Reconstruction of a random phase dynamics network from observations

Abstract

We consider networks of coupled phase oscillators of different complexity: KuramotoDaido-type networks, generalized Winfree networks, and hypernetworks with triple interactions. For these setups an inverse problem of reconstruction of the network connections and of the coupling function from the observations of the phase dynamics is addressed. We show how a reconstruction based on the minimization of the squared error can be implemented in all these cases. Examples include random networks with full disorder both in the connections and in the coupling functions, as well as networks where the coupling functions are taken from experimental data of electrochemical oscillators. The method can be directly applied to asynchronous dynamics of units, while in the case of synchrony, additional phase resettings are necessary for reconstruction.

Model reconstruction from temporal data for coupled oscillator networks

“…Although the approach of [17] outlined above is effective, it is limited by the requirement that the coupling function be known a priori. In [28], Pikovsky addresses this limitation by expressing Γ as a Fourier series so that it, too, can be estimated by solving an analogous optimization problem. However, a challenge remains.…”

“…If one represents the coupling function as in (3), then the system of equations is no longer linear as terms of the form A kj a n and A kj b n appear. Pikovsky [28] circumvents this issue by defining distinct coupling functions…”

Accelerated reference frames (ARFs) reveal networks from time series data

“…n N 0 i  -. Whereas exactly synchronous or phase-locked dynamics in principle can generally not reveal the complete network topology, inferring from transient dynamics towards synchrony or locking was so far restricted to driving-response settings with known signals [16] or to general model-free approaches using a large repertoire of functions [11,12,14,15]. While the former strategy allows to create linear mappings from recorded dynamics to network topology, the latter allows to infer links from transient dynamics following an unknown driving or perturbation.…”

“…Researchers routinely resort to indirect methods to infer the physical interactions from the networks' collective dynamics [9]. State-of-the-art approaches infer physical interactions via ODE modeling using large repertoire of functions [10][11][12][13][14]. Such approaches require the entire dynamics to admit a sparse representation in the chosen repertoire, which is difficult to satisfy if no prior information is provided.…”

“…This constraint comes about because in invariant states, the dynamics of each unit exhibits a strict functional dependence on the dynamics of the other units in the system such that the information contained in observed time series is insufficient for inferring interactions, no matter how many time series and how many data points per series are observed. As a consequence, practical approaches [14,[20][21][22][23][24] exploit transient dynamics towards synchronized or locked states such that the dynamics are not exactly invariant and do contain sufficient information for inferring network interactions, at least in principle. For simple classes of phase oscillators with constant, stateindependent frequencies exhibiting exactly phase-locked or fully synchronous dynamics, transforming time series to a frame of reference that is co-moving with the dynamics is often possible by subtracting a term −Ωt, containing the collective frequency Ω of the phase dynamics of each oscillator.…”