A Continuous Threshold Model of Cascade Dynamics

TL;DR

This paper introduces a continuous threshold model (CTM) for cascade dynamics in networks, analyzing how heterogeneity in agent thresholds influences whether cascades occur or are contained, especially near bifurcation points.

Contribution

The paper generalizes the linear threshold model to a continuous setting and analyzes cascade behavior in networks with heterogeneous thresholds, revealing conditions for cascades versus containment.

Findings

A subcritical bifurcation leads to cascades when threshold disparity is large.

System sensitivity peaks at bifurcation points, affecting cascade outcomes.

Heterogeneity in thresholds determines whether cascades are triggered or contained.

Abstract

We present a continuous threshold model (CTM) of cascade dynamics for a network of agents with real-valued activity levels that change continuously in time. The model generalizes the linear threshold model (LTM) from the literature, where an agent becomes active (adopts an innovation) if the fraction of its neighbors that are active is above a threshold. With the CTM we study the influence on cascades of heterogeneity in thresholds for a network comprised of a chain of three clusters of agents, each distinguished by a different threshold. The system is most sensitive to change as the dynamics pass through a bifurcation point: if the bifurcation is supercritical the response will be contained, while if the bifurcation is subcritical the response will be a cascade. We show that there is a subcritical bifurcation, thus a cascade, in response to an innovation if there is a large enough disparity between the thresholds of sufficiently large clusters on either end of the chain; otherwise the response will be contained.