Erosion of synchronization: Coupling heterogeneity and network structure

Abstract

We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.

The Kuramoto model in complex networks

“…However, it remains to be shown whether such conditions are necessary and sufficient to assure that complete or partial synchronization are attainable. Furthermore, it would be interesting to relate these findings with the recently observed phenomenon of erosion of synchronization, which consists in the loss of perfect synchronization due to coupling frustration [90,91].…”

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

“…Note that for σ increasingly large, the frustrations λ i become small so that perfect synchronisation requires fine-tuning to vanishing values. If there is any noise in practice, then tuning becomes difficult and perfect synchronisation becomes impossible leading to "erosion" effects [28,29]. Observe now that because of the boundedness of the sine in Eq.…”

“…This means that in this regime, the order parameter shows exact phase synchronisation as a consequence of the freedom in the λ i to tune them (or, alternately phrased, there is zero "erosion" of synchronisation in the sense of [28,29] because of the freedom to tune the λ i such that the "synchrony alignment function" -see also section 7 of [30] -exactly vanishes). The threshold coupling σ r for all oscillators to be recruited in this way must be distinguished from the critical coupling σ c usually defined as value where the ensemble or time average of the order parameter r, deviates from zero.…”

Synchronization of Heterogeneous Oscillators Under Network Modifications: Perturbation and Optimization of the Synchrony Alignment Function

“…Because this approach is based on a mathematical analysis, it is much more reliable than—yet in agreement with—known heuristics for enhancing synchronization such as implementing negative correlations between the frequencies of neighboring oscillators [9, 10, 52] or incorporating positive correlations between the oscillators’ degrees and natural frequency magnitudes [9, 52]. In addition to optimization, the SAF can be used to explore fundamental limitations on phase synchronization for systems with frustrated coupling—a phenomenon referred to as the erosion of synchronization [56, 54]. In continuing to develop this theoretical framework, we recently generalized the SAF to directed networks [53].…”