  Synchronization in the networks of coupled oscillators is a widely studied
topic in different areas. It is well-known that synchronization occurs if the
connectivity of the network dominates heterogeneity of the oscillators. Despite
extensive study on this topic, the quest for sharp closed-form synchronization
tests is still in vain. In this paper, we present an algorithm for finding the
Taylor expansion of the inverse Kuramoto map. We show that this Taylor series
can be used to obtain a hierarchy of increasingly accurate approximate tests
with low computational complexity. These approximate tests are then used to
estimate the threshold of synchronization as well as the position of the
synchronization manifold of the network.
