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Figures 12 13 and 14 show FHT and number of optima for 100 runs for each value of [MATH] on Small-World, scale-free and random networks respectively. All the three cases exhibit a similar behavior: positive values of [MATH] lead to slower exploration (i.e. smaller FHT) and higher diversity (i.e. bigger number of optima... |
As we did using rewiring parameter for Small-World networks in the previous section, we are able to decide the trade-off between exploration speed and diversity simply by tuning a single parameter. There is an important difference in tuning the algorithm using rewiring parameter ( [MATH] ) and weights update parameter ... |
VII Conclusions In this work we presented an investigation of the effects of network topology on the dynamics of a spatially-structured evolutionary algorithm for two combinatorial problems. In order to study the effects of the variations of network features, Small-World network models have been chosen due to the possi... |
The results on OneMax problem allow to depict the behavior of the selected network models with respect to their First Hitting Time and the generations needed to have the entire population composed by optimal individuals (FCT, First Convergence Time). A smaller APL resulted in faster exploration and convergence for all ... |
Simulations on Nmax problem allows to analyze with more accuracy the diversity of various algorithms, measured by the number of distinct optima found during the execution. The change of the rewiring factor [MATH] leads to a variation of the FHT in the range [MATH] (considering all the values of [MATH] ) with respect to... |
Observing both the results on OneMax and Nmax problems we can conclude that applying different graph models on ssEA we are able to obtain faster algorithms or algorithms able to explore a larger area of the solution space. We may correlate APL, which defines the speed of diffusion of information on a network, with the ... |
VII-A Future Work Although this work shed some light on the relationship between network structure and ssEAs performances many questions still remain open. As the problem of the diffusion and creation of the optima and the spreading of a rumor or an infectious disease have some common features Nekovee et al., 2007 , it... |
Moreover, it could be promising to better investigate the relationships between topological features, such as e.g APL and clustering coefficient, and algorith dynamics, possibly applying ssEAs on other common optimization problems and comparing them with classical EAs using explicit diversity maintaining methods (like ... |
Acknowledgment The authors would like to thank Dr. Andrea Gasparri for his precious comments. Matteo De Felice was born in Rome, Italy, in 1982. He received the Laurea Magistrale Degree in Informatics Engineering and the Ph.D. degree in Computer Science and Automation from the University of Rome “Roma Tre” in 2007 and ... |
From 2007 to 2010 he was working in the Energy Efficiency Department of ENEA (Italian Energy, New Technology and Environment Agency) and in 2011 he joined the Energy and Environment Modelling Unit in the same institution. His current research interests include neural networks, evolutionary computation and their applica... |
Dr. De Felice is a member of the IEEE Computational Intelligence Society and ACM Special Interest Group on Genetic and Evolutionary Computation. |
Sandro Meloni was born in Rome, Italy, in 1982. He carried out his undergraduate studies at the Department of Informatics and Automation of the University of Rome “Roma Tre”. He got his master degree in Computer Science in 2006. In October 2007 he started his Ph.D. at the same university dealing with Epidemic and Traff... |
Stefano Panzieri was born in Rome (Italy) on December 17th 1963. He took the Ph.D. in System Engineering in 1994 at University of Rome “La Sapienza”, From 1996 he is with the University of Rome “Roma Tre” as Associate Professor. His teachings are in the field of Automatic Control, Digital Control and Process Control wi... |
# Source: arxiv 1206.4893 # Title: Quantifying Self-Organization with Optimal Wavelets # Sections: all # Downloaded: 2026-03-03T01:58:42.898038+00:00 |
Quantifying Self-Organization with Optimal Wavelets Abstract The optimal wavelet basis is used to develop quantitative, experimentally applicable criteria for self-organization. The choice of the optimal wavelet is based on the model of self-organization in the wavelet tree. The framework of the model is founded on the... |
Self-organization, information theory, statistical complexity, wavelets pacs: 05.65. +b, 02.50.Tt, 89.75.Fb, 89.75.Kd In the most general sense, the term self-organization refers to the process or processes which cause the emergence of structures and organized behavior without the external influence. Measuring organiza... |
cosma2 feldman . Here we adhere to statistical description of the system and its configurations using the wavelet-domain decomposition and the properties of the wavelet tree (the graph of wavelet coefficients) |
hernandez mallat and statistical properties of the wavelet coefficients. The method is based on a parametric model for a wavelet tree distribution attributing hidden Markov (HM) variable to each node of the tree. The wavelet tree is considered as a self-organizing system by identifying hidden states of wavelet coeffici... |
The optimality of basis is essential for faithful representation of the original data (signal) and even more so for compression and denoising. The only systematic approach to this problem, founded on the microcannonical cascade formalism and applied to signals with microcannonical cascade processes, was presented in or... |
The wavelet transform decomposes a one dimensional spatial signal [MATH] in terms of shifted and dilated versions of a bandpass wavelet function [MATH] |
and shifted versions of a lowpass scaling function [MATH] hernandez mallat . For a signal of dyadic dimension [MATH] [MATH] length), the representation is |
[EQUATION] where [MATH] and [MATH] while [MATH] indexes dyadic scale of resolution (greater [MATH] correspond to higher resolution) and |
[MATH] indexes the spatial location. For a wavelet [MATH] centered at frequency [MATH] the detail coefficient [MATH] measures the signal content around place [MATH] and frequency [MATH] . Thus, we get a pyramid of detail coefficients in the form of the binary tree, presented in Fig. 1(a), in which each coefficient at a... |
[MATH] is used, starting numeration from the root of the tree. The label of predecessor for the node [MATH] is [MATH] . For random variables we use capital letters to denote the variable and lower case letters to denote realization of this variable. Wavelet decomposition of real-world data is sparse so that most of the... |
Sparsity of representation indicates that distribution of wavelet coefficients is non-Gaussian, typically much more peaky at zero and more spread elsewhere than a Gaussian crouse . A more suitable model of this density is a mixture of two Gaussians whose components corresponds to yin and yang states: |
[EQUATION] In the above expression, [MATH] denotes density function of the random variable that models detail coefficient of the node [MATH] , and [MATH] |
denotes distribution of hidden variable [MATH] whose values [MATH] or [MATH] correspond to the yin or yang states of the node. [MATH] is the number of components but model can be easily generalized to arbitrary number of hidden states. Gaussian density function of an argument [MATH] with mean [MATH] and variance [MATH]... |
Due to the wavelet tree structure, each node at the coarser scale has two successors at the finer one that share its spatial support. As a consequence, appearance of yang (yin) coefficient in a node very likely means that its successors will be yang (yin) coefficients. For that reason, hidden states tend to propagate a... |
crouse For [MATH] -state Gaussian mixture model for each wavelet coefficient ), HMM is determined with parameter model vector [EQUATION] |
using abbreviations [MATH] [MATH] Parameter estimation is performed by applying the maximum likelihood principle (ML) which is asymptotically efficient, unbiased and consistent as the number of observations increases. Direct ML estimation of the model parameters ( ) from the observed data is intractable since in estima... |
[MATH] . Yet, given the values of the states, ML estimator of [MATH] is simple (merely ML estimator of Gaussian means and variances). Therefore, we employ an iterative expectation maximization (EM) approach dempster , which jointly estimates both the model parameters [MATH] and probabilities for the hidden states |
[MATH] , given the observed coefficients [MATH] Due to the limited data available usually from only one or few signal observations random variables that have similar properties are modeled using a common distribution or common parameter set, the practice is known as |
tying rabiner . In order to ensure reliable parameter estimation we must share statistical information between related wavelet coefficients so we assume that all wavelet coefficients and state variables within a common scale are identically distributed, including identical parent-child state transition probabilities. C... |
crouse by developing a novel signal denoising method. Reconstructing the original signal all states with variances less then the noise variance are estimated to a single common value i.e. their informational content is completely lost.Having background noise of unknown power, all yin states of the data are essentially ... |
A paradigmatic approach to the emergence of self-organization phenomena, presented in cosma1 cosma2 and feldman begins with a dynamic random field on the network on which the random field of local causal states is constructed. To predict the original field either locally or globally, it is sufficient to know causal sta... |
grassberger the local statistical complexity is defined as the entropy of local causal state [EQUATION] If a spatially stationary process is dynamically autonomous from external influences self-organization takes place between time [MATH] and time [MATH] if and only if [MATH] |
cosma1 . Our aim is to perceive HMM from the viewpoint of self-organization giving the concept of self-organization specific physical interpretation within the model. Some semantic analogies of the terms used in cosma1 and cosma2 and the wavelet-domain HMM will be used in order to make the ideas more clear. First, it i... |
w-machine in analogy with the [MATH] machine presented in crutch1 and crutch2 . Random variable [MATH] represents the global causal state which contains minimal information for optimal prediction in the spatial domain. The proof follows from the EM algorithm which minimizes [MATH] so we have [MATH] . Knowledge of [MATH... |
only. The entropy of the wavelet tree may be expressed as [EQUATION] where [MATH] and [MATH] are differential entropies of continuous random variables. The extensive term [MATH] |
represents irreducible randomness that remains even after all correlations are subsumed. Addition of noise increases only this term while complexity [MATH] |
remains unaltered. Local complexity [MATH] has a specific physical interpretation - it is higher if the distribution of hidden yang an yin states in the node is more uniform. In that case, there is higher probability of yang coefficient appearance based on the persistence property in the nodes at the immediate neighbor... |
fulfills that goal for the complete tree. Higher global complexity means that yang states are more uniformly distributed within the tree allowing for more optimal preservation of background information. So, we define optimal representation of the data (signal) as the one which maximizes global complexity of the tree. W... |
cosma1 because global state is not determined from local states in only one time instant. This is the consequence of temporal irrelevance since prediction takes into consideration the complete signal, i.e. both the past and the future of the wavelet tree. Regardless of these differences, we demonstrate that optimality ... |
Derivation of the global complexity in terms of model parameters yields [EQUATION] This expression takes higher values if conditional variables [MATH] are more uniformly distributed i.e. if probability of changing state is higher. But in this case local states also tend to be more uniformly distributed so that local co... |
However, it is obvious that optimality of representation based on self-organization in the wavelet-tree implies optimal wavelet-based noise reduction. |
We illustrate the method in the context of dynamical systems by considering structure and randomness of the time series [MATH] , where [MATH] . The term [MATH] in Eq.( represents the measure of complexity (structure) and the conditional entropy |
[MATH] is the measure of randomness. Both are represented in Fig. 2 as a function of parameter [MATH] generated using the optimal, biorthogonal1.3, wavelet. The maximum complexity is attained for parameter value 3.5926, i.e. the value at which the deterministic chaos sets in. In Fig. 3 we present the complexity-entropy... |
For a given value of entropy multiple values of complexity are noticed indicating an intricate relationship between these two quantities. Not all complexity values are realizable for a particular entropy rate. Organization is evident in the diagram consisting of low and very high density regions exhibiting self-similar... |
We have argued that w-machine establishes relationship between information, prediction, retrodiction and denoising founded on the choice of the optimal wavelet and within the framework of statistical mechanics. Statistical complexity may be reliably calculated from data and at the same time noise may be removed in a hi... |
# Source: arxiv 1207.2589 # Title: Criticality and self-organization in branching processes: application to natural hazards # Sections: all # Downloaded: 2026-03-03T01:58:32.038577+00:00 |
Criticality and self-organization in branching processes: application to natural hazards Abstract 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... |
The Statistics of Natural Hazards Only fools, charlatans and liars predict earthquakes C. F. Richter Men, and women, have always been threatened by the dangers of Earth: volcanic eruptions, tsunamis, earthquakes, hurricanes, floods, etc. Sadly, still in the 21st century our societies have not been able to get rid of su... |
1.1 The Gutenberg-Richter law One of the first laws quantifying the occurrence of a natural hazard was proposed for earthquakes by the famous seismologists Beno Gutenberg and Charles F. Richter in the 1940’s, taking advantage from the recent development of the first magnitude scale by Richter himself. The Gutenberg-Ric... |
It is not possible to measure all earthquakes on our planet, but for some areas with very accurate seismic monitoring it has been found that the Gutenberg-Richter law holds down to magnitude minus 4 (Kwiatek et al.,, 2010 this corresponds to small rock cracks of a few centimeters in length (negative magnitudes are intr... |
Despite not being recognized or mentioned by Gutenberg and Ritchter in their original paper (1944), any reader with a minimum knowledge of probability and statistics will immediately realize that the Gutenberg-Richter law implies an exponential distribution of the magnitudes of earthquakes, i.e., |
[EQUATION] with [MATH] the probability density of [MATH] the parameter [MATH] taking a value close to 1, and the symbol [MATH] standing for proportionality (with the constant of proportionality ensuring proper normalization). |
But which is the meaning of the Gutenberg-Richter law, in addition to provide an easy-to-remember relationship between the relative abundances of earthquakes? The interpretation depends, of course, on the meaning of magnitude, which we have avoided to define. In fact, there is no a unique magnitude, but several of them... |
(Ben-Zion,, 2008 More in-depth understanding comes from the energy radiated by an earthquake, which is believed to be an exponential function of its magnitude |
(Kanamori and Brodsky,, 2004 that is, [EQUATION] with a proportionality factor close to 60 kJ (Utsu,, 1999 so, an increase by 1 in the magnitude implies an increase in energy by a factor [MATH] Thus, an earthquake of magnitude 9 radiates as much energy as 1000 earthquakes of magnitude 7, or as [MATH] of magnitude 5. |
One can reformulate then the Gutenberg-Richter law in terms of the energy. Indeed, the probability of an event is “independent” of the variable we use to describe it, and so, |
[EQUATION] with [MATH] the probability density of the energy. Using equation ( ), we can express [MATH] as a function of [MATH] [EQUATION] |
and differentiate to obtain [MATH] [EQUATION] so that equation ( ) reads: [EQUATION] Summarizing, this straightforward change of variables leads to |
[EQUATION] and this is just the so-called power-law distribution, or Pareto distribution (Newman,, 2005 , with exponent [MATH] around 1.67 when [MATH] is close to 1. Notice from equation ( ) that in order that [MATH] is a proper probability density function, it has to be defined above a minimum energy [MATH] otherwise ... |
cannot be measured (it is too small), this parameter is not important as it does not influence any properties of earthquakes. Figure |
displays the probability density of the seismic moment for worldwide shallow earthquakes (Kagan,, 2010 this variable is assumed to be proportional to the energy, but much easier to measure accurately (Kanamori and Brodsky,, 2004 and so, it should also be power-law distributed, with the same exponent. The straight line ... |
(Clauset et al.,, 2009 ; Peters et al.,, 2010 yields [MATH] Two important properties of power-law distributions are scale invariance (with some limitations due to the normalization condition) and divergence of the mean value (if the exponent [MATH] is below or equal to 2). These are explained in the Appendix. |
To conclude this subsection, let us mention that the power-law distribution of sizes is not a unique characteristic of earthquakes. It has been claimed that many other natural hazards are also power-law distributed, although with different exponents (and maybe with a lower or an upper cutoff): tsunamis (Burroughs and T... |
1.2 A first model for earthquake occurrence As far as we know, a first attempt to develop an earthquake model in order to explain the Gutenberg-Richter law was undertaken by Michio Otsuka in the early 1970’s (Otsuka,, 1971 1972 ; Kanamori and Mori,, 2000 He used as a metaphor the popular Chinese game of go, although we... |
Instead of playing domino, we are going to play a different game with their pieces. The idea is to make the domino pieces to topple, as in the well-known contests and attempts to break a Guinness world record, but with two important differences. First, the pieces are not put in a row, but, rather, they constitute a kin... |
Getting more concrete, Otsuka assumed that the tree representing the fault had a fixed number of branches at each position, or node, and that the toppling would propagate from each branch to the next element with a fixed probability [MATH] independently of any other variable. So, the number of propagating branches resu... |
The novel and original model in geophysics explained in this subsection, proposed by Otsuka in the 1970’s, was already known by a few mathematicians 100 years in advance. It will take us the next pages to explain the distribution of energy in this model. |
Branching Processes Besides gambling, many probabilists have been interested in reproduction G. Grimmett and D. Stirzaker Let us move to the Victorian (19th century) England. There, Sir Francis Galton, the polymath father of the statistical tools of correlation and regression, and cousin of Charles Darwin, was dedicate... |
(Watson and Galton,, 1875 In order to better understand the problem, he devised a null model in which the number of sons of each men was random (the abundance of women was not considered to be a limitation). Despite the apparent simplicity of the model, Galton was not able to solve it, and made a public call for help. ... |
2.1 Definition of the Galton-Watson process Let us consider “elements” that can generate other elements and so on. These elements may represent British aristocratic men that have some male descendants, (or, in a more fresh perspective, women from anywhere that give birth to her daughters, or, perhaps more properly, bac... |
[MATH] is independent from that of the other elements and all [MATH] are identically distributed, with probabilities [MATH] [MATH] [MATH] , with [MATH] |
(Harris,, 1963 (Naturally, the normalization condition imposes [MATH] .) The model starts with one single element, in what we call the zeroth generation of the process, as shown in Fig. . The [MATH] offsprings of this first element constitute the first generation. Let [MATH] denote the number of elements of the zeroth ... |
[EQUATION] with [MATH] where [MATH] corresponds to the number of offsprings of each element in the [MATH] generation. Equation ( ) can be used to simulate the process in a straightforward way and will be very important to its analytical treatment, in order to calculate the probability distribution of [MATH] , for any [... |
form a Markov chain, but this is not relevant for our purposes. And of course, Otsuka’s earthquake model is a particular case of the Galton-Watson process corresponding to a binomial distribution for [MATH] |
2.2 Generating functions An extremely convenient mathematical tool will be the probability generating function (Grimmett and Stirzaker,, 2001 For the random variable [MATH] this is, by definition, |
[EQUATION] where the brackets indicate expected value. The normalization condition guarantees that [MATH] is always defined at least in the [MATH] interval [MATH] although only the interval [MATH] will be of interest for us. Of course, the same definition applies to any other random variable; in the concrete case of [M... |
Very useful and straightforward properties will be, 1. [MATH] 2. [MATH] (by normalization); 3. [MATH] 4. [MATH] for [MATH] (non-decreasing function); |
5. [MATH] for [MATH] (non-convex function, “looking from above”); the primes denoting derivatives (left-hand derivatives at [MATH] ). Note that although we illustrate these properties with the variable [MATH] they are valid for the generating function of any other (discrete) random variable. So, the plot of a probabili... |
[MATH] or whether [MATH] This is natural, as the first case corresponds to a population that on average decreases from one generation to the next whereas in the second case the population grows, on average. |
Another property but not so straightforward is that the generating function of a sum of [MATH] independent identically distributed variables [MATH] |
(with [MATH] fixed) is the [MATH] -th power of the generating function of [MATH] that is, if [EQUATION] then [EQUATION] Indeed, [EQUATION] |
where we can factorize the expected values due to statistical independence among the [MATH] ’s. In general, if the random variables [MATH] were not identically distributed (but still independent), the generating function of their sum would be the product of their generating functions. The demonstration is essentially t... |
A following step is to consider that [MATH] is also a random variable, with generating function [MATH] . Then, [EQUATION] Note that equation ( 13 ) is just a generalization of equation ( 11 ), i.e., now we calculate the expected value of the powers of [MATH] depending on the values that [MATH] make take. In any case, i... |
[EQUATION] where the last equality is just the definition of the probability generating function of the random variable [MATH] , evaluated at [MATH] We stress that this is only valid for independent random variables. |
2.3 Distribution of number of elements per generation Going back to the Galton-Watson branching process, where we know that [MATH] we can identify [MATH] as [MATH] and [MATH] as [MATH] ; then equation ( 13 ) reads, |
[EQUATION] (dropping the subindex [MATH] ). As [MATH] it is straightforward to see by induction that the generating function of [MATH] is given by |
[EQUATION] where the superindex [MATH] denotes composition [MATH] times. This is valid for [MATH] ; for [MATH] we have, obviously, that |
[MATH] (because [MATH] with probability 1). In words, the generating function of the number of elements for each generation is obtained by the successive compositions of [MATH] This non-trivial result was first proved by Watson in 1874 (Harris,, 1963 |
2.4 Expected number of elements per generation Here we present an illuminating result, which will be useful at some point in the chapter. Although, in general, the successive compositions of the generation function leads to very complicated mathematical expressions, the moments of [MATH] can be computed in a simple way... |
[EQUATION] Let us then write [EQUATION] therefore, by induction, [EQUATION] Taking [MATH] and using that all the generating functions have to be 1 at that point, |
[EQUATION] So, when [MATH] the mean number of elements per generation decreases exponentially, whereas when [MATH] this number increases, constituting a stochastic realization of Malthusian growth. For this reason [MATH] is sometimes called the branching ratio. When [MATH] the average size of the population is constant... |
Another related issue is the one of the expected value of the number of elements per generation conditioned to the value of the previous generation, i.e., [MATH] As when [MATH] is fixed, [MATH] then, taking the expected value, |
[EQUATION] This result can be used to relate branching processes with martingales (Grimmett and Stirzaker,, 2001 but this does not have to bother us. |
2.5 The probability of extinction Extinction of the process is achieved when [MATH] for the first “time” (i.e., for the generation that yields [MATH] for the first [MATH] ). Then, all the subsequent [MATH] ’s are also zero, and extinction can be considered an “absorbing state”, in this sense. We now see that the probab... |
and is smaller than one for [MATH] This result, which may be referred to as the Galton-Watson-Haldane-Steffensen (criticality) theorem, was first proved by J. F. Steffensen, in the 1930’s (being unaware of the work by Galton and Watson, and later progress by Haldane). As Kendall, ( 1966 pointed out, after then, the sam... |
(Kendall,, 1975 Indeed, extinction may happen at the first generation, [MATH] or at the second, [MATH] , etc. All these extinction events are included in [MATH] , with [MATH] therefore, the probability of extinction [MATH] is given by |
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