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Research ArticleThe Minimum Spectral Radius of an Edge-Removed Network:A Hypercube Perspective
YingboWu,1 Tianrui Zhang,1 Shan Chen,2 and Tianhui Wang1
1School of Software Engineering, Chongqing University, Chongqing 400044, China2School of Mechanical Engineering, Chongqing University, Chongqing 400044, China
Correspondence should be addressed to Yingbo Wu; wyb@cqu.edu.cn
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.
The spectral radius minimization problem (SRMP), which aims to minimize the spectral radius of a network by deleting a givennumber of edges, turns out to be crucial to containing the prevalence of an undesirable object on the network. As the SRMP isNP-hard, it is very unlikely that there is a polynomial-time algorithm for it. As a result, it is proper to focus on the developmentof effective and efficient heuristic algorithms for the SRMP. For that purpose, it is appropriate to gain insight into the patternof an optimal solution to the SRMP by means of checking some regular networks. Hypercubes are a celebrated class of regularnetworks. This paper empirically studies the SRMP for hypercubes with two/three/four missing edges. First, for each of the threesubproblems of the SRMP, a candidate for the optimal solution is presented. Second, it is shown that the candidate is optimal forsmall-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes.The edgesin each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and henceis instructive in designing effective heuristic algorithms for the SRMP.
1. Introduction
Theepidemicmodeling is recognized as an effective approachto the understanding of propagation process of objects overa network [1, 2]. For instance, epidemic models help usunderstand the key factors that affect the prevalence ofmalware [3–8].The speed and extent of spread of an epidemicon a network depend largely on the structure of the network;whether the epidemic goes viral depends on whether thespectral radius of the network exceeds a threshold [9–14].Therefore, reducing the spectral radius of a network byremoving a set of edges is an effective approach to thecontainment of the prevalence of an undesirable epidemicon the network. The spectral radius minimization problem(SRMP) aims to remove a given number of edges of a networkso that the spectral radius of the resulting network attainsthe minimum. As the SRMP is NP-hard [15], it is muchunlikely that there be a polynomial-time algorithm for it. Asthus, a number of heuristic algorithms for the SRMP havebeen proposed [15–19]. In most situations, these heuristics
are ineffective, because they produce nonoptimal solutionsrather than optimal solutions. For the purpose of developingeffective heuristic algorithms for the SRMP, it is appropriateto gain insight into the pattern of an optimal solution tothe 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]. Dueto remarkable advantages in communication [22–25], faulttolerant communication [26–30], fault diagnosis [31–34],and parallel computation [35, 36], hypercubes have beenwidely adopted as the underlying interconnection network inmulticomputer systems [37]. To our knowledge, the SRMP forhypercubes 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 theSRMP, a candidate optimal solution is presented. Second,it is shown that the candidate is optimal for small-sizedhypercubes, and it is shown that the proposed candidate islikely to be optimal for medium-sized hypercubes. The edges
HindawiDiscrete Dynamics in Nature and SocietyVolume 2017, Article ID 1382980, 8 pageshttps://doi.org/10.1155/2017/1382980
2 Discrete Dynamics in Nature and Society
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Figure 1: Three examples of𝐻𝑛.
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Figure 2: The proposed candidate in𝐻𝑛.
in each candidate are evenly distributed over the network,which may be a common feature of all symmetric networksand hence is instructive in designing effective heuristicalgorithms for the SRMP.
The remaining materials are organized in this fashion:the preliminary knowledge is given in Section 2. Section 3presents the main results of this work. Finally, Section 4summarizes this work.
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Figure 3: Assume that the red edge in the upper-left 3D subcube of𝐻5is the first deleted edge and each of the remaining edges is a candidate
for the second deleted edge. The spectral radius of the surviving network formed by deleting each of the candidate edges from𝐻5is shown.
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Figure 4: The proposed candidate (red) versus 103 random candidates (blue).
4 Discrete Dynamics in Nature and Society
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Figure 5: The proposed candidate in𝐻𝑛.
2. Preliminaries
For fundamental knowledge on the spectral radius of anetwork, see [38, 39]. The SRMP is formulated as follows:given a network 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘, find aset of 𝑘 edges of 𝐺 so that the surviving network obtainedby removing the set of edges from the network achieves theminimum spectral radius.
An 𝑛-dimensional cube (𝑛-D cube, for short), denotedby 𝐻𝑛, is a network 𝐺 = (𝑉, 𝐸), where there is a one-to-one
correspondance 𝜙 from 𝑉 to the set of all 0-1 binary stringsof length 𝑛 so that node 𝑢 is adjacent to node V if and onlyif 𝜙(𝑢) differs from 𝜙(V) in exactly one bit position. In whatfollows, it is always assumed that the nodes of a hypercubehave been labelled with 0-1 strings in this way. See Figure 1for three small-sized hypercubes.
An 𝑛-D cube can also be defined in a recursive way asfollows. (1) A 0-D cube is a graph on a single node. (2) For𝑛 ≥ 1, an 𝑛-D cube is built from two copies of an (𝑛 − 1)-Dcube in this way: connect each node in one copy to the samenode in the other copy.
3. Main Results
This section considers the optimal scheme of deletingtwo/three/four edges from𝐻
𝑛, respectively.
3.1. Deleting Two Edges. Firstly, we consider a subproblem ofthe SRMP, denoted by SRMP-H2, for which two edges will bedeleted from a hypercube. Let us present a candidate for theoptimal solution to the SRMP-H2 as follows, where 𝑛 denotesthe dimension of the hypercube:
𝑒1= {0𝑛, 0𝑛−11} ,
𝑒2= {1𝑛, 1𝑛−10} .
(1)
Figure 2 shows the proposed candidate in𝐻2,𝐻3,𝐻4, and
𝐻5, respectively.For 2 ≤ 𝑛 ≤ 7, it follows by exhaustive search that the
proposed candidate is optimal. For instance, assume that thered edge in the upper-left 3D subcube of𝐻
5is the first deleted
edge, and each of the remaining edges is a candidate for
Discrete Dynamics in Nature and Society 5
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Figure 6: The proposed candidate (red) versus 103 random candidates (blue).
the second deleted edge. The spectral radius of the survivingnetwork formed by deleting each of the candidate edges from𝐻5is shown in Figure 3. It can be seen that the larger the
distance between the two edges, the smaller the spectralradius of the surviving network. At the extreme, the proposedcandidate is optimal.
For 8 ≤ 𝑛 ≤ 11, the proposed candidate is comparedwith 103 random candidates in terms of the spectral radiusof the surviving network; see Figure 4. It is concluded thatthe proposed candidate is optimal among these candidates.
Therefore, we propose the following conjecture.
Conjecture 1. For all 𝑛 ≥ 2, the proposed candidate is anoptimal solution to the SRMP-H2.
3.2. Deleting Three Edges. Secondly, we consider a subprob-lem of the SRMP problem, denoted by SRMP-H3, for whichthree edges will be removed from a hypercube. Let us presenta candidate for the optimal solution to the SRMP-H3 asfollows, where 𝑛 denotes the dimension of the hypercube,𝑐 = ⌊(𝑛 + 1)/3⌋:
𝑒1= {0𝑛, 0𝑛−11} ,
𝑒2= {0𝑛−2𝑐12𝑐, 0𝑛−2𝑐−112𝑐+1} ,
𝑒3= {1𝑛−𝑐−10𝑐+1, 1𝑛−𝑐0𝑐} .
(2)
Figure 5 shows the proposed candidate in𝐻3,𝐻4,𝐻5, and𝐻
6,
respectively.For 2 ≤ 𝑛 ≤ 6, it follows by exhaustive search that
the proposed candidate is optimal. For 7 ≤ 𝑛 ≤ 10, theproposed candidate is comparedwith 103 randomcandidates;see Figure 6. It is concluded that the proposed candidate isoptimal among these candidates. Therefore, we propose thefollowing conjecture.
Conjecture 2. For all 𝑛 ≥ 3, the proposed candidate is anoptimal solution to the SRMP-H3.
3.3. Deleting Four Edges. Finally, consider a subproblem ofthe SRMP, denoted by SRMP-H4, for which four edges willbe deleted from a hypercube. Let us present a candidate to theoptimal solution to the SRMP-H4 as follows, where 𝑛 denotesthe dimension of the hypercube, 𝑐 = ⌊(𝑛 − 1)/3⌋:
If 𝑛 ≡ 0, 2mod 3, then
𝑒1= {0𝑛, 0𝑛−11} ,
𝑒2= {0𝑛−2𝑐−212𝑐+2, 0𝑛−2𝑐−212𝑐+10} ,
𝑒3= {1𝑛−2𝑐−20𝑐1𝑐+10, 1𝑛−2𝑐−20𝑐1𝑐02} ,
𝑒4= {1𝑛−𝑐−20𝑐+11, 1𝑛−𝑐−20𝑐12} .
(3)
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Figure 7: The proposed candidate in𝐻𝑛.
If 𝑛 ≡ 1mod 3, then
𝑒1= {0𝑛, 0𝑛−11} ,
𝑒2= {0𝑛−2𝑐−112𝑐+1, 0𝑛−2𝑐−112𝑐0} ,
𝑒3= {1𝑛−2𝑐−10𝑐1𝑐+1, 1𝑛−2𝑐−10𝑐1𝑐0} ,
𝑒4= {1𝑛−𝑐−10𝑐+1, 1𝑛−𝑐−10𝑐1} .
(4)
Figure 7 shows the proposed candidate in𝐻3,𝐻4,𝐻5, and
𝐻6, respectively.For 3 ≤ 𝑛 ≤ 6, it follows by exhaustive search that the
proposed candidate is an optimal solution to the SRMP-H4problem. For 7 ≤ 𝑛 ≤ 10, the proposed candidate is comparedwith 103 random candidates; see Figure 8. It is concluded that
the proposed candidate is optimal among these candidates.Therefore, we propose the following conjecture.
Conjecture 3. For all 𝑛 ≥ 3, the proposed candidate is anoptimal solution to the SRMP-H4.
4. Summary
This paper has addressed the spectral radius minimizationproblem for hypercubes. Given the number of edges to bedeleted, a candidate for the optimal solution has been pre-sented. For small-sized hypercubes, the proposed candidatehas been shown to be optimal. Formedium-sized hypercubes,it has been shown that the proposed candidate is likely tobe optimal. Due to the symmetry of hypercubes, there aremultiple optimal solutions for each of the subproblems. The
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Figure 8: The proposed candidate (red) versus 103 random candidates (blue).
experimental results show that, up to isomorphism, all ofthe optimal solutions are identical. By observing the patternof the proposed candidate, it has been speculated that, forany symmetric network, the edges in an optimal solution arealways evenly distributed.
Towards this direction, some researches are yet to bedone. First, the proposed conjectures need a proof. Second,this work should be extended to asymmetric networks suchas the hypercube-like networks [40, 41], the small-world net-works [42], the scale-free networks [43, 44], and the generalnetworks [45, 46]. Last, the effectiveness of heuristics for thespectral radius minimization problem must be improved.
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper.
Acknowledgments
This work was supported by Science and Technology SupportProgram of China (Grant no. 2015BAF05B03), Natural Sci-ence Foundation of China (Grants nos. 61572006, 71301177),Basic and Advanced Research Program of Chongqing (Grantno. cstc2013jcyjA1658), and Fundamental Research Funds fortheCentral Universities (Grant no. 106112014CDJZR008823).
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