  A foremost challenge in modern network science is the inverse problem of
reconstruction (inference) of coupling equations and network topology from the
measurements of the network dynamics. Of particular interest are the methods
that can operate on real (empirical) data without interfering with the system.
One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett.99, 064101) was a
method suited for general limit-cycle oscillators, yielding both oscillators'
natural frequencies and coupling functions between them (phase equations) from
empirically measured time series. The present paper reviews the above method in
a way comprehensive to domain-scientists other than physics. It also presents
applications of the method to (i) detection of the network connectivity, (ii)
inference of the phase sensitivity function, (iii) approximation of the
interaction among phase-coherent chaotic oscillators, and (iv) experimental
data from a forced Van der Pol electric circuit. This reaffirms the range of
applicability of the method for reconstructing coupling functions and makes it
accessible to a much wider scientific community.

Chaotic Dynamics