Multiple timescale dynamics of network adaptation with constraints

Adaptive network dynamical systems describe the co-evolution of dynamical quantities on the nodes and the dynamics of the network connections themselves. For dense networks of many nodes, the resulting dynamics are typically high-dimensional. Here, we consider adaptive dynamical systems subject to constraints on network adaptation: Asymptotically, the adaptive dynamics of network connections evolves on a low-dimensional subset of possible connectivity. Such dimension reduction may be intrinsic to the adaptation rule or arise from an additional dynamical mechanism acting on a timescale distinct from that of network adaptation. We illustrate how network adaptation with various constraints influences the dynamics of Kuramoto oscillator networks and elucidates the role of multiple timescales in shaping the dynamics. Our results shed light on why one may expect effective low-dimensional adaptation dynamics in generally high-dimensional adaptive network dynamical systems.