A geometric scaling between collective organizations and interaction-space dimension

Abstract

The number of stable macroscopic organizations in complex systems is often much smaller than the large number of microscopic degrees of freedom would suggest. Yet theoretical approaches rarely address whether general limits constrain the diversity of admissible macroscopic organizations. We develop a geometric framework in which interactions among system components define a coarse-grained interaction space endowed with a metric structure. When this space has finite intrinsic dimensionality, geometric packing constraints impose bounds on the number of mutually distinguishable collective organizations. We derive a dimension-dependent scaling law showing that the number of stable macroscopic regimes grows polynomially with exponent equal to the intrinsic dimensionality of the interaction space. This implies that increasing microscopic complexity alone does not necessarily expand the range of macroscopic organizations. Instead, diversification requires an increase in the dimensionality of effective interactions. To illustrate our approach, we analyze an interacting system in which collective regimes correspond to regions of a low-dimensional parameter space describing effective interactions. In this setting, geometric packing constrains the number of robust organizations that the system can support. Overall, we argue that dimensionality of interaction space may act as a control parameter governing a variety of collective organization across physical and biological systems.