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

Huanshen Jia, Jianguo Qian

TL;DR

This work analyzes the edge-fault tolerance of regular graphs under random edge failures, defining EF tolerance $t_e(G,p)$ and MEF tolerance $t_e^M(G,p)$. It derives an upper bound $t_e(G,p) \le (1-p^d)^{i(G)}$ in terms of the independence number $i(G)$ and explores asymptotic behavior for large graphs and varying edge failure probability $p$. Through simulations on Hypercubes, Möbius Cubes, Ary Cubes, Circulant graphs, and random regular graphs, the study finds Möbius Cubes often maximize both tolerances, with MEF tolerance showing stronger sensitivity to graph structure than EF tolerance. The results suggest that, while some structured graphs offer higher fault tolerance, the EF tolerance is relatively robust to topology, and they raise open questions about phase-transition thresholds for random regular graphs and the role of symmetry in achieving maximal 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, Möbius Cubes, Ary-Cubes and Circulant graphs have also higher EF and MEF tolerance in general. In particular, the Möbius 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.

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

TL;DR

This work analyzes the edge-fault tolerance of regular graphs under random edge failures, defining EF tolerance and MEF tolerance . It derives an upper bound in terms of the independence number and explores asymptotic behavior for large graphs and varying edge failure probability . Through simulations on Hypercubes, Möbius Cubes, Ary Cubes, Circulant graphs, and random regular graphs, the study finds Möbius Cubes often maximize both tolerances, with MEF tolerance showing stronger sensitivity to graph structure than EF tolerance. The results suggest that, while some structured graphs offer higher fault tolerance, the EF tolerance is relatively robust to topology, and they raise open questions about phase-transition thresholds for random regular graphs and the role of symmetry in achieving maximal 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, Möbius Cubes, Ary-Cubes and Circulant graphs have also higher EF and MEF tolerance in general. In particular, the Möbius 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.
Paper Structure (5 sections, 9 theorems, 8 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 5 sections, 9 theorems, 8 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An upper bound on EF tolerance of regular graphs
  4. Simulation and numerical results
  5. Conclusion

Key Result

Theorem 2.1

( Max-Flow Min-Cut TheoremMenger) For any two vertices $u$ and $v$ in a graph $G$, the minimum size of an $(u, v)$-edge cut in $G$ equals the maximum number of edge-disjoint $(u, v)$-paths.

Figures (3)

  • Figure 1: A randomly generated 4-regular graph with 16 vertices.
  • Figure 2: The average probabilities ${\rm p}_f(A)$ (solid lines) and ${\rm p}^M_f(A)$ (dotted lines) with $f=0,1,\ldots,18$.
  • Figure 3: The values of $t_e(G,p)$ (solid lines) and $t_e^M(G,p)$ (dotted lines) of some graphs with $p=0,0.1,\ldots,0.8$.

Theorems & Definitions (15)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Definition 2.7
  • Lemma 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 5 more