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Haros graphs have been recently introduced as a set of graphs bijectively related to real numbers in the unit interval. Here we consider the iterated dynamics of a graph operator $\cal R$ over the set of Haros graphs. This operator was previously defined in the realm of graph-theoretical characterisation of low-dimensional nonlinear dynamics, and has a renormalization group (RG) structure. We find that the dynamics of $\cal R$ over Haros graphs is complex and includes unstable periodic orbits or arbitrary period and non-mixing aperiodic orbits, overall portraiting a chaotic RG flow. We identify a single RG stable fixed point whose basin of attraction is the set of rational numbers, associate periodic RG orbits with (pure) quadratic irrationals and aperiodic RG orbits with (non-mixing) families of non-quadratic algebraic irrationals and trascendental numbers. Finally, we show that the entropy gradients inside periodic RG orbits are constant. We discuss the possible physical interpretation of such chaotic RG flow and speculate on the entropy-constant periodic orbits as a possible confirmation of a (quantum field-theoretic) $c$-theorem applied inside the invariant set of a RG flow.

In this work, we explore and review linguistic laws, in the case of Catalan, going from prelinguistic to higher linguistic levels and addressing both speech and writing. We show evidence supporting the theory that linguistic laws are universal patterns in human language that are more robust in the oral corpus than in writing. This reinforces the “physical hypothesis,” which argues that linguistic laws could have a physiological and biophysical origin, and they are reflected in written texts as a consequence of speech symbolization. However, future work is necessary to increase empirical evidence by deeply analyzing other language corpora, propose new cognitive and physical models that clarify the mathematical formulation of some statistical patterns, find more evidence to explain the relationship between prelinguistic and higher linguistic levels, and understand the results reported up to date from a global interdisciplinary perspective of language theory. I.G.T. and A.H.-F. were supported by the project PRO2020-S03 (RCO03080 Lingüística Quantitativa) and PRO2021-S03HERNANDEZ by Institut d’Estudis Catalans. A.H-F. was also supported by the grant TIN2017-89244-R (MACDA) (Ministerio de Economía, Industria y Competitividad, Gobierno de España).

This paper discusses the applicability of Visibility Algorithms to detect faults in condition monitoring applications. The general purpose of Visibility Algorithms is to transform time series into graphs and study them through the characterization of their associated network. Degradation of a component results in changes to the network. This technique has been applied using a test rig of an aircraft fuel system to show that there is a correlation between the values of key metrics of visibility graphs and the severity of four failure modes. We compare the results of using Horizontal Visibility algorithms against Natural Visibility algorithms. The results also show how the Kullback-Leibler divergence and statistical entropy can be used to produce condition indicators. Experimental results show that there is little dispersion in the values of condition indicators, leading to a low probability of false positives and false negatives.

Tell Bartolo Luque and Fernando Ballesteros how far the Sun is from the Earth, and they will tell you the size of the Universe.

The origin and shape of metabolic scaling has been controversial since Kleiber found that basal metabolic rate of animals seemed to vary as a power law of their body mass with exponent 3/4, instead of 2/3, as a surface-to-volume argument predicts. The universality of exponent 3/4 -claimed in terms of the fractal properties of the nutrient network- has recently been challenged according to empirical evidence that observed a wealth of robust exponents deviating from 3/4. Here we present a conceptually simple thermodynamic framework, where the dependence of metabolic rate with body mass emerges from a trade-off between the energy dissipated as heat and the energy efficiently used by the organism to maintain its metabolism. This balance tunes the shape of an additive model from which different effective scalings can be recovered as particular cases, thereby reconciling previously inconsistent empirical evidence in mammals, birds, insects and even plants under a unified framework. This model is biologically motivated, fits remarkably well the data, and also explains additional features such as the relation between energy lost as heat and mass, the role and influence of different climatic environments or the difference found between endotherms and ectotherms.

In this paper we show how the cross-disciplinary transfer of techniques from Dynamical Systems Theory to Number Theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo $k$ are maximally chaotic and, while lacking forbidden patterns, display a non-trivial spectrum of Renyi entropies which suggest that every block of size $m>1$, while admissible, occurs with different probability. This non-uniform distribution of blocks for $m>1$ contrasts Dirichlet's theorem that guarantees equiprobability for $m=1$. We then explore in a similar fashion the sequence of prime gap residues. This sequence is again chaotic (positivity of Kolmogorov-Sinai entropy), however chaos is weaker as we find forbidden patterns for every block of size $m>1$. We relate the onset of these forbidden patterns with the divisibility properties of integers, and estimate the densities of gap block residues via Hardy-Littlewood $k$-tuple conjecture. We use this estimation to argue that the amount of admissible blocks is non-uniformly distributed, what supports the fact that the spectrum of Renyi entropies is again non-trivial in this case. We complete our analysis by applying the Chaos Game to these symbolic sequences, and comparing the IFS attractors found for the experimental sequences with appropriate null models., 18 pages, 20 figures

The origin and shape of metabolic scaling has been controversial since Kleiber found that basal metabolic rate of animals seemed to vary as a power law of their body mass with exponent 3/4, instead of 2/3, as a surface-to-volume argument predicts. The universality of exponent 3/4 -claimed in terms of the fractal properties of the nutrient network- has recently been challenged according to empirical evidence that observed a wealth of robust exponents deviating from 3/4. Here we present a conceptually simple thermodynamic framework, where the dependence of metabolic rate with body mass emerges from a trade-off between the energy dissipated as heat and the energy efficiently used by the organism to maintain its metabolism. This balance tunes the shape of an additive model from which different effective scalings can be recovered as particular cases, thereby reconciling previously inconsistent empirical evidence in mammals, birds, insects and even plants under a unified framework. This model is biologically motivated, fits remarkably well the data, and also explains additional features such as the relation between energy lost as heat and mass, the role and influence of different climatic environments or the difference found between endotherms and ectotherms., This work has been funded by projects AYA2013-48623-C2-2, FIS2013-41057-P, CGL2013-46862-C2-1-P and SAF2015-65878-R from the Spanish Ministerio de Economia y Competitividad and PrometeoII/2014/086, PrometeoII/2014/060 and PrometeoII/2014/065 from the Generalitat Valenciana (Spain). BL acknowledges funding from a Salvador de Madariaga fellowship, and L.L. acknowledges funding from EPSRC Early Career fellowship EP/P01660X/1.

Gravitation is one of the main forces of nature and is a fundamental factor for the habitability of the cosmos. Not only it is the main source of structures in a universe that, without it, would be only a faint soup of gases, but it is the main engine for creating stars and worlds, to make these planets inhabitable and to impose structural and functional limits on the organisms that can be developed in them.

Linguistic laws constitute one of the quantitative cornerstones of modern cognitive sciences and have been routinely investigated in written corpora, or in the equivalent transcription of oral corpora. This means that inferences of statistical patterns of language in acoustics are biased by the arbitrary, language-dependent segmentation of the signal, and virtually precludes the possibility of making comparative studies between human voice and other animal communication systems. Here we bridge this gap by proposing a method that allows to measure such patterns in acoustic signals of arbitrary origin, without needs to have access to the language corpus underneath. The method has been applied to six different human languages, recovering successfully some well-known laws of human communication at timescales even below the phoneme and finding yet another link between complexity and criticality in a biological system. These methods further pave the way for new comparative studies in animal communication or the analysis of signals of unknown code., Comment: Submitted for publication

A lo largo del siglo XX los estudios en lingüística cuantitativa han ido mostrando la aparición de leyes potenciales en las lenguas, primero en textos escritos y posteriormente en el habla. Son leyes que parecen ubicuas y robustas, pero ¿por qué aparecen en el lenguaje? ¿Son resultados espurios debidos a la arbitrariedad de la segmentación de las palabras, o realmente son universales de la comunicación compleja? ¿Podemos investigar la presencia de estas leyes en otros sistemas de comunicación animal de los que no conocemos el código? Los enfoques interdisciplinares y transdisciplinares en la lingüística y el estudio de los sistemas de comunicación se antojan imprescindibles.Se exponen a modo de ejemplo dos estudios recientes realizados sobre corpus acústicos de hasta dieciséis lenguas, mediante un método general de segmentación de señales (método de los umbrales). Exploramos aquí la posibilidad de que las leyes estadísticas que emergen en el lenguaje sean fruto de un sistema crítico auto-organizado, al igual que otros fenómenos presentes en la Naturaleza. El método de los umbrales que se presenta permite analizar cualquier tipo de señal sin necesidad de conocer su codificación o segmentación. Esto abre nuevos caminos en la investigación lingüística permitiendo entre otras cosas realizar estudios comparativos entre el lenguaje humano y otros sistemas de comunicación animal.

Horizontal visibility graphs (HVGs) are graphs constructed in correspondence with number sequences that have been introduced and explored recently in the context of graph-theoretical time series analysis. In most of the cases simple measures based on the degree sequence (or functionals of these such as entropies over degree and joint degree distributions) appear to be highly informative features for automatic classification and provide nontrivial information on the associated dynam- ical process, working even better than more sophisticated topological metrics. It is thus an open question why these seemingly simple measures capture so much information. Here we prove that, under suitable conditions, there exist a bijection between the adjacency matrix of an HVG and its degree sequence, and we give an explicit construction of such bijection. As a consequence, under these conditions HVGs are unigraphs and the degree sequence fully encapsulates all the information of these graphs, thereby giving a plausible reason for its apparently unreasonable effectiveness.

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

Prime numbers seem to be distributed among the natural numbers with no law other than that of chance; however, their global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes. In this paper, we show that a generalization of the well-known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in datasets, describes with astonishing precision the statistical distribution of leading digits in the prime number sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the non-trivial Riemann zeta zeros. We prove that the prime number theorem is, in the final analysis, responsible for these patterns.

We develop a probabilistic approach to optimum reserve design based on the species–area relationship. Specifically, we focus on the distribution of areas among a set of reserves maximizing biodiversity. We begin by presenting analytic solutions for the neutral case in which all species have the same colonization probability. The optimum size distribution is determined by the local-to-regional species richness ratio k. There is a critical kt ratio defined by the number of reserves raised to the scaling exponent of the species–area relationship. Below kt , a uniform area distribution across reserves maximizes biodiversity. Beyond kt , biodiversity is maximized by allocating a certain area to one reserve and uniformly allocating the remaining area to the other reserves. We proceed by numerically exploring the robustness of our analytic results when departing from the neutral assumption of identical colonization probabilities across species

Speech is a distinctive complex feature of human capabilities. In order to understand the physics underlying speech production, in this work, we empirically analyse the statistics of large human speech datasets ranging several languages. We first show that during speech, the energy is unevenly released and powerlaw distributed, reporting a universal robust Gutenberg–Richter-like law in speech. We further show that such ‘earthquakes in speech’ show temporal correlations, as the interevent statistics are again power-law distributed. As this feature takes place in the intraphoneme range, we conjecture that the process responsible for this complex phenomenon is not cognitive, but it resides in the physiological (mechanical) mechanisms of speech production. Moreover, we show that these waiting time distributions are scale invariant under a renormalization group transformation, suggesting that the process of speech generation is indeed operating close to a critical point. These results are put in contrast with current paradigms in speech processing, which point towards low dimensional deterministic chaos as the origin of nonlinear traits in speech fluctuations. As these latter fluctuations are indeed the aspects that humanize synthetic speech, these findings may have an impact in future speech synthesis technologies. Results are robust and independent of the communication language or the number of speakers, pointing towards a universal pattern and yet another hint of complexity in human speech.

A computational model is defined to explain the distribution of coin content of French people's purses as obtained from a sequence of surveys carried out in France during the years 2002 and 2003. The agreement between theoretical and experimental data is noticeable. The model predicts the purse content in both Dollar and Yen monetary areas.

Several order parameters have been considered to predict and characterize the transition between ordered and disordered phases in random Boolean networks, such as the Hamming distance between replicas or the stable core, which have been successfully used. In this work, we propose a natural and clear new order parameter: the temporal variance. We compute its value analytically and compare it with the results of numerical experiments. Finally, we propose a complexity measure based on the compromise between temporal and spatial variances. This new order parameter and its related complexity measure can be easily applied to other complex systems.

Random Boolean networks are among the best-known systems used to model genetic networks. They show an on–off dynamics and it is easy to obtain analytical results with them. Unfortunately very few genes are strictly on–off switched. On the other hand, continuous methods are in principle more suitable to capture the real behavior of the genome, but have difficulties when trying to obtain analytical results. In this work, we introduce a new model of random discrete network: random walk networks, where the state of each gene is changed by small discrete variations, being thus a natural bridge between discrete and continuous models.

This work is a tutorial in Molecular Evolution from the point of view of Physics. We discuss Eigen's model, a link between evolutionary theory and physics. We will begin by assuming the existence of (marco) molecules or replicators with the template property, that is, the capacity to self-replicate. According to this assumption, information will be randomly generated and destroyed by mutations in the code (i.e., errors in the copying process) and new bits of information will be fixed (made stable) by the existence of an external pressure on the system (i.e., selection), and the ability of the molecules to replicate themselves. Our aim is to build a model in order to describe molecular evolution from as general a standpoint as possible. As we will see, even very simple models from the theoretical point of view will have surprisingly deep consequences.

Random Boolean networks have been proposed as discrete models of genetic networks. Depending on the values of their control parameters, these networks fall by themselves in order or disorder phases. These networks are autonomous systems: no external inputs are considered. Nevertheless, in the real world the genetic networks are in5uenced by external signals. Many biological rhythms have 24-h periods related to sunlight, coupled with molecular clocks. In this work we study the response of Random Boolean Networks to analytical and non-analytical external periodic signals. The relationship between external and internal parameters for the determination of the dynamical behaviour of this networks is theoretically determined and simulations in agreement with these theoretical results are reported. c

Preliminary results in a simple model of disease spreading within a population of socially interacting mobile individuals are discussed. It is shown that when the traveling path of random walkers is varied from the local neighborhood to possible long-distance, a small-world effect appears affecting the percolation density that separates the endemic state from the disease-free state. © 2002 Published by Elsevier Science B.V.

When using γ-ray coded-mask cameras, one does not get a direct image as in classical optical cameras but the correlation of the mask response with the source. Therefore the data must be mathematically treated in order to reconstruct the original sky sources. Generally this reconstruction is based on linear methods, such as correlating the detector plane with a reconstruction array, or non-linear ones such as iterative or maximization methods (i.e. the EM algorithm). The latter have a better performance but they increase the computational complexity by taking a lot of time to reconstruct an image. Here we present a method for speeding up such kind of algorithms by making use of a neural network with a back-propagation learning rule.

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 Wolf's method for continuous systems., 16 pages, 5 eps-figures included, article submitted to Physica A

A variant of a simple method for chaos control is applied to achieve control in random Boolean networks (RBNs). It is shown that a RBN in the chaotic phase can be forced to behave periodically if a certain quenched fraction of the automata is given a fixed state (the system variables) every T time-steps. An analytic relationship between and T is derived using the size of the stable core as an order parameter.

A variant of a simple method for chaos control is applied to achieve control in random Boolean networks (RBN). It is shown that a RBN in the chaotic phase can be forced to behave periodically if a certain quenched fraction γ of the automata is given a fixed state (the system variables) every τ time steps. An analytic relationship between γ and τ is derived and numerically tested.

The standard method for determining the critical points in random networks (such as Boolean nets) is based on the Derrida approximation. This method leads to the critical points based on the so-called annealed approximation. In this paper, we present a very simple annealed approximation based on the study of damage spreading when single elements are modified. Several examples are analyzed.

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean field theory for the g-Sidon decision problem. We further improve the mean field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed., Submitted for publication

The type-I intermittency route to (or out of) chaos is investigated within the Horizontal Visibility graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct, according to the Horizontal Visibility algorithm, their associated graphs. We show how the alternation of laminar episodes and chaotic bursts has a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values of several network parameters. In particular, we predict that the characteristic power law scaling of the mean length of laminar trend sizes is fully inherited in 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 the block entropy over the degree distribution. 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 RG 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 exhibit extremal entropic properties., Comment: 8 figures

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 calculation of the probability distribution of connections in a stable network. Some consequences for genome stability are outlined.

Here we present a combinatorial decision problem, inspired by the celebrated quiz show called the countdown, that involves the computation of a given target number T from a set of k randomly chosen integers along with a set of arithmetic operations. We find that the probability of winning the game evidences a threshold phenomenon that can be understood in the terms of an algorithmic phase transition as a function of the set size k. Numerical simulations show that such probability sharply transitions from zero to one at some critical value of the control parameter, hence separating the algorithm's parameter space in different phases. We also find that the system is maximally efficient close to the critical point. We then derive analytical expressions that match the numerical results for finite size and permit us to extrapolate the behavior in the thermodynamic limit., Submitted for publication

Disregarding any underlying process (and therefore any physical, chemical, economical or whichever meaning of its mere numeric values), we can consider a time series just as an ordered set of values and play the naive mathematical game of turning this set into a different mathematical object with the aids of an abstract mapping, and see what happens: which properties of the original set are conserved, which are transformed and how, what can we say about one of the mathematical representations just by looking at the other... This exercise is of mathematical interest by itself. In addition, it turns out that time series or signals is a universal method of extracting information from dynamical systems in any field of science. Therefore, the preceding mathematical game gains some unexpected practical interest as it opens the possibility of analyzing a time series (i.e. the outcome of a dynamical process) from an alternative angle. Of course, the information stored in the original time series should be somehow conserved in the mapping. The motivation is completed when the new representation belongs to a relatively mature mathematical field, where information encoded in such a representation can be effectively disentangled and processed. This is, in a nutshell, a first motivation to map time series into networks.

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 [1] 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., 19 pages, 11 figures, accepted for publication in Chaos

We analyze the properties of the self-similar network obtained from the trajectories of unimodal maps at the transition to chaos via the horizontal visibility (HV) algorithm. We first show that this network is uniquely determined by the encoded sequence of positions in the dynamics within the Feigenbaum attractor and it is universal in that it is independent of the shape and nonlinearity of the maps in this class. We then find that the network degrees fluctuate at all scales with an amplitude that increases as the size of the network grows. This suggests the definition of a graph-theoretical Lyapunov exponent that measures the expansion rate of trajectories in network space. On good agreement with the map's counterpart, while at the onset of chaos this exponent vanishes, the subexponential expansion and contraction of network degrees can be fully described via a Tsallis-type scalar deformation of the expansion rate, that yields a discrete spectrum of non-null generalized exponents. We further explore the possibility of defining an entropy growth rate that describes the amount of information created along the trajectories in network space. Making use of the trajectory distributions in the map's accumulation point and the scaling properties of the associated network, we show that such entropic growth rate coincides with the spectrum of graph-theoretical exponents, what appears as a set of Pesin-like identities in the network., Comment: Accepted for publication in Physics Letters A