HNPP: Higher-order network-based personalized PageRank for detecting critical phase in complex biological systems.

Dynamic biological processes often undergo a critical transition, where the system shifts from one stable state to another with marked qualitative changes. Identifying such a critical state and its associated signaling molecules provides insight into the mechanisms of complex biological processes and allows timely intervention to avert catastrophic outcomes. However, existing critical point detection approaches are predominantly formulated on pairwise interactions, which insufficiently capture the nonlinear and higher-order dependencies inherent in high-dimensional biological data, thereby limiting their robustness and accuracy, especially in single-cell transcriptomic analyses. To address this challenge, we propose a new framework called higher-order network-based personalized PageRank (HNPP) to identify critical phases and signaling molecules at the single-cell level. By incorporating higher-order collaborative structures, HNPP captures many-body interaction patterns that extend beyond traditional pairwise relationships, enabling a more accurate characterization and quantification for the criticality of complex biological systems. The effectiveness of our proposed HNPP has been validated using a simulated dataset and six distinct real-world single-cell datasets. In addition, the results demonstrate that HNPP exhibits enhanced early-warning capability and higher accuracy compared to existing critical point detection methods. Furthermore, the computational findings are reinforced by functional analysis of the identified signaling molecules.