# References studying properties of a graph which are stable under finite perturbation

Let's say two locally finite, connected, undirected, infinite graphs are "finite perturbations" of each other if one can remove a finite subset from each and obtain isomorphic graphs (which are now possibly disconnected).

My question is: Is there any literature on properties which are stable under this type of perturbation?

For example, does anyone know of a reference for the following statements:

1. If a graph admits a non-constant bounded harmonic function then so does every finite perturbation of it.
2. If the simple random walk on a graph is recurrent then so is the simple random walk on any finite perturbation.
3. If the simple random walk on a graph has positive speed with positive probability, then the same holds for every finite perturbation of it.
4. If the simple random walk on a graph has positive speed almost surely, then the same holds for every finite perturbation of it.