# The shape of node reliability

## TL;DR

This paper investigates the mathematical properties of node reliability in networks with reliable edges and probabilistic nodes, revealing significant differences from other network robustness models.

## Contribution

It provides a detailed analysis of the analytic properties of node reliability functions, such as monotonicity and concavity, highlighting contrasts with other reliability models.

## Key findings

- Node reliability functions exhibit unique monotonicity and concavity properties.
- The study identifies fixed points and their implications for network robustness.
- Results contrast with properties of coherent set system-based reliability models.

## Abstract

Given a graph $G$ whose edges are perfectly reliable and whose nodes each operate independently with probability $p\in[0,1],$ the node reliability of $G$ is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce. We study analytic properties of the node reliability on the interval $[0,1]$ including monotonicity, concavity, and fixed points. Our results show a stark contrast between this model of network robustness and models that arise from coherent set systems (including all-terminal, two-terminal and K-terminal reliability).