Entropy is a powerful tool for the analysis of time series, as it allows describing the probability distributions of the possible state of a system, and therefore the information encoded in it. Nevertheless, important information may be codified also in the temporal dynamics, an aspect which is not usually taken into account. The idea of calculating entropy based on permutation patterns (that is, permutations defined by the order relations among values of a time series) has received a lot of attention in the last years, especially for the understanding of complex and chaotic systems. Permutation entropy directly accounts for the temporal information contained in the time series; furthermore, it has the quality of simplicity, robustness and very low computational cost. To celebrate the tenth anniversary of the original work, here we analyze the theoretical foundations of the permutation entropy, Entropy 2012, 14 1554 as well as the main recent applications to the analysis of economical markets and to the understanding of biomedical systems.

Many biological and man-made networked systems are characterized by the simultaneous presence of different sub-networks organized in separate layers, with links and nodes of qualitatively different types. While during the past few years theoretical studies have examined a variety of structural features of complex networks, the outstanding question is whether such features are characterizing all single layers, or rather emerge as a result of coarse-graining, i.e. when going from the multilayered to the aggregate network representation. Here we address this issue with the help of real data. We analyze the structural properties of an intrinsically multilayered real network, the European Air Transportation Multiplex Network in which each commercial airline defines a network layer. We examine how several structural measures evolve as layers are progressively merged together. In particular, we discuss how the topology of each layer affects the emergence of structural properties in the aggregate network.

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. The occurrence of a first-order phase transition to synchronization of an ensemble of networked phase oscillators was reported, so far, for very particular network architectures. Here, we show how a sharp, discontinuous transition can occur, instead, as a generic feature of networks of phase oscillators. Precisely, we set conditions for the transition from unsynchronized to synchronized states to be first-order, and demonstrate how these conditions can be attained in a very wide spectrum of situations. We then show how the occurrence of such transitions is always accompanied by the spontaneous setting of frequency-degree correlation features. Third, we show that the conditions for abrupt transitions can be even softened in several cases. Finally, we discuss, as a possible application, the use of this phenomenon to express magnetic-like states of synchronization.

In this Letter we identify the general rules that determine the synchronization properties of interconnected networks. We study analytically, numerically and experimentally how the degree of the nodes through which two networks are connected influences the ability of the whole system to synchronize. We show that connecting the high-degree (low-degree) nodes of each network turns out to be the most (least) effective strategy to achieve synchronization. We find the functional relation between synchronizability and size for a given network-of-networks, and report the existence of the optimal connector link weights for the different interconnection strategies. Finally, we perform an electronic experiment with two coupled star networks and conclude that the analytical results are indeed valid in the presence of noise and parameter mismatches. [5,6] or interaction between modules [7,8], can be obtained, sometimes with counter-intuitive results. Similarly, while synchronization in complex networks has been widely studied [9], very few works have investigated synchronization in N oN s. Huang et al. [10] showed that when two networks interact through random connections an exact balance between the weight of internal links in a network and the weight of links between networks results in greater synchronization between the two networks. It has also been shown that for multiple interacting networks, random connections between distant networks increase the synchronization of the complete N oN [11].Real networks exhibit high heterogeneity of the node degree, with hubs (i.e., high-degree) and peripheral (i.e., lowdegree) nodes [12]. What happens if connector links between the networks, termed inter-links, are not randomly created, but are instead chosen according to a particular connection strategy? Carlson et al. [13] analyzed the influence that lowdegree nodes may have on the collective dynamics of networks. Wang et al. [14] showed that when two neuron clusters get connected, both the heterogeneity of the network and the degree (i.e. number of connections) of the connector nodes, i.e. the nodes reached by inter-links, influence the coherent behavior of the whole system. A recent study demonstrated that the proper selection of connector nodes has strong implications on structural (centrality) and dynamical properties (spreading or population dynamics) occurring in a N oN [15].In this Letter, we study in a systematic way how connector nodes between a group of networks with heterogeneous topology affect synchronization and stability of the resulting N oN , and provide general rules for electing in a non-random fashion the connector nodes that maximize the synchronizability.The stability of the synchronized state of a group of coupled identical dynamical units is given by the corresponding Master Stability Function (MSF) [16]. For a given dynamical system and coupling form, the stability of synchronization depends on the second lowest eigenvalue λ 2 , usually called the spectral gap or algebraic connectivity, and the largest ei...

Competitive interactions represent one of the driving forces behind evolution and natural selection in biological and sociological systems 1,2 . For example, animals in an ecosystem may vie for food or mates; in a market economy, firms may compete over the same group of customers; sensory stimuli may compete for limited neural resources in order to enter the focus of attention. Here, we derive rules based on the spectral properties of the network governing the competitive interactions between groups of agents organized in networks. In the scenario studied here the winner of the competition, and the time needed to prevail, essentially depend on the way a given network connects to its competitors and on its internal structure. Our results allow assessing the extent to which real networks optimize the outcome of their interaction, but also provide strategies through which competing networks can improve on their situation. The proposed approach is applicable to a wide range of systems that can be modeled as networks 3 .Often, the outcome of a competitive process between agents is affected not only by their direct competitors but also by the specific network of connections in which they operate. Complex networks theory offers a large number of topological measures 3 , which can be derived in a simple way from the adjacency matrix G, containing the information on network connectivity. These measures can then be used to explain important dynamical and functional properties such as robustness 4,5 , synchronization 6 , spreading 7 or congestion 8,9 .Hitherto, the emphasis has been on the properties of single isolated networks. Typically, however, networks interact with other networks, while simultaneously retaining their original identity. Recently, a few studies examined how global structural properties or dynamical processes taking place on them are affected by the existence of connected networks, showing for instance that robustness 10-13 , synchronization 14 , cooperation 15,16 , transport 17 or epidemic spreading 18-21 change dramatically when considering a network of networks 22 . Interestingly, while certain types of network interdependencies may enhance vulnerability with respect to the case of isolated networks 10 , the addition of links between networks may also hinder cascading processes, such as failures in a power grid, on interconnected networks 23 . This holistic view requires a broad redefinition of classical network parameters 24 . Even more importantly, one of the major challenges lies in the identification of guidelines for how to best link networks 25 .If one considers the outcome of interaction from the perspective of a single network, which is either forced into or evaluates the benefits from interacting with another one, an important challenge arises: "How can I make the most of my interaction with another network?" To answer this question, we consider two separate networks competing for given limited resources, which in general are related to the structural properties of the network and to the out...

By combining complex network theory and data mining techniques, we provide objective criteria for optimization of the functional network representation of generic multivariate time series. In particular, we propose a method for the principled selection of the threshold value for functional network reconstruction from raw data, and for proper identification of the network's indicators that unveil the most discriminative information on the system for classification purposes. We illustrate our method by analysing networks of functional brain activity of healthy subjects, and patients suffering from Mild Cognitive Impairment, an intermediate stage between the expected cognitive decline of normal aging and the more pronounced decline of dementia. We discuss extensions of the scope of the proposed methodology to network engineering purposes, and to other data mining tasks.

One contribution of 12 to a Theme Issue 'Complex network theory and the brain'.

In vitro primary cultures of dissociated invertebrate neurons from locust ganglia are used to experimentally investigate the morphological evolution of assemblies of living neurons, as they self-organize from collections of separated cells into elaborated, clustered, networks. At all the different stages of the culture's development, identification of neurons' and neurites' location by means of a dedicated software allows to ultimately extract an adjacency matrix from each image of the culture. In turn, a systematic statistical analysis of a group of topological observables grants us the possibility of quantifying and tracking the progression of the main network's characteristics during the self-organization process of the culture. Our results point to the existence of a particular state corresponding to a small-world network configuration, in which several relevant graph's micro- and meso-scale properties emerge. Finally, we identify the main physical processes ruling the culture's morphological transformations, and embed them into a simplified growth model qualitatively reproducing the overall set of experimental observations.