Generalized Langevin Models of Linear Agent-Based Systems: Strategic Influence Through Environmental Coupling

Agent-based models typically treat systems in isolation, discarding environmental coupling as either computationally prohibitive or dynamically irrelevant. We demonstrate that this neglect misses essential physics: environmental degrees of freedom create memory effects that fundamentally alter system dynamics. By systematically transforming linear update rules into exact generalized Langevin equations, we show that unobserved environmental agents manifest as memory kernels whose timescales and coupling strengths are determined by the environmental interaction spectrum. Network topology shapes this memory structure in distinct ways: small-world rewiring drives dynamics toward a single dominant relaxation mode, while fragmented environments sustain multiple persistent modes corresponding to isolated subpopulations. We apply this framework to covert influence operations where adversaries manipulate target populations exclusively via environmental intermediaries. The steady-state response admits a random-walk interpretation through hitting probabilities, revealing how zealot opinions diffuse through the environment to shift system agent opinions toward the zealot mean - even when zealots never directly contact targets.