A concentration phenomenon for $h$-extra edge-connectivity reliability analysis of enhanced hypercubes $Q_{n,2}$ with exponentially many faulty links

TL;DR

This work investigates the refinement of network reliability via the $h$-extra edge-connectivity $\lambda_h$ for the $(n,2)$-enhanced hypercube $Q_{n,2}$. By leveraging the auxiliary functions $\xi_m(G)$ and $ex_m(G)$ and analyzing the $m$-subset structure of $Q_{n,2}$, the authors derive explicit expressions and identify critical sequences $m_{n,r}$ that yield plateaus at $\lambda_h(Q_{n,2})=2^{n-1}$. The core result shows a concentration phenomenon: for $n\ge 9$ and $h$ in $\left\lceil\frac{11\times2^{n-1}}{48}\right\rceil \le h \le 2^{n-1}$, the $h$-extra edge-connectivity equals $2^{n-1}$, with $\,\lambda_h$ also equal to $\xi_h$ at certain boundary values; the bounds are tight and the asymptotic proportion of $h$ with concentration tends to $37/48$ (about 77.083%). These findings enhance the reliability interpretation for large-scale $Q_{n,2}$ networks and pave the way for similar analyses in the more general $Q_{n,k}$ families.

Abstract

Reliability assessment of interconnection networks is critical to the design and maintenance of multiprocessor systems. The $(n, k)$-enhanced hypercube $Q_{n,k}$, as a variation of the hypercube $Q_{n}$, was proposed by Tzeng and Wei in 1991. As an extension of traditional edge-connectivity, $h$-extra edge-connectivity of a connected graph $G,$ $λ_h(G),$ is an essential parameter for evaluating the reliability of interconnection networks. This article intends to study the $h$-extra edge-connectivity of the $(n,2)$-enhanced hypercube $Q_{n,2}$. Suppose that the link malfunction of an interconnection network $Q_{n,2}$ does not isolate any subnetwork with no more than $h-1$ processors, the minimum number of these possible faulty links concentrates on a constant $2^{n-1}$ for each integer $\lceil\frac{11\times2^{n-1}}{48}\rceil \leq h \leq 2^{n-1}$ and $n\geq 9$. That is, for about $77.083\%$ of values where $h\leq2^{n-1},$ the corresponding $h$-extra edge-connectivity of $Q_{n,2}$, $λ_h(Q_{n,2})$, presents a concentration phenomenon. Moreover, the lower and upper bounds of $h$ mentioned above are both tight.

Figures (6)

• Figure 1: The values of $\xi_h(Q_{9,2})$ and $\lambda_h(Q_{9,2})$.
• Figure 2: $Q_{3,1}$ (i.e. $FQ_3$), $Q_{3,2}$ and $Q_{4,2}$.
• Figure 3: The bitmaps of adjacency matrix of $Q_{n,2}$ for $4 \le n \le 7$.
• Figure 4: Induced subgraphs $Q_{4,2}[L_{4}^{4}]$, $Q_{4,2}[L_{6}^{4}]$ and $Q_{4,2}[L_{8}^{4}]$.
• Figure 5: The comparison of the sizes of $h$-extra edge-cuts in $Q_{5,2}$ between the simulation and our results.

Theorems & Definitions (12)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 3.1
• Lemma 3.2