Impact of local network characteristics on network reconstruction

Inferring connectivity of an oscillatory network via the phase dynamics reconstruction

We review an approach for reconstructing oscillatory networks' undirected and directed connectivity from data. The technique relies on inferring the phase dynamics model. The central assumption is that we observe the outputs of all network nodes. We distinguish between two cases. In the first one, the observed signals represent smooth oscillations, while in the second one, the data are pulse-like and can be viewed as point processes. For the first case, we discuss estimating the true phase from a scalar signal, exploiting the protophase-to-phase transformation. With the phases at hand, pairwise and triplet synchronization indices can characterize the undirected connectivity. Next, we demonstrate how to infer the general form of the coupling functions for two or three oscillators and how to use these functions to quantify the directional links. We proceed with a different treatment of networks with more than three nodes. We discuss the difference between the structural and effective phase connectivity that emerges due to high-order terms in the coupling functions. For the second case of point-process data, we use the instants of spikes to infer the phase dynamics model in the Winfree form directly. This way, we obtain the network's coupling matrix in the first approximation in the coupling strength.

Asymmetry in the Kuramoto model with nonidentical coupling

Synchronization and desynchronization of coupled oscillators appear to be the key property of many physical systems. It is believed that to predict a synchronization (or desynchronization) event, the knowledge on the exact structure of the oscillatory network is required. However, natural sciences often deal with observations where the coupling coefficients are not available. In the present paper we suggest a way to characterize synchronization of two oscillators without the reconstruction of coupling. Our method is based on the Kuramoto chain with three oscillators with constant but nonidentical coupling. We characterize coupling in this chain by two parameters: the coupling strength s and disparity σ. We give an analytical expression of the boundary s_{max} of synchronization occurred when s>s_{max}. We propose asymmetry A of the generalized order parameter induced by the coupling disparity as a new characteristic of the synchronization between two oscillators. For the chain model with three oscillators we present the self-consistent inverse problem. We explore scaling properties of the asymmetry A constructed for the inverse problem. We demonstrate that the asymmetry A in the chain model is maximal when the coupling strength in the model reaches the boundary of synchronization s_{max}. We suggest that the asymmetry A may be derived from the phase difference of any two oscillators if one pretends that they are edges of an abstract chain with three oscillators. Performing such a derivation with the general three-oscillator Kuramoto model, we show that the crossover from the chain to general network of oscillators keeps the interrelation between the asymmetry A and synchronization. Finally, we apply the asymmetry A to describe synchronization of the solar magnetic field proxies and discuss its potential use for the forecast of solar cycle anomalies.

Statistical inference links data and theory in network science

The number of network science applications across many different fields has been rapidly increasing. Surprisingly, the development of theory and domain-specific applications often occur in isolation, risking an effective disconnect between theoretical and methodological advances and the way network science is employed in practice. Here we address this risk constructively, discussing good practices to guarantee more successful applications and reproducible results. We endorse designing statistically grounded methodologies to address challenges in network science. This approach allows one to explain observational data in terms of generative models, naturally deal with intrinsic uncertainties, and strengthen the link between theory and applications.

Theoretical models and structures recovered from measured data serve for analysis of complex networks. The authors discuss here existing gaps between theoretical methods and real-world applied networks, and potential ways to improve the interplay between theory and applications.

Network structure from a characterization of interactions in complex systems

Many natural and man-made complex dynamical systems can be represented by networks with vertices representing system units and edges the coupling between vertices. If edges of such a structural network are inaccessible, a widely used approach is to identify them with interactions between vertices, thereby setting up a functional network. However, it is an unsolved issue if and to what extent important properties of a functional network on the global and the local scale match those of the corresponding structural network. We address this issue by deriving functional networks from characterizing interactions in paradigmatic oscillator networks with widely-used time-series-analysis techniques for various factors that alter the collective network dynamics. Surprisingly, we find that particularly key constituents of functional networks—as identified with betweenness and eigenvector centrality—coincide with ground truth to a high degree, while global topological and spectral properties—clustering coefficient, average shortest path length, assortativity, and synchronizability—clearly deviate. We obtain similar concurrences for an empirical network. Our findings are of relevance for various scientific fields and call for conceptual and methodological refinements to further our understanding of the relationship between structure and function of complex dynamical systems.