Computing Betweenness Centrality in Link Streams Cl emence Magnien - PowerPoint PPT Presentation

Computing Betweenness Centrality in Link Streams Cl´ emence Magnien joint work with Fr´ ed´ eric Simard and Matthieu Latapy July 2020 clemence.magnien@lip6.fr – july 2020 – 1/10

Link Streams – definitions link stream L = ( T , V , E ) T = [ α, ω ] ⊂ R , V finite set, E ⊆ T × V ⊗ V ( t , uv ) ∈ E ⇔ u and v are linked at time t link segment [ i , j ] × { uv } ⊆ E with [ i , j ] maximal here: finite number of link segments (incl. singletons) temporal node ( t , u ) ∈ T × V clemence.magnien@lip6.fr – july 2020 – 2/10

Link Streams – definitions link stream L = ( T , V , E ) T = [ α, ω ] ⊂ R , V finite set, E ⊆ T × V ⊗ V ( t , uv ) ∈ E ⇔ u and v are linked at time t link segment [ i , j ] × { uv } ⊆ E with [ i , j ] maximal here: finite number of link segments (incl. singletons) temporal node ( t , u ) ∈ T × V a b c d e 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 time exs: [9 , 11] × { de } , { 16 } × { de } , [23 , 24] × { de } , [30 , 31] × { de } clemence.magnien@lip6.fr – july 2020 – 2/10

(Shortest Fastest) Paths a b c d e 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 time clemence.magnien@lip6.fr – july 2020 – 3/10

(Shortest Fastest) Paths a b c d e 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 time path ( x , u ) − → ( y , v ): length: k duration: t k − t 1 v 0 , t 1 , v 1 , t 2 , v 2 , . . . t k , v k shortest paths fastest paths such that u = v 0 , v k = v , → shortest fastest paths (sfp) x ≤ t 1 ≤ t 2 ≤ · · · ≤ t k ≤ y , ֒ and ( t i , v i − 1 v i ) ∈ E for all i clemence.magnien@lip6.fr – july 2020 – 3/10

(Shortest Fastest) Paths a b c d e 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 time path ( x , u ) − → ( y , v ): length: k duration: t k − t 1 v 0 , t 1 , v 1 , t 2 , v 2 , . . . t k , v k shortest paths fastest paths such that u = v 0 , v k = v , → shortest fastest paths (sfp) x ≤ t 1 ≤ t 2 ≤ · · · ≤ t k ≤ y , ֒ and ( t i , v i − 1 v i ) ∈ E for all i some paths from (0 , a ) to (26 , e ): a , 2 , b , 4 , c , 6 , d , 9 , e fastest path , length 4, duration 7, not shortest a , 9 , c , 18 , d , 23 , e shortest path , length 3, duration 14, not fastest a , 2 , b , 4 , c , 6 , d , 9 , e shortest fastest path (sfp) clemence.magnien@lip6.fr – july 2020 – 3/10

(Shortest Fastest) Paths a b c d e 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 time path ( x , u ) − → ( y , v ): length: k duration: t k − t 1 v 0 , t 1 , v 1 , t 2 , v 2 , . . . t k , v k shortest paths fastest paths such that u = v 0 , v k = v , → shortest fastest paths (sfp) x ≤ t 1 ≤ t 2 ≤ · · · ≤ t k ≤ y , ֒ and ( t i , v i − 1 v i ) ∈ E for all i some paths from (0 , a ) to (26 , e ): a , 2 , b , 4 , c , 6 , d , 9 , e fastest path , length 4, duration 7, not shortest a , 9 , c , 18 , d , 23 , e shortest path , length 3, duration 14, not fastest a , 2 , b , 4 , c , 6 , d , 9 , e shortest fastest path (sfp) a , 2 , b , 5 , c , 6 , d , 9 , e too ⇒ infinity of sfp clemence.magnien@lip6.fr – july 2020 – 3/10

Betweenness Centrality in Link Streams graphs: link streams: vertex, temporal vertex, shortest paths, shortest fastest paths, all u and v all ( t , u ) and ( t ′ , v ) shortest paths shortest fastest paths clemence.magnien@lip6.fr – july 2020 – 4/10

Betweenness Centrality in Link Streams graphs: link streams: vertex, temporal vertex, shortest paths, shortest fastest paths, all u and v all ( t , u ) and ( t ′ , v ) shortest paths shortest fastest paths � σ (( i , u ) , ( j , w ) , ( t , v )) � B ( t , v ) = d i d j σ (( i , u ) , ( j , w )) i ∈ T , j ∈ T u ∈ V , w ∈ V � �� � fraction of all sfp ( i , u ) − → ( j , w ) that involve ( t , v ) clemence.magnien@lip6.fr – july 2020 – 4/10

Example u x 2 < a ≤ b < c ≤ d < 7 and v 10 < e ≤ f < g ≤ h < 15 y w 2 a k b c l d 7 10 e m f g n h 15 time contribution of u and w to B ( t , v ) with t ∈ [ b , c ] ? two families of sfp from u to w : ◮ u , 2 , x , k , v , ℓ, y , 7 , w with k ∈ [ a , b ] and ℓ ∈ [ c , d ] ( blue family ) ( b − a ) · ( d − c ) sfp ◮ u , 10 , x , m , v , n , y , 15 , w with m ∈ [ e , f ] and n ∈ [ g , h ] ( green family ) ( f − e ) · ( h − g ) sfp clemence.magnien@lip6.fr – july 2020 – 5/10

Example u x 2 < a ≤ b < c ≤ d < 7 v and 10 < e ≤ f < g ≤ h < 15 y w 2 a k b c l d 7 10 e m f g n h 15 time contribution of u and w to B ( t , v ) with t ∈ [ b , c ] ? sfp from ( i , u ) to ( j , w ) : ◮ blue ones if i ∈ [0 , 2] and j ∈ [7 , 15[ ◮ both blue and green ones i ∈ [0 , 2] and j ∈ [15 , 17] ◮ green ones if i ∈ ]2 , 10] and j ∈ [15 , 17] ◮ no sfp for all others i and j clemence.magnien@lip6.fr – july 2020 – 5/10

Example u x 2 < a ≤ b < c ≤ d < 7 v and 10 < e ≤ f < g ≤ h < 15 y w 2 a k b c l d 7 10 e m f g n h 15 time contribution of u and w to B ( t , v ) with t ∈ [ b , c ] ? sfp from ( i , u ) to ( j , w ) : ◮ blue ones if i ∈ [0 , 2] and j ∈ [7 , 15[ → . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . all involve ( t , v ) ֒ ◮ both blue and green ones i ∈ [0 , 2] and j ∈ [15 , 17] → . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . blue fraction involve ( t , v ) ֒ ( b − a ) · ( d − c ) ( b − a ) · ( d − c )+( f − e ) · ( h − g ) ◮ green ones if i ∈ ]2 , 10] and j ∈ [15 , 17] → . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .none involve ( t , v ) ֒ ◮ no sfp for all others i and j clemence.magnien@lip6.fr – july 2020 – 5/10

Example u x 2 < a ≤ b < c ≤ d < 7 v and 10 < e ≤ f < g ≤ h < 15 y w 2 a k b c l d 7 10 e m f g n h 15 time contribution of u and w to B ( t , v ) with t ∈ [ b , c ] ? � 2 � 15 � 2 � 17 1 d j d i + blue fraction d j d i 0 7 0 15 = 16 + 4 · blue fraction ( b − a ) · ( d − c ) blue fraction = ( b − a ) · ( d − c )+( f − e ) · ( h − g ) clemence.magnien@lip6.fr – july 2020 – 5/10

Example – what if a = b ? u x v y w m n 2 c ℓ d 7 10 e f g h 15 time a=b how many blue paths? green paths ? clemence.magnien@lip6.fr – july 2020 – 6/10

Example – what if a = b ? u x v y w m n 2 c ℓ d 7 10 e f g h 15 time a=b how many blue paths? green paths ? blue paths: u , 2 , x , a , v , ℓ, y , 7 , w with ℓ ∈ [ c , d ] ֒ → volume ( d − c ), dimension 1 green paths: u , 10 , x , m , v , n , y , 15 , w with m ∈ [ e , f ] and n ∈ [ g , h ] → volume ( f − e ) · ( h − g ), dimension 2 ֒ clemence.magnien@lip6.fr – july 2020 – 6/10

Example – what if a = b ? u x v y w m n 2 c ℓ d 7 10 e f g h 15 time a=b how many blue paths? green paths ? blue paths: u , 2 , x , a , v , ℓ, y , 7 , w with ℓ ∈ [ c , d ] ֒ → volume ( d − c ), dimension 1 green paths: u , 10 , x , m , v , n , y , 15 , w with m ∈ [ e , f ] and n ∈ [ g , h ] → volume ( f − e ) · ( h − g ), dimension 2 ֒ fraction involving ( t , v ) ? 0 except if e = f or g = h ... clemence.magnien@lip6.fr – july 2020 – 6/10

Volumes of Shortest Paths Identify times of beginning and end of fastest paths latency pairs from u to w : ( s i , a i ) clemence.magnien@lip6.fr – july 2020 – 7/10

Volumes of Shortest Paths Identify times of beginning and end of fastest paths latency pairs from u to w : ( s i , a i ) → algorithm for volumes of sfp from u to w : ֒ shortest paths within each latency pair clemence.magnien@lip6.fr – july 2020 – 7/10

Volumes of Shortest Paths Identify times of beginning and end of fastest paths latency pairs from u to w : ( s i , a i ) → algorithm for volumes of sfp from u to w : ֒ shortest paths within each latency pair using BFS-like from event time to event time: i,u i,u t t’ t t’ t,x t’,y t’,w t’,w path ( i , u ) − → ( t , x ) → ( t ′ , y ) path ( i , u ) − then path ( t , x ) − → ( t ′ , w ) then jump y → w at t ′ clemence.magnien@lip6.fr – july 2020 – 7/10

Volumes of Shortest Paths Identify times of beginning and end of fastest paths latency pairs from u to w : ( s i , a i ) → algorithm for volumes of sfp from u to w : ֒ shortest paths within each latency pair using BFS-like from event time to event time: i,u i,u t t’ t t’ t,x t’,y t’,w t’,w path ( i , u ) − → ( t , x ) → ( t ′ , y ) path ( i , u ) − then path ( t , x ) − → ( t ′ , w ) then jump y → w at t ′ + operation on volumes (considering dimension) clemence.magnien@lip6.fr – july 2020 – 7/10

Global Scheme for B ( t , v ) t,v clemence.magnien@lip6.fr – july 2020 – 8/10

Global Scheme for B ( t , v ) u t,v w for all u and w : clemence.magnien@lip6.fr – july 2020 – 8/10

Global Scheme for B ( t , v ) < a−s = a−s = a−s = a−s s a > a−s < a−s > a−s < a−s u t,v w for all u and w : compute latency pairs for u and w clemence.magnien@lip6.fr – july 2020 – 8/10