Diversity of information pathways drives scaling and sparsity in real-world networks

Empirical complex systems must differentially respond to external perturbations and, at the same time, internally distribute information to coordinate their components. While networked backbones help with the latter, they limit the components' individual degrees of freedom and reduce their collective dynamical range. Here, we show that real-world networks are formed to optimize the gain in information flow and loss in response diversity. Encoding network states as density matrices, we demonstrate that such a trade-off mathematically resembles the thermodynamic efficiency characterized by heat and work in physical systems. Our findings explain, analytically and numerically, the sparsity and the empirical scaling law observed in hundreds of real-world networks across multiple domains. We show, through numerical experiments in synthetic and biological networks, that ubiquitous topological features such as modularity and small-worldness emerge to optimize the above trade-off for middle- to large-scale information exchange between system's units. Our results highlight that the emergence of some of the most prevalent topological features of real-world networks have a thermodynamic origin.