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

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.

Figures (3)

• Figure 3: Bifurcation diagram of NOD dynamics \ref{['eq:NOD-dyn']} for a single agent with respect to bifurcation parameter $b$. Thick solid blue and dashed red lines correspond to stable and unstable fixed points, respectively. The point marked $SN$ corresponds to the Saddle-Node bifurcation that marks the tipping point to activation. Values $b^*$ and $z^*$ are the activation threshold and the value achieved at the tipping point, respectively. Faint gray arrows represent the direction of the flow in each region.
• Figure 4: Simulated cascade in a random instance of LTM, and two NOD relaxations, one with $\gamma\approx0.3$ (small) and the other $\gamma\approx0.7$ (large). Both, with $k=1.1$. Color bars correspond to times of activation, with color at bottom corresponding with the initial time and the color at the top corresponding to the time at the activation of the last node in cascade. The seed set consists of $4$ nodes colored dark blue.
• Figure 5: Cascades in the NOD model for distributed subthreshold inputs. $a)$ Network structure, nodes are identified by color. This is the same network used in \ref{['fig:dist-input']}, but in this case the thresholds for each node are chosen to be equal $\tau_i=0.5$, leading to homogeneous $b_i^*$ in the NOD relaxation. $b)$ Top subplot shows opinion states ($\boldsymbol{z}(t)$) of agents across time. Bottom subplot shows the applied time-varying input $b$ for each agent. The threshold for all agents is $b_i^*\approx 0.02$, shown as a dashed red line. In this case, a subthreshold input of $b_i=0.01$ is applied simultaneously to each agent as a step function for $50$ time-steps. $c)$ Similar to previous. In this case, the input step function is applied with a delay selected uniformly randomly from the interval $[0,50].$

Theorems & Definitions (14)

• Definition 1
• Remark 1
• Remark 2
• Proposition 1
• proof
• Definition 2
• Proposition 2
• proof
• Theorem 1
• proof