Temporal profiles of avalanches on networks

TL;DR

This paper derives a universal equation for the average temporal profile of avalanches on networks, revealing how network topology influences avalanche symmetry and providing methods to identify critical cascade dynamics.

Contribution

It introduces a Markov branching process approach to model avalanche shapes on networks, highlighting the impact of heavy-tailed degree distributions at criticality.

Findings

Nonsymmetric avalanche shapes occur in networks with heavy-tailed degree distributions.

The derived equation predicts avalanche shape behavior at criticality.

Numerical simulations support the theoretical predictions.

Abstract

An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the temporal profiles that characterize the growth and decay of avalanches of fixed duration. At the critical point of the dynamics the average avalanche shapes for different durations can be rescaled so that they collapse onto a single universal curve. We apply Markov branching process theory to derive a simple equation governing the average avalanche shape for cascade dynamics on networks. Analysis of the equation at criticality demonstrates that nonsymmetric average avalanche shapes (as observed in some experiments) occur for certain combinations of dynamics and network topology; specifically, on networks with heavy-tailed degree distributions. We give examples using numerical simulations of models for information spreading, neural dynamics, and behaviour adoption and we propose simple experimental tests to quantify whether cascading systems are in the critical state.