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

Semra Gunduc, David J. Butts, Michael S. Murillo

TL;DR

This work shows that unobserved environmental agents manifest as memory kernels whose timescales and coupling strengths are determined by the environmental interaction spectrum, and applies this framework to covert influence operations where adversaries manipulate target populations exclusively via environmental intermediaries.

Abstract

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.

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

TL;DR

This work shows that unobserved environmental agents manifest as memory kernels whose timescales and coupling strengths are determined by the environmental interaction spectrum, and applies this framework to covert influence operations where adversaries manipulate target populations exclusively via environmental intermediaries.

Abstract

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.
Paper Structure (25 equations, 3 figures)

This paper contains 25 equations, 3 figures.

Figures (3)

  • Figure 1: Illustration of graph structure and resulting modal memory dynamics. Top row shows adjacency matrices of different graph configurations. Each matrix is partitioned into system--system (upper-left, dark gray), bath--bath (lower-right, blue), and symmetric system--bath coupling blocks (off-diagonal, light gray), dots represent individual edges. Bottom row shows the corresponding modal memory dynamics. All graphs have $N=300$ nodes and mean degree $k=6$. Left two panels: Watts--Strogatz small--world graphs with rewiring probabilities $p_{\mathrm{ws}} = 0.1$ and $p_{\mathrm{ws}} = 0.5$, showing that increasing $p_{\mathrm{ws}}$ spreads edges (dots are edges) more globally, enhancing system-memory coupling and accelerating memory dynamics. Right three panels: echo--chamber bath configurations with fixed $p_{\mathrm{ws}} = 0.1$ and varying system-bath interaction density $f \in \{0.1, 0.3, 1.0\}$ and no interaction between bath blocks, demonstrating that higher interaction density produces stronger and more widespread memory modes.
  • Figure 2: Eigenvalue spectra $\mathrm{Re}(\lambda_k)$ and mode coupling strengths $\|C_k\|_F$. (left) Single bath system: one dominant eigenvalue governs the dynamics while the others decay. (right) Echo-chamber (block-diagonal) bath: each isolated bath block retains its own dominant eigenvalue and corresponding mode, so multiple modes remain simultaneously dominant. Note the change of scale.
  • Figure 3: Zealot influence on system agents. In the top row zealots opinions are sampled from a single uniform distribution, while in the bottom they are sampled from two disjoint uniform distributions. All system agents are sampled from a single uniform distribution. Opinion dynamics are shown in the left column, and steady-state opinion distributions are in the right column. Over time the influence of the zealots on system agents is seen by the tightening and shifting of the system agent opinion distribution. At steady-state the the system agent opinions are distributed around the mean zealot opinion.