Synergistic Motifs in Gaussian Systems

Abstract

High-order interdependencies are central features of complex systems, yet a mechanistic explanation for their emergence remains elusive. Currently, it is unknown under what conditions high-order interdependencies, quantified by the information-theoretic construct of synergy, arise in systems governed by pairwise interactions. We solve this problem by providing precise sufficient and necessary conditions for when synergy prevails over low-order interdependencies in the weak interaction regime, namely, we prove that antibalanced (highly frustrated) correlational structures in Gaussian systems are sufficient for synergy-dominance and that antibalanced interaction motifs in Ornstein-Uhlenbeck processes are necessary for synergy-dominance. We validate the applicability of these analytical insights beyond the weak interaction regime, as well as in Ising, oscillatory, and empirical networks from multiple domains. Our results demonstrate that pairwise interactions can give rise to synergistic information in the absence of explicit high-order mechanisms, and highlight structural balance theory as an instrumental conceptual framework to study high-order interdependencies.

Figures (14)

• Figure 1: Examples of synergistic motifs of pairwise correlations that ensure synergy-dominance. These structures are either antibalanced or only contain closed walks $w$ of even length with positive weight $\sigma(w)$. Missing edges indicate uncorrelated relationships.
• Figure 2: Antibalanced interaction motifs are necessary for synergy-dominance in dynamical Gaussian systems with strong interactions. Mean $\Omega(\boldsymbol{X})$ as a function of the number of antibalanced triangles in the interaction matrix. Lower bounds indicate the lowest $\Omega(\textbf{X})$ encountered in our numerical exploration across all configurations with a specific no. of antibalanced triangles. Each mean $\Omega(\textbf{X})$ is coloured according to its mean balance-energy value (Eq. \ref{['eqn:structural-energy']}). Lower bounds are coloured with the energy value of the configuration that resulted in the lowest recorded $\Omega(\textbf{X})$, not the highest recorded energy (not shown). Each $A\in\mathbb{R}^{8\times8}$ has $\rho(A) \approx 0.88$ (see SM SM).
• Figure 3: Antibalanced interaction motifs characterize synergy-dominated complex systems with pairwise coupling. Plots show $\Omega$ versus the number of antibalanced triangles in the pairwise coupling matrix. (a) For the Ising model we compute $\Omega$ exactly for each possible coupling configuration ($\beta=0.25$). Darker markers indicate multiple configurations with identical values of $\Omega$ and no. of antibalanced triangles (b) For the SL network of oscillators we report the mean $\Omega$ (circle) over $100$ initial conditions and the minimum $\Omega$ (star) found (lower bound) across configurations with a specific no. of antibalanced triangles. See SM SM for implementation details.
• Figure 4: Structural balance-energy predicts synergy: the most synergistic $N$-plets maximise $U$ and positive $U$ guarantees synergy-dominance.(a)$N$-plets from a multivariate time-series of $9$ foreign-exchange (FX) rate logarithmic returns. (b)$N$-plets from resting-state fMRI functional-connectivity matrices ($229$ brain regions).
• Figure S5: Mean relative error between $\Omega$ computed using Eq. ($1$) in the main text and Eq. \ref{['eqn:o-info-full-walk-expansion']} up to walks of length $k=k_{max}=8$ for different spectral radii $\rho(W)$.