Publisher's Note: Entropy of weighted recurrence plots [Phys. Rev. E 90, 042919 (2014)].

This corrects the article DOI: 10.1103/PhysRevE.90.

Traveling phase waves in asymmetric networks of noisy chaotic attractors.

We explore identical Rössler systems organized into two equally sized groups, among which differing positive and negative in- and out-coupling strengths are allowed. With this asymmetric coupling, we analyze patterns in the phase dynamics that coexist with chaotic amplitudes. We specifically investigate traveling phase waves where the oscillators settle on a new rhythm different from their own.

Thermodynamic characterization of networks using graph polynomials.

In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy.

Collective dynamics in two populations of noisy oscillators with asymmetric interactions.

We study two intertwined globally coupled networks of noisy Kuramoto phase oscillators that have the same natural frequency but differ in their perception of the mean field and their contribution to it. Such a give-and-take mechanism is given by asymmetric in- and out-coupling strengths which can be both positive and negative. We uncover in this minimal network of networks intriguing patterns of discordance, where the ensemble splits into two clusters separated by a constant phase lag.

Effects of assortative mixing in the second-order Kuramoto model.

In this paper we analyze the second-order Kuramoto model in the presence of a positive correlation between the heterogeneity of the connections and the natural frequencies in scale-free networks. We numerically show that discontinuous transitions emerge not just in disassortative but also in strongly assortative networks, in contrast with the first-order model. We also find that the effect of assortativity on network synchronization can be compensated by adjusting the phase damping.

Universality in the spectral and eigenfunction properties of random networks.

By the use of extensive numerical simulations, we show that the nearest-neighbor energy-level spacing distribution P(s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ≡αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that the Brody distribution characterizes well P(s) in the transition from α=0, when the vertices in the network are isolated, to α=1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality.

Entropy of weighted recurrence plots.

The Shannon entropy of a time series is a standard measure to assess the complexity of a dynamical process and can be used to quantify transitions between different dynamical regimes. An alternative way of quantifying complexity is based on state recurrences, such as those available in recurrence quantification analysis. Although varying definitions for recurrence-based entropies have been suggested so far, for some cases they reveal inconsistent results.

Cluster explosive synchronization in complex networks.

The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter we demonstrate that the nodes in a second-order Kuramoto model perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call cluster explosive synchronization. We provide a rigorous analytical treatment using a mean-field analysis in uncorrelated networks.