Graph and Hypergraph Theories Applied to Dynamic Protein-Protein Interaction Network Analysis, and Deep-Learning Frameworks for Protein Complex Network Prediction.

Protein interactions form large-scale networks known as protein-protein interaction networks (PPINs) or protein complex networks (PCNs). Extracting meaningful structural frameworks from these molecular relationships through mathematical modeling enables a deeper understanding of biological processes. Although static protein network models have provided valuable insights into the organization of PPINs, they are limited in their ability to capture the dynamic and cooperative nature of protein complexes. This review begins by introducing fundamental concepts in graph and hypergraph theory, with an emphasis on centrality measures. We then discuss the evolution of PPIN analysis from static representations to dynamic graph- and hypergraph-based frameworks. Specifically, we review dynamic PPINs and the challenges associated with their interpolation, dynamic centrality measures, and network models capable of representing multi-node relationships that have been applied to PPINs. Finally, we highlight recent advances in machine learning and deep learning approaches that integrate interaction data with functional annotations, sequence information, and cellular context to predict novel interactions and reconstruct transient protein complexes. Taken together, dynamic PPIN modeling combined with experimental validation provides an integrated framework for understanding coordinated protein functions in cellular processes and across biological systems as well as supporting drug development.