Variational principles for hierarchical and modular architectures in complex networks

The structure of real-world networks often reflects a balance between efficient information integration and the preservation of functional diversity. Hierarchical and modular architectures naturally support this trade-off, enabling fast local interactions while maintaining long-term diversity across scales. Yet, a unified theoretical framework that explains the emergence and functionality of such architectures remains elusive. In this work, we address this problem through a variational principle based on the density matrix formalism. This framework defines the von Neumann entropy for networks and establishes a complete information thermodynamics, which has already been shown to predict the emergence of sparsity in complex networks. Firstly, we characterize hierarchy as a spectral property of the random-walk Laplacian, relating it to the asymmetry of the characteristic dynamical time scales. Secondly, we analyze the thermodynamic efficiency of hierarchically organized networks, showing that hierarchy reshapes the balance between short- and long-time performance and ultimately enhances integrated efficiency. These results lay the groundwork for a deeper understanding of why hierarchy ubiquitously emerges in complex systems and open new avenues to explore its role in shaping dynamics across biological, social, and technological networks.