Saito Yuto, Sasahara Hampei, Imura Jun-Ichi
Graduate School of Engineering, Institute of Science Tokyo, Tokyo, 152-8552, Japan.
Sci Rep. 2025 Jul 1;15(1):21177. doi: 10.1038/s41598-025-08338-y.
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.
早期预警信号（EWS）对于预测复杂系统中的临界转变（CTs）至关重要。在高维网络系统中，动态网络标记（DNM）理论已被开发出来，通过在临界转变之前立即检测特定子网络中的显著波动来获取早期预警信号。在数学上，DNM节点被表征为与线性近似系统的主导特征值相对应的右特征向量的非零元素。虽然DNM理论已证明其有效性，特别是在生物学应用中，但传统方法仅限于单层网络，无法考虑包括细胞间相互作用在内的层次结构。为了解决这一局限性，我们将DNM理论扩展到异构层次网络，通过理论和数值方法分析其在临界转变之前的行为。我们的研究结果表明，更强的相互作用需要更多数量的测量子网络，但能够更精确地识别DNM节点。这些结果突出了采样策略在检测临界转变中的关键作用，并有助于为层次网络建立更全面的DNM理论。
