The contagion model with social dependency

Empirical evidence demonstrates that contagion relies on social relationships, and the level of social dependency varies for different contagious entities (e.g., diseases or information). To unravel the influence of social dependency on the contagion dynamics, we introduce a social dependency coefficient and present a contagion model with the memory of non-redundant influence on complex networks, which bridges the simple and complex contagions. In this model, individuals exist in one of three states: susceptible, infected, or recovered. Susceptible individuals become infected when the cumulative non-redundant effects they have received (represented by a belief function) exceed their thresholds. By percolation method and mean-field theory, we find that low social dependency can expand the size of final recovered population, yet this effect is not continuous. Specifically, the level of social dependency can be categorized into three intervals based on the critical transmission probability. In the low-dependency interval, contagious entities can spread widely at a low transmission probability. In the medium dependency interval, the critical transmission probability increases stepwise with the social dependency. In the high-dependency interval, the population is free from large outbreaks of contagion at any transmission probability. Besides, the results are not qualitatively affected by the heterogeneous network structure and the theoretical predictions are consistent with the simulation results.