Graph reliability evaluation via random $K$-out-of-$N$ systems

The present study was concerned with network failure problems for simple connected undirected graphs. A connected graph becomes unconnected through edge failure, under the assumptions that only edges can fail and each edge has an identical failure distribution. The main purpose of the present study was to show recurrent relations with respect to the number of edges in graph generation procedures. To this end, simple connected undirected graphs were correlated to random $K$-out-of-$N$ systems, and key features of such systems were applied. In addition, some simple graph cases and examples were analyzed.