Tree networks of real-world data: analysis of efficiency and spatiotemporal scales

Hierarchical tree structures are common in many real-world systems, from tree roots and branches to neuronal dendrites and biologically inspired artificial neural networks, as well as in technological networks for organizing and searching complex datasets of high-dimensional patterns. Within the class of hierarchical self-organized systems, we investigate the interplay of structure and function, associated with the emergence of complex tree structures in disordered environments. Using an algorithm that creates and searches trees of real-world patterns, our work stands at the intersection of statistical physics, machine learning, and network theory. We resolve the network properties over multiple phase transitions and across a continuity of scales, using the von Neumann entropy, its generalized susceptibility, and the recent definition of thermodynamic-like quantities, such as work, heat, and efficiency. We show that scale-invariance, i.e. power-law Laplacian spectral density, is a key feature to construct trees capable of combining fast information flow and sufficiently rich internal representation of information, enabling the system to achieve its functional task efficiently. Moreover, the complexity of the environmental conditions the system has adapted to is encoded in the value of the exponent of the power-law spectral density, inherently related to the network spectral dimension, and directly influencing the traits of those functionally efficient networks. Thereby, we provide a novel metric to estimate the complexity of high-dimensional datasets.