Maximal Intervals of Decrease and Inflection Points for Node Reliability

TL;DR

This paper investigates the behavior of node reliability in graphs, revealing that the number of intervals where reliability decreases and the inflection points can be arbitrarily large within the probability range.

Contribution

It demonstrates that unlike other reliability measures, node reliability can have an unbounded number of decreasing intervals and inflection points, highlighting complex behavior.

Findings

Number of maximal decreasing intervals is unbounded.

There can be arbitrarily many inflection points.

Node reliability exhibits complex, non-monotonic behavior.

Abstract

The \textit{node reliability} of a graph $G$ is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce, given that the edges are perfectly reliable but each node operates independently with probability $p\in[0,1]$. We show that unlike many other notions of graph reliability, the number of maximal intervals of decrease in $[0,1]$ is unbounded, and that there can be arbitrarily many inflection points in the interval as well.