A Continuous-Time and State-Space Relaxation of the Linear Threshold Model with Nonlinear Opinion Dynamics
This work addresses the problem of modeling complex contagions more accurately for researchers in social networks and dynamical systems, though it appears incremental as it relaxes an existing model rather than proposing a new paradigm.
The authors tackled the limitations of the Linear Threshold Model (LTM) in capturing multiple time-scales and subthreshold signaling in collective behavior propagation by introducing a continuous-time and state-space relaxation based on Nonlinear Opinion Dynamics (NOD), proving that activation in the discrete LTM guarantees activation in the continuous NOD relaxation under appropriate parameters and establishing computable conditions for equivalence.
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.