Mathematical Analysis for Non-reciprocal-interaction-based Model of Collective Behavior

In many natural and social systems, collective behaviors emerge as a consequence of non-reciprocal interaction between their constituents. As a first step towards understanding the core principle that underlies these phenomena, we previously proposed a minimal model of collective behavior based on non-reciprocal interactions by drawing inspiration from friendship formation in human society, and demonstrated via simulations that various non-trivial patterns emerge by changing parameters. In this study, a mathematical analysis of the proposed model wherein the system size is small is performed. Through the analysis, the mechanism of the transition between several patterns is elucidated.