Centrality of shortest paths: Algorithms and complexity results

TL;DR

The paper investigates the problem of maximizing various centrality measures over shortest paths in graphs. It introduces a polynomial-time BFS/Dijkstra-inspired algorithm for the most degree-central shortest path in unweighted graphs, and proves NP-hardness for the weighted degree case, while also addressing betweenness and closeness centralities. Betweenness-central shortest paths are shown to be solvable in polynomial time, whereas closeness-central shortest paths are NP-hard in both weighted and unweighted graphs. Computational experiments demonstrate substantial speedups over previous approaches, with practical insights on when to prefer degree versus betweenness centrality in network design and defense scenarios.

Abstract

The degree centrality of a node, defined as the number of nodes adjacent to it, is often used as a measure of importance of a node to the structure of a network. This metric can be extended to paths in a network, where the degree centrality of a path is defined as the number of nodes adjacent to it. In this paper, we reconsider the problem of finding the most degree-central shortest path in an unweighted network. We propose a polynomial algorithm with the worst-case running time of $O(|E||V|^2Δ(G))$, where $|V|$ is the number of vertices in the network, $|E|$ is the number of edges in the network, and $Δ(G)$ is the maximum degree of the graph. We conduct a numerical study of our algorithm on synthetic and real-world networks and compare our results to the existing literature. In addition, we show that the same problem is NP-hard when a weighted graph is considered. Furthermore, we consider other centrality measures, such as the betweenness and closeness centrality, showing that the problem of finding the most betweenness-central shortest path is solvable in polynomial time and finding the most closeness-central shortest path is NP-hard, regardless of whether the graph is weighted or not.

Figures (4)

• Figure 1: Illustration of \ref{['lem:most-central-path']}. The path $\langle s, p_1, \ldots, p_{k - 1} \rangle$ must be a most degree-central shortest path between $s$ and $p_{k - 1}$ if the path $\langle s, p_1, \ldots, p_{k}, t\rangle$ is a most degree-central shortest path between $s$ and $t$.
• Figure 2: Example of a weighted graph $G$ used in proof of \ref{['thm:np-complete']}. The blue circle vertices and edges indicate the initial unweighted graph $H$, and the red square vertices and dashed edges indicate the additional vertices and edges with edge weight $|U|$.
• Figure 3: Augmenting a weighted graph.
• Figure 4: Illustration of graph used in the proof of \ref{['lem:most-betweenness-shortest-path']}.

Theorems & Definitions (27)

• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof