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.Brownian motion ͉ complex systems ͉ fractals

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(k)=(1/3)(2/3)(k-2), independent of the probability distribution from which the series was generated. Accordingly, visibility graphs with other P(k) 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.

Baryon Acoustic Oscillations (BAO) provide a "standard ruler" of known physical length, making it one of the most promising probes of the nature of dark energy. The detection of BAO as an excess of power in the galaxy distribution at a certain scale requires measuring galaxy positions and redshifts. "Transversal" (or "angular") BAO measure the angular size of this scale projected in the sky and provide information about the angular distance. "Line-of-sight" (or "radial") BAO require very precise redshifts, but provide a direct measurement of the Hubble parameter at different redshifts, a more sensitive probe of dark energy. The main goal of this paper is to show that it is possible to obtain photometric redshifts with enough precision (σ z ) to measure BAO along the line of sight. There is a fundamental limitation as to how much one can improve the BAO -2measurement by reducing σ z . We show that σ z ∼ 0.003(1 + z) is sufficient: a much better precision will produce an oversampling of the BAO peak without a significant improvement on its detection, while a much worse precision will result in the effective loss of the radial information. This precision in redshift can be achieved for bright, red galaxies, featuring a prominent 4000Å break, by using a filter system comprising about 40 filters, each with a width close to 100Å, covering the wavelength range from ∼ 4000Å to ∼ 8000Å, supplemented by two broad-band filters similar to the SDSS u and z bands. We describe the practical implementation of this idea, a new galaxy survey project, PAU * , to be carried out with a telescope/camera combination with an etendue about 20 m 2 deg 2 , equivalent to a 2 m telescope equipped with a 6 deg 2 -FoV camera, and covering 8000 sq. deg. in the sky in four years. We expect to measure positions and redshifts for over 14 million red, early-type galaxies with L > L ⋆ and i AB 22.5 in the redshift interval 0.1 < z < 0.9, with a precision σ z < 0.003(1 + z). This population has a number density n 10 −3 Mpc −3 h 3 galaxies within the 9 (Gpc/h) 3 volume to be sampled by our survey, ensuring that the error in the determination of the BAO scale is not limited by shot-noise. By itself, such a survey will deliver precisions of order 5% in the dark-energy equation of state parameter w, if assumed constant, and can determine its time derivative when combined with future CMB measurements. In addition, PAU will yield high-quality redshift and low-resolution spectroscopy for hundreds of millions of other galaxies, including a very significant high-redshift population. The data set produced by this survey will have a unique legacy value, allowing a wide range of astrophysical studies.Subject headings: large-scale structure of universe -cosmological parameters * Physics of the Accelerating Universe (PAU): http://www.ice.cat/

The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.

Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [B. Luque et al., PLoS ONE 6, 9 (2011)] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here, we provide an in depth description of the HV treatment of the Feigenbaum scenario, together with analytical derivations that relate to the degree distributions, mean distances, clustering coefficients, etc., associated to the bifurcation cascades and their accumulation points. We describe how the resultant families of graphs can be framed into a renormalization group scheme in which fixed-point graphs reveal their scaling properties. These fixed points are then re-derived from an entropy optimization process defined for the graph sets, confirming a suggested connection between renormalization group and entropy optimization. Finally, we provide analytical and numerical results for the graph entropy and show that it emulates the Lyapunov exponent of the map independently of its sign.