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[EQUATION] i.e., by the infinite iteration of the point [MATH] through the generating function [MATH] (using the key property that the probability of a zero value is the value of the generating function at zero, and equation ( 16 ) again). |
We now calculate the iteration [MATH] In the interval [MATH] the function [MATH] is non-decreasing and non-convex, taking values from [MATH] to 1. If the slope of [MATH] at [MATH] , given by |
[MATH] , is smaller than or equal to 1, then [MATH] only crosses (or reaches) the diagonal at [MATH] (otherwise, [MATH] would need to be convex somewhere), and the iteration of the point [MATH] ends at the point [MATH] (which is the attractor, see Fig. ). Therefore, |
[EQUATION] i.e., extinction is unavoidable if [MATH] There is a trivial exception, though, associated to [MATH] (and zero for the rest); this is an extremely boring situation indeed. In this case, [MATH] , and therefore [MATH] which means, obviously, that the probability of extinction is zero. |
If the slope of [MATH] at [MATH] is [MATH] (which only can happen for a non-linear generating function, [MATH] ), then [MATH] has to cross the diagonal at a point [MATH] smaller than one, which is the attractive solution to which the iteration tends, see Fig. again. In mathematical language, |
[EQUATION] where [EQUATION] The demonstration is elaborated in the Appendix. Summarizing, [EQUATION] with [MATH] except in the trivial case [MATH] which has [MATH] but yields [MATH] |
Equation ( 26 ) clearly shows that, in general, the point [MATH] separates two distinct behaviors: extinction for sure for [MATH] |
and the possibility of non-extinction (non-sure extinction) for [MATH] . Therefore, [MATH] constitutes a critical case separating these behaviors, called therefore subcritical ( [MATH] and supercritical ( [MATH] ). It is instructive to point out that, as [MATH] is always a solution of |
[MATH] , Watson concluded, incorrectly, that the population always gets extinct, no matter the value of [MATH] (Kendall,, 1966 2.6 The probability of extinction for the binomial distribution |
For the sake of illustration we will consider a simple concrete example, a binomial distribution (Ross,, 2002 ; Grimmett and Stirzaker,, 2001 |
[EQUATION] This assumes that each element has only a fixed number of trials [MATH] to generate other elements, and any of these [MATH] trials has a constant probability [MATH] of being successful. The generating function turns out to be, using the binomial theorem |
[EQUATION] Let us consider the simple case with [MATH] and define [MATH] As we know, the probability of extinction will come from the smallest solution in [MATH] of |
[EQUATION] So, [EQUATION] but the square root can be written as [MATH] and then, [EQUATION] Therefore, the smallest root depends on whether [MATH] |
is below or above [MATH] [EQUATION] As for the binomial distribution [MATH] (Ross,, 2002 the critical case [MATH] corresponds obviously to [MATH] in agreement with the behavior of [MATH] |
2.7 No stability of the population Although this subsection contains an interesting result to better understand the behavior of the Galton-Watson process, it can be skipped as it is not connected to the rest of the chapter. In fact, the iteration of the point [MATH] |
shows what happens to the whole generating function of [MATH] when [MATH] Indeed, in the same way as in subsection 2.5, [EQUATION] |
whereas [EQUATION] except for [MATH] which always fulfills [MATH] see Fig. ). Note that a flat generating function corresponds to probabilities equal to zero, except for the zero value, i.e., |
[EQUATION] In this way, for [MATH] we have that [MATH] and the population gets extinct; but for [MATH] we have found [MATH] having any other finite value of [MATH] a zero probability, this means that [MATH] goes to infinite, when [MATH] with probability [MATH] that is, [MATH] cannot remain positive and bounded. The onl... |
2.8 Non-equilibrium phase transition Let us analyze in more detail what happens around the “transition point” [MATH] As we just have seen, recall equation ( 25 ), the extinction probability is given by the solution of [MATH] When [MATH] the only solution in [MATH] is [MATH] |
(except in the trivial case [MATH] ). When [MATH] we have to take the smallest solution of [MATH] in [MATH] In terms of the non-extinction probability, [MATH] we need to look for the largest [MATH] that is solution of |
[EQUATION] in the range [MATH] We explore the case of [MATH] close to 1, for which [MATH] is close to zero, and, using the binomial theorem, we can expand |
[MATH] which yields [EQUATION] where we have introduced the mean [MATH] and the second factorial moment [MATH] (which we assume exists). Therefore, up to second order in [MATH] we need to solve |
[EQUATION] It is immediate that one solution of equation ( 38 ) is [MATH] and one can realize that this solution is exact up to any order in [MATH] The other solution is [MATH] , but we must pay attention to the value of [MATH] , which can be written as [MATH] with [MATH] i.e., the variance. Existence of [MATH] and [MA... |
[EQUATION] (using the formula for the geometric series), therefore, [MATH] around zero means [MATH] around one, and we can write the second solution as |
[EQUATION] which is only in the range of interest for [MATH] In conclusion, we have [EQUATION] valid in the limit of small [MATH] . For [MATH] this limit is equivalent to [MATH] The separate case [MATH] is only achieved in the trivial situation where |
[MATH] (otherwise, the mean cannot approach one). In this way, we obtain a behavior that is the one corresponding to a continuous phase transition in thermodynamic equilibrium. Identifying [MATH] with a control parameter (as temperature, or more properly, the inverse of temperature) and [MATH] with an order parameter (... |
at the transition point [MATH] , with [EQUATION] For magnetic systems, [MATH] corresponds to the so-called Curie temperature. For the Galton-Watson branching process we can extract from equation ( 41 ) that |
[EQUATION] where we assume that the variance of [MATH] does not go to zero at the transition point. We can compare the previous general result, |
[MATH] , for [MATH] above but close to 1, with the result we found for the binomial distribution with [MATH] (see equation ( 32 )), for which |
[EQUATION] when [MATH] Using that in this case [MATH] and [MATH] (see Ross, ( 2002 ), [EQUATION] because [MATH] for [MATH] So, equations ( 32 ) and ( 41 ) agree close to the transition point. Figure shows also how they disagree as [MATH] increases. |
Finally, for completeness, we can play with the pathological case given by [MATH] Let us consider first the following model, [MATH] [MATH] (and zero otherwise), with [MATH] Then, [MATH] , and we know that [MATH] Next, let us consider |
[MATH] [MATH] (and zero otherwise), giving [MATH] In this case, [MATH] always, yielding a discontinuous, or first order phase transition. |
2.9 Distribution of the total size of the population: binomial distribution and rooted trees Our main interest will now be to calculate the total size [MATH] of the population, summing across all generations, i.e., |
[EQUATION] this corresponds to the total number of individuals that have ever been born, the total number of neutrons participating in a nuclear chain reaction, or the energy released during an event in an earthquake model. |
Let us go back to the concrete binomial case, [EQUATION] The size distribution can be calculated using elementary probability and combinatorics. One needs to take advantage of the representation of a branching process as a tree (which is a connected graph with no loops). Each element is associated to a node, and branch... |
[MATH] nodes is [MATH] (Christensen and Moloney,, 2005 Therefore, a particular tree of size [MATH] comes with a probability [MATH] , and the probability [MATH] of having an undefined tree of size [MATH] is obtained by summing for all possible trees of size [MATH] In the case [MATH] the number of trees with [MATH] nodes... |
[EQUATION] see the Appendix for its calculation. Then, [EQUATION] It can be checked, using the generating function of the Catalan numbers, that this expression is normalized for [MATH] but not for [MATH] in fact, |
[EQUATION] see the Appendix again. Nevertheless, the exact expression we have obtained for [MATH] does not teach us anything about the behavior of this function (unless one has a great intuition about the behavior of the binomial coefficients). In this regard, Stirling’s approximation is of great help (Christensen and ... |
[EQUATION] see the Appendix once more. The symbol [MATH] is nothing else than the [MATH] number. So, for large sizes we can apply the approximation to [MATH] |
and also to [MATH] [EQUATION] Therefore, the binomial coefficient turns out to be, [EQUATION] and the Catalan number, replacing [MATH] |
[EQUATION] This is an exponential increasing function of [MATH] and the term [MATH] does not seem to play any role, asymptotically. However, introducing the factor [MATH] , we go back to equation ( 49 ), getting |
[EQUATION] Notice that [MATH] is no larger than [MATH] so the exponential term becomes decreasing, except for [MATH] , where it disappears. We can go one step further, by writing, |
[EQUATION] with the characteristic size defined as [EQUATION] and finally equation ( 55 ) reads, [EQUATION] So, for [MATH] large, but substantially smaller than [MATH] the size probability mass function is a power law, with exponent [MATH] For larger [MATH] , the exponential decay dominates. The exception is the critic... |
Another critical exponent arises for the divergence of the characteristic size [MATH] . Introducing the deviation with respect to the critical point, |
[MATH] , one can write, [EQUATION] and so, close to the critical point (for small [MATH] ), [EQUATION] (using the formula of the geometric series), then |
[EQUATION] (using the Taylor expansion of the logarithm at point 1) and [EQUATION] Therefore, the characteristic size [MATH] diverges at the critical point as a power law, with an exponent equal to 2. This allows to write the asymptotic formula ( [MATH] large) for the size distribution in a simpler form, close to the c... |
[EQUATION] Hence, after this perhaps long but worthwhile digression, we are able to say something about the energy distribution in Otsuka’s model, which the reader will have already noted is a particular case of the Galton-Watson process. If one takes [MATH] the resulting energy distribution has an exponential tail, wi... |
(the probability of having an earthquake of size larger than [MATH] is ridiculously small). This is the subcritical case. On the other hand, if [MATH] there are two types of earthquakes, first, those similar to the subcritical ones, with a size limited by the scale defined by [MATH] and second, infinite or never-ending... |
The agreement between the model and real earthquakes is qualitative but not quantitative, as the model leads to [MATH] whereas for earthquakes |
[MATH] In the next subsection we will explain that the model value of 3/2 is rather robust and other versions of the Galton-Watson process lead to the same exponent. This discrepancy has been explored in detail by Kagan, ( 2010 who argues that there are a series of technical artifacts that make increase the value of th... |
2.10 Generating function of the total size of the population In order to advance further in the understanding of branching processes, our little story carries us to the U.S. during the Second World War. While soldiers were fighting in the field and civilians were suffering the horrors of war, a group of scientists gath... |
Hawkins and Ulam showed, among other things, that the generating function [MATH] of the total size of the population, [MATH] , fulfills, in the non-supercritical case, |
[EQUATION] where, as usual, [MATH] is the generating function of the number of offsprings of an individual element. What follows in this subsection is based in their work for the Manhattan Project (Hawkins and Ulam,, 1944 ; Ulam,, 1990 but our derivation is somewhat simpler. What we call total size of the population wi... |
First, it is convenient to consider the size from generation 1 to [MATH] (excluding by now the zero generation). This is [EQUATION] |
with probabilities [MATH] and a generating function [MATH] A size [MATH] in generations from 1 to [MATH] can be decomposed into a size [MATH] in the first generation, with probability [MATH] and a size [MATH] in the remaining [MATH] generations (from [MATH] to [MATH] ), but starting with [MATH] elements; this has a pro... |
[EQUATION] except for [MATH] , where [MATH] If we multiply by [MATH] and sum for all [MATH] , from [MATH] to [MATH] we will obtain on the left hand side the generating function of [MATH] , which turns out to be |
[EQUATION] The term inside the square brackets is the generating function of the size from [MATH] to [MATH] generations but, instead of starting with one single element (the usual [MATH] ), starting with [MATH] elements ( [MATH] ). As these [MATH] parents are independent of each other, the resulting size will be the su... |
[EQUATION] Substituting into equation ( 67 ), this leads to [EQUATION] where we have introduced the definition of [MATH] If we want to include the zero generation in the size, we need to add an independent variable with generating function [MATH] |
(as [MATH] takes the value 1 with probability 1), and then, the generating function of the size from generation [MATH] to [MATH] |
is the product [MATH] This leads to [EQUATION] Coming back to the total size, [EQUATION] the corresponding generating function is [MATH] If the probability of extinction is one, i.e., if the system is not supercritical, this is the same as [MATH] and therefore we have |
[EQUATION] So, the desired generating function is the solution of this equation, with [MATH] known. We will not be able to solve it in general; however, notice that this is not necessary in order to get the moments of [MATH] Differentiating equation ( 72 ) with respect [MATH] one obtains |
[EQUATION] and taking [MATH] and isolating, [EQUATION] which goes to infinity as [MATH] goes to 1, that is, at the critical point. Of course, as we have mentioned, the result is not applicable in the supercritical case, [MATH] where the population can growth to infinite with a non-zero probability. Further differentiat... |
The same result could have been obtained directly, as [EQUATION] where the last equality only holds in the subcritical case, otherwise, [MATH] goes to infinity. |
In a few cases, the equation for [MATH] allows to easily obtain a solution. Revisiting the binomial example with [MATH] , for which |
[MATH] , one gets [EQUATION] from where [EQUATION] with [MATH] Using the Taylor expansion for the square root term (see the Appendix), |
[EQUATION] and recognizing the Catalan numbers [MATH] there, we get (see the Appendix), [EQUATION] where we also realize that only the minus sign before the square root leads to a true generating function. Therefore, the coefficients of [MATH] lead to |
[EQUATION] for [MATH] . This result is exactly the same as the one we obtained previously in a different manner (see equation ( 49 )), although in this way we do not need to count trees, as the Catalan numbers arise directly in the series expansion (in fact, we do not even need to know them). |
We confirm that the results for Otsuka’s binomial model yield a size exponent equal to 3/2. But it would be desirable to test the robustness of such exponent value, as, after all, the model is a crude simplification of reality, and we would like that modifications of the model do not lead to a totally different behavio... |
2.11 Self-organized branching process At this point we are ready to accept the agreement, not only qualitative but, following Kagan’s remarks (Kagan,, 2010 , also quantitative, between a critical branching process and earthquake occurrence. So, in order to tune the model to reality we just need to take [MATH] (in Otsuk... |
But we can try to go one step farther and ask: why do we find that the tectonic systems (and other geosystems related to natural catastrophes) are always keeping a delicate balance between a subcritical and a supercritical state, i.e., in an apparent critical state? Can the coincidence be just fortuitous? In the reprod... |
However, this evolutionary scenario is not applicable to a tectonic system, where, when the process (the earthquake) gets extinct, a new one will start sooner or later. Rather, the situation would be analogous to finding all magnetic materials on Earth at the onset of magnetization, which would mean that their temperat... |
(Sornette,, 1992 ; Pruessner and Peters,, 2006 Zapperi et al., ( 1995 propose a model in this line. They start with a standard branching process but introduce some important modifications: |
They limit the number of generations to a maximum [MATH] , so [MATH] After the extinction of the process (which is obviously certain when the number of generations is limited), the parameters of the process change for the next realization, in such a way that for subcritical cases ( [MATH] ), the mean [MATH] of the numb... |
In order to be more concrete, let us consider the usual binomial distribution with only 0, 1, or 2 possible offsprings and a probability [MATH] that each reproductive trial is successful. Then we already know that [MATH] [MATH] , and [MATH] |
correspond to the subcritical, critical, and supercritical cases, respectively. The dynamics proposed by Zapperi and coauthors relies on the activity that reaches the “boundary” of the system (defined by the last generation, [MATH] ), which is [MATH] , changing the probability [MATH] through the following formula |
[EQUATION] with [MATH] a discrete time index counting the number of realizations of the process (do not confuse with [MATH] ) and |
[MATH] the maximum number of possible elements, i.e., the number of branches of the underlying complete tree. Thus, if the activity does not reach the boundary, |
[MATH] is zero and the parameter [MATH] is increased by [MATH] this is a very small number in the limit of very large systems [MATH] ). On the other hand, if the activity at the boundary is greater than one, [MATH] is decreased by [MATH] |
We already know that the expected value of [MATH] is [MATH] with [MATH] the mean of the offspring distribution ( [MATH] in our particular binomial model). Let us introduce a noise term, [MATH] , which takes into account the fluctuations of [MATH] with respect its mean, i.e., [MATH] Obviously, by construction, [MATH] If... |
[EQUATION] This is a discrete dynamical system, or a map, for which a fixed point [MATH] exists, [MATH] Moreover, the fixed point is attractive, as |
[MATH] (Alligood et al.,, 1997 , due to [MATH] Taking into account the value of the standard deviation of [MATH] (Harris,, 1963 , it can be shown that the noise term [MATH] will have a vanishing effect in the limit of very large systems, and then the stochastic evolution will lead the system towards the deterministic f... |
This spontaneous evolution of a system towards a particular organized state is referred to as self-organization. It is clear now that what Zapperi et al. introduced is a branching process that self-organizes towards a critical state. Nevertheless, the particular dynamics they propose seems a bit arbitrary. How can this... |
2.12 Self-organized criticality and sandpile models In fact, the self-organized branching process introduced by Zapperi et al., ( 1995 was naturally embedded in the previous notion of self-organized criticality (SOC), invented by Bak and coworkers in the 1980’s (Bak,, 1996 ; Jensen,, 1998 ; Christensen and Moloney,, 20... |
(Bak et al.,, 1987 The fact that earthquakes (and other hazards) were a manifestation of self-organized criticality was a fortunate by-product, pointed by |
Ito and Matsuzaki, ( 1990 Sornette and Sornette, ( 1989 , and Bak and Tang, ( 1989 shortly after the introduction of the SOC concept, see also the review of Main, ( 1996 Nowadays, natural hazards are one of the main applications of SOC, despite the original lack of attention by Bak et al., ( 1987 As we have seen throug... |
The metaphor used by Bak in order to illustrate his ideas was that of a pile of sand (Bak,, 1996 We have to recognize that the sandpile we are going to consider is a bit esoteric; in fact, there is a clear correspondence between the model and a pile only in one dimension (the one-dimensional model corresponds to a pile... |
(Newman,, 2011 So, consider a system consisting in a large number of elements, such that each element can store a certain number of discrete packages (or particles), but when this limit is surpassed the packages are released to other elements – the neighbors. The situation is analogous to what happens in a Ministry off... |
To be specific, let us consider that each element can store at most one package; if some extra package arrives to it, the element releases two packages to some other units, taken randomly (either among all other elements, what defines random neighbors or among the [MATH] nearest neighbors in a [MATH] dimensional square... |
[EQUATION] where [MATH] counts the number of packages of element [MATH] and [MATH] denotes two of its neighbors. Obviously, this process can give rise to an avalanche in the transference of packages, which only stops when all elements have no more than one package. In that case, the system is perturbed by the addition ... |
[EQUATION] where [MATH] denotes a randomly selected unit. The system also releases packages outside (or to the garbage can, in the bureaucrats picture); in a [MATH] dimensional lattice this happens when a boundary element selects as a neighbor an external element; in a fully random-neighbor system this happen just with... |
The simple rules of the model make that the total number of packages in the system, [MATH] evolves, from the addition of one package to the next, accordingly to |
[EQUATION] where drop is the number of packages that are expelled from the system. The key parameter of this model is [MATH] defined, for each element, as the probability that its number of packages is equal to one (so they are at the onset of instability). But in a mean field description all elements are uncorrelated ... |
we obtain [EQUATION] which we can recognize as essentially equation ( 81 ), the one introduced by Zapperi et al., ( 1995 in the self-organized branching process. We have already realized that this equation provides a feedback mechanism of the number of packages into the toppling (branching) probability (early identific... |
Both in the limit of an infinite dimension lattice or in a fully random neighbor system one realizes that the evolution of an avalanche corresponds to a set of propagating non-interacting packages (as the probability that the activity comes back to an element is vanishingly small), and therefore the activity evolves as... |
Recapitulating, self-organized criticality offers a coherent framework for the understanding of earthquakes and many other natural hazards mentioned in the first section. Indeed, both phenomena (SOC and earthquakes) show a highly non-linear response, where a small and slow perturbation or driving (the addition of grain... |
Main, ( 1996 mentions additional characteristics of seismicity present in SOC models, namely, stress drops that are small in comparison with the regional tectonic stress field and the existence of seismicity induced or triggered by relatively small stress perturbations. All this makes SOC a very plausible mechanism for... |
as the so-called OFC model (Olami et al.,, 1992 See also Main, ( 1996 However, as far as we know, the authentic hallmark of SOC, the existence of an underlying second-order (continuous) phase transition, has not been found in earthquakes. The very nature of SOC makes almost impossible to identify such an abrupt change ... |
The same reasoning applies to other natural hazards, for which, at least, sandpile-like models are abundant in the literature, and their classification as SOC systems is plausible (Jensen,, 1998 The case of hurricanes is still not clear (Corral,, 2010 whereas for tsunamis we can state that their power-law distribution |
(Burroughs and Tebbens,, 2005 does not arise from a SOC mechanism, as they are not slowly driven (rather, they are violently driven by earthquakes, landslides and meteorite impacts). |
Finally, it is worth mentioning that there is another connection between branching processes and earthquakes. Instead of using the branching to model the propagation of individual earthquakes, it is used for the way in which one earthquake triggers other earthquakes, i.e., aftershocks, following the so-called Omori law... |
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