A polynomial algorithm for the most degree-central shortest path problem

The degree centrality of a node, defined to be the number of nodes adjacent to it, can be 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 to be 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 $O(|V|^3(\Delta(G))^2)$, where $|V|$ is the number of vertices 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.