The Informational Cost of Structure: Representational Complexity in Networked Dynamical Systems

Abstract

How much information is required to represent a dynamical system in terms of an interaction structure and an evolution rule? We address this question using algorithmic information theory. We introduce Representational Complexity, the excess description length of a structure-plus-rule model relative to the shortest possible description of the dynamics itself. This intrinsic description defines a universal lower bound: no exact structural representation can be more concise. If arbitrary rules are allowed, graphs, hypergraphs, and other formalisms can all reach this bound by shifting information between structure and dynamics, so expressiveness alone cannot distinguish them. Meaningful differences arise only when scientific modeling restricts the admissible structures and rules. Within this setting, we identify conditions under which graph and hypergraph descriptions are informationally equivalent, and show how graph-preferred, hypergraph-preferred, and mixed regimes can emerge when those conditions are relaxed. Because Kolmogorov complexity is not computable, we complement the formal results with explicit description-length estimates. Our framework reframes the choice of network representation as a question of informational cost and mechanistic transparency rather than universal expressive power.