Scalable resetting algorithms for synchronization of pulse-coupled oscillators over rooted directed graphs

We study the problem of robust global synchronization of pulse-coupled oscillators (PCOs) over directed graphs. It is known that when the digraphs are strongly connected, global synchronization can be achieved by using a class of deterministic binary set-valued resetting controllers (Poveda and Teel, 2019). However, for large-scale networks, these algorithms are not scalable because some of their tuning parameters have upper bounds of the order [Formula presented], where N is the number of agents. This paper resolves this scalability issue by presenting several new results about global synchronization of PCOs with more general network topologies using the frameworks of deterministic and stochastic hybrid dynamical systems. First, we establish that similar deterministic binary resetting algorithms can achieve robust global and fixed-time synchronization in any rooted acyclic digraph. Moreover, in this case, we show that the synchronization dynamics are now scalable as the tuning parameters of the algorithm are network independent, i.e., of order O(1). However, the algorithms cannot be further extended to all rooted digraphs. We establish this new impossibility result by introducing a counter-example with a particular rooted digraph for which global synchronization cannot be achieved, irrespective of the tuning parameters. Nevertheless, we show that if the binary resetting algorithms are modified by accommodating an Erdös–Renýi type random graph model, then the resulting stochastic resetting dynamics will guarantee global synchronization almost surely for all rooted digraphs and, moreover, the tuning parameters of the dynamics are network independent. Stability and robustness properties of the resetting algorithms are studied using the tools from set-valued hybrid dynamical systems. Numerical simulations are provided at the end of the paper for demonstration of the main results.