Bartolo Luque

The architecture of mutualistic networks minimizes competition and increases biodiversity

The main theories of biodiversity either neglect species interactions or assume that species interact randomly with each other. However, recent empirical work has revealed that ecological networks are highly structured, and the lack of a theory that takes into account the structure of interactions precludes further assessment of the implications of such network patterns for biodiversity. Here we use a combination of analytical and empirical approaches to quantify the influence of network architecture on the number of coexisting species. As a case study we consider mutualistic networks between plants and their animal pollinators or seed dispersers. These networks have been found to be highly nested, with the more specialist species interacting only with proper subsets of the species that interact with the more generalist. We show that nestedness reduces effective interspecific competition and enhances the number of coexisting species. Furthermore, we show that a nested network will naturally emerge if new species are more likely to enter the community where they have minimal competitive load. Nested networks seem to occur in many biological and social contexts, suggesting that our results are relevant in a wide range of fields.

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.

The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion

Fractional Brownian motion (fBm) has been used as a theoretical framework to study real-time series appearing in diverse scientific fields. Because of its intrinsic nonstationarity and long-range dependence, its characterization via the Hurst parameter, H, requires sophisticated techniques that often yield ambiguous results. In this work we show that fBm series map into a scale-free visibility graph whose degree distribution is a function of H. Concretely, it is shown that the exponent of the power law degree distribution depends linearly on H. This also applies to fractional Gaussian noises (fGn) and generic f−β noises. Taking advantage of these facts, we propose a brand new methodology to quantify long-range dependence in these series. Its reliability is confirmed with extensive numerical simulations and analytical developments. Finally, we illustrate this method quantifying the persistent behavior of human gait dynamics.

Time series irreversibility: a visibility graph approach

We propose a method to measure real-valued time series irreversibility which combines two different tools: the horizontal visibility algorithm and the Kullback-Leibler divergence. This method maps a time series to a directed network according to a geometric criterion. The degree of irreversibility of the series is then estimated by the Kullback-Leibler divergence (i.e. the distinguishability) between the inand outdegree distributions of the associated graph. The method is computationally efficient and does not require any ad hoc symbolization process. We find that the method correctly distinguishes between reversible and irreversible stationary time series, including analytical and numerical studies of its performance for: (i) reversible stochastic processes (uncorrelated and Gaussian linearly correlated), (ii) irreversible stochastic processes (a discrete flashing ratchet in an asymmetric potential), (iii) reversible (conservative) and irreversible (dissipative) chaotic maps, and (iv) dissipative chaotic maps in the presence of noise. Two alternative graph functionals, the degree and the degree-degree distributions, can be used as the Kullback-Leibler divergence argument. The former is simpler and more intuitive and can be used as a benchmark, but in the case of an irreversible process with null net current, the degree-degree distribution has to be considered to identify the irreversible nature of the series.

Lyapunov exponents in random Boolean networks

A new order parameter approximation to random boolean networks (RBN) is introduced, based on the concept of Boolean derivative. A statistical argument involving an annealed approximation is used, allowing to measure the order parameter in terms of the statistical properties of a random matrix. Using the same formalism, a Lyapunov exponent is calculated, allowing to provide the onset of damage spreading through the network and how sensitive it is to minimal perturbations. Finally, the Lyapunov exponents are obtained by means of different approximations: through distance method and a discrete variant of the Wolfs method for continuous systems.

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.

Phase transitions and antichaos in generalized Kauffman networks

The critical properties of Kauffman networks with different input connectivities are studied. A general condition for stability is derived and used as a constraint (the antichaos constraint) in the catculation of the probabibty distribution of connections in a stable network. Some consequences for genome stability are outlined. Random Boolean networks (RBN) [ l-31 were proposed by KaufFman as simple (yet reasonable) models of genetic systems. In these networks, a set of N binary elements is used. The state of each element at a given time step t, is given by Sj( t) E (0,l f (i = 1, ---, N). The dynamical state of each &(t) is updated (synchronously) by means of a Boolean function L&. Each element receives inputs from exactly K elements, and we have a dynamical system defined from &(t+ 1 )=&[&,(t>, &z(t), -=, si,(t) I (1) Using this theoretical framework, several methods of statistical mechanics can be applied [ 4-71. A very impo~ant result of these studies was the existence of a phase tr~sition in the dynamical behaviour of RBN [ 41. Using K as a key parameter, it was shown that for Kc= 2 a phase transition occurs separating the socalled chaotic (K> Kc) and frozen (Kc Kc) phases. Here by “chaos’ we do not mean low-dimensional deterministic chaos but a phase where damage spreading takes place (i.e. the appearance of changes in global d~amical behaviour caused by transiently altering the activity of a single binary variable). At K& nets crystallize spontaneous order, and several properties involving the number of attractors, cycle lengths or the stability against minimal and structural perturbations seem to be in agreement with what one can expect about the real genome [ l&9]. The critical point Kc was analytically determined by Derrida and his colleages [ 4-61 in a set of remarkable theoretical studies. Using the so-called annealed approximation, they analysed the evolution in the overlaps of two randomly chosen RBN of a given connectivity K. The connections and the Boolean functions are chosen at random among the elements of the set 5

Visibility Algorithms: A Short Review

(N) of all the Boolean functions of connectivity K. Using two con~gurations, say C,(t)=(&p(t), . . ..sp(t)) ) which are also randomly taken from the set U(N) of all the possible N-strings (clearly # S’(N) L= 2 N), we have A&(t) common spins; let Nai,( t+ I ) be the net overlap after one iteration of C, (t) and C, ( t ) under ( 1) . Then a new set of connections and Boolean functions is again taken from ~~(~), and a new iteration is performed over C1 (t-i1) and C,( t+ 1). A 037%9601/95/

Dynamical small-world behavior in an epidemical model of mobile individuals

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