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.

翻译：节点的度中心性定义为与其相邻的节点数量，可用于衡量节点对网络结构的重要性。该度量可推广至网络中的路径，其中路径的度中心性定义为与其相邻的节点数量。本文重新考虑了在无权网络中寻找最多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$O(|V|^3(\Delta(G))^2)$，其中$|V|$为网络中的顶点数，$\Delta(G)$为图的最大度。我们基于合成网络与现实世界网络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，我们证明了在加权图情形下该问题为NP难问题。