  To understand and control the dynamics of coupled oscillators, it is
important to reveal the structure of the interaction network from observed
data. While various techniques have been developed for inferring the network of
asynchronous systems, it remains challenging to infer the network of
synchronized oscillators without external stimulations. In this study, we
develop a method for non-invasively inferring the network of synchronized
and/or de-synchronized oscillators. An approach to network inference would be
to fit the data to a set of differential equations describing the dynamics of
phase oscillators. However, we show that this method fails to infer the true
network due to the problems that arise when we use short-time phase
differences. Therefore, we propose a method based on the circle map, which
describes the phase change in one oscillatory cycle. We demonstrate the
efficacy of the proposed method through the successful inference of the network
structure from simulated data of limit cycle oscillator models. Our method
provides a unified and concise framework for network estimation for a wide
class of oscillator systems.
