The giant component threshold for random regular graphs with edge faults

Abstract

Let G be a given graph (modelling a communication network) which we assume suffers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 - p). In particular we are interested in robustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our result is: If the degree d is fixed then p = 1/(d - 1) is a threshold probability for the existence of a linear-sized component in a faulty version of almost all random regular graphs. We show: If each edge of an arbitrary graph G with maximum degree bounded above by d is present with, probability p = Λ/(d - 1) where Λ < 1 is fixed then the faulted version of G has only components whose size is at most logarithmic in the number of nodes with high probability. If on the other hand G is a random regular graph with degree d and p = Λ/(d - 1) where Λ > 1 then for almost all G the faulted version of G has a linear-sized component with high probability. Note that these results imply some kind of optimality of random regular graphs among the class of graphs with the same degree bound. The theme is: Use the known expansion properties of almost all random regular graphs to obtain strong robustness results. This has not been done systematically before.