Functional mesoscale organization of complex networks

The network density matrix (NDM) framework, enabling an information-theoretic and multiscale treatment of network flow, has been gaining momentum over the last decade. Benefiting from the counterparts of physical functions such as free energy and entropy, NDM's applications range from estimating how nodes influence network flows across scales the centrality of nodes at the local level to explaining the emergence of structural and functional order. Here, we introduce a generalized notion of the network internal energy $E_\tau$, where $\tau$ denotes a temporal hyperparameter allowing for multi-resolution analysis, showing how it measures the leakage of dynamical correlations from arbitrary partitions, where the minimally leaky subsystems have minimal $E_\tau$. Moreover, we analytically demonstrate that $E_\tau$ reduces to the well-known modularity function at the smallest temporal scale $\tau = 0$. We investigate this peculiar resemblance by comparing the communities minimizing $E_\tau$, with those detected by widely used methods like multiscale modularity and Markov stability. Our work provides a detailed analytical and computational picture of network generalized internal energy, and explores its effectiveness in detecting communities in synthetic and empirical networks within a unifying framework.