Fernando J. Ballesteros

From time series to complex networks: The visibility graph

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 methods 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.

MEASURING BARYON ACOUSTIC OSCILLATIONS ALONG THE LINE OF SIGHT WITH PHOTOMETRIC REDSHIFTS: THE PAU SURVEY

Baryon Acoustic Oscillations (BAOs) provide a standard ruler of known physical length, making it one of the most promising probes of the nature of dark energy (DE). The detection of BAOs as an excess of power in the galaxy distribution at a certain scale requires measuring galaxy positions and redshifts. Transversal (or angular) BAOs measure the angular size of this scale projected in the sky and provide information about the angular distance. Line-of-sight (or radial) BAOs require very precise redshifts, but provide a direct measurement of the Hubble parameter at different redshifts, a more sensitive probe of DE. The main goal of this paper is to show that it is possible to obtain photometric redshifts with enough precision (? z ) to measure BAOs along the line of sight. There is a fundamental limitation as to how much one can improve the BAO measurement by reducing ? z . We show that ? 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 4000 ? break, by using a filter system comprising about 40 filters, each with a width close to 100 ?, covering the wavelength range from ~4000 to ~8000 ?, supplemented by two broad-band filters similar to the Sloan Digital Sky Survey u and z bands. We describe the practical implementation of this idea, a new galaxy survey project, PAU16Physics of the Accelerating Universe (PAU): http://www.ice.cat/pau., to be carried out with a telescope/camera combination with an etendue about 20 m2 deg2, equivalent to a 2 m telescope equipped with a 6 deg2 field of view camera, and covering 8000 deg2 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 iAB 22.5 in the redshift interval 0.1 < z < 0.9, with a precision ? z < 0.003(1 + z). This population has a number density n 10?3 Mpc?3 h 3 galaxies within the 9 Gpc3 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 cosmic microwave background 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.

Feigenbaum Graphs: A Complex Network Perspective of Chaos

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.

Analytical properties of horizontal visibility graphs in the Feigenbaum scenario.

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.

Self-Organized Critical Random Boolean Networks

Standard random Boolean networks display an order-disorder phase transition. We add to the standard random Boolean networks a disconnection rule that couples the control and order parameters. In this way, the system is driven to the critical line transition. Under the influence of perturbations the system points out self-organized critical behavior. Several numerical simulations have been done and compared with a proposed analytical treatment.

Quasiperiodic Graphs: Structural Design, Scaling and Entropic Properties

A novel class of graphs, here named quasiperiodic, are constructed via application of the Horizontal Visibility algorithm to the time series generated along the quasiperiodic route to chaos. We show how the hierarchy of mode-locked regions represented by the Farey tree is inherited by their associated graphs. We are able to establish, via Renormalization Group (RG) theory, the architecture of the quasiperiodic graphs produced by irrational winding numbers with pure periodic continued fraction. Finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy.

KIC 8462852: Will the Trojans return in 2021?

KIC 8462852 stood out among more than 100,000 stars in the Kepler catalogue because of the strange features of its light curve: a wide, asymmetric dimming taking up to 15 per cent of the light at D793 and a period of multiple, narrow dimmings happening approximately 700 days later. Several models have been proposed to account for this abnormal behaviour, most of which require either unlikely causes or a finely-tuned timing. We aim at offering a relatively natural solution, invoking only phenomena that have been previously observed, although perhaps in larger or more massive versions. We model the system using a large, ringed body whose transit produces the first dimming and a swarm of Trojan objects sharing its orbit that causes the second period of multiple dimmings. The resulting orbital period is

Random Boolean networks response to external periodic signals

T\approx12

NGC 1600: Cluster or Field Elliptical?

years, with a semi-major axis

Where to Look

a\approx6