Galen J. Wilkerson
Understanding how network structure constrains and enables information processing is a central problem in the statistical mechanics of interacting systems. Here we study random networks across the structural percolation transition and analyze how connectivity governs realizable input-output transformations under cascade dynamics. Using Erdos-Renyi networks as a minimal ensemble, we examine structural, functional, and information-theoretic observables as functions of mean degree. We find that the emergence of the giant connected component coincides with a sharp transition in realizable information processing: complex input-output response functions become accessible, functional diversity increases rapidly, output entropy rises, and directed information flow, quantified by transfer entropy, extends beyond local neighborhoods. We term this coincidence of structural, functional, and informational transitions functional percolation, referring to a sharp expansion of the space of realizable input-output functions at the percolation threshold. Near criticality, networks exhibit a Pareto-optimal tradeoff between functional complexity and diversity, suggesting that percolation criticality may provide a general organizing principle of information processing capacity in systems with local interactions and propagating influences.

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