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

Jiayuan Zhong, Xuerong Gu, Dandan Ding, Qiao Wei, Bowen Niu, Ting Tao, Pei Chen and Rui Liu

PLOS Computational Biology, 2026, vol. 22, issue 7, 1-21

Abstract: 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.Author summary: In complex biological processes, such as embryonic development and disease progression, there exists a critical phase or tipping point preceding the transition, where a considerable qualitative shift occurs. Accurate identification of such critical phases and their associated signaling molecules is essential for understanding the underlying mechanisms of biological processes and enabling timely interventions to prevent adverse outcomes. However, existing methods mainly rely on pairwise gene relationships, potentially overlooking higher-order interactions among multiple genes, and often exhibit limited robustness and effectiveness when applied to noisy and sparse single-cell data. To address this challenge, we developed HNPP, a higher-order network-based personalized PageRank framework for detecting critical phases from single-cell data. By incorporating higher-order collaborative structures through simplicial complexes, HNPP captures many-body interaction patterns beyond conventional pairwise relationships, providing a more accurate characterization of critical dynamics in complex biological systems. We validated the proposed method using one simulated dataset and six real-world single-cell datasets spanning embryonic development and disease progression. Our results show that HNPP identifies critical phases more effectively than existing methods, providing a useful tool for studying dynamic biological transitions.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:plo:pcbi00:1014475

DOI: 10.1371/journal.pcbi.1014475

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