Detecting existence of a hidden mediator between a pair of individual time series.

Collective behaviors in groups emerge from complex webs of direct and indirect interactions among individuals. While pairwise interactions are fundamental to understanding group dynamics, each agent's historical trajectory confounds causal inference, making it challenging to disentangle direct interactions from those mediated by hidden intermediaries. Here, we address the question: given observational tracking data from only a single pair of agents, can we determine whether their interaction is direct or mediated by an unseen third agent? We propose a framework based on modified transfer entropy analysis across delay times to detect the presence of a hidden intermediary. Our approach reveals a distinct signature: direct interactions exhibit a consistently decreasing modified transfer entropy with increasing delay time, whereas indirect interactions deviate from this trend. This method provides a simple yet versatile tool for uncovering hidden structures in complex systems, with broad implications for networked dynamics in biological, social, and artificial systems.