harmonic function

A real or complex-valued function $f:V\to ℝ$ or $f:V\to ℂ$ defined on the vertices $V$ of a graph $G=(V,E)$ is called harmonic at $v\in V$ if its value at $v$ is its average value at the neighbours of $v$:

$$f(v)=\frac{1}{\mathrm{deg}(v)}\sum _{\{u,v\}\in E}f(u).$$

It is called harmonic except on $A$, for some $A\subseteq V$, if it is harmonic at each $v\in V\setminus A$, and harmonic if it is harmonic at each $v\in V$.

Any harmonic $f:{\mathbb{Z}}^n\to ℝ$, where ${\mathbb{Z}}^n$ is the $n$-dimensional grid, is if below (or above). However, this is not necessarily true on other graphs.