Dynamical network markers for heterogeneous hierarchical networks.

Early-warning signals (EWS) are crucial for predicting critical transitions (CTs) in complex systems. In high-dimensional network systems, dynamical network marker (DNM) theory has been developed to obtain EWS by detecting significant fluctuations in specific subnetworks immediately before a CT. Mathematically, DNM nodes are characterized as the non-zero elements of the right eigenvector corresponding to the dominant eigenvalue of the linearly approximated system. While DNM theory has demonstrated effectiveness, particularly in biological applications, conventional approaches are limited to monolayer networks and fail to account for hierarchical structures including cell-to-cell interactions. To address this limitation, we extend DNM theory to heterogeneous hierarchical networks, analyzing their behavior before CTs through both theoretical and numerical approaches. Our findings reveal that stronger interactions necessitate a larger number of measured subnetworks but enable more precise identification of DNM nodes. These results highlight the critical role of sampling strategies in detecting CTs and contribute to a more comprehensive DNM theory for hierarchical networks.