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| In applied probability theory, the '''Simon model''' is a class of [[stochastic model]]s that results in a [[power-law]] distribution function. It was proposed by [[Herbert A. Simon]]<ref name=simon>Simon, H. A., 1955, Biometrika 42, 425.</ref> to account for the wide range of empirical [[Frequency distribution|distributions]] following a power-law. It models the dynamics of a system of elements with associated counters (e.g., words and their frequencies in texts, or nodes in a network and their connectivity <math>k</math>). In this model the dynamics of the system is based on constant growth via addition of new elements (new instances of words) as well as incrementing the counters (new occurrences of a word) at a rate proportional to their current values.
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| == Description ==
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| To model this type of network growth as described above, Bornholdt and Ebel<ref name=BE>Bornholdt, S. and H. Ebel, Phys. Rev. E 64 (2001) 035104(R).</ref> considered a network with <math>n</math> nodes, and each node with connectivities <math>k_i</math>, <math>i = 1, \ldots, n</math>. These nodes
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| form classes <math>[k]</math> of <math>f(k)</math> nodes with identical connectivity <math>k</math>.
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| Repeat the following steps:
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| (i) With probability <math>\alpha</math> add a new node and attach a link to it from an arbitrarily chosen node.
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| (ii) With probability <math>1-\alpha</math> add one link from an arbitrary node to a node <math>j</math> of class <math>[k]</math> chosen with probability <math>P_{\text{new link to class $[k]$}} \propto k f(k)</math>.
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| For this stochastic process, Simon found a stationary solution exhibiting [[power-law]] scaling, <math>P(k) \propto k^{- \gamma}</math>, with exponent <math>\gamma = 1 + \frac{1}{1- \alpha}.</math> | |
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| == Properties ==
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| (i) [[Barabási-Albert (BA) model]] can be mapped to the subclass <math>\alpha= 1/2</math> of Simon's model, when using the simpler probability for a node being
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| connected to another node <math>i</math> with connectivity <math>k_i</math> <math>P_{\text{new link to $i$}} \propto k_i </math> (same as the preferential attachment at [[BA model]]). In other words, the Simon model describes a general class of stochastic processes that can result in a [[scale-free network]], appropriate to capture [[Zipf's law|Pareto and Zipf's laws]].
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| (ii) The only free parameter of the model <math>\alpha</math> reflects the relative
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| growth of number of nodes versus the number of links. In general <math>\alpha</math> has small values; therefore, the scaling exponents can be predicted to be <math>\gamma\approx 2</math>. For instance, Bornholdt and Ebel<ref name="BE"/> studied the linking dynamics of World Wide Web, and predicted the scaling exponent as <math>\gamma \approx 2.1</math>, which was consistent with observation.
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| (iii) The interest in the scale-free model comes from its ability to describe the topology of complex networks. The Simon model does not have an underlying network structure, as it was designed to describe events whose frequency follows a [[power-law]]. Thus network measures going beyond the [[degree distribution]] such | |
| as the [[average path length]], [http://austria.phys.nd.edu/netwiki/index.php/Graph_Spectra spectral properties], and [[clustering coefficient]], cannot be obtained from this mapping.
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| The Simon model is related to [[generalized scale-free model]]s with growth and preferential attachment properties. For more reference, see.<ref name=BA>Barabási, A.-L., and R. Albert, Statistical mechanics of complex networks, Reviews of Modern Physics, Vol 74, page 47-97, 2002.</ref><ref name=AM>Amaral, L. A. N., A. Scala, M. Barthelemy, and H. E. Stanley, 2000, Proc. Natl. Acad. Sci. U.S.A. '''97''', 11149.</ref>
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| == References ==
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| <references/>
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| [[Category:Power laws]]
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