Paolo Lapo Cerni — Università di Padova #
Emergence of hierarchical modularity in empirical interconnected systems #
Hierarchical modularity is a ubiquitous feature of complex systems, from biological and neural networks to social and technological infrastructures, yet the general physical principle underlying its emergence remains unclear. Here we show that hierarchical interactions can arise as the optimal architecture for multiscale information processing in sparse interconnected systems. Using the network density matrix formalism, we define a scale-dependent thermodynamic efficiency that quantifies the trade-off between information transport, dynamical trapping, and response diversity across propagation scales. We then introduce an integrated efficiency, obtained by aggregating this quantity over all scales, as a macroscopic variational principle for network organization. Analytical arguments and numerical experiments show that hierarchical modular architectures maximize integrated efficiency. Therefore, such architectures are not merely a descriptive hallmark of complexity, but the emergent outcome of a general information-thermodynamic optimization principle governing multiscale dynamics in complex systems.