General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Community Detection by Decomposing a Graph into Relaxed Cliques Fabio Furini, Timo Gschwind, Stefan Irnich, Roberto Wolfler Calvo LAMSADE, Université Paris-Dauphine
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Outline General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Social Network Analysis (SNA) ◮ Relationships between members of a network can be encoded by an undirected graph G = ( V , E ) ◮ vertices ( V ) represent the members of the network ◮ edges ( E ) indicate the existence of a relationship ◮ Community Detection aims at clustering the members into communities such that: ◮ relatively few edges are in the cutsets ◮ but relatively many are internal edges. ◮ The clustering is intended to reveal hidden or reproduce known features of the network ◮ In this talk we present a framework for community detection when the internal structure of the community is important
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results First case study: Dolphins – Social Network ◮ 62 bottlenose dolphins living in Doubtful Zig Sound, New Zealand Ripplefluke Wave MN23 DN21 Feather DN16 Quasi Jet Gallatin Mus Web TR82 Upbang SN90 Number1 Notch SN89 Beescratch Knit ◮ An edge indicates that a pair of dolphins were Zap DN63 seen together more often than expected SN100 MN60 CCL PL Vau (love? ) Oscar Double Haecksel Beak SN9 Cross TR77 Topless Trigger ◮ After dolphin SN100 left the place for some SN96 TR99 Kringel Five Jonah time, the dolphins separated into two groups MN83 Grin Fish TSN103 SN4 indicated with the two colors MN105 Patchback Bumper Hook Scabs Thumper SN63 ◮ Question 1: where did SN100 go? Stripes SMN5 Shmuddel Whitetip Zipfel ◮ Question 2: is it possible to predict this split? TR120 Fork TSN83 TR88
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results First case study: Dolphins – Social Network ◮ 62 bottlenose dolphins living in Doubtful Zig Sound, New Zealand Ripplefluke Wave MN23 DN21 Feather DN16 Quasi Jet Gallatin Mus Web TR82 Upbang SN90 Number1 Notch SN89 Beescratch Knit ◮ An edge indicates that a pair of dolphins were Zap DN63 seen together more often than expected SN100 MN60 CCL PL Vau (love? ) Oscar Double Haecksel Beak SN9 Cross TR77 Topless Trigger ◮ After dolphin SN100 left the place for some SN96 TR99 Kringel Five Jonah time, the dolphins separated into two groups MN83 Grin Fish TSN103 SN4 indicated with the two colors MN105 Patchback Bumper Hook Scabs Thumper SN63 ◮ Question 1: where did SN100 go? Stripes SMN5 Shmuddel Whitetip Zipfel ◮ Question 2: is it possible to predict this split? TR120 Fork TSN83 TR88
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results First case study: Dolphins – Social Network ◮ 62 bottlenose dolphins living in Doubtful Zig Sound, New Zealand Ripplefluke Wave MN23 DN21 Feather DN16 Quasi Jet Gallatin Mus Web TR82 Upbang SN90 Number1 Notch SN89 Beescratch Knit ◮ An edge indicates that a pair of dolphins were Zap DN63 seen together more often than expected SN100 MN60 CCL PL Vau (love? ) Oscar Double Haecksel Beak SN9 Cross TR77 Topless Trigger ◮ After dolphin SN100 left the place for some SN96 TR99 Kringel Five Jonah time, the dolphins separated into two groups MN83 Grin Fish TSN103 SN4 indicated with the two colors MN105 Patchback Bumper Hook Scabs Thumper SN63 ◮ Question 1: where did SN100 go? Stripes SMN5 Shmuddel Whitetip Zipfel ◮ Question 2: is it possible to predict this split? TR120 Fork TSN83 TR88
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results First case study: Dolphins – Social Network ◮ 62 bottlenose dolphins living in Doubtful Zig Sound, New Zealand Ripplefluke Wave MN23 DN21 Feather DN16 Quasi Jet Gallatin Mus Web TR82 Upbang SN90 Number1 Notch SN89 Beescratch Knit ◮ An edge indicates that a pair of dolphins were Zap DN63 seen together more often than expected SN100 MN60 CCL PL Vau (love? ) Oscar Double Haecksel Beak SN9 Cross TR77 Topless Trigger ◮ After dolphin SN100 left the place for some SN96 TR99 Kringel Five Jonah time, the dolphins separated into two groups MN83 Grin Fish TSN103 SN4 indicated with the two colors MN105 Patchback Bumper Hook Scabs Thumper SN63 ◮ Question 1: where did SN100 go? Stripes SMN5 Shmuddel Whitetip Zipfel ◮ Question 2: is it possible to predict this split? TR120 Fork TSN83 TR88
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results First case study: Dolphins – Social Network ◮ 62 bottlenose dolphins living in Doubtful Zig Sound, New Zealand Ripplefluke Wave MN23 DN21 Feather DN16 Quasi Jet Gallatin Mus Web TR82 Upbang SN90 Number1 Notch SN89 Beescratch Knit ◮ An edge indicates that a pair of dolphins were Zap DN63 seen together more often than expected SN100 MN60 CCL PL Vau (love? ) Oscar Double Haecksel Beak SN9 Cross TR77 Topless Trigger ◮ After dolphin SN100 left the place for some SN96 TR99 Kringel Five Jonah time, the dolphins separated into two groups MN83 Grin Fish TSN103 SN4 indicated with the two colors MN105 Patchback Bumper Hook Scabs Thumper SN63 ◮ Question 1: where did SN100 go? Stripes SMN5 Shmuddel Whitetip Zipfel ◮ Question 2: is it possible to predict this split? TR120 Fork TSN83 TR88
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Motivation Network Analysis: Graphs representing real networks have structures! → community structure or clustering karate football Important applications in many networked systems from biology, sociology, computer science, engineering, economics, politics, linguistics, etc.
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Motivation Network Analysis: 1. Community Detection [Fortunato(2010)] ◮ Partition graphs into vertex subsets ◮ Few edges between subsets, many internal edges ◮ Maximize modularity [Newman and Girvan(2004)] � | E ( V i ) | p � � µ = Q ( V 1 , V 2 , . . . , V p ) = − exp ( V i ) | E | i = 1 exp ( V i ) is the expected fraction of inner-cluster edges ◮ Subsets have no specific structure 2. Relaxed Cliques [Pattillo et al.(2013a)] ◮ Subgraphs with a specific structure ◮ Find maximum cardinality/weight relaxed clique
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Motivation Network Analysis: 1. Community Detection [Fortunato(2010)] ◮ Partition graphs into vertex subsets ◮ Few edges between subsets, many internal edges ◮ Maximize modularity [Newman and Girvan(2004)] � | E ( V i ) | p � � µ = Q ( V 1 , V 2 , . . . , V p ) = − exp ( V i ) | E | i = 1 exp ( V i ) is the expected fraction of inner-cluster edges ◮ Subsets have no specific structure 2. Relaxed Cliques [Pattillo et al.(2013a)] ◮ Subgraphs with a specific structure ◮ Find maximum cardinality/weight relaxed clique
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Motivation Network Analysis: 1. Community Detection [Fortunato(2010)] ◮ Partition graphs into vertex subsets ◮ Few edges between subsets, many internal edges ◮ Maximize modularity [Newman and Girvan(2004)] � | E ( V i ) | p � � µ = Q ( V 1 , V 2 , . . . , V p ) = − exp ( V i ) | E | i = 1 exp ( V i ) is the expected fraction of inner-cluster edges ◮ Subsets have no specific structure 2. Relaxed Cliques [Pattillo et al.(2013a)] ◮ Subgraphs with a specific structure ◮ Find maximum cardinality/weight relaxed clique
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Dolphins case study: Maximizing Modularity (a) Real-world split: modularity µ ( G ) = 0 . 3735; (b) Decomposition with maximum modularity: µ = 0 . 5285 Partitioning into cliques is also not a good idea!
General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results Cliques and Clique Relaxations A clique S forms an extreme subset in the following senses: Degree Every node i ∈ S has maximum degree (= | S | − 1) Distance The distance dist ( i , j ) between any two nodes i , j ∈ S is minimal (=1) Density G [ S ] has maximum edge density (=1); Connectivity The vertex connectivity κ ( G [ S ]) is maximum (= | S | − 1) Analysis of large, complex networks : ◮ Cliques can model cohesive substructures, e.g., subgroups. ◮ However, requirements of a clique were found too restrictive!
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