Information Flow in a Boolean Network Model of Collective Behavior

In animal groups, leaders have often been proposed to be those individuals who possess additional knowledge about their surroundings, such as the location of a food source or a potential predator. Understanding how this information propagates through the group to shape collective response is an important step to elucidate the evolutionary basis of leadership. In this paper, we study a Boolean model of collective behavior, in which a single leader interacts with a group of followers in a binary decision-making process. Through an analytical treatment of the associated Markov chain, we establish closed-form solutions for the transition probability matrix and the stationary distribution, as functions of the noise in the decision-making process and the size of the group. We leverage these expressions to quantify information transfer within the group, measured through the information-theoretic construct of transfer entropy. We find that information transfer depends nonlinearly on the group size and noise. For low noise intensities, the system is nearly deterministic, such that no information is shared within the group; an equivalent effect is observed for large noise intensities, which mask the information transfer. We determine the existence of critical noise intensities at which the leader maximizes information transfer to a follower or followers maximize information sharing between each other for a given group size. These analytical findings suggest that noise might have a positive role in collective behavior, facilitating the transfer of knowledge within the group, from leaders to followers.