Probability model of edge-fault tolerance for regular graphs with respect to edge connectivity

TL;DR

This paper models the probability of edge-fault tolerance in regular graphs, deriving bounds and analyzing how different graph structures influence network robustness under link failures.

Contribution

It introduces a probabilistic framework for edge-fault tolerance, deriving an upper bound for regular graphs and comparing structural effects on network resilience.

Findings

Hypercubes, M"{o}bius Cubes, Ary-Cubes, and Circulant graphs have higher EF and MEF tolerance.

M"{o}bius Cube exhibits the highest EF and MEF tolerance among tested graphs.

EF tolerance is less affected by graph structure compared to MEF tolerance.

Abstract

We consider the probability model of edge-fault tolerance of a network in the sense of connectivity with link faults. Using graph-theoretical notation, we define the edge-fault (EF) and Menger-type edge-fault (MEF) tolerances of a graph as the probabilities that the graph is connected and strongly Menger edge-connected when each edge has a certain failure probability, respectively. We derive an upper bound on the EF tolerance for regular graphs, which reveals an asymptotical behavior when graphs and edge failure probability are large enough. We also perform a simulation experiment on a number of randomly generated regular graphs and some typically well-used graphs. The numerical results show that, in addition to their well-structured properties for networks, Hypercubes, M\"{o}bius Cubes, Ary-Cubes and Circulant graphs have also higher EF and MEF tolerance in general. In particular, the M\"{o}bius Cube has both the highest EF and MEF tolerance among all involved graphs. The numerical results also hint that, in contrast to MEF tolerance, the EF tolerance of regular graphs is not strongly effected by the graph structure.