arXiv subjects

Alvaro Corral

Publications and source records attributed to Alvaro Corral.

At least 19 recordsLinked to original sources

arXiv2024

Universal finite-time scaling in the transcritical, saddle-node, and pitchfork discrete and continuous bifurcations

Bifurcations are one of the most remarkable features of dynamical systems. Corral et al. [Sci. Rep. 8(11783), 2018] showed the existence of scaling laws describing the transient (finite-time) dynamics in discrete dynamical systems close to a bifurcation point, following an approach that was valid for the transcritical as well as for the saddle-node bifurcations. We reformulate those previous results and extend them to other discrete and continuous bifurcations, remarkably the pitchfork bifurcation. In contrast to the previous work, we obtain a finite-time bifurcation diagram directly from the scaling law, without a necessary knowledge of the stable fixed point. The derived scaling laws provide a very good and universal description of the transient behavior of the systems for long times and close to the bifurcation points.

cond-mat.stat-mech↗

arXiv2023

Moments of undersampled distributions: Application to the size of epidemics

The total number of fatalities of an epidemic outbreak is a dramatic but extremely informative quantity. Knowledge of the statistics of this quantity allows the calculation of the mean total number of fatalities conditioned to the fact that the outbreak has surpassed a given number of fatalities, which is very relevant for risk assessment. However, the fact that the total number of fatalities seems to be characterized by a power-law tailed distribution with exponent (of the survival function) smaller than one poses an important theoretical difficulty, due to the non-existence of a mean value for such distributions. Cirillo and Taleb [Nature Phys. 16, 606 (2020)] propose a transformation from a so-called dual variable, which displays a power-law tail, to the total number of fatalities, which becomes bounded by the total world population. Here, we (i) show that such a transformation is ad hoc and unphysical; (ii) propose alternative transformations and distributions (also ad hoc); (iii) argue that the right framework for this problem is statistical physics, through finite-size scaling; and (iv) demonstrate that the real problem is not the non-existence of the mean value for power-law tailed distributions but the fact that the tail of the different theoretical distributions (which is what distinguishes one model from the other) is far from being well sampled with the available number of empirical data. Our results are also valid for many other hazards displaying (apparent) power-law tails in their size.

physics.soc-ph↗

arXiv2022

Finite-size scaling of human-population distributions over fixed-size cells and its relation to fractal spatial structure

Using demographic data of high spatial resolution for a region in the south of Europe, we study the population over fixed-size spatial cells. We find that, counterintuitively, the distribution of the number of inhabitants per cell increases its variability when the size of the cells is increased. Nevertheless, the shape of the distributions is kept constant, which allows us to introduce a scaling law, analogous to finite-size scaling, with a scaling function reasonably well fitted by a gamma distribution. This means that the distribution of the number of inhabitants per cell is stable or invariant under addition with neighboring cells (plus rescaling), defying the central-limit theorem, due to the obvious dependence of the random variables. The finite-size scaling implies a power-law relations between the moments of the distribution and its scale parameter, which are found to be related with the fractal properties of the spatial pattern formed by the population. The match between theoretical predictions and empirical results is reasonably good.

physics.soc-ph↗

arXiv2020

Finite-size scaling versus dual random variables and shadow moments in the size distribution of epidemics

I discuss and criticize the use of "dual transformations" to calculate "shadow moments" in the distribution of epidemic fatalities. I show how this transformation is arbitrary and not only yields an unjustified functional form for the distribution of fatalities but turns out to be highly unphysical (see Fig. 1 in the manuscript). Therefore, the expected number of fatalities calculated from the "shadow moments" approach lacks any theoretical support. This is of fundamental importance to assess risk from pandemics. I argue that the natural way to address this sort of problems in statistical physics is by means of finite-size scaling.

physics.soc-ph↗

arXiv2020

Tail of the distribution of fatalities in epidemics

The size that an epidemic can reach, measured in terms of the number of fatalities, is an extremely relevant quantity. It has been recently claimed [Cirillo & Taleb, Nature Physics 2020] that the size distribution of major epidemics in human history is "extremely fat-tailed", i.e., asymptotically a power law, which has important consequences for risk management. Reanalyzing this data, we find that, although the fatality distribution may be compatible with a power-law tail, these results are not conclusive, and other distributions, not fat-tailed, could explain the data equally well. As an example, simulation of a log-normally distributed random variable provides synthetic data whose statistics are undistinguishable from the statistics of the empirical data. Theoretical reasons justifying a power-law tail as well as limitations in the current data are also discussed.

physics.soc-ph↗

arXiv2020

Scientific comment on "Tail risk of contagious diseases"

Cirillo and Taleb [Nature Phys. 16, 606-613 (2020)] study the size of major epidemics in human history in terms of the number of fatalities. Using the figures from 72 epidemics, from the plague of Athens (429 BC) to the COVID-19 (2019-2020), they claim that the resulting fatality distribution is ``extremely fat-tailed'', i.e., asymptotically a power law. This has important consequences for risk, as the mean value of the fatality distribution becomes infinite. Reanalyzing the same data, we find that, although the data may be compatible with a power-law tail, these results are not conclusive, and other distributions, not fat-tailed, could explain the data equally well. Simulation of a log-normally distributed random variable provides synthetic data whose statistics are undistinguishable from the statistics of the empirical data.

physics.soc-ph↗

arXiv2019

The brevity law as a scaling law, and a possible origin of Zipf's law for word frequencies

An important body of quantitative linguistics is constituted by a series of statistical laws about language usage. Despite the importance of these linguistic laws, some of them are poorly formulated, and, more importantly, there is no unified framework that encompasses all them. This paper presents a new perspective to establish a connection between different statistical linguistic laws. Characterizing each word type by two random variables, length (in number of characters) and absolute frequency, we show that the corresponding bivariate joint probability distribution shows a rich and precise phenomenology, with the type-length and the type-frequency distributions as its two marginals, and the conditional distribution of frequency at fixed length providing a clear formulation for the brevity-frequency phenomenon. The type-length distribution turns out to be well fitted by a gamma distribution (much better than with the previously proposed lognormal), and the conditional frequency distributions at fixed length display power-law-decay behavior with a fixed exponent $\alpha\simeq 1.4$ and a characteristic-frequency crossover that scales as an inverse power $\delta\simeq 2.8$ of length, which implies the fulfilment of a scaling law analogous to those found in the thermodynamics of critical phenomena. As a by-product, we find a possible model-free explanation for the origin of Zipf's law, which should arise as a mixture of conditional frequency distributions governed by the crossover length-dependent frequency.

physics.soc-ph↗

arXiv2019

From Boltzmann to Zipf through Shannon and Jaynes

The word-frequency distribution provides the fundamental building blocks that generate discourse in language. It is well known, from empirical evidence, that the word-frequency distribution of almost any text is described by Zipf's law, at least approximately. Following Stephens and Bialek [Phys. Rev. E 81, 066119, 2010], we interpret the frequency of any word as arising from the interaction potential between its constituent letters. Indeed, Jaynes' maximum-entropy principle, with the constrains given by every empirical two-letter marginal distribution, leads to a Boltzmann distribution for word probabilities, with an energy-like function given by the sum of all pairwise (two-letter) potentials. The improved iterative-scaling algorithm allows us finding the potentials from the empirical two-letter marginals. Appling this formalism to words with up to six letters from the English subset of the recently created Standardized Project Gutenberg Corpus, we find that the model is able to reproduce Zipf's law, but with some limitations: the general Zipf's power-law regime is obtained, but the probability of individual words shows considerable scattering. In this way, a pure statistical-physics framework is used to describe the probabilities of words. As a by-product, we find that both the empirical two-letter marginal distributions and the interaction-potential distributions follow well-defined statistical laws.

physics.soc-ph↗

arXiv2019

Truncated lognormal distributions and scaling in the size of naturally defined population clusters

Using population data of high spatial resolution for a region in the south of Europe, we define cities by aggregating individuals to form connected clusters. The resulting cluster-population distributions show a smooth decreasing behavior covering six orders of magnitude. We perform a detailed study of the distributions, using state-of-the-art statistical tools. By means of scaling analysis we rule out the existence of a power-law regime in the low-population range. The logarithmic-coefficient-of-variation test allows us to establish that the power-law tail for high population, characteristic of Zipf's law, has a rather limited range of applicability. Instead, lognormal fits describe the population distributions in a range covering from a few dozens individuals to more than one million (which corresponds to the population of the largest cluster).

physics.soc-ph↗

arXiv2019

Time window to constrain the corner value of the global seismic-moment distribution

It is well accepted that, at the global scale, the Gutenberg-Richter (GR) law describing the distribution of earthquake magnitude or seismic moment has to be modified at the tail to properly account for the most extreme events. It is debated, though, how much additional time of earthquake recording will be necessary to properly constrain this tail. Using the global CMT catalog, we study how three modifications of the GR law that incorporate a corner-value parameter are compatible with the size of the largest observed earthquake in a given time window. Current data lead to a rather large range of parameter values (e.g., corner magnitude from 8.6 to 10.2 for the so-called tapered GR distribution). Updating this estimation in the future will strongly depend on the maximum magnitude observed, but, under reasonable assumptions, the range will be substantially reduced by the end of this century, contrary to claims in previous literature.

physics.data-an↗

arXiv2019

The distinct flavors of Zipf's law in the rank-size and in the size-distribution representations, and its maximum-likelihood fitting

In the last years, researchers have realized the difficulties of fitting power-law distributions properly. These difficulties are higher in Zipf's systems, due to the discreteness of the variables and to the existence of two representations for these systems, i.e., two versions about which is the random variable to fit. The discreteness implies that a power law in one of the representations is not a power law in the other, and vice versa. We generate synthetic power laws in both representations and apply a state-of-the-art fitting method (based on maximum-likelihood plus a goodness-of-fit test) for each of the two random variables. It is important to stress that the method does not fit the whole distribution, but the tail, understood as the part of a distribution above a cut-off that separates non-power-law behavior from power-law behavior. We find that, no matter which random variable is power-law distributed, the rank-size representation is not adequate for fitting, whereas the representation in terms of the distribution of sizes leads to the recovery of the simulated exponents, may be with some bias.

physics.data-an↗

arXiv2018

Finite-time scaling in local bifurcations

Finite-size scaling is a key tool in statistical physics, used to infer critical behavior in finite systems. Here we use the analogous concept of finite-time scaling to describe the bifurcation diagram at finite times in discrete dynamical systems. We analytically derive finite-time scaling laws for two ubiquitous transitions given by the transcritical and the saddle-node bifurcation, obtaining exact expressions for the critical exponents and scaling functions. One of the scaling laws, corresponding to the distance of the dynamical variable to the attractor, turns out to be universal. Our work establishes a new connection between thermodynamic phase transitions and bifurcations in low-dimensional dynamical systems, and opens new avenues to identify the nature of dynamical shifts in systems for which only short time series are available.

nlin.AO↗

arXiv2018

Dependence of exponents on text length versus finite-size scaling for word-frequency distributions

Some authors have recently argued that a finite-size scaling law for the text-length dependence of word-frequency distributions cannot be conceptually valid. Here we give solid quantitative evidence for the validity of such scaling law, both using careful statistical tests and analytical arguments based on the generalized central-limit theorem applied to the moments of the distribution (and obtaining a novel derivation of Heaps' law as a by-product). We also find that the picture of word-frequency distributions with power-law exponents that decrease with text length [Yan and Minnhagen, Physica A 444, 828 (2016)] does not stand with rigorous statistical analysis. Instead, we show that the distributions are perfectly described by power-law tails with stable exponents, whose values are close to 2, in agreement with the classical Zipf's law. Some misconceptions about scaling are also clarified.

physics.data-an↗

arXiv2018

Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects

We revisit the problem of Brownian diffusion with drift in order to study finite-size effects in the geometric Galton-Watson branching process. This is possible because of an exact mapping between one-dimensional random walks and geometric branching processes, known as the Harris walk. In this way, first-passage times of Brownian particles are equivalent to sizes of trees in the branching process (up to a factor of proportionality). Brownian particles that reach a distant boundary correspond to percolating trees, and those that do not correspond to non-percolating trees. In fact, both systems display a second-order phase transition between "insulating" and "conducting" phases, controlled by the drift velocity in the Brownian system. In the limit of large system size, we obtain exact expressions for the Laplace transforms of the probability distributions and their first and second moments. These quantities are also shown to obey finite-size scaling laws.

cond-mat.stat-mech↗

arXiv2016

Exact derivation of a finite-size-scaling law and corrections to scaling in the geometric Galton-Watson process

The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.

cond-mat.stat-mech↗

arXiv2015

Testing Universality in Critical Exponents: the Case of Rainfall

One of the key clues to consider rainfall as a self-organized critical phenomenon is the existence of power-law distributions for rain-event sizes. We have studied the problem of universality in the exponents of these distributions by means of a suitable statistic whose distribution is inferred by several variations of a permutational test. In contrast to more common approaches, our procedure does not suffer from the difficulties of multiple testing and does not require the precise knowledge of the uncertainties associated to the power-law exponents. When applied to seven sites monitored by the Atmospheric Radiation Measurement Program the test lead to the rejection of the universality hypothesis, despite the fact that the exponents are rather close to each other.

physics.data-an↗

arXiv2014

Log-log Convexity of Type-Token Growth in Zipf's Systems

It is traditionally assumed that Zipf's law implies the power-law growth of the number of different elements with the total number of elements in a system - the so-called Heaps' law. We show that a careful definition of Zipf's law leads to the violation of Heaps' law in random systems, and obtain alternative growth curves. These curves fulfill universal data collapses that only depend on the value of the Zipf's exponent. We observe that real books behave very much in the same way as random systems, despite the presence of burstiness in word occurrence. We advance an explanation for this unexpected correspondence.

physics.soc-ph↗

arXiv2014

Finite-size scaling of survival probability in branching processes

Branching processes pervade many models in statistical physics. We investigate the survival probability of a Galton-Watson branching process after a finite number of generations. We reveal the finite-size scaling law of the survival probability for a given branching process ruled by a probability distribution of the number of offspring per element whose standard deviation is finite, obtaining the exact scaling function as well as the critical exponents. Our findings prove the universal behavior of branching processes concerning the survival probability.

cond-mat.stat-mech↗