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 rushed.kanawati@lipn.univ-paris13.fr AAFD’14, Villetaneuse, 29 April 2014 source: muxviz 1 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion P LAN I NTRODUCTION 1 Community Detection in Multiplex Networks 2 Applying monoplex community detection algorithms Adaptation of monoplex community detection algorithms Community evaluation approaches Challenges 3 Conclusion 4 2 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion B ACKGROUND 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 3 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion C OMPLEX NETWORKS : E XAMPLES DBLP: co-authorship network Co-rating of films: MovieLens Mubarak’s resignation on twitter (Gephi.org) Temporal access similarity of places: Bourget 4 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion M ULTIPLEX N ETWORK Definition A set of actors related by different types of relations Motivation ◮ Real networks are dynamic . ◮ Real networks are heterogeneous . ◮ Nodes are usually qualified by a set of attributes. Source: muxviz 5 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion M ULTIPLEX NETWORKS : R ELATED TERMS Recommended readings ✏ S. Mikko Kivel¨ a et. al. . Multilayer Networks . arXiv:1309.7233, March 2014 6 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion M ULTIPLEX N ETWORKS : E XAMPLES Airlines Europe Network DBLP co-auorship network D4D dataset (cˆ ote d’ivoire) Source: muxviz Source: muxviz Source: muxviz 7 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion M ULTIPLEX N ETWORK : N OTATIONS G = < V , E 1 , . . . , E α : E k ⊆ V × V ∀ k ∈ { 1 , . . . , α } > ◮ V : set of nodes (a.k.a. vertices, actors, sites) ◮ E k : set of edges of type k (a.k.a. ties, links, bonds) Notations ◮ A [ k ] Adjacency Matrix of slice k : a [ k ] � = 0 si les nœuds ( v i , v j ) ∈ E k , 0 otherwise. ij ◮ n = | V | ◮ m k = | E k | . We have often m ∼ n ◮ Neighbor’s of v in slice k : Γ( v ) [ k ] = { x ∈ V : ( x , v ) ∈ E k } . v = � Γ( v ) [ k ] � ◮ Node degree in slice k : d k 8 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion C OMMUNITY ? Some definitions ◮ A dense subgraph loosely coupled to other modules in the network ◮ A community is a set of nodes seen as one by nodes outside the community ◮ A subgraph where almost all nodes are linked to other nodes in the community. 9 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion C OMMUNITY ? Some definitions ◮ A dense subgraph loosely coupled to other modules in the network ◮ A community is a set of nodes seen as one 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 10 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion C OMMUNITY DETECTION IN MULTIPLEX NETWORKS Definition ? What is a dense subgraph in a multiplex network ? [BCG11] 11 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion C OMMUNITY 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. 12 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion L AYER AGGREGATION Principle ◮ A ← F ( A [ 1 ] , . . . , A [ α ] ) ◮ Apply a monoplex community detection algorithm on graph having A as adjacency matrix Aggregation functions ∃ 1 ≤ l ≤ α : A [ l ] � 1 ij � = 0 α A ij = 1 w k A [ k ] A ij = � ij 0 otherwise α k = 1 A ij = � { d : A [ d ] A ij = sim ( v i , v j ) � = 0 } � ij 13 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion K- UNIFORM HYPERGRAPH TRANSFORMATION Principle ◮ A k-uniform hypergraph is a hypergraph in which the cardinality of 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) 14 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion E NSEMBLE CLUSTERING APPROACHES Principle ◮ Apply a monoplex algorithm on each layer of the multiplex ◮ Apply clustering ensemble on obtained α clusterings. 15 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion E NSEMBLE CLUSTERING : APPROACHES CSPA: Cluster-based Similarity Partitioning Algorithm ◮ Let K be the number of basic models, C i ( x ) be the cluster in model i to which x belongs. K � δ ( C i ( v ) , C i ( u )) ◮ Define a similarity graph on objects : sim ( v , u ) = i = 1 K ◮ Cluster the obtained graph : Isolate connected components after prunning edges Apply community detection approach ◮ Complexity : O ( n 2 kr ) : n # objects, k # of clusters, r # of clustering solutions 16 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion CSPA : E XEMPLE from Seifi, M. Cœurs stables de communaut´ es dans les graphes de terrain. Th` ese de l’universit´ e Paris 6, 2012 17 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion E NSEMBLE 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 ) 18 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion E NSEMBLE 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 ( nk 2 r 2 ) 19 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion G ROUP - BASED APPROCHES Principle Search for special (dense) subgraphs: ◮ k-clique ◮ n-clique ◮ γ -dense clique ◮ K-core 20 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion G ROUP - BASED APPROCHES from Symeon Papadopoulos, Community Detection in Social Media, CERTH-ITI, 22 June 2011 21 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion E XAMPLE : C LIQUE PERCOLATION 22 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion E XAMPLE : C LIQUE PERCOLATION ◮ Suits fairly dense graph. ◮ Cohesive group concept are not yet generalized to multiplex networks 23 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion N ETWORK - 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 24 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion M ODULARITY OPTIMIZATION APPROACHES Modularity: a partition quality criteria ( A ij − λ × d i d j Q ( P ) = 1 � � 2 m ) 2 m c ∈P i , j ∈ c Figure: For λ = 1, Q = ( 15 + 6 ) − ( 11 . 25 + 2 . 56 ) = 0 . 275 25 25 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion M ODULARITY OPTIMIZATION APPROACHES ◮ Applying classical optimization algorithms (ex. Genetic algorithms [Piz12]). ◮ Applying hierarchical clustering and select the level with Q max (ex. Walktrap [PL06]) ◮ Divisive approach : Girvan-Newman algorithm [GN02] ◮ Greedy optimization : Louvain algorithm [BGL08] ◮ . . . 26 / 44
I NTRODUCTION Community Detection in Multiplex Networks Challenges Conclusion M ULTIPLEX M ODULARITY [MRM + 10] Generalized modularity ◮ d [ k ] i d [ k ] Q multiplex ( P ) = 1 j � � A [ s ] δ kl + δ ij C kl ij − λ k ij 2 m [ k ] 2 µ c ∈ P i , j ∈ c k , l : 1 → α m [ k ] + C l ◮ µ = � jk j ∈ V k , l : 1 → α ◮ C kl ij Inter slice coupling = 0 ∀ i � = j 27 / 44
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