Temporal Betweenness Centrality on Shortest Walks Variants

TL;DR

This work advances temporal betweenness centrality by systematically comparing passive and active shortest-walk variants in temporal graphs, and by providing tighter, Brandes-style algorithmic bounds that scale as $O(n^2\,m\,T + n^2\,T)$ for passive walks and $O(n^2\,T^2 + n\,m\,T)$ for active walks. It unifies three cost-criteria—shortest, shortest $k$-restless, and shortest foremost—across both walk types (with strict/non-strict variants), and introduces a predecessor-graph framework that enables efficient, per-source recurrence computations of node-time betweenness contributions. The authors also offer an open-source C++ implementation and demonstrate, on real datasets, that temporal rankings align closely with static aggregations while revealing distinct time-centered insights for the active variant; notably, for $sh$-$fm$, the top nodes are well-predicted using the first $10\%$ of interactions. These results yield practical benefits for understanding time-resolved network influence and provide a foundation for scalable temporal-centrality analyses, with open questions remaining for active shortest-foremost paths and extensions to continuous-time formalisms.

Abstract

Betweenness centrality has been extensively studied since its introduction in 1977 as a measure of node importance in graphs. This measure has found use in various applications and has been extended to temporal graphs with time-labeled edges. Recent research by Buss et al. and Rymar et al. has shown that it is possible to compute the shortest path betweenness centrality of all nodes in a temporal graph in $O(n^3\,T^2)$ and $O(n^2\,m\,T^2)$ time, respectively, where $T$ is the maximum time, $m$ is the number of temporal edges, and $n$ is the number of nodes. These approaches considered paths that do not take into account contributions from intermediate temporal nodes. In this paper, we study the classical temporal betweenness centrality paths that we call \textit{passive} shortest paths, as well as an alternative variant that we call \textit{active} shortest paths, which takes into account contributions from all temporal nodes. We present an improved analysis of the running time of the classical algorithm for computing betweenness centrality of all nodes, reducing the time complexity to $O(n\,m\,T+ n^2\,T)$. Furthermore, for active paths, we show that the betweenness centrality can be computed in $O(n\,m\,T+ n^2\,T^2)$. We also show that our results hold for different shortest paths variants. Finally, we provide an open-source implementation of our algorithms and conduct experiments on several real-world datasets. We compare the results of the two variants on both the node and time dimensions of the temporal graph, and we also compare the temporal betweenness centrality to its static counterpart. Our experiments suggest that for the shortest foremost variant looking only at the first $10\%$ of the temporal interaction is a very good approximation for the overall top ranked nodes.

Figures (5)

• Figure 1: A temporal graph having nodes $V = \{a,b,c,d\}$ and $T = 7$ with arrows representing the set $\mathcal{E}$. The walk $W = [(a,b,1), (b,c,5)]$ can be denoted as $W = a \overset{1}{\rightarrow} b \overset{5}{\rightarrow} c$. (left) Walks are passive then $\mathcal{V}(W) = [(a,1),(b,1),(c,5) ]$ marked in red. (right) Walks are active then $\mathcal{V}(W) = [ (a,1),(b,1),(b,2),(b,3),(b,4),(b,5),(c,5)]$ marked in blue.
• Figure 2: The predecessor graphs of shortest paths from node $a$ on the temporal graph of Figure \ref{['fig:ex']}. (left) the walks are considered active and (right) the walks are considered passive.
• Figure 3: Distribution of time centrality values $B(t)$ for the datasets. Each column represents a dataset. (1st row) correspond to the distribution of $B(t)$ for passive shortest, (2nd row) active shortest, (3rd row) passive shortest restless, (4th row) active shortest restless and (5rd row) passive shortest foremost. The x-axis represents the renormalized life time of the temporal graph and the y-axis represents the values of $B(t)$ grouped into $20$ bars.
• Figure 4: Heatmap of betweenness centrality of $B(v)$ comparisons of the datasets. (left) Kendall-tau rank correlation rankings and (right) intersection rate of the top $20$ nodes. In the figure act stands for active variant, pas for passive variant, sh-rl_act active shortest restless, sh-rl_pas passive shortest restless and stat for the static betweenness centrality on the aggregated graph.
• Figure 5: Each column corresponds to a dataset. (1st row) correspond to the distribution of $B(t)$ for passive shortest, (2nd row) active shortest, (3rd row) passive shortest restless, (4th row) active shortest restless and (5rd row) passive shortest foremost. Each graph has on its x-axis the $\mu$ values and on its y-axis the size of the intersection between top $10$ ranked nodes of nodes $B_G(v)$ of the temporal graph $G$ and top $10$ ranked nodes of the $B_{G^{\leq \mu}}(v)$.

Theorems & Definitions (38)

• Definition 1: Temporal walk
• Definition 2: Visited temporal nodes
• Example 1: Motivation example for the study of active walks
• Remark 1
• Definition 3: Set of shortest walks
• Remark 2
• Definition 4
• Definition 5
• Definition 6: Betweenness centrality of a temporal node
• Definition 7: Overall betweenness of a node and overall betweenness of a time