Designing Cluster Consensus on Higher-order Interaction Networks

This paper examines the cluster consensus design problem on higher-order interaction networks. Specifically, the higher-order interaction mechanism is captured by matrix-weighted networks that allow the interdependency across the dimensions of the agents' states, and the matrix-valued weight matrices A(ij) is an element of R-dxd associated with specific edges are further assumed to share the same nullspace for design purposes. Under mild assumptions on network connectivity, we first examine the case that the nullspace of positive semi-definite edges is spanned by a nonzero vector xi is an element of R-d and show that the predictable cluster consensus can be achieved, which is eventually located in the 1-dimensional linear space determined by span {xi} and the average of agents' initial states. Moreover, the transient state of agents in each cluster can also be explicitly characterized. Namely, the derivative of the average state of agents in each cluster is perpendicular to span {xi}. To generalize the above results, we proceed to examine the case that the nullspace of positive semi-definite edges is spanned by more than one linearly independent d-dimensional vector, in which case, analogous results can be obtained, and the explicit geometric interpretation is also provided.