Quantifying Influence and Information Transfer in a Modified Vicsek Model with Non-reciprocal Interaction

Abstract

Understanding information transfer among individuals is fundamental to revealing collective dynamics of complex systems. Information transfers are quantified by information-theoretical measures and are often correlated with the concept of influence. However, a clear, quantitative definition of influence remains lacking. Here, we introduce a modified Vicsek model that allows a quantitative definition of influence. The model incorporates non-reciprocal interactions and exhibits three distinct collective phase transitions. At the pairwise level, we find quasi-linear relations between influence and transfer entropy at fixed noise strengths and a Boltzmann sigmoidal relation between influence and normalized transfer entropy below maximum noise strength; we reveal that noise on influencers enhances information transfer, whereas noise on followers suppresses information transfer. At the collective level, we find that both influence and normalized transfer entropy identify the same transition points across three phase transitions and that noise-induced phase transitions are associated with changes in the relative importance of influencers' presents or followers' presents on followers' futures. Finally, we use our model to assess partial information decomposition methods and identify two methods most suitable for analyzing our system, one based on pointwise surprisal changes and the other on secret key agreement. Our work is a first step in differentiating the concept of influence from information transfer, provides a concrete testbed for methods emerging from the growing field of information-theoretical causality quantification, and offers new insights into the dynamics of complex systems.