Synergistic Motifs in Gaussian Systems

TL;DR

This paper establishes precise conditions under which high-order interdependencies, measured by synergy, emerge in Gaussian and related systems, revealing that pairwise interactions can generate synergy without explicit high-order mechanisms.

Contribution

It provides the first rigorous conditions linking antibalanced structures to synergy dominance in Gaussian systems and validates these insights across various network types.

Findings

Antibalanced structures are sufficient for synergy in Gaussian systems.

Antibalanced motifs are necessary for synergy in Ornstein-Uhlenbeck processes.

Pairwise interactions can produce high-order interdependencies without explicit mechanisms.

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.