Homophily in competing behavior spreading among the heterogeneous population with higher-order interactions

Competing behavior spreading dynamics occur not only through pairwise interactions but also through higher-order collective interactions. The simplicial complex is widely adopted to describe the co-existence of pairwise and higher-order interactions. Previous studies have demonstrated that heterogeneous populations and the homophily effects are crucial in shaping the spreading pattern and phase transition. There is still a lack of a theoretical study for competing spread when higher-order interactions, heterogeneous populations, and homophily effects are all considered at the same time. We propose a mathematical model for the competing behaviors A and B to study the effects of homophily on heterogeneous populations with higher-order interactions. The heterogeneity population consists of three groups. Agents who only adopt behavior A or B are denoted as ΩA and ΩB, respectively. Agents in ΩAB may adopt one of two behaviors. To capture the competing behavior dynamics, we offer a theoretical Microscopic Markov Chain Approach (MMCA). We find that increasing 1-simplex transmission rate contributed to the spread of both two behaviors. The saddle point of the system is investigated and it is shown that the observed coexistence is caused by the average result of multiple experiments, revealing that there is still no coexistence present under our model. Decreasing the proportion of the population ΩAB would lead to a significant decrease in the final adopted density of the system. Due to the existence of groups that only adopt behavior A or B, there are always adopted individuals in the system. In addition, the final adopted density is almost consistent across different homophily effects when the two behaviors interact symmetrically. When the proportion of ΩA remains constant, the final adopted density of behavior A decreases significantly as the proportion of ΩB increases, whereas the final adopted density of behavior B remains almost constant. Also, When the proportion of ΩAB is fixed, an increase in the proportion of population ΩA (ΩB) makes the final adopted density of behavior A (behavior B) to increase with it.