  We show that a large class of pulse coupled oscillators converge with high
probability from random initial conditions on a large class of graphs with time
delays. Our analysis combines previous local convergence results, probabilistic
network analysis, and a new classification scheme for Type II phase response
curves to produce rigorous lower bounds for convergence probabilities based on
network density. These bounds are then used to develop a simple, fast and
rigorous computational analytic technique. These results suggest new methods
for the analysis of pulse coupled oscillators, and provide new insights into
the operation of biological Type II phase response curves and also the design
of decentralized and minimal clock synchronization schemes in sensor nets.
