Quantifying influence and information transfer in a modified Vicsek model with nonreciprocal interactions

Understanding information transfer among individuals is fundamental to revealing the collective dynamics of complex systems. Information transfer has been quantified using various information-theoretic tools and assigned the concept of influence. However, information-theoretic measures are inherently statistical, not causal, and influence in the context of causal inference implies a causal relation, so equating influence with information transfer creates conceptual confusion and interpretational challenges. Here, we introduce an influence-based Vicsek model with nonreciprocal interactions to distinguish influence from information transfer and examine their relationship. At the pairwise level, for fixed noise strengths, influence and transfer entropy exhibit quasilinear relations; for fixed interaction weights, influence and transfer entropy exhibit nonlinear relations because noise on influencers enhances information transfer, whereas noise on followers suppresses the overall information transfer. At the collective level, we find that both influence and normalized transfer entropy form two-branched relations that clearly identify the transition points across three distinct phase transitions. These transition points reveal a different aspect of the collective dynamics not captured by classical order parameters: 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 distinguishing the concept of influence from information transfer in a physical model system, provides a concrete testbed for methods emerging from the growing field of information theory-based causal inference, and offers new insights into the dynamics of complex systems.