Statistical Physics of Information Flow in Complex Interdependent Systems

Natural and artificial complex systems consist of many interacting units, like neurons in the brain or individuals in societies, organized into non-trivial configurations and structures that often resemble networks. Recently, the adoption of network density matrix framework has been witnessed in various computational and analytical domains dealing with structural and relational data. This relatively novel framework, grounded in the intricate physics of linear or non-linear responses to stochastic perturbations, has unraveled the multiscale properties of various complex systems, from cells to brains and social systems. This is mostly achieved thanks to the macroscopic descriptors obtained from density matrices. For instance, how diverse a system responds to external disturbances can be captured by the network Von Neumann entropy, and the network free energy proxies how fast signals transport in the system. Here, we delve deep into the theoretical underpinnings of the network density matrix framework. Drawing from the foundational principles, we highlight its relevance for analyzing empirical systems within the framework of modern network science. Each chapter of this thesis will dissect either the theory or a particular application, showcasing the versatility and depth of the framework aiming to build a comprehensive guide for fellow researchers, professionals, and academics considering the potential of network density matrices in their respective domains. More specifically, in Chapter 1, we show how density matrices can be derived from a statistical field theory, the maximum entropy principle, and a generalized framework. In Chapter 3, we analyze how robust real-world systems are against damage through the lens of our theory. In Chapter 4, we use network density matrices to identify the groups of components that form functional modules in biological networks. Finally, we dedicate Chapter 5 to a thermodynamics-like formalism obtained from the network density matrices that predicts the macroscopic properties emerging from network formation.