Characterization of matroidal connectivity of regular networks

Abstract

High performance computing system, which takes an interconnection network as its infrastructure topology, has been utilized in scientific computing, big data analysis as well as artificial intelligence. With the rapid growth of infrastructure topology (interconnection network) in high performance computing system, the probability of network element malfunction increases dramatically. Generally, the reliability of interconnection network is measured by the traditional (edge) connectivity, which is limited by the minimum degree of the network. In order to overcome the defect, the matroidal connectivity has been newly proposed to extend the edge connectivity. For the regular network G, let Bk(G)={B1,B2,…,Bk} be a fixed edge partition ordered sequence of G and χk=(x1,x2,…,xk) be a sequence of k integers. For any D⊆E(G), if |D∩Bi|≤xi(i∈[k]) and G−D is disconnected, then D is named matroidal cut set of G. The matroidal connectivity of G, denoted by λ(G,Bk(G)), is the minimum size of matroidal cut set of G. In this work, we characterize the matroidal connectivity of a class of n-dimensional regular networks Gn. To be more specific, we show that λ(Gn,Bn(Gn))=(k−l+1)f(l+j), where f(n) is a function about n, and l,k,n are positive integers (l≥1), j=n−k. As empirical analysis, we directly determine the matroidal connectivity of some well-known interconnection networks, such as hypercube Qn, star graph STn and alternating group graph AGn.