In this work we present a simple and fast computational method, the visibility algorithm , that converts a time series into a graph. The constructed graph inherits several properties of the series in its structure. Thereby, periodic series convert into regular graphs, and random series do so into random graphs. Moreover, fractal series convert into scale-free networks, enhancing the fact that power law degree distributions are related to fractality, something highly discussed recently. Some remarkable examples and analytical tools are outlined to test the method's reliability. Many different measures, recently developed in the complex network theory, could by means of this new approach characterize time series from a new point of view.

The visibility algorithm has been recently introduced as a mapping between time series and complex networks. This procedure allows us to apply methods of complex network theory for characterizing time series. In this work we present the horizontal visibility algorithm, a geometrically simpler and analytically solvable version of our former algorithm, focusing on the mapping of random series (series of independent identically distributed random variables). After presenting some properties of the algorithm, we present exact results on the topological properties of graphs associated with random series, namely, the degree distribution, the clustering coefficient, and the mean path length. We show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series since any random series maps to a graph with an exponential degree distribution of the shape $P\left(k\right)=\left(1/3\right){\left(2/3\right)}^{k-2}$, independent of the probability distribution from which the series was generated. Accordingly, visibility graphs with other $P\left(k\right)$ are related to nonrandom series. Numerical simulations confirm the accuracy of the theorems for finite series. In a second part, we show that the method is able to distinguish chaotic series from independent and identically distributed (i.i.d.) theory, studying the following situations: (i) noise-free low-dimensional chaotic series, (ii) low-dimensional noisy chaotic series, even in the presence of large amounts of noise, and (iii) high-dimensional chaotic series (coupled map lattice), without needs for additional techniques such as surrogate data or noise reduction methods. Finally, heuristic arguments are given to explain the topological properties of chaotic series, and several sequences that are conjectured to be random are analyzed. DOI: http://dx.doi.org/10.1103/PhysRevE.80.046103 © 2009 The American Physical Society

Using Network Science to Analyse Football Passing Networks: Dynamics, Space, Time, and the Multilayer Nature of the Game

Objective: To evaluate the feasibility and reproducibility of fetal lung texture analysis using a novel automatic quantitative ultrasound analysis and to assess its correlation with gestational age. Methods: Prospective cross-sectional observational study. To evaluate texture features, 957 left and right lung images in a 2D four-cardiac-chamber view plane were previously delineated from fetuses between 20 and 41 weeks of gestation. Quantification of lung texture was performed by the Automatic Quantitative Ultrasound Analysis (AQUA) software to extract image features. A standard learning approach composed of feature transformation and a regression model was used to evaluate the association between texture features and gestational age. Results: The association between weeks of gestation and fetal lung texture quantified by the AQUA software presented a Pearson correlation of 0.97. The association was not influenced by delineation parameters such as region of interest (ROI) localization, ROI size, right/left lung selected or sonographic parameters such as ultrasound equipment or transducer used. Conclusions: Fetal lung texture analysis measured by the AQUA software demonstrated a strong correlation with gestational age. This supports further research to explore the use of this technology to the noninvasive prediction of fetal lung maturity.