Geometric unfolding of synchronization dynamics on networks

Abstract

We study the synchronized state in a population of network-coupled, heterogeneous oscillators. In particular, we show that the steady-state solution of the linearized dynamics may be written as a geometric series whose subsequent terms represent different spatial scales of the network. Specifically, each additional term incorporates contributions from wider network neighborhoods. We prove that this geometric expansion converges for arbitrary frequency distributions and for both undirected and directed networks provided that the adjacency matrix is primitive. We also show that the error in the truncated series grows geometrically with the second largest eigenvalue of the normalized adjacency matrix, analogously to the rate of convergence to the stationary distribution of a random walk. Last, we derive a local approximation for the synchronized state by truncating the spatial series, at the first neighborhood term, to illustrate the practical advantages of our approach.

Self-organized explosive synchronization in complex networks: Emergence of synchronization bombs

“…Second, note that the normalized Adjacency matrix, Â, is a stochastic row sum and its spectra is bounded in µ( Â) ∈ [−1, 1], with the largest eigenvalue µ max ( Â) = 1 if the network is connected. The remaining of the spectra follows Wigner's semicircle law for random networks, becoming narrower as the link density increases, and it deviates from the random case in the presence of modules (shifting towards positive eigenvalues) or bipartite-like structures (shifting towards negative eigenvalues) [26]. Thus, from Fig.…”

“…(3). In the following we analyze the emergence of several structural and dynamical patterns that are usually associated with explosive transitions [17,26].…”

“…In principle, one needs all the spectral information of the network to estimate the value of r. However, we can leverage recent results on the geometric expansion of Eq. ( 14) [26], where it is shown that the linearized solution can be expressed as a sum of contributions from increasingly further neighborhoods. This way, the local approximation of synchrony [26] is obtained by truncating the expansion at its second term (taking into account the effect of the nearest neighbors of the nodes), leading to…”

“…Further analyses showed that this is only one of the possible mechanisms that inhibit the emergence of a large synchronization cluster and it was found that, by imposing frequency anti-correlations among connected units in the form of frequency gaps [24] or adaptive anti-Hebbian rules for the weights [25], explosive transitions occur as the coupling constant is tuned. Recently, it has been found that these degree and frequency correlations associated to explosive behavior also optimize the global phase synchronization in the system [26,27]. Furthermore, explosive transitions can also appear in multilayer and dynamically coupled systems [28,29] and they can be enhanced by the presence of noise [30] and higher-order -beyond pair-wise-interactions [31].…”

“…We first present our model of the synchronization bomb for an ensemble of Kuramoto oscillators. We introduce the optimal local rule for connecting or disconnecting units, derived from the truncated expansion of the linearized dynamics [26] under the assumptions of maximizing global synchrony with local information, and explore the basic mechanisms and phenomenology of the synchrony-driven percolation process. Second, we analyze the explosive fingerprints that emerge on the underlying structure, in the form of degree-frequency correlations, dissasortative dynamical and structural patterns, and a delayed percolation threshold.…”