Scale-free behavior and universality in random fragmentation and aggregation

Banavar, Jayanth R.; Rios, Paolo De Los; Flammini, Alessandro; Holter, Neal S.; Maritan, Amos

doi:10.1103/physreve.69.036123

Cited by 7 publications

(7 citation statements)

References 16 publications

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“…2B, inset). Power-laws in the stationary distribution of random segmentation and multiplicative processes have been reported before [9] and can be obtained by introducing slight modifications, such as reflecting boundaries, frozen segments, merging or reset events [16,17,18]. In contrast, the approximate scaling of S d (n) in Eq.…”

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confidence: 92%

Zipf’s Law in the Popularity Distribution of Chess Openings

Blasius

Tönjes

2009

Phys. Rev. Lett.

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We perform a quantitative analysis of extensive chess databases and show that the frequencies of opening moves are distributed according to a power law with an exponent that increases linearly with the game depth, whereas the pooled distribution of all opening weights follows Zipf's law with universal exponent. We propose a simple stochastic process that is able to capture the observed playing statistics and show that the Zipf law arises from the self-similar nature of the game tree of chess. Thus, in the case of hierarchical fragmentation the scaling is truly universal and independent of a particular generating mechanism. Our findings are of relevance in general processes with composite decisions.

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confidence: 92%

Zipf’s Law in the Popularity Distribution of Chess Openings

Blasius

Tönjes

2009

Phys. Rev. Lett.

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“…Our model is based on the idea of aggregation [19] and it is very similar to one recently and independently introduced in Ref. [20,21] in a different context, and analyzed partly by Alava and Dorogovtsev [22].…”

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confidence: 99%

Scale-free networks with an exponent less than two

2006

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We study scale free simple graphs with an exponent of the degree distribution γ less than two. Generically one expects such extremely skewed networks -which occur very frequently in systems of virtually or logically connected units -to have different properties than those of scale free networks with γ > 2: The number of links grows faster than the number of nodes and they naturally posses the small world property, because the diameter increases by the logarithm of the size of the network and the clustering coefficient is finite. We discuss a simple prototype model of such networks, inspired by real world phenomena, which exhibits these properties and allows for a detailed analytical investigation. [7]. Indeed in each of these systems nodes -web pages or actors -are linked -by hyperlinks or collaboration in the same movie -to a number k of other nodes, which is called the degree of the node [27], and which obeys a power law distribution P (k) ∼ k −γ . In many cases (table I) the exponent γ of such a distribution is larger than two which its occurrence has been related to some interaction mechanismsuch as preferential attachment [3] -in simplified models.Scale-free networks with an exponent γ < 2 have received less attention, despite of their widespread appearance (table I), in the peer-to-peer Gnutella network [28] [8, 9], outgoing E-mails network [10], traffic in networks [11], co-authorship network in high energy physics [12] and in the network of dependency among software packages [13,14].The aim of this letter is to show that simple graphs with γ < 2 have markedly different properties than simple graphs with γ > 2. We shall do this first on the basis of general arguments and then using a prototype model motivated by the above mentioned real networks. This model reproduces all the discussed generic properties. Furthermore we show that its generalization to a weighted network exhibits non-trivial statistical properties.

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“…This may help account for the power laws previously observed in many contexts even without preferential attachment 13 . A related discussion can be found in 14 .…”

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confidence: 99%

A Random Categorization Model for Hierarchical Taxonomies

D’Amico

Rabadán

Kleban

2017

Sci Rep

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A taxonomy is a standardized framework to classify and organize items into categories. Hierarchical taxonomies are ubiquitous, ranging from the classification of organisms to the file system on a computer. Characterizing the typical distribution of items within taxonomic categories is an important question with applications in many disciplines. Ecologists have long sought to account for the patterns observed in species-abundance distributions (the number of individuals per species found in some sample), and computer scientists study the distribution of files per directory. Is there a universal statistical distribution describing how many items are typically found in each category in large taxonomies? Here, we analyze a wide array of large, real-world datasets – including items lost and found on the New York City transit system, library books, and a bacterial microbiome – and discover such an underlying commonality. A simple, non-parametric branching model that randomly categorizes items and takes as input only the total number of items and the total number of categories is quite successful in reproducing the observed abundance distributions. This result may shed light on patterns in species-abundance distributions long observed in ecology. The model also predicts the number of taxonomic categories that remain unrepresented in a finite sample.

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Scale-free behavior and universality in random fragmentation and aggregation

Cited by 7 publications

References 16 publications

Zipf’s Law in the Popularity Distribution of Chess Openings

Zipf’s Law in the Popularity Distribution of Chess Openings

Scale-free networks with an exponent less than two

A Random Categorization Model for Hierarchical Taxonomies

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