Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 1382980, 8 pages

https://doi.org/10.1155/2017/1382980

## The Minimum Spectral Radius of an Edge-Removed Network: A Hypercube Perspective

^{1}School of Software Engineering, Chongqing University, Chongqing 400044, China^{2}School of Mechanical Engineering, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Yingbo Wu; nc.ude.uqc@byw

Received 15 December 2016; Revised 17 February 2017; Accepted 28 March 2017; Published 19 April 2017

Academic Editor: Yong Deng

Copyright © 2017 Yingbo Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The spectral radius minimization problem (SRMP), which aims to minimize the spectral radius of a network by deleting a given number of edges, turns out to be crucial to containing the prevalence of an undesirable object on the network. As the SRMP is NP-hard, it is very unlikely that there is a polynomial-time algorithm for it. As a result, it is proper to focus on the development of effective and efficient heuristic algorithms for the SRMP. For that purpose, it is appropriate to gain insight into the pattern of an optimal solution to the SRMP by means of checking some regular networks. Hypercubes are a celebrated class of regular networks. This paper empirically studies the SRMP for hypercubes with two/three/four missing edges. First, for each of the three subproblems of the SRMP, a candidate for the optimal solution is presented. Second, it is shown that the candidate is optimal for small-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes. The edges in each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and hence is instructive in designing effective heuristic algorithms for the SRMP.

#### 1. Introduction

The epidemic modeling is recognized as an effective approach to the understanding of propagation process of objects over a network [1, 2]. For instance, epidemic models help us understand the key factors that affect the prevalence of malware [3–8]. The speed and extent of spread of an epidemic on a network depend largely on the structure of the network; whether the epidemic goes viral depends on whether the spectral radius of the network exceeds a threshold [9–14]. Therefore, reducing the spectral radius of a network by removing a set of edges is an effective approach to the containment of the prevalence of an undesirable epidemic on the network. The spectral radius minimization problem (SRMP) aims to remove a given number of edges of a network so that the spectral radius of the resulting network attains the minimum. As the SRMP is NP-hard [15], it is much unlikely that there be a polynomial-time algorithm for it. As thus, a number of heuristic algorithms for the SRMP have been proposed [15–19]. In most situations, these heuristics are ineffective, because they produce nonoptimal solutions rather than optimal solutions. For the purpose of developing effective heuristic algorithms for the SRMP, it is appropriate to gain insight into the pattern of an optimal solution to the SRMP by means of checking some regular networks. Recently, Yang et al. [20] studied the SRMP for 2D tori.

Hypercubes are a class of regular networks [21]. Due to remarkable advantages in communication [22–25], fault tolerant communication [26–30], fault diagnosis [31–34], and parallel computation [35, 36], hypercubes have been widely adopted as the underlying interconnection network in multicomputer systems [37]. To our knowledge, the SRMP for hypercubes is still unsolved.

This paper addresses three subproblems of the SRMP, where two/three/four edges are removed from a hypercube, respectively. First, for each of the three subproblems of the SRMP, a candidate optimal solution is presented. Second, it is shown that the candidate is optimal for small-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes. The edges in each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and hence is instructive in designing effective heuristic algorithms for the SRMP.

The remaining materials are organized in this fashion: the preliminary knowledge is given in Section 2. Section 3 presents the main results of this work. Finally, Section 4 summarizes this work.

#### 2. Preliminaries

For fundamental knowledge on the spectral radius of a network, see [38, 39]. The SRMP is formulated as follows: given a network and a positive integer , find a set of edges of so that the surviving network obtained by removing the set of edges from the network achieves the minimum spectral radius.

An -dimensional cube (-D cube, for short), denoted by , is a network , where there is a one-to-one correspondance from to the set of all 0-1 binary strings of length so that node is adjacent to node if and only if differs from in exactly one bit position. In what follows, it is always assumed that the nodes of a hypercube have been labelled with 0-1 strings in this way. See Figure 1 for three small-sized hypercubes.