Pairwise interactions between individuals are taken as fundamental drivers of collective behavior responsible for group cohesion and decision-making. While an individual directly influences only a few neighbors, over time indirect influences penetrate a much larger group. The abiding question is how this spread of influence comes to affect the collective. One or a few individuals are often identified as leaders, being more influential than others. Transfer entropy and time-delayed mutual information are used to identify underlying asymmetric interactions, such as leader-follower classification in aggregated individuals--cells, birds, fish, and animals. However, these conflate distinct functional modes of information flow between individuals. Computing information measures conditioning on multiple agents requires the proper sampling of a probability distribution whose dimension grows exponentially with the number of agents being conditioned on. Employing simple models of interacting self-propelled particles, we examine the pitfalls of using time-delayed mutual information and transfer entropy to quantify the strength of influence from a leader to a follower. Surprisingly, one must be wary of these pitfalls even for two interacting particles. As an alternative we decompose transfer entropy and time-delayed mutual information into intrinsic, shared, and synergistic modes of information flow. The result not only properly reveals the underlying effective interactions, but also facilitates a more detailed diagnosis of how individual interactions lead to collective behavior. This exposes the role of individual and group memory in collective behaviors. In addition, we demonstrate in a multi-agent system how knowledge of the decomposed information modes between a single pair of agents reveals the nature of many-body interactions without conditioning on additional agents.
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