Expected distance based on random walks

By considering a graph as a network of resistances, Klein and Randić (J Math Chem 12(1):81–95, 1993) proposed the definition of a distance measure. Indeed, if each edge of the graph represents a resistance of 1Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \varOmega $$\end{document}, the equivalent resistance of the graph between each pair of vertices may be used as a distance. Based upon random walks in graphs, Stephenson and Zelen (Soc Netw 11(1):1–37, 1989) built a computational model to find the probability that each edge is used. From a mathematical point of view, both articles are based upon exactly the same model and the link between random walks and the electrical representation was established by Newman (Soc Netw 27(1):39–54, 2005) when defining an alternative to Freeman’s (Sociometry 40:35–41, 1977, Soc Netw 1(3):215–239, 1979) betweenness centrality based upon random walks. In the present paper, the similitude between these two processes is exploited to propose a new random walks based distance measure that may be defined as the expected length of a walk between any pair of vertices. We call it the expected distance and we prove that it is actually a distance. From this new definition, the RW Index is proposed that sums the expected walks lengths between pairs of vertices exactly in the same way as the Wiener index sums the shortest paths distances or the Kirchhoff index sums the equivalent resistances. We compare the three indices and establish the vertex and the edge decompositions for both. We compute some value of the RW index for some families of graphs and conjecture the upper and lower bounds of the RW index.