Network hierarchy entropy for quantifying graph dissimilarity

Abstract Quantifying subtle structural differences between networks remains a critical challenge across diverse scientific disciplines. Traditional network comparison methods often overlook the crucial role of edges and their interactions with nodes, thereby limiting their ability to capture complex structural dissimilarity governed by node-edge interplay. Here, we introduce a dissimilarity measure based on network hierarchy entropy, defined via the cross-entropy between node-level and edge-level distance distributions. This measure captures multiscale structural complexity by integrating hierarchical information encoded in shortest-path distributions across nodes and edges. Extensive experiments on synthetic and empirical networks show that this measure effectively discriminates fine-grained variations between networks with identical mesoscopic structures and robustly tracks evolving topologies in dynamic networks. It achieves 74.62% classification accuracy in distinguishing enzyme from non-enzyme proteins, comparable to state-of-the-art supervised learning models but without requiring feature engineering.