Fig 10 - uploaded by Francesc Font-Clos
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Polish mathematician Stanislaw Ulam, together with the current version of the first page of Ref. Hawkins and Ulam (1944), after being unclassified for public release. The work, in which important formulas for branching processes are derived, was done as a part of the Manhattan project.
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The statistics of natural catastrophes contains very counter-intuitive
results. Using earthquakes as a working example, we show that the energy
radiated by such events follows a power-law or Pareto distribution. This means,
in theory, that the expected value of the energy does not exist (is infinite),
and in practice, that the mean of a finite set...
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Collective organization in matter plays a significant role in its expressed physical properties. Typically, it is detected via an order parameter, appropriately defined for each given system’s observed emergent patterns. Recent developments in information theory, however, suggest quantifying collective organization in a system- and phenomenon-agnos...
Citations
... Remarkable examples are lasers, oscillating chemical reactions, activity of genes, animal or plant populations, and epidemic spreading. 1,2 Trying to strength the connection between branching processes [3][4][5] and thermodynamic phase transitions, [6][7][8] as well as to better understand the emergence of critical behavior in the infinite system-size limit, Garcia-Millan et al. 9,10 studied the renowned Galton-Watson process. They analytically established the existence of finite-size scaling 11 and universality 8 in the relation between the probability of survival at different finite times and the mean offspring number per individual when the offspring distribution is Chaos ARTICLE pubs.aip.org/aip/cha ...
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.
... where we use Equation (18) and the property ∑ all (i) π (i) = 1. ...
... An interesting research thread is the relationship between the Motzkin number and Catalan number recurrences, resulting from the rules defining the simple dynamics of toy models of earthquakes, with inverse power distributions with various exponents (not only ̸ = 3 2 , but also arbitrary in some range) [11]. Another relation (however, of a different nature) of Catalan numbers to Self-Organized dynamical systems related to the earthquake model (namely the Otsuka model, being a branching process) is described in [18]. ...
Motivated by a simple model of earthquake statistics, a finite random discrete dynamical system is defined in order to obtain Catalan number recurrence by describing the stationary state of the system in the limit of its infinite size. Equations describing dynamics of the system, represented by partitions of a subset of {1,2,…,N}, are derived using basic combinatorics. The existence and uniqueness of a stationary state are shown using Markov chains terminology. A well-defined mean-field type approximation is used to obtain block size distribution and the consistency of the approach is verified. It is shown that this recurrence asymptotically takes the form of Catalan number recurrence for particular dynamics parameters of the system.
... A branching process describes how an ancestor element can generate descendant elements (Harris, 1963) -e.g. an active neuron activating other neurons. The branching parameter σ captures the expected number of offspring elements from a given ancestor (Corral & Font-Clos, 2012). Interestingly, branching processes undergo a phase transition when σ = 1, at which point avalanches can span all scales. ...
... Another hallmark of critical systems is a branching parameter (σ) close to 1 (see Methods). The branching parameter describes the expected number of offsprings that an ancestor produces, and when σ~1 the system undergoes a phase transition and avalanches can span all scales (Corral & Font-Clos, 2012). We calculate σ by estimating the mean ratio of descendants to ancestors in spontaneous data (see Methods), and find σ slightly below 1 in spontaneous activity (σ = 0.93 ± 0.03) ( Figure 3E). ...
Neuronal activity propagates through the network during seizures, engaging brain dynamics at multiple scales. Such propagating events can be described through the avalanches framework, which can relate spatiotemporal activity at the microscale with global network properties. Interestingly, propagating avalanches in healthy networks are indicative of critical dynamics, where the network is organised to a phase transition, which optimises certain computational properties. Some have hypothesised that the pathological brain dynamics of epileptic seizures are an emergent property of microscale neuronal networks collectively driving the brain away from criticality. Demonstrating this would provide a unifying mechanism linking microscale spatiotemporal activity with emergent brain dysfunction during seizures. Here, we investigated the effect of drug-induced seizures on critical avalanche dynamics, using in vivo whole-brain 2-photon imaging of GCaMP6s larval zebrafish (males and females) at single neuron resolution. We demonstrate that single neuron activity across the whole brain exhibits a loss of critical statistics during seizures, suggesting that microscale activity collectively drives macroscale dynamics away from criticality. We also construct spiking network models at the scale of the larval zebrafish brain, to demonstrate that only densely connected networks can drive brain-wide seizure dynamics away from criticality. Importantly, such dense networks also disrupt the optimal computational capacities of critical networks, leading to chaotic dynamics, impaired network response properties and sticky states, thus helping to explain functional impairments during seizures. This study bridges the gap between microscale neuronal activity and emergent macroscale dynamics and cognitive dysfunction during seizures.
Significance Statement
Epileptic seizures are debilitating and impair normal brain function. It is unclear how the coordinated behaviour of neurons collectively impairs brain function during seizures. To investigate this we perform fluorescence microscopy in larval zebrafish, which allows for the recording of whole-brain activity at single-neuron resolution. Using techniques from physics, we show that neuronal activity during seizures drives the brain away from criticality, a regime that enables both high and low activity states, into an inflexible regime that drives high activity states. Importantly, this change is caused by more connections in the network, which we show disrupts the ability of the brain to respond appropriately to its environment. Therefore, we identify key neuronal network mechanisms driving seizures and concurrent cognitive dysfunction.
... As in the Galton-Watson (discrete-time) model [23][24][25], it has a fixed point solution, η * , ...
Epidemics unfold by means of a spreading process from each infected individual to a random number of secondary cases. It has been claimed that the so-called superspreading events in COVID-19 are governed by a power-law tailed distribution of secondary cases, with no finite variance. Using a continuous-time branching process, we show that for such power-law superspreading the survival probability of an outbreak as a function of time and the basic reproductive number fulfills a "finite-time scaling" law (analogous to finite-size scaling) with universal-like characteristics only dependent on the power-law exponent. This clearly shows how the phase transition separating a subcritical and a supercritical phase emerges in the infinite-time limit (analogous to the thermodynamic limit). We quantify the counterintuitive hazards infinite-variance superspreading poses and conclude that superspreading only leads to new phenomenology in the infinite-variance case.
... Fortunately, physical insights can shed light on this problem. Assuming a simple (meanfield) model in which infections propagate following a Galton-Watson stochastic branching process [25][26][27], and if the number of fatalities is a fixed fraction of the number of infections, identifying the branching ratio with the basic reproductive number, R 0 , it is immediate to see that R 0 < 1 leads to rather small epidemics (few number of fatalities), whereas for R 0 > 1 two scenarios are possible starting from a single individual, again, few fatalities (as given by an exponential tail for x), or an infinite number of fatalities (in an infinite system). ...
... The situation is that of self-organized critical phenomena [1,28]. Under these circumstances, one would expect a power-law tail for large x, with an exponent α = 1/2 [27]. ...
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.
... It turns out that the solution to the physical interpretation of the GR law has a price to be paid: the power-law distribution, when 1 + β is smaller than 2 (which is indeed the case), is not "well behaved", in the sense that the mean value of the seismic moment becomes infinite. The reason is that, for power-law distributed seismic moments, events in the tail of the distribution, despite having very small probability, bring an enormous contribution to seismicmoment release [7], and the seismic-moment sample mean does not converge, no matter how large the number of data is, due to the inapplicability of the law of large numbers to power-law distributions [8] such as that in Eq (1). In consequence, as when extended to the whole range of earthquake sizes the GR law is unphysical, the tail of the distribution of seismic moment must deviate from the GR power-law shape [9]. ...
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.
... It turns out that the solution to the physical interpretation of the GR law has a price to be paid: the power-law distribution, when 1 + β is smaller than 2 (which is indeed the case), is not "well behaved", in the sense that the mean value of the seismic moment becomes infinite. The reason is that, for power-law distributed seismic moments, events in the tail of the distribution, despite having very small probability, bring an enormous contribution to seismic-moment release [7], and the seismic-moment sample mean does not converge, no matter how large the number of data is, due to the inapplicability of the law of large numbers to power-law distributions [8] such as that in Eq. (1). In consequence, as when extended to the whole range of earthquake sizes the GR law is unphysical, the tail of the distribution of seismic moment must deviate from the GR power-law shape [9]. ...
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.
... Divergence of moments has the annoying consequence that some important results of probability theory do not hold, as the law of large numbers when ≤ 2 (Corral, 2015;Shiryaev, 1996). So, in this case one cannot estimate the mean of the distribution from the sample mean, simply because the mean of the distribution is infinite and the sample mean cannot converge to this value (Corral & Font-Clos, 2013). If 2 < ≤ 3, the standard central limit theorem does not apply and the sample mean neither follows Gaussian statistics nor has a finite variability (Bouchaud & Georges, 1990). ...
... In practice, the disadvantage is that the real underlying theoretical distribution f(x) is unknown. For instance, simple branching processes (bp; which are mean field models by construction) yield discrete probability distributions for their total number of elements x, which (close to their critical point and for large x) lead to truncated gamma tails, f bp (x) ∝ x −3/2 e −x/ (Corral & Font-Clos, 2013); but branching processes with finite-size effects lead to more complicated functional forms for the tail, even in the critical case . Beyond mean field, little is known, and of course, real systems are more complicated than any model. ...
... For earthquakes and wildfires it is recognized that one feasible mechanism can be self-organized criticality (SOC) (Bak, 1996;Bak et al., 1987), as both phenomena are characterized by an activity front that propagates (somewhat fast) through a substrate, in an avalanche-like manner. The activation can be modeled as a branching process, but these processes only yield power law distributions if they are precisely at their critical point (Corral & Font-Clos, 2013). So the self-organization mechanism ensures that criticality is achieved "spontaneously" in these systems, through a balance between driving and dissipation, and so the substrate has to be at the onset of instability (all the time). ...
The size or energy of diverse structures or phenomena in geoscience appears to follow power law distributions. A rigorous statistical analysis of such observations is tricky, though. Observables can span several orders of magnitude, but the range for which the power law may be valid is typically truncated, usually because the smallest events are too tiny to be detected and the largest ones are limited by the system size. We revisit several examples of proposed power law distributions dealing with potentially damaging natural phenomena. Adequate fits of the distributions of sizes are especially important in these cases, given that they may be used to assess long‐term hazard. After reviewing the theoretical background for power law distributions, we improve an objective statistical fitting method and apply it to diverse data sets. The method is described in full detail, and it is easy to implement. Our analysis elucidates the range of validity of the power law fit and the corresponding exponent and whether a power law tail is improved by a truncated lognormal. We confirm that impact fireballs and Californian earthquakes show untruncated power law behavior, whereas global earthquakes follow a double power law. Rain precipitation over space and time and tropical cyclones show a truncated power law regime. Karst sinkholes and wildfires, in contrast, are better described by truncated lognormals, although wildfires also may show power law regimes. Our conclusions only apply to the analyzed data sets but show the potential of applying this robust statistical technique in the future.
... Note that this is in disagreement with the rate-and-state formulation [12]. field limit, 1 + β < = 1 + 2b < /3 1.49 ± 0.05 3/2) [10]. It has been argued that this is the true value one should observe in general if it not were for a series of artifacts and biases in the measurement of earthquake sizes [27]. ...
Coulomb-stress theory has been used for years in seismology to understand how earthquakes trigger each other. Whenever an earthquake occurs, the stress field changes, and places with positive increases are brought closer to failure. Earthquake models that relate earthquake rates and Coulomb stress after a main event, such as the rate-and-state model, assume that the distribution of earthquake magnitudes is not affected by the change in the Coulomb stress. We apply several statistical analyses to the aftershock sequence of the Landers earthquake (California, USA, 1992, moment magnitude 7.3), to show that the distribution of magnitudes is sensitive to the sign of the Coulomb-stress increase; in particular, the value of the Gutenberg-Richter law is significantly decreased for events that received a decrease in the Coulomb stress. These events have a distribution of focal mechanisms very close to the one of the previous-to-mainshock seismicity, whereas the events with a positive increase of the stress are characterized by a much larger proportion of strike-slip events.
... Here, p is assumed to be homogeneous across time and users. An analytic solution to (2) is intractable for our relevant case [31] for the same reasons that apply to Eq (1). Notably, in our case, the response rate depends on the depth of the tree, q ¼q � f ðgÞ. ...
Online communities, which have become an integral part of the day-to-day life of people and organizations, exhibit much diversity in both size and activity level; some communities grow to a massive scale and thrive, whereas others remain small, and even wither. In spite of the important role of these proliferating communities, there is limited empirical evidence that identifies the dominant factors underlying their dynamics. Using data collected from seven large online platforms, we observe a relationship between online community size and its activity which generally repeats itself across platforms: First, in most platforms, three distinct activity regimes exist—one of low-activity and two of high-activity. Further, we find a sharp activity phase transition at a critical community size that marks the shift between the first and the second regime in six out of the seven online platforms. Essentially, we argue that it is around this critical size that sustainable interactive communities emerge. The third activity regime occurs above a higher characteristic size in which community activity reaches and remains at a constant and higher level. We find that there is variance in the steepness of the slope of the second regime, that leads to the third regime of saturation, but that the third regime is exhibited in six of the seven online platforms. We propose that the sharp activity phase transition and the regime structure stem from the branching property of online interactions.
