# Persistent Threshold Dynamics with Recovery in Complex Networks

## TL;DR

This paper analyzes how complex networks exhibit persistent cascade and recovery dynamics, identifying phases of collapse or activity based on trigger and recovery rates, with an analytical framework validated by simulations.

## Contribution

It introduces an analytical framework to characterize critical behavior and temporal evolution of cascade-recovery dynamics in quasi-robust networks.

## Key findings

- Network can be in collapsing or active phase depending on trigger and recovery rates.
- Analytical predictions match agent-based simulation results.
- Framework predicts temporal evolution of network activity.

## Abstract

Threshold rules of spreading in binary-state networks lead to cascades. We study persistent cascade-recovery dynamics on quasi-robust networks, i.e., networks which are robust against small trigger but may collapse for larger one. It is observed that depending on the relative rate of triggering and recovery, the network falls into one of the two dynamical phases: collapsing or active phase. We devise an analytical framework which characterizes not only the critical behavior but also the temporal evolution of network activity in both phases. Agent-based simulation results show good agreement with the analytical calculations, indicating strong predicative power of our method for persistent cascade dynamics in complex networks.