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 $O(|E||V|^2\Delta(G))$, 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.

翻译：节点的度中心性定义为与该节点相邻的节点数量，常被用作衡量节点对网络结构重要性的指标。该度量可扩展至网络中的路径，其中路径的度中心性定义为与该路径相邻的节点数量。本文重新考虑了在无权重网络中寻找最度中心最短路径的问题。我们提出了一种多项式算法，其最坏情况运行时间为$O(|E||V|^2\Delta(G))$，其中$|V|$为网络中的顶点数，$|E|$为网络中的边数，$\Delta(G)$为图的最大度数。我们在合成网络和真实网络上对该算法进行了数值研究，并将结果与现有文献进行了比较。此外，我们证明当考虑加权图时，相同问题是NP难的。