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.



References studying properties of a graph which are stable under finite perturbation
The references for these can be found in Doyle and Snell's deathless classic (which is [available for free on arXiv.org](https://arxiv.org/abs/math/0001057). ). Section 2.4 is particularly *a propos.*


*Doyle, Peter G.; Snell, J.Laurie*, Random walks and electric networks, The Carus Mathematical Monographs, 22. Washington, D. C.: The Mathematical Association of America. Distr. by John Wiley & Sons, New York etc. XIII, 159 p. £ 22.00 (1984). [ZBL0583.60065](https://zbmath.org/?q=an:0583.60065).


