Investigation on the influence of heterogeneous synergy in contagion processes on complex networks.

Synergistic contagion in a networked system occurs in various forms in nature and human society. While the influence of network's structural heterogeneity on synergistic contagion has been well studied, the impact of individual-based heterogeneity on synergistic contagion remains unclear. In this work, we introduce individual-based heterogeneity with a power-law form into the synergistic susceptible-infected-susceptible model by assuming the synergistic strength as a function of individuals' degree and investigate this synergistic contagion process on complex networks. By employing the heterogeneous mean-field (HMF) approximation, we analytically show that the heterogeneous synergy significantly changes the critical threshold of synergistic strength σc that is required for the occurrence of discontinuous phase transitions of contagion processes. Comparing to the synergy without individual-based heterogeneity, the value of σc decreases with degree-enhanced synergy and increases with degree-suppressed synergy, which agrees well with Monte Carlo prediction. Next, we compare our heterogeneous synergistic contagion model with the simplicial contagion model [Iacopini et al., Nat. Commun. 10, 2485 (2019)], in which high-order interactions are introduced to describe complex contagion. Similarity of these two models are shown both analytically and numerically, confirming the ability of our model to statistically describe the simplest high-order interaction within HMF approximation.