Algorithmic Aspects of Temporal Betweenness Sebastian Buß Hendrik Molter Rolf Niedermeier Maciej Rymar Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Germany Algorithmic Aspects of Temporal Graphs III
Betweenness Centrality: Motivation I “How important is Berlin main station as a hub for the public transportation network?” Hendrik Molter, TU Berlin 2 / 20 Algorithmic Aspects of Temporal Betweenness
Betweenness Centrality: Motivation II Transportation Networks Routing Networks Protein Networks Social Networks Hendrik Molter, TU Berlin 3 / 20 Algorithmic Aspects of Temporal Betweenness
Betweenness Centrality: Definition Betweenness of a vertex v in a graph G = ( V , E ) : “How likely is a shortest path to pass through vertex v ?” Betweenness Centrality σ sz ( v ) C B ( v ) = ∑ σ sz s � = v � = z σ sz : # shortest paths from s to z σ sz ( v ) : # shortest paths from s to z via v A remark on motivation: Betweenness assumes information travels along optimal paths! In many scenarious unrealistic → Random walk based centralities (e.g. PageRank). Hendrik Molter, TU Berlin 4 / 20 Algorithmic Aspects of Temporal Betweenness
Betweenness Centrality: Example v Darker colors indicate higher betweenness centrality. σ sz ( v ) C B ( v ) = ∑ = 42 . σ sz s � = v � = z Hendrik Molter, TU Berlin 5 / 20 Algorithmic Aspects of Temporal Betweenness
Betweenness Centrality: Background Ulrik Brandes, Prof. for Social Networks at First formally described by Linton Freeman ETH Zürich in 1977 Published “blueprint” for all modern Freeman was Prof. for Sociology at UC betweenness algorithms in 2001 (J. Math. Irvine and founder of the journal “Social Sociol.): Brandes’ algorithm Networks” Main achievement: improved running time Measure for quantifying the control on the for sparse graphs, linear space requirement communication in a social network Hendrik Molter, TU Berlin 6 / 20 Algorithmic Aspects of Temporal Betweenness
Temporal Graphs and Temporal Paths: Definition Temporal Graph Temporal ( s , z ) -Path A temporal graph G = ( V , ( E i ) i ∈ [ τ ] ) is a Sequence of time edges forming a path from vertex set V with a list of edge sets E 1 ,..., E τ s to z that have: over V , where τ is the lifetime of G . increasing time stamps (strict). non-decreasing time stamps (non-strict). 2 3 G : 3 s 1 1 2 z 1 3 2 4 Not a temporal path. 1 , 2 s z G 1 : G 2 : G 3 : 1 1 3 4 Non-strict temporal path. (Not strict.) s z 1 2 3 4 Temporal path (both strict and non-strict). Hendrik Molter, TU Berlin 7 / 20 Algorithmic Aspects of Temporal Betweenness
Optimal Temporal Paths Method: Replace shortest path by “optimal” temporal path . Problem: Which temporal path from s to z is optimal? 1 5 s z 1 2 3 4 5 3 4 Shortest temporal paths use the minimum number of edges. Foremost temporal paths have a minimum arrival time. Fastest temporal paths have a minimum difference between starting and arrival time. Most well-motivated: Foremost temporal paths. Hendrik Molter, TU Berlin 8 / 20 Algorithmic Aspects of Temporal Betweenness
Temporal Betweenness Centrality Temporal Betweenness Variants: (strict vs. non-strict) × (# optimality criteria) = six temporal betweenness variants { Strict, Non-Strict } { Shortest, Foremost, Fastest } Temporal Betweeness What is known: Temporal Betweenness has been already been studied extensively. Many more combinations and considerations are possible and have been made. Most approaches use static expansions and use known algorithms for static betweenness. Question: Which variants are computable with a “temporal version” of Brandes’ algorithm? Hendrik Molter, TU Berlin 9 / 20 Algorithmic Aspects of Temporal Betweenness
Main Ideas of Brandes’ Algorithm Recall: C B ( v ) = ∑ s � = v � = t σ st ( v ) / σ st Main Idea: Cleverly sum up “dependencies” δ s • ( v ) in a modified BFS without calculating them explicitly. C B ( v ) = ∑ δ s • ( v ) , where δ s • ( v ) = ∑ σ st ( v ) / σ st . s ∈ V \{ v } t ∈ V \{ v } Use a recursive formula for δ s • ( v ) based on “successors” in shortest paths starting at s . s v w z Vertex w is a “successor” of v (with respect to s ). (Presumably) Necessary conditions for this approach: Counting shortest/optimal paths is easy. “Successor relation” is acyclic. Hendrik Molter, TU Berlin 10 / 20 Algorithmic Aspects of Temporal Betweenness
Obstacles for Tractability I: Hard Counting Observation Computing betweenness values is at least as hard as counting optimal paths. Static case: 1 1 1 1 Known facts 1 s t 1 Counting shortest paths from s to z can be done in poly time. 1 1 1 Counting all paths from s to z is #P-hard [Valiant 79]. 1 Corollary Counting non-strict foremost or fastest temporal paths is #P-hard. Hendrik Molter, TU Berlin 11 / 20 Algorithmic Aspects of Temporal Betweenness
Obstacles for Tractability II: Cyclic Successors Foremost / Fastest temporal paths: Observations: w 2 3 w is a successor of v s z 2 v is a successor of w 2 v 3 Shortest Temporal Paths: 2 7 w 1 8 w is a successor of v 3,6 s z v is a successor of w 5 4 v Hendrik Molter, TU Berlin 12 / 20 Algorithmic Aspects of Temporal Betweenness
Cyclic Successors: Solution Main idea: Consider “vertex appearances” instead of vertices. Define: Vertex appearance ( v , t ) is “visited” by a temporal path if the path arrives in vertex v at time t . s v z t Temporal Betweenness Centrality of Vertex Appearances ∄ temp. ( s , z ) -path 0 C ( ⋆ ) B ( v , t ) = ∑ δ ( ⋆ ) sz ( v , t ) , where δ ( ⋆ ) sz ( v , t ) = σ ( ⋆ ) sz ( v , t ) otherwise s � = v � = z σ ( ⋆ ) sz σ ( ⋆ ) σ ( ⋆ ) sz : # ⋆ -temp. paths from s to z sz ( v , t ) : # ⋆ -temp. paths from s to z via ( v , t ) ⋆ ∈ { foremost, fastest, shortest } Hendrik Molter, TU Berlin 13 / 20 Algorithmic Aspects of Temporal Betweenness
Brandes Algorithm for Temporal Betweenness Main recipe to transfer Brandes algorithm to the temporal setting: Define dependencies in an analogous way. Show a recursive formula for the dependencies similar to the static case. Main differences to the static setting: Counting optimal paths can be #P-hard in the temporal setting (foremost & fastest). � Presumably impossible to adapt Brandes for these optimality criteria. Successors can behave “cyclicly”. � Necessary to consider vertex appearances. Walks can be optimal. Hendrik Molter, TU Berlin 14 / 20 Algorithmic Aspects of Temporal Betweenness
Temporal Betweenness: Tractable Variants for Foremost Temp. Paths Shortest Foremost Temporal Paths Strict Prefix-Foremost Temporal Paths Shortest temporal paths among all foremost Foremost temporal paths where every prefix temporal paths. is also a foremost temporal paths. Techniques for shortest temporal paths Lemma [Wu et al. 16] directly adaptable. A prefix foremost temporal path always exists. w 1 4 Techniques for shortest temporal paths adaptable with some modifications. s z 2 3 v 4 Hendrik Molter, TU Berlin 15 / 20 Algorithmic Aspects of Temporal Betweenness
Temporal Betweenness: Prefix Foremost Temporal Paths Observation Counting non-strict prefix foremost temporal paths is #P-hard. Observation Time steps at which vertices are visited by prefix foremost temporal paths (starting from s ) are unique. w 1 4 s z 2 3 4 v For temporal betweenness based on strict prefix foremost temporal paths, we can consider vertices (instead of vertex appearances). � Faster and more space efficient algorithm. Hendrik Molter, TU Berlin 16 / 20 Algorithmic Aspects of Temporal Betweenness
Our Results Table of theoretical results: strict non-strict O ( n 3 · τ 2 ) time, O ( n · τ + M ) space Shortest Foremost #P-hard Fastest #P-hard O ( n · M · log M ) time, O ( n + M ) space Prefix-foremost #P-hard O ( n 3 · τ 2 ) time, O ( n · τ + M ) space Shortest foremost τ : lifetime n : # vertices M : # time edges Main Messages from Empirical Evaluation: Two prefix-foremost and shortest foremost variants produce similar vertex rankings based on betweenness score. Prefix-foremost betweenness can be computed much faster. Hendrik Molter, TU Berlin 17 / 20 Algorithmic Aspects of Temporal Betweenness
Temporal Betweenness Experiments I: Running Time highschool-2011 highschool-2012 highschool-2013 primaryschool hospital-ward hypertext infectious karlsruhe facebook-like Blue: non-strict shortest (foremost) Time is on a log-scale, ranges from few betweenness. minutes up to three hours. Green: strict shortest (foremost) “infectious” and “karlsruhe” not solved for betweenness. (non-)strict shortest (foremost) Red: strict prefix foremost betweenness. betweenness within three hours. Hendrik Molter, TU Berlin 18 / 20 Algorithmic Aspects of Temporal Betweenness
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