Algorithmic Aspects of Temporal Betweenness

Buß Sebastian
Buß Sebastian
Rymar Maciej
Rymar Maciej

KDD '20: The 26th ACM SIGKDD Conference on Knowledge Discovery and Data Mining Virtual Event CA USA July, 2020, pp. 2084-2092, 2020.

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We investigated several variants of temporal betweenness centrality based on the various optimization criteria for temporal paths

Abstract:

The betweenness centrality of a graph vertex measures how often this vertex is visited on shortest paths between other vertices of the graph. In the analysis of many real-world graphs or networks, betweenness centrality of a vertex is used as an indicator for its relative importance in the network. In recent years, a growing number of rea...More

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Introduction
  • Graph metrics such as betweenness centrality are studied and applied in many application areas, including social and technological network analysis [19, 27], wireless routing [5], machine learning [24], and neuroscience [30].
  • A path may be optimal if it minimizes the number of edges (“shortest”), the arrival time (“foremost”), or the overall travel time (“fastest”).
  • For any of these path types the authors can define and study a variant of temporal betweenness centrality.
  • The authors consider two subtypes of foremost paths, namely shortest foremost, and prefix-foremost paths
Highlights
  • Graph metrics such as betweenness centrality are studied and applied in many application areas, including social and technological network analysis [19, 27], wireless routing [5], machine learning [24], and neuroscience [30]
  • We investigate ways to extend the approach of Brandes’ algorithm [2] for static graphs to variants of temporal betweenness based on types of optimal temporal paths for which the counting problem is not intractable
  • We investigated several variants of temporal betweenness centrality based on the various optimization criteria for temporal paths
  • We have shown a surprising discrepancy in their computational complexity: while some variations are #P-hard, others can be computed in polynomial time
  • An intuitive explanation for this behavior might be that our algorithms strongly rely on a recursive formulation for the so-called temporal dependencies, which in turn requires that the predecessor relation of optimal temporal paths is acyclic and that prefixes of optimal temporal paths are optimal
  • For all optimal temporal path types for which we show computational hardness of the corresponding counting problem, one of the two mentioned requirements is not given
Results
  • The authors discuss the results of the experiments. The authors analyzed the influence of the temporal betweenness type on the running time, the distribution of the temporal betweenness values, and the ranking of the ten vertices with the hightest temporal betweenness values.

    Running time.
  • The authors analyzed the influence of the temporal betweenness type on the running time, the distribution of the temporal betweenness values, and the ranking of the ten vertices with the hightest temporal betweenness values.
  • Running time.
  • As indicated by the theoretical running time bounds, Algorithm 1 is several orders of magnitudes slower than Algorithm 2.
  • Algorithm 1 solved all instances except for “karlsruhe” and “infectious” within 45 minutes (keep in mind that Algorithm 1 computes shortest and foremost shortest temporal be-.
  • Sh (Fm) 6.08 · 101 1.82 · 102 2.44 · 103 8.94 · 102 9.82 · 101
Conclusion
  • The authors investigated several variants of temporal betweenness centrality based on the various optimization criteria for temporal paths.
  • The authors found that counting foremost, and fastest paths, is #P-hard, and in turn, the computation of the corresponding betweenness centrality scores is #P-hard as well.
  • One can count shortest and shortest foremost temporal paths in polynomial time both for strict and non-strict paths.
  • For all optimal temporal path types for which the authors show computational hardness of the corresponding counting problem, one of the two mentioned requirements is not given
Summary
  • Introduction:

    Graph metrics such as betweenness centrality are studied and applied in many application areas, including social and technological network analysis [19, 27], wireless routing [5], machine learning [24], and neuroscience [30].
  • A path may be optimal if it minimizes the number of edges (“shortest”), the arrival time (“foremost”), or the overall travel time (“fastest”).
  • For any of these path types the authors can define and study a variant of temporal betweenness centrality.
  • The authors consider two subtypes of foremost paths, namely shortest foremost, and prefix-foremost paths
  • Results:

    The authors discuss the results of the experiments. The authors analyzed the influence of the temporal betweenness type on the running time, the distribution of the temporal betweenness values, and the ranking of the ten vertices with the hightest temporal betweenness values.

    Running time.
  • The authors analyzed the influence of the temporal betweenness type on the running time, the distribution of the temporal betweenness values, and the ranking of the ten vertices with the hightest temporal betweenness values.
  • Running time.
  • As indicated by the theoretical running time bounds, Algorithm 1 is several orders of magnitudes slower than Algorithm 2.
  • Algorithm 1 solved all instances except for “karlsruhe” and “infectious” within 45 minutes (keep in mind that Algorithm 1 computes shortest and foremost shortest temporal be-.
  • Sh (Fm) 6.08 · 101 1.82 · 102 2.44 · 103 8.94 · 102 9.82 · 101
  • Conclusion:

    The authors investigated several variants of temporal betweenness centrality based on the various optimization criteria for temporal paths.
  • The authors found that counting foremost, and fastest paths, is #P-hard, and in turn, the computation of the corresponding betweenness centrality scores is #P-hard as well.
  • One can count shortest and shortest foremost temporal paths in polynomial time both for strict and non-strict paths.
  • For all optimal temporal path types for which the authors show computational hardness of the corresponding counting problem, one of the two mentioned requirements is not given
Tables
  • Table1: An overview of the computational complexity of the temporal betweenness variants we consider. Here, n refers to the number of vertices, M refers to the total number of time edges, and T refers to the number of time steps
  • Table2: Statistics for the data sets used in our experiments. The lifetime T of a graph is the difference between the largest and smallest time stamp on an edge in the graph. The resolution r indicates how often edges were measured. The last three columns state the running times in seconds of our implementation. A -1 indicates that the instance was not solved within three hours
  • Table3: Kendall’s tau correlation measure for each pair of vertex rankings induced by the different temporal betweenness versions. Values close to 1 indicate strong agreement and values close to -1 indicate strong disagreement between the rankings
  • Table4: Size of the intersection of each pair of sets containing the ten vertices with highest temporal betweenness values
Download tables as Excel
Related work
  • There is an enormous amount of work on the concept of betweenness centrality in static graphs, as already indicated by the huge citation numbers concerning Brandes path-breaking algorithm [2]. Betweenness centrality was defined in 1977 by Freeman [7]. We refrain from further discussing the static case which is already discussed in many textbooks.

    The theory of temporal graphs is comparatively young [12, 13, 14, 20] but strongly grows in many directions. We focus our discussion of related work on temporal walks, paths, and the computation of temporal betweenness centrality. Bui-Xuan et al [3] did an early work on algorithms that find optimal temporal paths (called “journeys” there). In particular, they presented algorithms for shortest, fastest, and foremost temporal paths. Afterwards, Wu et al [32] provided state-of-the-art algorithms for optimal temporal paths. Based on breadth-first search which finds shortest paths in static graphs, Wu et al showed that shortest, foremost, fastest and reverse-foremost (strict) temporal paths can be found in a similar fashion. Himmel et al [11] and Casteigts et al [4] expanded on the work of Wu et al and studied a more complex variation of temporal paths and walks with constraints on the waiting time in each vertex. Himmel et al [11] contributed efficient algorithms to find optimal temporal walks but Casteigts et al [4] showed that finding optimal temporal paths is NP-hard in settings with upper bounds on the waiting time.
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