The Resistance Distance is Meaningless for Large Random Geometric Graphs

We study convergence properties of the resistance distance on random geometric graphs for increasing sample size. It turns out that the suitably scaled resistance distance between two fixed points converges to a non-trivial limit. However, this limit no longer takes into account global properties of the graph, as for example the cluster structure. Quite to the opposite, the limit distance function is rather meaningless. As our simulations show, this phenomenon can already be observed for rather small sample sizes. Thus, we discourage the usage of the resistance distance for learning purposes on the type of graph we have analyzed so far.