Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements. Important related issues covered in this work comprise the representation of the evolution of complex networks in terms of trajectories in several measurement spaces, the analysis of the correlations between some of the most traditional measurements, perturbation analysis, as well as the use of multivariate statistics for feature selection and network classification. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the proper application and interpretation of measurements.

Synchronization of an ensemble of oscillators is an emergent phenomenon present in several complex systems, ranging from social and physical to biological and technological systems. The most successful approach to describe how coherent behavior emerges in these complex systems is given by the paradigmatic Kuramoto model. This model has been traditionally studied in complete graphs. However, besides being intrinsically dynamical, complex systems present very heterogeneous structure, which can be represented as complex networks. This report is dedicated to review main contributions in the field of synchronization in networks of Kuramoto oscillators. In particular, we provide an overview of the impact of network patterns on the local and global dynamics of coupled phase oscillators. We cover many relevant topics, which encompass a description of the most used analytical approaches and the analysis of several numerical results. Furthermore, we discuss recent developments on variations of the Kuramoto model in networks, including the presence of noise and inertia. The rich potential for applications is discussed for special fields in engineering, neuroscience, physics and Earth science. Finally, we conclude by discussing problems that remain open after the last decade of intensive research on the Kuramoto model and point out some promising directions for future research.

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Many real-world systems can be studied in terms of pattern recognition tasks, so that proper use (and understanding) of machine learning methods in practical applications becomes essential. While many classification methods have been proposed, there is no consensus on which methods are more suitable for a given dataset. As a consequence, it is important to comprehensively compare methods in many possible scenarios. In this context, we performed a systematic comparison of 9 well-known clustering methods available in the R language assuming normally distributed data. In order to account for the many possible variations of data, we considered artificial datasets with several tunable properties (number of classes, separation between classes, etc). In addition, we also evaluated the sensitivity of the clustering methods with regard to their parameters configuration. The results revealed that, when considering the default configurations of the adopted methods, the spectral approach tended to present particularly good performance. We also found that the default configuration of the adopted implementations was not always accurate. In these cases, a simple approach based on random selection of parameters values proved to be a good alternative to improve the performance. All in all, the reported approach provides subsidies guiding the choice of clustering algorithms.

A central issue in ecology is the defi nition and identifi cation of keystone species, i.e. species that are relatively more important than others for maintaining community structure and ecosystem functioning. Network theory has been pointed out as a robust theoretical framework to enhance the operationality of the keystone species concept. We used the concept of centrality as a proxy for a species ' relative importance for the structure of seed dispersal networks composed of either frugivorous bats or birds and their food-plants. Centrality was expected to be determined mainly by dietary specialization, but also by body mass and geographic range size. Across 15 Neotropical datasets, only specialized frugivore species reached the highest values of centrality. Furthermore, the centrality of specialized frugivores varied widely within and among networks, whereas that of secondary and opportunistic frugivores was consistently low. A mixed-eff ects model showed that centrality was best explained by dietary specialization, but not by body mass or range size. Furthermore, the relationship between centrality and those three ecological correlates diff ered between bat -and bird -fruit networks. Our fi ndings suggest that dietary specialization is key to understand what makes a frugivore species a keystone in seed dispersal networks, and that taxonomic identity also plays a signifi cant role. Specialized frugivores may play a central role in network structuring and ecosystem functioning, which has important implications for conservation and restoration.

Pattern recognition has been employed in a myriad of industrial, commercial and academic applications. Many techniques have been devised to tackle such a diversity of applications. Despite the long tradition of pattern recognition research, there is no technique that yields the best classification in all scenarios. Therefore, as many techniques as possible should be considered in high accuracy applications. Typical related works either focus on the performance of a given algorithm or compare various classification methods. In many occasions, however, researchers who are not experts in the field of machine learning have to deal with practical classification tasks without an in-depth knowledge about the underlying parameters. Actually, the adequate choice of classifiers and parameters in such practical circumstances constitutes a long-standing problem and is one of the subjects of the current paper. We carried out a performance study of nine well-known classifiers implemented in the Weka framework and compared the influence of the parameter configurations on the accuracy. The default configuration of parameters in Weka was found to provide near optimal performance for most cases, not including methods such as the support vector machine (SVM). In addition, the k-nearest neighbor method frequently allowed the best accuracy. In certain conditions, it was possible to improve the quality of SVM by more than 20% with respect to their default parameter configuration.

Spreading processes have been largely studied in the literature, both analytically and by means of large-scale numerical simulations. These processes mainly include the propagation of diseases, rumors and information on top of a given population. In the last two decades, with the advent of modern network science, we have witnessed significant advances in this field of research. Here we review the main theoretical and numerical methods developed for the study of spreading processes on complex networked systems. Specifically, we formally define epidemic processes on single and multilayer networks and discuss in detail the main methods used to perform numerical simulations. Throughout the review, we classify spreading processes (disease and rumor models) into two classes according to the nature of time: (i) continuous-time and (ii) cellular automata approach, where the second one can be further divided into synchronous and asynchronous updating schemes. Our revision includes the heterogeneous mean-field, the quenched-mean field, and the pair quenched mean field approaches, as well as their respective simulation techniques, emphasizing similarities and differences among the different techniques. The content presented here offers a whole suite of methods to study epidemiclike processes in complex networks, both for researchers without previous experience in the subject and for experts. 6

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the SIS and SIR dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasi-stationary state method we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: if the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we verified the barrier effect, i.e., for three-layer configuration, when the layer with the largest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems opening new possibilities for the study of spreading processes.Comment: Revised version. 25 pages and 18 figure