  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.
