Recently, professor Shing-Tung Yau -- the president of BIMSA -- together with research faculty members Rongling Wu, Jie Wu et al., have published an article titled "Hypernetwork modelling and topology of high-order interactions for complex systems" in PNAS. This work develops a generalized statistical mechanics model to reconstruct hypernetworks and uses GLMY homology to analyze their topological architecture, offering a new perspective on the statistical mechanical understanding of high-order interactions in complex systems.

High-order interactions (HOI) is a core aspect of complex systems. However, existing network models can only detect paired interactions, and a general model capturing HOI is not yet known. Toward this problem, the aforementioned work integrates evolutionary game theory and behavioural ecology into a unified statistical mechanical framework, in order to reconstruct bidirectional, signed, and and weighted hypernetworks. These hypernetworks are able to characterize how constituent agents are influenced by their own strategies, the strategies of coexisting agents, and strategies of interactions between other agents, as well as how directed pairwise interactions are influenced by individual agents.

This work uses a newly developed tool in algebraic topology, called GLMY homology, in order to dissect the topological architecture of nodes, links and hyperlinks in hypernetworks. The integration of statistical mechanics and GLMY homology provides a generic tool for unveiling hidden patterns in complex systems across a wide spectrum of physical and biological scenarios.

(figure; caption: hexad hypernetwork)

(figure; The GLMY homology dissection of microbial hypernetworks)

The definition of a  hypernetwork can distinguish between how pairwise interactions modulate a third agent (active HOI) and how the altered state of each agent in turn governs interactions between other agents (passive HOI). The simultaneous occurance of active and passive HOI drives complex systems to exhibit mutiple time and space scales.

In this work, the newly developed hypernetwork reconstruction was applied to model hexa-species microbial communities, and techniques from GLMY homology was used to dissect the topological architecture of these hypernetworks. It was found that pairwise interactions and HOIs play distinct roles in shaping community behavior and dynamics. The statistical relevance of the hypernetwork model is validated using a series of in vitro mono-, co-, and tricultural experiments based on three bacterial species.

In summary, the hypernetwork model established in this study offers a more effective approach to dissecting the topological architecture and mechanisms of interspecies interactions underlying community behaviors. By integrating the hypernetwork model with GLMY homology, this framework provides a comprehensive understanding of highly complex communities (such as the gut microbiota), providing crucial insights for advancements across physical and biological contexts.