Environment-Driven Emergence of Higher-Order Collective Behavior

TL;DR

This study shows that higher-order dependencies, captured by the O-information $\Omega$, can arise solely from a shared stochastic environment acting on three variables, even in the absence of direct interactions. It derives a no-go theorem showing that time-invariant coupling to a common environment cannot generate synergy, while time-dependent environmental coupling—and its interplay with deterministic interactions—induces transitions between redundancy ($\Omega>0$) and synergy ($\Omega<0$). The work reveals a geometric partition of correlation space into regions supporting redundancy or synergy and demonstrates that synergy occupies larger regions, making environment-driven higher-order organization a distinct mechanism beyond traditional pairwise interactions. The findings highlight the importance of disentangling environmental mediation from interaction-based effects to understand and control collective behavior in complex systems.

Abstract

Collective behavior is commonly attributed to direct interactions among system components. Using a minimal stochastic model, we show that higher-order collective structure can instead emerge from shared stochastic environments, even in the absence of interactions. Quantified via the O-information, environmental fluctuations induce both redundant and synergistic dependencies, with the latter occupying larger regions of the correlation space. We establish a no-go theorem showing that time-independent coupling between the system variables and a shared stochastic environment rules out synergistic higher-order behavior. Crucially, this constraint can be overcome dynamically: transitions between redundancy and synergy arise from time-dependent environmental coupling or from the nontrivial interplay between shared environments and direct interactions. Together, these results identify environmental mediation as a distinct mechanism of higher-order collective organization beyond the conventional interaction-centric paradigm.

Figures (4)

• Figure 1: System composed of three elements with state variables $z_{1}(t)$, $z_{2}(t)$, and $z_{3}(t)$. The evolution of all elements is affected by a shared global environment represented by a stochastic process $W(t)$ and by local environment factors $W_{1}(t)$, $W_{2}(t)$, and $W_{3}(t)$. The functions $\left\{f_{k}(t)\right\}_{k=1}^{3}$ quantify the coupling strength between each element and the global environment. The dynamics of the first element include a deterministic interaction term $m \! \left( \boldsymbol{z} \right)$ capturing the interaction with itself and the other two elements.
• Figure 2: Phase diagram of redundancy (green) and synergy (red) in triplet correlation space. (a,b) Two-dimensional slices of the correlation space $\left( \rho_{12}, \rho_{13}, \rho_{23} \right) \! \in \! \left[ -1, 1\right]^{3}$ at fixed $\rho_{23}$, showing the sign of the O-information: redundant regions ($\Omega \! > \! 0$) and synergistic regions ($\Omega \! < \! 0$). $\Omega$ is not defined in white regions where $g^{\ast} \leq 0$. Triangles show the sign of the pairwise environment-mediated correlations between the dynamical variables $\{z_{1}, z_{2}, z_{3}\}$. Positive correlations, $\rho_{kl}>0$, between $z_{k}$ and $z_{l}$ are marked by continuous black edges. Dotted blue edges mark negative correlations. (c) O-information for a system with constant parameters ($\boldsymbol{\mu} \! = \! \left( 0.5, 0.6, 0.4 \right)$, $\boldsymbol{\theta} \! = \! \left( 0, 0.25, 0.15 \right)$ and $\boldsymbol{f}(t) \! = \! \boldsymbol{\varphi} \! = \! (0.2, 0.2, 0.2)$) yielding the correlations marked by a star in panel (b). The analytical prediction (stars) is compared with numerical simulations (circles). The mean and $95\%$ confidence interval (CI) of the numerical simulations are indicated by the dashed line and shaded region, respectively.
• Figure 3: System with time-varying coupling to the shared environment. (a) Temporal variation of the coupling strength to the shared environment for each variable, as indicated in the legend. (b) Time evolution of the pairwise correlation coefficients. (c) Analytical and numerical results for the O-information. Simulations were run with parameters $\boldsymbol{\mu} \! = \! \left( 0.5, 0.6, 0.4 \right)$, $\boldsymbol{\theta} \! = \! \left( 0, 0.1, 0.1 \right)$, $\boldsymbol{\varphi} \! = \! \left( 0.1, 0.1, 0.1 \right)$, $\boldsymbol{\alpha} \! = \! \left( 0, 0, 1 \right)$ and $\boldsymbol{\beta} \! = \! \left( 0, 1, 0.1 \right)$, using $2 \! \times \! 10^{4}$ trajectories, and $4 \! \times \! 10^{3}$ time steps.
• Figure 4: Effect of deterministic interactions on the behavior of systems with constant coupling to a shared environment. Time evolution of the O-information for several values of the deterministic couplings $m_{1}$, $m_{2}$, and $m_{3}$, indicated by the symbols in the legends. Panels (a) and (b) correspond to positive and negative values of the deterministic couplings, respectively. The solid black lines show a baseline for $\Omega(t)$ defined by the corresponding systems without deterministic interactions (i.e., with $m \! \left( \boldsymbol{z} \right) \! = \! 0$). Simulations were run with parameters $\boldsymbol{\mu} \! = \! \left( 0.5, 0.6, 0.4 \right)$, $\boldsymbol{\theta} \! = \! \left( 0, 0.25, 0.15 \right)$ and $\boldsymbol{f}(t) \! = \! \boldsymbol{\varphi} \! = \! \left( 0.2, 0.6, 1 \right)$, using $2\times10^{4}$ trajectories, and $4\times10^{3}$ time steps.