Foundations of Data Science

and then reaching a from y before reaching b. That is
p x = ∑ y
p xy p y .
Hence, p x is the same harmonic function as the voltage function v x and v and p satisfy the
same boundary conditions at a and b.. Thus, they are identical functions. The probability
of a walk starting at x reaching a before reaching b is the voltage v x .
Probabilistic interpretation of current
In a moment, we will set the current into the network at a to have a value which we will
equate with one random walk. We will then show that the current i xy is the net frequency
with which a random walk from a to b goes through the edge xy before reaching b. Let
u x be the expected number of visits to vertex x on a walk from a to b before reaching b.
Clearly u b = 0. Every time the walk visits x, x not equal to a, it must come to x from
some vertex y. Thus, the number of visits to x before reaching b is the sum over all y of
the number of visits u y to y before reaching b times the probability p yx of going from y
to x. For x not equal to b or a
u x = ∑ y≠b
u y p yx .
Since u b = 0 and c x p xy = c y p yx
u x = ∑ all y
u y
c x p xy
c y
and hence ux
c x
= ∑ u y
c y
p xy . It follows that ux
c x
is harmonic with a and b as the boundary
y
where the boundary conditions are u b = 0 and u a equals some fixed value. Now, u b
c b
= 0.
Setting the current into a to one, fixed the value of v a . Adjust the current into a so that
v a equals ua
c a
. Now ux
c x
and v x satisfy the same boundary conditions and thus are the same
harmonic function. Let the current into a correspond to one walk. Note that if the walk
starts at a and ends at b, the expected value of the difference between the number of times
the walk leaves a and enters a must be one. This implies that the amount of current into
a corresponds to one walk.
Next we need to show that the current i xy is the net frequency with which a random
walk traverses edge xy.
(
ux
i xy = (v x − v y )c xy = − u )
y c xy c xy
c xy = u x − u y = u x p xy − u y p yx
c x c y c x c y
The quantity u x p xy is the expected number of times the edge xy is traversed from x to y
and the quantity u y p yx is the expected number of times the edge xy is traversed from y to
x. Thus, the current i xy is the expected net number of traversals of the edge xy from x to y.
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