Foundations of Data Science

ε of the steady state distribution.
In these examples, we have chosen simple probability distributions. The methods extend
to more complex situations.
5.5 Electrical Networks and Random Walks
In the next few sections, we study the relationship between electrical networks and
random walks on undirected graphs. The graphs have nonnegative weights on each edge.
A step is executed by picking a random edge from the current vertex with probability
proportional to the edge’s weight and traversing the edge.
An electrical network is a connected, undirected graph in which each edge (x, y) has
a resistance r xy > 0. In what follows, it is easier to deal with conductance defined as the
reciprocal of resistance, c xy = 1
r xy
, rather than resistance. Associated with an electrical
network is a random walk on the underlying graph defined by assigning a probability
p xy = cxy
c x
to the edge (x, y) incident to the vertex x, where the normalizing constant c x
equals ∑ c xy . Note that although c xy equals c yx , the probabilities p xy and p yx may not be
y
equal due to the normalization required to make the probabilities at each vertex sum to
one. We shall soon see that there is a relationship between current flowing in an electrical
network and a random walk on the underlying graph.
Since we assume that the undirected graph is connected, by Theorem 5.2 there is
a unique stationary probability distribution.The stationary probability distribution is π
where π x = cx
c 0
where c 0 = ∑ c x . To see this, for all x and y
x
π x p xy = c x
c 0
c xy
c x
= c xy
c 0
= c y
c 0
c yx
c y
= π y p yx
and hence by Lemma 5.3, π is the unique stationary probability.
Harmonic functions
Harmonic functions are useful in developing the relationship between electrical networks
and random walks on undirected graphs. Given an undirected graph, designate
a nonempty set of vertices as boundary vertices and the remaining vertices as interior
vertices. A harmonic function g on the vertices is one in which the value of the function
at the boundary vertices is fixed to some boundary condition and the value of g at any
interior vertex x is a weighted average of the values at all the adjacent vertices y, with
weights p xy satisfying ∑ y p xy = 1 for each x. Thus, if at every interior vertex x for some
set of weights p xy satisfying ∑ y p xy = 1, g x = ∑ g y p xy , then g is an harmonic function.
y
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