Let a network be represented by a simple graph $\mathcal{G}$ with $n$ vertices. A common approach to investigate properties of a network is to use the adjacency matrix $A=[a_{ij}]_{i,j=1}^n\in\R^{n\times n}$ associated with the graph $\mathcal{G}$, where $a_{ij}>0$ if there is an edge pointing from vertex $v_i$ to vertex $v_j$, and $a_{ij}=0$ otherwise. Both $A$ and its positive integer powers reveal important properties of the graph. This paper proposes to study properties of a graph $\mathcal{G}$ by also using the path length matrix for the graph. The $(ij)^{th}$ entry of the path length matrix is the length of the shortest path from vertex $v_i$ to vertex $v_j$; if there is no path between these vertices, then the value of the entry is $\infty$. Powers of the path length matrix are formed by using min-plus matrix multiplication and are important for exhibiting properties of $\mathcal{G}$. We show how several known measures of communication such as closeness centrality, harmonic centrality, and eccentricity are related to the path length matrix, and we introduce new measures of communication, such as the harmonic $K$-centrality and global $K$-efficiency, where only (short) paths made up of at most $K$ edges are taken into account. The sensitivity of the global $K$-efficiency to changes of the entries of the adjacency matrix also is considered.

翻译：设一个网络由具有$n$个顶点的简单图$\mathcal{G}$表示。研究网络属性的常用方法是使用与图$\mathcal{G}$关联的邻接矩阵$A=[a_{ij}]_{i,j=1}^n\in\R^{n\times n}$，其中若存在从顶点$v_i$指向顶点$v_j$的边，则$a_{ij}>0$，否则$a_{ij}=0$。矩阵$A$及其正整数幂揭示了图的重要属性。本文提出同时利用图的路径长度矩阵来研究图$\mathcal{G}$的属性。路径长度矩阵的第$(ij)$个元素是从顶点$v_i$到顶点$v_j$的最短路径长度；若这些顶点之间不存在路径，则该元素值为$\infty$。路径长度矩阵的幂通过最小-加法矩阵乘法形成，对于展示$\mathcal{G}$的属性至关重要。我们证明了若干已知通信度量（如紧密度中心性、调和中心性和离心率）与路径长度矩阵的关系，并引入了新的通信度量，如调和$K$-中心性和全局$K$-效率，其中仅考虑由至多$K$条边构成的（短）路径。还考虑了全局$K$-效率对邻接矩阵元素变化的敏感性。