  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.
