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78-470: WPSL may refer to: Weighted planar stochastic lattice , a mathematical structure sharing some of the properties both of lattices and of graphs Women's Premier Soccer League WPSL (AM) , a radio station (1590 AM) licensed to Port St. Lucie, Florida, United States National Pro Fastpitch , formerly known as the Women's Pro Softball League Topics referred to by

156-403: A x − k {\displaystyle f(x)=ax^{-k}} , scaling the argument x {\displaystyle x} by a constant factor c {\displaystyle c} causes only a proportionate scaling of the function itself. That is, where ∝ {\displaystyle \propto } denotes direct proportionality . That is, scaling by

234-460: A power law . Besides, unlike regular lattices, the sizes of its cells are not equal; rather, the distribution of the area size of its blocks obeys dynamic scaling , whose coordination number distribution follows a power-law. The construction process of the WPSL can be described as follows. It starts with a square of unit area which we regard as an initiator. The generator then divides the initiator, in

312-422: A relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent : one quantity varies as a power of another. The change is independent of the initial size of those quantities. For instance, the area of a square has a power law relationship with the length of its side, since if the length is doubled, the area is multiplied by 2 , while if

390-575: A weighted planar stochastic lattice (WPSL) is a structure that has properties in common with those of lattices and those of graphs . In general, space-filling planar cellular structures can be useful in a wide variety of seemingly disparate physical and biological systems. Examples include grain in polycrystalline structures, cell texture and tissues in biology, acicular texture in martensite growth, tessellated pavement on ocean shores, soap froths and agricultural land division according to ownership etc. The question of how these structures appear and

468-406: A common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. The Tweedie distributions are a family of statistical models characterized by closure under additive and reproductive convolution as well as under scale transformation. Consequently, these models all express a power-law relationship between the variance and the mean. These models have

546-572: A constant c {\displaystyle c} simply multiplies the original power-law relation by the constant c − k {\displaystyle c^{-k}} . Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f ( x ) {\displaystyle f(x)} and x {\displaystyle x} , and

624-505: A full characterization of the tail behavior of many well-known probability distributions, including power-law distributions, distributions with other types of heavy tails, and even non-heavy-tailed distributions. Bundle plots do not have the disadvantages of Pareto Q–Q plots, mean residual life plots and log–log plots mentioned above (they are robust to outliers, allow visually identifying power laws with small values of α {\displaystyle \alpha } , and do not demand

702-423: A fundamental role as foci of mathematical convergence similar to the role that the normal distribution has as a focus in the central limit theorem . This convergence effect explains why the variance-to-mean power law manifests so widely in natural processes, as with Taylor's law in ecology and with fluctuation scaling in physics. It can also be shown that this variance-to-mean power law, when demonstrated by

780-442: A hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income). Among them are: A broken power law is a piecewise function , consisting of two or more power laws, combined with a threshold. For example, with two power laws: The pieces of a broken power law can be smoothly spliced together to construct

858-402: A lattice, namely the weighted planar stochastic lattice. For instance, unlike a network or a graph, it has properties of lattices as its sites are spatially embedded. On the other hand, unlike lattices, its dual (obtained by considering the center of each block of the lattice as a node and the common border between blocks as links) display the property of networks as its degree distribution follows

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936-419: A more convergent estimate than the maximum likelihood method. It has been applied to study probability distributions of fracture apertures. In some contexts the probability distribution is described, not by the cumulative distribution function , by the cumulative frequency of a property X , defined as the number of elements per meter (or area unit, second etc.) for which X  >  x applies, where x

1014-411: A plane into contiguous and non-overlapping cells. For instance, Voronoi diagram and Apollonian packing are formed by partitioning or tiling of a plane into contiguous and non-overlapping convex polygons and disks respectively. Regular planar lattices like square lattices, triangular lattices, honeycomb lattices, etc., are the simplest example of the cellular structure in which every cell has exactly

1092-402: A power law relationship, as many non power-law distributions will appear as straight lines on a log–log plot. This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in

1170-450: A power-law distribution of the form to the data x ≥ x min {\displaystyle x\geq x_{\min }} , where the coefficient α − 1 x min {\displaystyle {\frac {\alpha -1}{x_{\min }}}} is included to ensure that the distribution is normalized . Given a choice for x min {\displaystyle x_{\min }} ,

1248-438: A significant bias in α ^ {\displaystyle {\hat {\alpha }}} , while choosing it too large increases the uncertainty in α ^ {\displaystyle {\hat {\alpha }}} , and reduces the statistical power of our model. In general, the best choice of x min {\displaystyle x_{\min }} depends strongly on

1326-486: A small finite sample-size bias of order O ( n − 1 ) {\displaystyle O(n^{-1})} , which is small when n  > 100. Further, the standard error of the estimate is σ = α ^ − 1 n + O ( n − 1 ) {\displaystyle \sigma ={\frac {{\hat {\alpha }}-1}{\sqrt {n}}}+O(n^{-1})} . This estimator

1404-450: A small set of universality classes. Similar observations have been made, though not as comprehensively, for various self-organized critical systems, where the critical point of the system is an attractor . Formally, this sharing of dynamics is referred to as universality , and systems with precisely the same critical exponents are said to belong to the same universality class . Scientific interest in power-law relations stems partly from

1482-977: A smoothly broken power law. There are different possible ways to splice together power laws. One example is the following: ln ⁡ ( y y 0 + a ) = c 0 ln ⁡ ( x x 0 ) + ∑ i = 1 n c i − c i − 1 f i ln ⁡ ( 1 + ( x x i ) f i ) {\displaystyle \ln \left({\frac {y}{y_{0}}}+a\right)=c_{0}\ln \left({\frac {x}{x_{0}}}\right)+\sum _{i=1}^{n}{\frac {c_{i}-c_{i-1}}{f_{i}}}\ln \left(1+\left({\frac {x}{x_{i}}}\right)^{f_{i}}\right)} where 0 < x 0 < x 1 < ⋯ < x n {\displaystyle 0<x_{0}<x_{1}<\cdots <x_{n}} . When

1560-429: A wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of theory of large deviations (also called extreme value theory ), which considers the frequency of extremely rare events like stock market crashes and large natural disasters . It is primarily in the study of statistical distributions that

1638-436: Is a slowly varying function , which is any function that satisfies lim x → ∞ L ( r x ) / L ( x ) = 1 {\displaystyle \lim _{x\rightarrow \infty }L(r\,x)/L(x)=1} for any positive factor r {\displaystyle r} . This property of L ( x ) {\displaystyle L(x)} follows directly from

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1716-542: Is a variable real number. As an example, the cumulative distribution of the fracture aperture, X , for a sample of N elements is defined as 'the number of fractures per meter having aperture greater than x . Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope). Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow

1794-486: Is accumulated, they continue to grow. These power-law probability distributions are also called Pareto-type distributions, distributions with Pareto tails, or distributions with regularly varying tails. A modification, which does not satisfy the general form above, with an exponential cutoff, is In this distribution, the exponential decay term e − λ x {\displaystyle \mathrm {e} ^{-\lambda x}} eventually overwhelms

1872-588: Is also a power-law function, but with a smaller scaling exponent. For data, an equivalent form of the cdf is the rank-frequency approach, in which we first sort the n {\displaystyle n} observed values in ascending order, and plot them against the vector [ 1 , n − 1 n , n − 2 n , … , 1 n ] {\displaystyle \left[1,{\frac {n-1}{n}},{\frac {n-2}{n}},\dots ,{\frac {1}{n}}\right]} . Although it can be convenient to log-bin

1950-523: Is an active area of research in statistics; see below. A power-law x − k {\displaystyle x^{-k}} has a well-defined mean over x ∈ [ 1 , ∞ ) {\displaystyle x\in [1,\infty )} only if k > 2 {\displaystyle k>2} , and it has a finite variance only if k > 3 {\displaystyle k>3} ; most identified power laws in nature have exponents such that

2028-431: Is assumed here that a random sample is obtained from a probability distribution, and that we want to know if the tail of the distribution follows a power law (in other words, we want to know if the distribution has a "Pareto tail"). Here, the random sample is called "the data". Pareto Q–Q plots compare the quantiles of the log-transformed data to the corresponding quantiles of an exponential distribution with mean 1 (or to

2106-467: Is close to 0, because Pareto Q–Q plots are not designed to identify distributions with slowly varying tails. On the other hand, in its version for identifying power-law probability distributions, the mean residual life plot consists of first log-transforming the data, and then plotting the average of those log-transformed data that are higher than the i -th order statistic versus the i -th order statistic, for i  = 1, ...,  n , where n

2184-517: Is convenient to assume a lower bound x m i n {\displaystyle x_{\mathrm {min} }} from which the law holds. Combining these two cases, and where x {\displaystyle x} is a continuous variable, the power law has the form of the Pareto distribution where the pre-factor to α − 1 x min {\displaystyle {\frac {\alpha -1}{x_{\min }}}}

2262-502: Is different from Wikidata All article disambiguation pages All disambiguation pages Weighted planar stochastic lattice Starting with a square, say of unit area, and dividing randomly at each step only one block, after picking it preferentially with respect to ares, into four smaller blocks creates weighted planar stochastic lattice (WPSL) . Essentially it is a disordered planar lattice as its block size and their coordination number are random. In applied mathematics,

2340-409: Is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially. The median does exist, however: for a power law x , with exponent ⁠ k > 1 {\displaystyle k>1} ⁠ , it takes the value 2 x min , where x min

2418-449: Is distributed according to a power-law known as the Pareto distribution (for example, the net worth of Americans is distributed according to a power law with an exponent of 2). On the one hand, this makes it incorrect to apply traditional statistics that are based on variance and standard deviation (such as regression analysis ). On the other hand, this also allows for cost-efficient interventions. For example, given that car exhaust

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2496-406: Is equivalent to the popular Hill estimator from quantitative finance and extreme value theory . For a set of n integer-valued data points { x i } {\displaystyle \{x_{i}\}} , again where each x i ≥ x min {\displaystyle x_{i}\geq x_{\min }} , the maximum likelihood exponent is the solution to

2574-453: Is one of the most active applications of network science, being used to foresee the spread of influenza or to contain Ebola. The WPSL can be a good candidate for applying epidemic like models since it has the properties of graph or network and the properties of traditional lattice as well. Power law In statistics , a power law is a functional relationship between two quantities, where

2652-488: Is the normalizing constant . We can now consider several properties of this distribution. For instance, its moments are given by which is only well defined for m < α − 1 {\displaystyle m<\alpha -1} . That is, all moments m ≥ α − 1 {\displaystyle m\geq \alpha -1} diverge: when α ≤ 2 {\displaystyle \alpha \leq 2} ,

2730-426: Is the minimum value for which the power law holds. The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the critical exponents of

2808-474: Is the size of the random sample. If the resultant scatterplot suggests that the plotted points tend to stabilize about a horizontal straight line, then a power-law distribution should be suspected. Since the mean residual life plot is very sensitive to outliers (it is not robust), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots. Log–log plots are an alternative way of graphically examining

2886-626: The method of expanding bins , implies the presence of 1/ f noise and that 1/ f noise can arise as a consequence of this Tweedie convergence effect. Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or Pareto Q–Q plots ), mean residual life plots and log–log plots . Another, more robust graphical method uses bundles of residual quantile functions. (Please keep in mind that power-law distributions are also called Pareto-type distributions.) It

2964-422: The area of the initiator equal to one. In step two, we pick one of the four blocks preferentially with respect to their areas. Consider that we pick the block 3 {\displaystyle 3} and apply the generator onto it to divide it randomly into four smaller blocks. Thus the label 3 {\displaystyle 3} is now redundant and hence we recycle it to label the top left corner while

3042-437: The average and all higher-order moments are infinite; when 2 < α < 3 {\displaystyle 2<\alpha <3} , the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the central moment estimators (like the mean and the variance) for diverging moments will never converge – as more data

3120-441: The best known power-law functions in nature. The power-law model does not obey the treasured paradigm of statistical completeness. Especially probability bounds, the suspected cause of typical bending and/or flattening phenomena in the high- and low-frequency graphical segments, are parametrically absent in the standard model. One attribute of power laws is their scale invariance . Given a relation f ( x ) =

3198-431: The cdfs of the data and the power law with exponent α {\displaystyle \alpha } , respectively. As this method does not assume iid data, it provides an alternative way to determine the power-law exponent for data sets in which the temporal correlation can not be ignored. This criterion can be applied for the estimation of power-law exponent in the case of scale-free distributions and provides

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3276-503: The collection of much data). In addition, other types of tail behavior can be identified using bundle plots. In general, power-law distributions are plotted on doubly logarithmic axes , which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) cumulative distribution (ccdf) that is, the survival function , P ( x ) = P r ( X > x ) {\displaystyle P(x)=\mathrm {Pr} (X>x)} , The cdf

3354-429: The constant C is a scaling factor to ensure that the total area is 1, as required by a probability distribution. More often one uses an asymptotic power law – one that is only true in the limit; see power-law probability distributions below for details. Typically the exponent falls in the range 2 < α < 3 {\displaystyle 2<\alpha <3} , though not always. More than

3432-410: The continuous version should not be applied to discrete data, nor vice versa. Further, both of these estimators require the choice of x min {\displaystyle x_{\min }} . For functions with a non-trivial L ( x ) {\displaystyle L(x)} function, choosing x min {\displaystyle x_{\min }} too small produces

3510-416: The data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided. The survival function, on the other hand, is more robust to (but not without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a survival function representation is favored over that of

3588-753: The ease with which certain general classes of mechanisms generate them. The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems; see also universality above. The ubiquity of power-law relations in physics is partly due to dimensional constraints , while in complex systems , power laws are often thought to be signatures of hierarchy or of specific stochastic processes . A few notable examples of power laws are Pareto's law of income distribution, structural self-similarity of fractals , scaling laws in biological systems , and scaling laws in cities . Research on

3666-591: The estimation of the power-law exponent, which does not assume independent and identically distributed (iid) data, uses the minimization of the Kolmogorov–Smirnov statistic , D {\displaystyle D} , between the cumulative distribution functions of the data and the power law: with where P e m p ( x ) {\displaystyle P_{\mathrm {emp} }(x)} and P α ( x ) {\displaystyle P_{\alpha }(x)} denote

3744-422: The exponent α {\displaystyle \alpha } (Greek letter alpha , not to be confused with scaling factor a {\displaystyle a} used above) is greater than 1 (otherwise the tail has infinite area), the minimum value x min {\displaystyle x_{\text{min}}} is needed otherwise the distribution has infinite area as x approaches 0, and

3822-480: The final results of any observable quantities. Note that a i {\displaystyle a_{i}} is the area of the i {\displaystyle i} th block which can be well regarded as the probability of picking the i {\displaystyle i} th block. These probabilities are naturally normalized ∑ i a i = 1 {\displaystyle \sum _{i}a_{i}=1} since we choose

3900-417: The first step, randomly with uniform probability into four smaller blocks. In the second step and thereafter, the generator is applied to only one of the blocks. The question is: How do we pick that block when there is more than one block? The most generic choice would be to pick preferentially according to their areas so that the higher the area the higher the probability to be picked. For instance, in step one,

3978-589: The frequencies of words in most languages, frequencies of family names , the species richness in clades of organisms, the sizes of power outages , volcanic eruptions, human judgments of stimulus intensity and many other quantities. Empirical distributions can only fit a power law for a limited range of values, because a pure power law would allow for arbitrarily large or small values. Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among

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4056-800: The function is plotted as a log-log plot with horizontal axis being ln ⁡ x {\displaystyle \ln x} and vertical axis being ln ⁡ ( y / y 0 + a ) {\displaystyle \ln(y/y_{0}+a)} , the plot is composed of n + 1 {\displaystyle n+1} linear segments with slopes c 0 , c 1 , . . . , c n {\displaystyle c_{0},c_{1},...,c_{n}} , separated at x = x 1 , . . . , x n {\displaystyle x=x_{1},...,x_{n}} , smoothly spliced together. The size of f i {\displaystyle f_{i}} determines

4134-426: The generator divides the initiator randomly into four smaller blocks. Let us label their areas starting from the top left corner and moving clockwise as a 1 , a 2 , a 3 {\displaystyle a_{1},a_{2},a_{3}} and a 4 {\displaystyle a_{4}} . But of course the way we label is totally arbitrary and will bear no consequence to

4212-439: The identification of power-law probability distributions using random samples has been proposed. This methodology consists of plotting a bundle for the log-transformed sample . Originally proposed as a tool to explore the existence of moments and the moment generation function using random samples, the bundle methodology is based on residual quantile functions (RQFs), also called residual percentile functions, which provide

4290-403: The length is tripled, the area is multiplied by 3 , and so on. The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares , cloud sizes, the foraging pattern of various species, the sizes of activity patterns of neuronal populations,

4368-562: The log likelihood function becomes: The maximum of this likelihood is found by differentiating with respect to parameter α {\displaystyle \alpha } , setting the result equal to zero. Upon rearrangement, this yields the estimator equation: where { x i } {\displaystyle \{x_{i}\}} are the n {\displaystyle n} data points x i ≥ x min {\displaystyle x_{i}\geq x_{\min }} . This estimator exhibits

4446-406: The mean is well-defined but the variance is not, implying they are capable of black swan behavior. This can be seen in the following thought experiment: imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest person entering the room, with a monthly income of about 1 billion US$ . What happens to the average income in the room? Income

4524-440: The most reliable techniques are often based on the method of maximum likelihood . Alternative methods are often based on making a linear regression on either the log–log probability, the log–log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent. For real-valued, independent and identically distributed data, we fit

4602-418: The name "power law" is used. In empirical contexts, an approximation to a power-law o ( x k ) {\displaystyle o(x^{k})} often includes a deviation term ε {\displaystyle \varepsilon } , which can represent uncertainty in the observed values (perhaps measurement or sampling errors) or provide a simple way for observations to deviate from

4680-416: The origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics , computer science , linguistics , geophysics , neuroscience , systematics , sociology , economics and more. However, much of the recent interest in power laws comes from the study of probability distributions : The distributions of

4758-400: The particular form of the lower tail, represented by L ( x ) {\displaystyle L(x)} above. More about these methods, and the conditions under which they can be used, can be found in . Further, this comprehensive review article provides usable code (Matlab, Python, R and C++) for estimation and testing routines for power-law distributions. Another method for

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4836-399: The pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended. There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield unbiased and consistent answers . Some of

4914-447: The plot tend to converge to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published. A disadvantage of these plots is that, in order for them to provide reliable results, they require huge amounts of data. In addition, they are appropriate only for discrete (or grouped) data. Another graphical method for

4992-399: The power-law behavior at very large values of x {\displaystyle x} . This distribution does not scale and is thus not asymptotically as a power law; however, it does approximately scale over a finite region before the cutoff. The pure form above is a subset of this family, with λ = 0 {\displaystyle \lambda =0} . This distribution is

5070-429: The power-law function (perhaps for stochastic reasons): Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated power function is possible: p ( x ) = C x − α {\displaystyle p(x)=Cx^{-\alpha }} for x > x min {\displaystyle x>x_{\text{min}}} where

5148-417: The quantiles of a standard Pareto distribution) by plotting the former versus the latter. If the resultant scatterplot suggests that the plotted points asymptotically converge to a straight line, then a power-law distribution should be suspected. A limitation of Pareto Q–Q plots is that they behave poorly when the tail index α {\displaystyle \alpha } (also called Pareto index)

5226-488: The requirement that p ( x ) {\displaystyle p(x)} be asymptotically scale invariant; thus, the form of L ( x ) {\displaystyle L(x)} only controls the shape and finite extent of the lower tail. For instance, if L ( x ) {\displaystyle L(x)} is the constant function, then we have a power law that holds for all values of x {\displaystyle x} . In many cases, it

5304-694: The rest of three new blocks are labelled a 5 , a 6 {\displaystyle a_{5},a_{6}} and a 7 {\displaystyle a_{7}} in a clockwise fashion. In general, in the j {\displaystyle j} th step, we pick one out of 3 j − 2 {\displaystyle 3j-2} blocks preferentially with respect to area and divide randomly into four blocks. The detailed algorithm can be found in Dayeen and Hassan and Hassan, Hassan, and Pavel. This process of lattice generation can also be described as follows. Consider that

5382-436: The same size and the same coordination number. The planar Voronoi diagram , on the other hand, has neither a fixed cell size nor a fixed coordination number. Its coordination number distribution is rather Poissonian in nature. That is, the distribution is peaked about the mean where it is almost impossible to find cells which have significantly higher or fewer coordination number than the mean. Recently, Hassan et al proposed

5460-452: The same term [REDACTED] This disambiguation page lists articles associated with the title WPSL . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=WPSL&oldid=751305444 " Categories : Disambiguation pages Broadcast call sign disambiguation pages Hidden categories: Short description

5538-594: The sharpness of splicing between segments i − 1 , i {\displaystyle i-1,i} . A power law with an exponential cutoff is simply a power law multiplied by an exponential function: In a looser sense, a power-law probability distribution is a distribution whose density function (or mass function in the discrete case) has the form, for large values of x {\displaystyle x} , where α > 1 {\displaystyle \alpha >1} , and L ( x ) {\displaystyle L(x)}

5616-429: The straight-line on the log–log plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws. Thus, accurately fitting and validating power-law models

5694-490: The substrate is a square of unit area and at each time step a seed is nucleated from which two orthogonal partitioning lines parallel to the sides of the substrate are grown until intercepted by existing lines. It results in partitioning the square into ever smaller mutually exclusive rectangular blocks. Note that the higher the area of a block, the higher is the probability that the seed will be nucleated in it to divide that into four smaller blocks since seeds are sown at random on

5772-443: The substrate. It can also describes kinetics of fragmentation of two-dimensional objects. Before 2000 epidemic models , for instance, were studying by applying them on regular lattices like square lattice assuming that everyone can infect everyone else in the same way. The emergence of a network-based framework has brought a fundamental change, offering a much much better pragmatic skeleton than any time before. Today epidemic models

5850-454: The system. Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach criticality —can be shown, via renormalization group theory, to share the same fundamental dynamics. For instance, the behavior of water and CO 2 at their boiling points fall in the same universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by

5928-413: The tail of a distribution using a random sample. Taking the logarithm of a power law of the form f ( x ) = a x k {\displaystyle f(x)=ax^{k}} results in: which forms a straight line with slope k {\displaystyle k} on a log-log scale. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for

6006-456: The transcendental equation where ζ ( α , x m i n ) {\displaystyle \zeta (\alpha ,x_{\mathrm {min} })} is the incomplete zeta function . The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for α ^ {\displaystyle {\hat {\alpha }}} are not equivalent, and

6084-489: The understanding of their topological and geometrical properties have always been an interesting proposition among scientists in general and physicists in particular. Several models prescribe how to generate cellular structures. Often these structures can mimic directly the structures found in nature and they are able to capture the essential properties that we find in natural structures. In general, cellular structures appear through random tessellation , tiling, or subdivision of

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