Foundations of Data Science

Effective resistance and escape probability
Set v a = 1 and v b = 0. Let i a be the current flowing into the network at vertex a and
out at vertex b. Define the effective resistance r eff between a and b to be r eff = va
i a
and
the effective conductance c eff to be c eff = 1
r eff
. Define the escape probability, p escape , to
be the probability that a random walk starting at a reaches b before returning to a. We
now show that the escape probability is c eff
c a
. For convenience, assume that a and b are
not adjacent. A slight modification of the argument suffices for the case when a and b are
adjacent.
i a = ∑ (v a − v y )c ay
y
Since v a = 1,
i a = ∑ ∑ c ay
c ay − c a v y
c
y
y a
[
= c a 1 − ∑ ]
p ay v y .
y
For each y adjacent to the vertex a, p ay is the probability of the walk going from vertex
a to vertex y. Earlier we showed that v y is the probability of a walk starting at y going
to a before reaching b. Thus, ∑ p ay v y is the probability of a walk starting at a returning
y
to a before reaching b and 1 − ∑ y
p ay v y is the probability of a walk starting at a reaching
b before returning to a. Thus, i a = c a p escape . Since v a = 1 and c eff = ia
v a
, it follows that
c eff = i a . Thus, c eff = c a p escape and hence p escape = c eff
c a
.
For a finite connected graph, the escape probability will always be nonzero. Now
consider an infinite graph such as a lattice and a random walk starting at some vertex
a. Form a series of finite graphs by merging all vertices at distance d or greater from a
into a single vertex b for larger and larger values of d. The limit of p escape as d goes to
infinity is the probability that the random walk will never return to a. If p escape → 0, then
eventually any random walk will return to a. If p escape → q where q > 0, then a fraction
of the walks never return. Thus, the escape probability terminology.
5.6 Random Walks on Undirected Graphs with Unit Edge Weights
We now focus our discussion on random walks on undirected graphs with uniform
edge weights. At each vertex, the random walk is equally likely to take any edge. This
corresponds to an electrical network in which all edge resistances are one. Assume the
graph is connected. We consider questions such as what is the expected time for a random
walk starting at a vertex x to reach a target vertex y, what is the expected time until the
random walk returns to the vertex it started at, and what is the expected time to reach
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