Expected distance based on random walks
Abstract
By considering a graph as a network of resistances, Klein and
Randi c [14] proposed the definition of a distance measure. Indeed, if each edge
of the graph represents a resistance of 1Ω, 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 [17] 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 [16] when defining an alternative to Freeman’s betweenness cen-
trality [9, 10] based upon random walks.
In the present paper, the similitude between these two processes is ex-
ploited 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.
Domains
Operations Research [cs.RO]
Origin : Files produced by the author(s)