All-terminal hypernetwork reliability synthesis of a kind of semi-deterministic hypergraphs

Abstract

Network reliability plays an important role in analysis, synthesis and detection of real-world networks. In this paper, we first propose the concept of hypernetwork reliability, which generalizes the concept of network reliability. The model for hypernetwork reliability studies consists of a hypergraph with perfect reliable vertices and equal and independent hyperedge failure probability [Formula: see text]. The measure of reliability is defined as the probability that a hypergraph is connected. Let [Formula: see text] be an [Formula: see text]-uniform hypergraph with the number of vertices [Formula: see text] and the number of hyperedges [Formula: see text], where every hyperedge connects [Formula: see text] vertices. We confirm the possibility of the existence of a fixed hypergraph that is optimal or least for all hyperedges same survival possible [Formula: see text]. It is simple to verify that such hypergraph exists if [Formula: see text]. For a kind of 2-regular 3-uniform hypergraphs, we calculate the upper and lower bounds on the all-terminal reliability, and describe the class of hypergraphs that reach the boundary.