Foundations of Data Science

There are important connections between electrical networks and random walks on
undirected graphs. Choose two vertices a and b. For reference purposes let the voltage
v b equal zero. Attach a current source between a and b so that the voltage v a equals
one. Fixing the voltages at v a and v b induces voltages at all other vertices along with a
current flow through the edges of the network. The analogy between electrical networks
and random walks is the following. Having fixed the voltages at the vertices a and b, the
voltage at an arbitrary vertex x equals the probability of a random walk starting at x
reaching a before reaching b. If the voltage v a is adjusted so that the current flowing into
vertex a corresponds to one walk, then the current flowing through an edge is the net
frequency with which a random walk from a to b traverses the edge.
Probabilistic interpretation of voltages
Before showing that the voltage at an arbitrary vertex x equals the probability of a
random walk starting at x reaching a before reaching b, we first show that the voltages
form a harmonic function. Let x and y be adjacent vertices and let i xy be the current
flowing through the edge from x to y. By Ohm’s law,
i xy = v x − v y
r xy
= (v x − v y )c xy .
By Kirchhoff’s law the currents flowing out of each vertex sum to zero.
∑
i xy = 0
y
Replacing currents in the above sum by the voltage difference times the conductance
yields
∑
(v x − v y )c xy = 0
or
y
∑
v x c xy = ∑
y
y
v y c xy .
Observing that ∑ c xy = c x and that p xy = cxy
c x
, yields v x c x = ∑ v y p xy c x . Hence,
y
y
v x = ∑ v y p xy . Thus, the voltage at each vertex x is a weighted average of the voltages
at the adjacent vertices. Hence the voltages form a harmonic function with {a, b} as
y
the boundary.
Let p x be the probability that a random walk starting at vertex x reaches a before b.
Clearly p a = 1 and p b = 0. Since v a = 1 and v b = 0, it follows that p a = v a and p b = v b .
Furthermore, the probability of the walk reaching a from x before reaching b is the sum
over all y adjacent to x of the probability of the walk going from x to y in the first step
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