# Heterogeneous micro-structure of percolation in sparse networks

## TL;DR

This paper investigates the heterogeneous responses of individual nodes in sparse networks to edge removal, revealing variability in node connectivity and cluster sizes, with implications for understanding cascading failures and epidemic spread.

## Contribution

It introduces a message-passing framework to analyze node heterogeneity in percolation, linking it to non-backtracking centrality and providing analytical results for large random graphs.

## Key findings

- Node-to-node heterogeneity is significant near the percolation threshold.
- Non-backtracking centrality captures heterogeneity in node responses.
- Analytical approximations are provided for Erdős-Rényi graphs.

## Abstract

We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the probability of a particular node to remain part of the giant component, and in the expected size of small clusters containing that node. In the vicinity of the percolation threshold, weakly non-linear analysis reveals that node-to-node heterogeneity is captured by the recently introduced notion of non-backtracking centrality. We supplement these results for fixed finite networks by a population dynamics approach to analyse random graph models in the infinite system size limit, also providing closed-form approximations for the large mean degree limit of Erd\H{o}s-R\'enyi random graphs. Interpreted in terms of the application of percolation to real-world processes, our results shed light on the heterogeneous exposure of different nodes to cascading failures, epidemic spread, and information flow.