A Recursive Formula for the Reliability of a r-Uniform Complete Hypergraph and Its Applications

Ke Zhang,Lixin Dong,Zhonglin Ye,Haixing Zhao

doi:10.1155/2018/3131087

Ke Zhang, Lixin Dong + Show 2 more

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https://doi.org/10.1155/2018/3131087

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Abstract

The reliability polynomial R(S,p) of a finite graph or hypergraph S=(V,E) gives the probability that the operational edges or hyperedges of S induce a connected spanning subgraph or subhypergraph, respectively, assuming that all (hyper)edges of S fail independently with an identical probability q=1-p. In this paper, we investigate the probability that the hyperedges of a hypergraph with randomly failing hyperedges induce a connected spanning subhypergraph. The computation of the reliability for (hyper)graphs is an NP-hard problem. We provide recurrence relations for the reliability of r-uniform complete hypergraphs with hyperedge failure. Consequently, we determine and calculate the number of connected spanning subhypergraphs with given size in the r-uniform complete hypergraphs.

Highlights

As we all know, the topological structure of a real-world complex system is often described by a graph
We find that K63 does not contain the connected subhypergraph with 2 edges
We find that the spanning hypertree of runiform complete hypergraph is a hypertree and a spanning subhypergraph, but the reverse is not true

Summary

Introduction

The topological structure of a real-world complex system is often described by a graph. We provide a new method to calculate the number of connected spanning subhypergraphs in Knr. The runiform complete hypergraphs have long been in the focus of hypernetwork reliability research for different reasons. In [26] Gilbert presented a recursive algorithm for the calculation of the all-terminal reliability of a complete graph Kn by fixing a vertex V ∈ V and considered the relationships between all connected subgraphs containing V of order k and corresponding subgraphs of order n − k, the probability R(Kn, p) is exactly equivalent to the probability that the induced spanning subgraph is connected for n vertices set We substantially generalize this idea and propose the result in hypergraphs based on above analogies, and the research result from Gilbert is a special case of the research achievements proposed by this paper. The equality holds if and only if H − E has r components for any edge E which belongs to ε(H)

A Recursive Formula for the Reliability of a r-Uniform Complete Hypergraph

Enumeration of Spanning Hypertrees in r-Uniform Complete Hypergraphs

Conclusions and Open Problems