction, . . .

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CHALLENGES

◮ Metrics in multiplex networks : Degree, shortest path, clustering coefficient [BNL13] Random walks , Centralities [DSGA13] ◮ Scale-problem: Parallel implementations & ensemble approaches. ◮ Evaluation problem (not specific to multiplex)

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CONCLUSIONS

◮ Networks are everywhere (ou presque) ◮ Community detection can help in different analysis and application tasks ◮ Multiplex networks provide a rich representation of real-world interaction systems ◮ Promising approaches : Local approaches (propagation, seed-centric) ◮ A lot of work to reformulate basic network concepts for multiplex settings. ◮ Problems : Evaluation and interpretation of computed communities.

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PUBLICIT´

E !

Mining Multiples Networks@Cason’2014, 30 July 2014 Porto Deadline 15 May http://lipn.fr/mnm14

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PUBLICIT´

E !

Dynamic network Mining Workshop@ECML’14, 19 Sep. 2014, Nancy Deadline : 20 June 2014 http://lipn.fr/dynak2

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PUBLICIT´

E !

MARAMI’2014: French Conference on Network Analysis, 15-17 Oct. Paris Deadline : 13 June 2014 Special session for doctoral students: http://lipn.fr/jfgg14

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That’s all folks ! Questions ?

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BIBLIOGRAPHY I

Michele Berlingerio, Michele Coscia, and Fosca Giannotti, Finding and characterizing communities in multidimensional networks, ASONAM, IEEE Computer Society, 2011, pp. 490–494. Vincent D Blondel, Jean-loup Guillaume, and Etienne Lefebvre, Fast unfolding of communities in large networks, 1–12. Federico Battiston, Vincenzo Nicosia, and Vito Latora, Metrics for the analysis of multiplex networks, CoRR abs/1308.3182 (2013). Manlio De Domenico, Albert Sol´ e, Sergio G´

  • mez, and Alex Arenas, Random walks on multiplex networks, CoRR

abs/1306.0519 (2013).

  • B. H. Good, Y.-A. de Montjoye, and A. Clauset., The performance of modularity maximization in practical contexts., Physical

Review E (2010), no. 81, 046106.

  • M. Girvan and M. E. J. Newman, Community structure in social and biological networks, PNAS 99 (2002), no. 12, 7821–7826.

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BIBLIOGRAPHY II

Manel Hmimida and Rushed Kanawati, A seed-centric algorithm for community detection in multiplex networks, Submitted to First european conference on social network analysis (Barcelona), July 2014. Rushed Kanawati, Seed-centric approaches for community detection in complex networks, 6th international conference on Social Computing and Social Media (Crete, Greece) (Gabriele Meiselwitz, ed.), vol. LNCS 8531, Springer, June 2014, pp. 197–208. Andrea Lancichinetti and Santo Fortunato, Limits of modularity maximization in community detection, CoRR abs/1107.1 (2011). Peter J Mucha, Thomas Richardson, Kevin Macon, Mason A Porter, and Jukka-Pekka Onnela, Community structure in time-dependent, multiscale, and multiplex networks, Science 328 (2010), no. 5980, 876–878. Clara Pizzuti, A multiobjective genetic algorithm to find communities in complex networks, IEEE Trans. Evolutionary Computation 16 (2012), no. 3, 418–430. 43 / 44

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BIBLIOGRAPHY III

Manisha Pujari and Rushed Kanawati, Link prediction in multiplex bibliographical networks, International Journal of Complex Systems in Science 2 (2013). Pascal Pons and Matthieu Latapy, Computing communities in large networks using random walks, J. Graph Algorithms Appl. 10 (2006), no. 2, 191–218. Zied Yakoubi and Rushed Kanawati, Licod: Leader-driven approaches for community detection, Vietnam Journal of Computer Science (2014), 30 pages, submitted. 44 / 44