# Correlations between thresholds and degrees: An analytic approach to model attacks and failure cascades

## TL;DR

This paper develops an analytic approach to understand how correlations between node degrees and thresholds influence cascade dynamics and network robustness, enabling better predictions of attack outcomes and failure propagation.

## Contribution

It introduces a tractable analytic method in the thermodynamic limit to incorporate degree-threshold correlations into cascade models, advancing beyond traditional approximations.

## Key findings

- Correlations significantly affect cascade thresholds and dynamics.
- Irregular phase transitions can occur due to degree-threshold correlations.
- Different cascade speeds impact control strategies.

## Abstract

Two node variables determine the evolution of cascades in random networks: a node's degree and threshold. Correlations between both fundamentally change the robustness of a network, yet, they are disregarded in standard analytic methods as local tree or heterogeneous mean field approximations because of the bad tractability of order statistics. We show how they become tractable in the thermodynamic limit of infinite network size. This enables the analytic description of node attacks that are characterized by threshold allocations based on node degree. Using two examples, we discuss possible implications of irregular phase transitions and different speeds of cascade evolution for the control of cascades.