Dynamics, geometry and C* algebras seminar

Speaker: : Eddie Nijholt (UIUC)

Title: Hidden Symmetry in Network Dynamical Systems

Abstract: Network dynamical systems are ubiquitous in both nature and engineering, yet they remain poorly understood. An important reason for this is that network structure is not preserved by most coordinate transformations. As a result, most established techniques from dynamical systems theory are not suited for analysing such systems. We show that for large classes of networks there is a way around this problem. In these cases, the network structure may be realised as a kind of hidden symmetry. This means that the network dynamical systems are precisely those vector fields that respect certain semi-conjugacies, either from the system to itself or involving some larger “lift” of the network. As a result, the study of network dynamical systems becomes that of equivariant (i.e. symmetrical) systems. In particular, this description allows one to adapt many important techniques from dynamical systems theory to the network setting. Moreover, it motivates an interesting generalisation of classical equivariant dynamics, moving from (compact) group symmetries to more exotic algebraic structures, such as monoids, categories and quivers.