MACHEN - Modeling, Analyzing, and Controlling Higher-ordEr Networks

While our knowledge of complex systems has grown, it often simplifies complex to pairwise interactions. This proposal advances our understanding by formalizing higher-order systems beyond conventional pairwise relationships. For example, social interactions often occur within groups; reactions in chemistry often require many reactants to generate their products; in gene expression, many genes may be required to express a given phenotype. To incorporate these types of interactions and overcome the pairwise limitation, we propose the use of higher-order networks, which aim to introduce a comprehensive language that formalizes higher-order systems, creating a general framework for modeling dynamic processes that incorporate data-rich structures and hopefully unifies both hypergraphs and simplicial complexes. Furthermore, we will explore innovative strategies for controlling these systems, using both open-loop and closed-loop approaches. Ultimately, the goal is to provide a fundamental control theory of higher-order systems that provides the necessary toolbox for studying, understanding, and controlling these systems. Some fundamental questions are: What are such systems' natural and general mathematical representations? What are their structural and dynamical bounds? Can we control/observe these systems? How can we design efficient control strategies? Despite its theoretical focus, this proposal is interdisciplinary, with potential applications in various domains. This proposal embarks on a timely endeavor to develop a control theory for higher-order systems. This endeavor is consistent with the PI's expertise, an electrical engineering background combined with his studies and contributions to the field of complex systems, and, more recently, his contributions to seminal papers on the study of higher-order systems, which motivated him to propose and pursue this line of research.