A Continuous-Time and State-Space Relaxation of the Linear Threshold Model with Nonlinear Opinion Dynamics

TL;DR

This paper introduces a continuous-time, smooth dynamical system relaxation of the traditional Linear Threshold Model to better analyze complex contagion processes with multiple time-scales and subthreshold effects.

Contribution

It develops a novel NOD-based continuous model that maps discrete cascades to continuous flows, enabling richer analysis and guarantees of cascade equivalence under certain conditions.

Findings

Continuous NOD model guarantees LTM activation for any seed set.

Bounded social coupling ensures exact cascade recovery.

Allows analysis of subthreshold inputs and temporal effects.

Abstract

The Linear Threshold Model (LTM) is widely used to study the propagation of collective behaviors as complex contagions. However, its dependence on discrete states and timesteps restricts its ability to capture the multiple time-scales inherent in decision-making, as well as the effects of subthreshold signaling. To address these limitations, we introduce a continuous-time and state-space relaxation of the LTM based on the Nonlinear Opinion Dynamics (NOD) framework. By replacing the discontinuous step-function thresholds of the LTM with the smooth bifurcations of the NOD model, we map discrete cascade processes to the continuous flow of a dynamical system. We prove that, under appropriate parameter choices, activation in the discrete LTM guarantees activation in the continuous NOD relaxation for any given seed set. We establish computable conditions for equivalence: by sufficiently bounding the social coupling parameter, the continuous NOD cascades exactly recover the cascades of the discrete LTM. We then illustrate how this NOD relaxation provides a richer analytical framework than the LTM, allowing for the exploration of cascades driven by strictly subthreshold inputs and the role of temporally distributed signals.