🤖 AI Summary
This paper investigates the problem of identifying the “most central shortest path.” Theoretically, it establishes the first complete complexity landscape: shortest paths maximizing degree centrality are polynomial-time solvable in unweighted graphs, whereas degree-centrality maximization in weighted graphs—and closeness-centrality maximization in all graphs—are NP-hard; betweenness-centrality maximization remains polynomial-time solvable. Methodologically, the paper proposes the first algorithm for unweighted graphs with time complexity $O(|E||V|^2Delta(G))$, integrating breadth-first search, adjacency-structure traversal, and rigorous graph-theoretic analysis. Extensive experiments validate its efficiency on both synthetic and real-world networks. The main contributions are: (i) precisely delineating the computational boundaries of central shortest path problems; (ii) resolving the open question of polynomial-time solvability for degree-centrality maximization; and (iii) systematically characterizing complexity dichotomies across multiple centrality measures.
📝 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 $O(|E||V|^2Delta(G))$, where $|V|$ is the number of vertices in the network, $|E|$ is the number of edges in the network, and $Delta(G)$ 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.