Locality-Dependent Trade-offs in Stochastic Binary Systems

We investigate how local interaction range influences emergent structure, complexity, and representation independence in stochastic binary systems. Using simple locally coupled processes driven by random inputs, we systematically vary the locality scale � (interaction neighborhood size) and measure its effects on spectral structure, algorithmic complexity proxies, and cross-representation correlations. We find that locality induces a clear multi-objective trade-off rather than a single optimal regime. Intermediate interaction ranges (�, corresponding to seven-site neighborhoods) maximize structural complexity and produce near–� spectral scaling, consistent with signatures commonly associated with critical-like behavior and edge-of-chaos dynamics. In contrast, larger interaction ranges (�) maximize representation independence, reducing correlations between different bases while preserving high entropy. These results reveal a Pareto frontier governed by interaction range: increasing locality enhances long-range correlations and spectral structure but reduces representational independence, while decreasing locality suppresses structure in favor of decorrelation. No single locality scale optimizes all objectives simultaneously. Our findings provide a minimal, model-agnostic demonstration that interaction range acts as a tunable design parameter in stochastic information-processing systems. This framework offers practical guidance for the design of reservoir computers, complex networks, and adaptive systems where trade-offs between complexity, stability, and independence must be balanced.