Graphs of experimental results given in concentration of trace metals versus time in days

The impact of CO2 leakage on solubility and distribution of trace metals in seawater and sediment has been studied in lab scale chambers. Seven metals (Al, Cr, Ni, Pb, Cd, Cu, and Zn) were investigated in membrane-filtered seawater samples, and DGT samplers were deployed in water and sediment during the experiment. During the first phase (16 days), "dissolved" (<0.2 µm) concentrations of all elements increased substantially in the water. The increase in dissolved fractions of Al, Cr, Ni, Cu, Zn, Cd and Pb in the CO2 seepage chamber was respectively 5.1, 3.8, 4.5, 3.2, 1.4, 2.3 and 1.3 times higher than the dissolved concentrations of these metals in the control. During the second phase of the experiment (10 days) with the same sediment but replenished seawater, the dissolved fractions of Al, Cr, Cd, and Zn were partly removed from the water column in the CO2 chamber. DNi and DCu still increased but at reduced rates, while DPb increased faster than that was observed during the first phase. DGT-labile fractions (MeDGT) of all metals increased substantially during the first phase of CO2 seepage. DGT-labile fractions of Al, Cr, Ni, Cu, Zn, Cd and Pb were respectively 7.9, 2.0, 3.6, 1.7, 2.1, 1.9 and 2.3 times higher in the CO2 chamber than that of in the control chamber. AlDGT, CrDGT, NiDGT, and PbDGT continued to increase during the second phase of the experiment. There was no change in CdDGT during the second phase, while CuDGT and ZnDGT decreased by 30% and 25%, respectively in the CO2 chamber. In the sediment pore water, DGT labile fractions of all the seven elements increased substantially in the CO2 chamber. Our results show that CO2 leakage affected the solubility, particle reactivity and transformation rates of the studied metals in sediment and at the sediment-water interface. The metal species released due to CO2 acidification may have sufficiently long residence time in the seawater to affect bioavailability and toxicity of the metals to biota.

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.

Feigenbaum graphs at the onset of chaos

We analyze the properties of networks obtained from the trajectories of unimodal maps at the transi- tion to chaos via the horizontal visibility (HV) algorithm. We find that the network degrees fluctuate at all scales with amplitude that increases as the size of the network grows, and can be described by a spectrum of graph-theoretical generalized Lyapunov exponents. We further define an entropy growth rate that describes the amount of information created along paths in network space, and find that such en- tropy growth rate coincides with the spectrum of generalized graph-theoretical exponents, constituting a set of Pesin-like identities for the network.

Detecting periodicity with horizontal visibility graphs

The horizontal visibility algorithm was recently introduced as a mapping between time series and networks. The challenge lies in characterizing the structure of time series (and the processes that generated those series) using the powerful tools of graph theory. Recent works have shown that the visibility graphs inherit several degrees of correlations from their associated series, and therefore such graph theoretical characterization is in principle possible. However, both the mathematical grounding of this promising theory and its applications are in its infancy. Following this line, here we address the question of detecting hidden periodicity in series polluted with a certain amount of noise. We first put forward some generic properties of horizontal visibility graphs which allow us to define a (graph theoretical) noise reduction filter. Accordingly, we evaluate its performance for the task of calculating the period of noisy periodic signals, and compare our results with standard time domain (autocorrelation) methods. Finally, potentials, limitations and applications are discussed.

Horizontal visibility graphs generated by type-I intermittency

The type-I intermittency route to (or out of) chaos is investigated within the horizontal visibility (HV) graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct their associatedHVgraphs.We showhowthe alternation of laminar episodes and chaotic bursts imprints a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values for several network parameters. In particular, we predict that the characteristic power-law scaling of the mean length of laminar trend sizes is fully inherited by the variance of the graph degree distribution, in good agreement with the numerics. We also report numerical evidence on how the characteristic power-law scaling of the Lyapunov exponent as a function of the distance to the tangent bifurcation is inherited in the graph by an analogous scaling of block entropy functionals defined on the graph. Furthermore, we are able to recast the full set of HV graphs generated by intermittent dynamics into a renormalization-group framework, where the fixed points of its graph-theoretical renormalization-group flow account for the different types of dynamics.We also establish that the nontrivial fixed point of this flow coincides with the tangency condition and that the corresponding invariant graph exhibits extremal entropic properties.

Quasiperiodic graphs at the onset of chaos

We examine the connectivity fluctuations across networks obtained when the horizontal visibility (HV) algorithm is used on trajectories generated by nonlinear circle maps at the quasiperiodic transition to chaos. The resultant HV graph is highly anomalous as the degrees fluctuate at all scales with amplitude that increases with the size of the network. We determine families of Pesin-like identities between entropy growth rates and generalized graph-theoretical Lyapunov exponents. An irrational winding number with pure periodic continued fraction characterizes each family. We illustrate our results for the so-called golden, silver, and bronze numbers

Quasiperiodic Graphs: Structural Design, Scaling and Entropic Properties

A novel class of graphs, here named quasiperiodic, are const ructed 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 Far ey 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. And finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy