  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 set-valued reset controllers. However, for large-scale networks,
these algorithms are not scalable because some of their tuning parameters have
upper bounds of the order of O(1/N), where N is the number of agents. This
paper resolves this scalability issue by presenting several new results in the
context of global synchronization of PCOs with more general network topologies
using deterministic and stochastic hybrid dynamical systems. First, we
establish that similar deterministic 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
counterexample with a particular rooted digraph for which global
synchronization cannot be achieved, irrespective of the tuning of the reset
rule. Nevertheless, we show that if the resetting algorithms are modified by
accommodating an Erdos-Renyi type random graph model, then the resulting
stochastic resetting dynamics will guarantee global synchronization almost
surely for all rooted digraphs and, moreover, the tunable 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.
