Complexity Flow and Emergent Topology in Neural Networks

Investigating the emergence and flow of complexity in neural networks allows us to uncover new insights into the behavior of machine learning models. In this work, we apply statistical and field-theoretic methods to study convolutional neural networks through the lens of complex systems. We represent trained networks as graphs and investigate how their topological and spectral properties evolve under training, drawing analogies to renormalization group (RG) flow and phase transitions in disordered systems. Using the generalized Rosenzweig-Porter (gRP) model as a physical analog, we explore correlations in the network's weight matrices via graph metrics, scaling analyses, and multi-point correlators inspired by spin-glass and quantum information theory. This approach allows us to search for signatures of universality or critical behavior in learning dynamics. Our framework aims to uncover whether neural networks exhibit emergent phases or critical behavior as a function of training depth and sparsification. Preliminary results and key findings will highlight signatures of complexity flow, potential universality classes, and their implications for both computational and physical models of learning.