Tensor Vector Product-Based Dynamical Systems

The tensor vector product-based dynamical system (TVPDS) representation was pioneered in 2021 by Chen et al. [IEEE Trans. Netw. Sci. Eng.] as a novel framework for characterizing the multidimensional state dynamics of hypergraphs. Hypergraphs are generalizations of graphs where hyperedges can connect more than two nodes. The evolution of a TVPDS is characterized by the tensor vector product between a dynamic tensor and a state vector, which in fact can be equivalently reformulated in the form of homogeneous polynomial dynamical systems. By utilizing polynomial systems theory and tensor algebra, the system-theoretic properties of TVPDSs, such as stability, controllability, and observability, are investigated. In particular, the controllability and observability of hypergraphs are explored through the lens of the TVPDS representation.