I am interested which "relevant" infinite graphs G=(V,E) (or in my use cases called networks) are out there that are spherical symmetric,
i.e. for a fixed origin 0 and each pair $v,w \in V$ with the same distance n from the origin (=length of smallest possible path) there is an automorphism $\phi:V \to V$ on G that fixes the origin and $\phi(v) = w$.
So far I would say that d-regular trees fullfil this definition and a modified version of $Z^n$ where one adds, starting from the origin, straight lines in each direction in $Z^n$.
My motivation for this comes from the following theorem about the transience of such graphs.
Let $\partial V_n = \{v \in V: d(0,v)=n\}$ be all vertices with distance n from 0 and $E_n$ all edges between $V_n$ and $V_{n+1}$. Then a Markov chain on G is transient iff. $\sum_n \frac{1}{E_n} < \infty$.