In this section the endpoints of edges of a considered graph can have an order, making it a **directed graph**. A graph with positive function/vector defined on its edges is called **weighted graph**.

**Random walk** of a particle on a graph G with discrete time is the following process:

- At moment
*t = 0*the particle occupies a vertex*v*of*G*. - At moment
*t = n+1*the particle moves to a neighbor of its position at the moment*t = n*w/probability proportional to the weight of the edge connecting/pointing to the neighbor.

Choosing a subset of vertices of a graph as **boundary**, the **harmonic measure** of a subset *S* of the boundary is the function/vector on vertices of *G* that equals the probability that a particle, starting its random walk at a vertex *p*, occupies a boundary vertex in the set *S* before a boundary vertex not in *S*.

It follows from the definition that the harmonic measure at *p* of a single boundary node *b* equals to the sum over the finite paths through interior from *p* to *b*:

or

Note, that an edge or a vertex may appear multiple times in a path.

The **Brownian motion** is a continuous/limiting analog of the random walk. It follows from the averaging property of the operator that the hitting probabilities of a particle under Brownian motion are described by harmonic functions, defined in the previous section. The harmonic functions are conformaly invariant.