2010, Donner, R.V., Zou, Y., Donges, J.F., Marwan, N., Kurths, J.

This paper presents a new approach for analysing the structural properties of time series from complex systems. Starting from the concept of recurrences in phase space, the recurrence matrix of a time series is interpreted as the adjacency matrix of an associated complex network, which links different points in time if the considered states are closely neighboured in phase space. In comparison with similar network-based techniques the new approach has important conceptual advantages, and can be considered as a unifying framework for transforming time series into complex networks that also includes other existing methods as special cases. It has been demonstrated here that there are fundamental relationships between many topological properties of recurrence networks and different nontrivial statistical properties of the phase space density of the underlying dynamical system. Hence, this novel interpretation of the recurrence matrix yields new quantitative characteristics (such as average path length, clustering coefficient, or centrality measures of the recurrence network) related to the dynamical complexity of a time series, most of which are not yet provided by other existing methods of nonlinear time series analysis. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

2014, Klimm, F., Borge-Holthoefer, J., Wessel, N., Kurths, J., Zamora-Lopez, G.

The analysis of complex networks is devoted to the statistical characterization of the topology of graphs at different scales of organization in order to understand their functionality. While the modular structure of networks has become an essential element to better apprehend their complexity, the efforts to characterize the mesoscale of networks have focused on the identification of the modules rather than describing the mesoscale in an informative manner. Here we propose a framework to characterize the position every node takes within the modular configuration of complex networks and to evaluate their function accordingly. For illustration, we apply this framework to a set of synthetic networks, empirical neural networks, and to the transcriptional regulatory network of the Mycobacterium tuberculosis. We find that the architecture of both neuronal and transcriptional networks are optimized for the processing of multisensory information with the coexistence of well-defined modules of specialized components and the presence of hubs conveying information from and to the distinct functional domains.

2014, Traxl, D., Boers, N., Kurths, J.

The effects of white noise and global coupling strength on the maximum degree of synchronization in complex networks are explored. We perform numerical simulations of generic oscillator models with both linear and non-linear coupling functions on a broad spectrum of network topologies. The oscillator models include the Fitzhugh-Nagumo model, the Izhikevich model and the Kuramoto phase oscillator model. The network topologies range from regular, random and highly modular networks to scale-free and small-world networks, with both directed and undirected edges. We then study the dependency of the maximum degree of synchronization on the global coupling strength and the noise intensity. We find a general scaling of the synchronizability, and quantify its validity by fitting a regression model to the numerical data.

2017, Mitra, C., Kittel, T., Choudhary, A., Kurths, J., Donner, R.V.

Maintaining the synchronous motion of dynamical systems interacting on complex networks is often critical to their functionality. However, real-world networked dynamical systems operating synchronously are prone to random perturbations driving the system to arbitrary states within the corresponding basin of attraction, thereby leading to epochs of desynchronized dynamics with a priori unknown durations. Thus, it is highly relevant to have an estimate of the duration of such transient phases before the system returns to synchrony, following a random perturbation to the dynamical state of any particular node of the network. We address this issue here by proposing the framework of single-node recovery time (SNRT) which provides an estimate of the relative time scales underlying the transient dynamics of the nodes of a network during its restoration to synchrony. We utilize this in differentiating the particularly slow nodes of the network from the relatively fast nodes, thus identifying the critical nodes which when perturbed lead to significantly enlarged recovery time of the system before resuming synchronized operation. Further, we reveal explicit relationships between the SNRT values of a network, and its global relaxation time when starting all the nodes from random initial conditions. Earlier work on relaxation time generally focused on investigating its dependence on macroscopic topological properties of the respective network. However, we employ the proposed concept for deducing microscopic relationships between topological features of nodes and their respective SNRT values. The framework of SNRT is further extended to a measure of resilience of the different nodes of a networked dynamical system. We demonstrate the potential of SNRT in networks of Rössler oscillators on paradigmatic topologies and a model of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics illustrating the conceivable practical applicability of the proposed concept.

2014, Schultz, P., Heitzig, J., Kurths, J.

To analyse the relationship between stability against large perturbations and topological properties of a power transmission grid, we employ a statistical analysis of a large ensemble of synthetic power grids, looking for significant statistical relationships between the single-node basin stability measure and classical as well as tailormade weighted network characteristics. This method enables us to predict poor values of single-node basin stability for a large extent of the nodes, offering a node-wise stability estimation at low computational cost. Further, we analyse the particular function of certain network motifs to promote or degrade the stability of the system. Here we uncover the impact of so-called detour motifs on the appearance of nodes with a poor stability score and discuss the implications for power grid design.

2012, Ge, T., Cui, Y., Lin, W., Kurths, J., Liu, C.

In this paper, we propose a new approach to characterize time series with noise perturbations in both the time and frequency domains by combining Granger causality and complex networks. We construct directed and weighted complex networks from time series and use representative network measures to describe their physical and topological properties. Through analyzing the typical dynamical behaviors of some physical models and the MIT-BIH 7 human electrocardiogram data sets, we show that the proposed approach is able to capture and characterize various dynamics and has much potential for analyzing real-world time series of rather short length.