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

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.

Connectivity-preserving distributed algorithms for removing links in directed networks

“…Since the link removal problem is NP-hard, most of the work focuses on heuristics and approximations of the problem. That is, they focused on developing strategies to approximate and compute a suboptimal solution to this problem for both unidirectional and bidirectional networks, see for example (Bishop & Shames, 2011;Van Mieghem et al, 2011;Chen et al, 2016;Milanese et al, 2010;Yang et al, 2016;Wu et al, 2017). Among these work, an effective and scalable algorithm based on eigenvalue sensitivity analysis is proposed in (Chen et al, 2016) to minimize the dominant eigenvalue of an adjacency matrix by removing a fraction of links from a directed network.…”