Stochastic modeling of cascade dynamics: A unified approach for simple and complex contagions across homogeneous and heterogeneous threshold distributions on networks

The COVID-19 pandemic has underscored the importance of understanding, forecasting, and avoiding infectious processes, as well as the necessity for understanding the diffusion and acceptance of preventative measures. Simple contagions, like virus transmission, can spread with a single encounter, while complex contagions, such as preventive social measures (e.g., wearing masks, social distancing), may require multiple interactions to propagate. This disparity in transmission mechanisms results in differing contagion rates and contagion patterns between viruses and preventive measures. Furthermore, the dynamics of complex contagions are significantly less understood than those of simple contagions. Stochastic models, integrating inherent variability and randomness, offer a way to elucidate complex contagion dynamics. This paper introduces a stochastic model for both simple and complex contagions and assesses its efficacy against ensemble simulations for homogeneous and heterogeneous threshold configurations. The model provides a unified framework for analyzing both types of contagions, demonstrating promising outcomes across various threshold setups on Erds-Rényi graphs.

Figure 1

Results for the time evolution of the spreading density $d\left(t\right)$ for different seed sizes $d\left(0\right)$ for homogeneous threshold configurations $T\in \{1,2,3,4,5\}$ with the solid line representing the averages over the ensemble simulations and the dashed line representing the result of the stochastic model.

Figure 2

Results for the time evolution of the spreading densities $d^T\left(t\right)$ of each equally sized subset $B_T$ with thresholds $T\in \{1,2,3,4,5\}$ (heterogeneous case) starting from initial seed sizes $d\left(0\right)\in \{0.01,0.02,0.05,0.1\}$ with the solid line being the averages over the multiple simulation runs and the dashed line the result of the stochastic model. Disappearing dashed lines correspond to a perfect match between the simulation runs and the stochastic model.

Figure 3

Results for the time evolution of the spreading density $d\left(t\right)$ for different seed sizes $d\left(0\right)$ for the heterogeneous threshold configuration with equal amount of $T\in \{1,2,3,4,5\}$ with the dashed gray lines being the average spreading density over multiple simulation runs and the red line the result of the stochastic model. The blue area marks the confidence intervals around the stochastic solution determined via Eqs. (10) and (11).