Making Temporal Betweenness Computation Faster and Restless

TL;DR

The paper addresses the challenge of computing temporal betweenness centrality under various optimality criteria, including restless walks, and breaks new ground with an algorithm that achieves $O(nM)$ time for shortest and SFo paths while using $O(n+M)$ space in the non-restless case. It further extends to restless scenarios, yielding a polynomial-time solution for multiple criteria with complexity $O(nM)$. Empirical results show dramatic speedups over prior exact methods (up to about $250\times$ faster) and enable exact computations on graphs with over a million temporal edges, surpassing the previous approximate capabilities. A coherent framework for multiple betweenness notions via cost/target-cost structures supports fast, exact counting across Fast, Foremost, Shortest, Shortest-Fastest, and related variants, with case studies on diverse public-transport networks highlighting both consistent centrality patterns and the influence of waiting constraints. The work thus delivers scalable, exact temporal centrality tools with practical impact for large, time-evolving networks and lays out clear avenues for extending the framework to broader constraints and transfer-heavy settings.

Abstract

Buß et al [KDD 2020] recently proved that the problem of computing the betweenness of all nodes of a temporal graph is computationally hard in the case of foremost and fastest paths, while it is solvable in time O(n 3 T 2 ) in the case of shortest and shortest foremost paths, where n is the number of nodes and T is the number of distinct time steps. A new algorithm for temporal betweenness computation is introduced in this paper. In the case of shortest and shortest foremost paths, it requires O(n + M ) space and runs in time where M is the number of temporal edges, thus significantly improving the algorithm of Buß et al in terms of time complexity (note that T is usually large). Experimental evidence is provided that our algorithm performs between twice and almost 250 times better than the algorithm of Buß et al. Moreover, we were able to compute the exact temporal betweenness values of several large temporal graphs with over a million of temporal edges. For such size, only approximate computation was possible by using the algorithm of Santoro and Sarpe [WWW 2022]. Maybe more importantly, our algorithm extends to the case of restless walks (that is, walks with waiting constraints in each node), thus providing a polynomial-time algorithm (with complexity O(nM )) for computing the temporal betweenness in the case of several different optimality criteria. Such restless computation was known only for the shortest criterion (Rymar et al [JGAA 2023]), with complexity O(n 2 M T 2 ). We performed an extensive experimental validation by comparing different waiting constraints and different optimisation criteria. Moreover, as a case study, we investigate six public transit networks including Berlin, Rome, and Paris. Overall we find a general consistency between the different variants of betweenness centrality. However, we do measure a sensible influence of waiting constraints, and note some cases of low correlation for certain pairs of criteria in some networks.

Figures (7)

• Figure 1: An example of a temporal graph, where $n=8$, $M=13$, $T=11$. The lower part of the label of each temporal edge indicates its availability time $\tau$ and its traversal time $\lambda$ (hence, the arrival time of the temporal edge is $\tau+\lambda$). The (overlined) upper part of the label of each temporal edge indicates its position in the $E^{\mathrm{arr}}$ list. The $E^{\mathrm{dep}}$, $E_{\mathrm{node}}^{\mathrm{dep}}$, $E^{\mathrm{arr}}_{\mathrm{dep}}$ lists also refer to the (overlined) indexes of $E^{\mathrm{arr}}$. The underlined indices, instead, indicate the position of the corresponding edge in $E^{\mathrm{arr}}$ into the the list $E_{\mathrm{node}}^{\mathrm{dep}}$ of its tail.
• Figure 2: The quartiles of the weighted Kendall $\tau$ over 14 networks, for all pairs of betweenness measures, with $\beta=2400$.
• Figure 3: The average weighted Kendall $\tau$ over 14 networks, for all pairs of betweenness measures, as a function of the value of $\beta$.
• Figure 4: The weighted Kendall $\tau$ between the SFa and the SFo betweenness rankings (solid) and between the SFo betweenness and the betweenness rankings (dashed), for each public transport network and for $\beta=300,600,1200,2400,\infty$.
• Figure 5:

Theorems & Definitions (5)

• Lemma 5
• Theorem 6
• Theorem 8
• proof
• proof