Algorithmic Challenges in Link Streams: the case of clique - PowerPoint PPT Presentation

Introduction Maximal cliques in link streams Link stream edition problems Algorithmic Challenges in Link Streams: the case of clique computations Cl´ emence Magnien work in collaboration with Tiphaine Viard, Matthieu Latapy, Phan Thi Ha Duong, Binh-Minh Bui-Xuan, Pierre Meyer ComplexNetworks(.fr) LIP6 (CNRS, Sorbonne Universit´ e) first.last@lip6.fr July 9th, 2018 logobas C. Magnien 1/21

Introduction Maximal cliques in link streams Link stream edition problems Outline Introduction 1 Maximal cliques in link streams 2 Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams with durations Link stream edition problems 3 logobas C. Magnien 2/21

Introduction Maximal cliques in link streams Link stream edition problems Link streams Models of temporal interactions L = ( T , V , E ) T = [ α ; ω ] V set of nodes One link = ( t , uv ) E ⊆ T × V ⊗ V set of links a Two cases of interest b instantaneous link c streams d link streams with 0 2 4 time durations logobas C. Magnien 3/21

Introduction Maximal cliques in link streams Link stream edition problems Link streams Models of temporal interactions L = ( T , V , E ) T = [ α ; ω ] V set of nodes One link = ( t , uv ) E ⊆ T × V ⊗ V set of links a Two cases of interest b instantaneous link c streams d link streams with 0 2 4 time durations logobas C. Magnien 3/21

Introduction Maximal cliques in link streams Link stream edition problems Link streams Models of temporal interactions L = ( T , V , E ) T = [ α ; ω ] V set of nodes One link = ( b , e , uv ) E ⊆ T × V ⊗ V set of links a Two cases of interest b instantaneous link c streams d link streams with 0 2 4 time durations logobas C. Magnien 3/21

Introduction Maximal cliques in link streams Link stream edition problems Definitions Extensions of graph definitions Paths (Strongly) Connected components Betweenness Centrality Cores and shells . . . Extensions of algorithms ? logobas C. Magnien 4/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Outline Introduction 1 Maximal cliques in link streams 2 Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams with durations Link stream edition problems 3 logobas C. Magnien 5/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Clique (in a graph) X ⊆ V Induced subgraph : all possible links exist logobas C. Magnien 6/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Clique (in a graph) X ⊆ V Induced subgraph : all possible links exist Maximal clique : not included in any other clique logobas C. Magnien 6/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Clique (in a graph) X ⊆ V Induced subgraph : all possible links exist Maximal clique : not included in any other clique logobas C. Magnien 6/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Clique (in a graph) X ⊆ V Induced subgraph : all possible links exist Maximal clique : not included in any other clique logobas C. Magnien 6/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems ∆-clique in instantaneous link streams ( X , [ b ; e ]) ⊆ V × T Induced sub-stream : all possible links exist all the time All the time : at least every ∆ Maximal if is not included in any other Examples for ∆ = 3 : logobas Signatures of distributed applications, meetings, . . . C. Magnien 7/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems ∆-clique in instantaneous link streams ( X , [ b ; e ]) ⊆ V × T Induced sub-stream : all possible links exist all the time All the time : at least every ∆ Maximal if is not included in any other Examples for ∆ = 3 : 4 6 4 6 2 0 2 8 a a b b c c logobas Signatures of distributed applications, meetings, . . . C. Magnien 7/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Cliques in link streams with duration ( X , [ b ; e ]) , ⊆ V × T Induced sub-stream : all possible links exist all the time Maximal if is not included in any other a b c d 0 2 4 6 8 time logobas C. Magnien 8/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Cliques in link streams with duration ( X , [ b ; e ]) , ⊆ V × T Induced sub-stream : all possible links exist all the time Maximal if is not included in any other a b c d 0 2 4 6 8 time logobas C. Magnien 8/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Outline Introduction 1 Maximal cliques in link streams 2 Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams with durations Link stream edition problems 3 logobas C. Magnien 9/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Enumerate maximal ∆-cliques in a link stream Naive algorithm Queue Q for all ( t , uv ) ∈ E , Discovered cliques ( { u , v } , [ t , t ]) is a ∆-clique − → Q X, b, e e’ > e ? While Q � = ∅ : X {u} ? U b’<b ? pop C from Q : X {u} , b, e X, b, e’ X, b’, e U if a node or time can be added − → Q otherwise C is maximal logobas C. Magnien 10/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Enumerate maximal ∆-cliques in a link stream Naive algorithm Queue Q for all ( t , uv ) ∈ E , Discovered cliques ( { u , v } , [ t , t ]) is a ∆-clique − → Q X, b, e While Q � = ∅ : e’ > e ? X {u} ? U pop C from Q : X {u} , b, e X, b, e’ U if a node or time can be added Is maximal − → Q otherwise C is maximal logobas C. Magnien 10/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Time extension ∆ = 4 a b c d 0 2 4 6 time logobas C. Magnien 11/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Time extension ∆ = 4 a b c d 0 2 4 6 time for all links : latest occurrence earliest such occurrence logobas C. Magnien 11/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Time extension ∆ = 4 a b c d 0 2 4 6 time for all links : latest occurrence earliest such occurrence logobas add ∆ C. Magnien 11/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Sketch of proof (1) 1 Initially, all elements of Q are ∆-cliques 2 one step : transforms a ∆-clique into (several) ∆-cliques 3 the output contains only maximal ∆-cliques logobas C. Magnien 12/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Sketch of proof (2) All maximal ∆-cliques of L are in the output Let C = ( X , [ b , e ]) be an arbitrary maximal ∆-clique. ( s , uv ) : earliest link of C C 0 = ( { u , v } , [ s , s ]) C 1 = ( { u , v } , [ s , s + ∆]) . . . (add nodes) C k = ( X , [ s , s + ∆]) . . . (increase time on the right) C e = ( X , [ s , e ]) C = ( X , [ b , e ]) logobas C. Magnien 13/21

Introduction Maximal ∆-cliques in instantaneous link stream Maximal cliques in link streams Maximal cliques in link streams with durations Link stream edition problems Complexity O (2 n n 2 m 3 + 2 n n 3 m 2 ) Interesting observations No relation between n and m small n , large m − → reasonable running time 2 n : All subsets of nodes In practice : of nodes linked at the same time − → Running time increases with ∆ logobas C. Magnien 14/21