Parameter-free community detection by an operator on eigenvectors of the network adjacency matrix

Community detection is an important problem in network science, potentially enabling structure discovery in key datasets. Existing computational algorithms for detecting communities rely on optimization procedures that require prior knowledge on the number of communities or several free parameters that must be manually tuned. Here, we introduce a fully analytical, parameter-free method for detecting communities in weighted, directed networks, using only the eigenspectrum of the network adjacency matrix. This approach includes a new and robust technique for estimating the number of communities. We prove theoretical guarantees for successful community detection in networks with clustered connectivity and demonstrate the robustness of these guarantees through simulations across a range of noise. We then demonstrate how this mathematical construction can inform the design of specific dynamics in networks of nonlinear oscillators. Taken together, these results open a new path for community detection grounded in mathematical operations on the network adjacency matrix and its eigenspectrum, as an alternative to optimization-based algorithms.