“Simon model”的意思、由来-开放百科全书

In applied probability theory, the Simon model is a class of stochastic models that results in a power-law distribution function. It was proposed by Herbert A. Simon^[1] to account for the wide range of empirical 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

). 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.

Description

To model this type of network growth as described above, Bornholdt and Ebel^[2] considered a network with

nodes, and each node with connectivities

. These nodes

(i) With probability

add a new node and attach a link to it from an arbitrarily chosen node.

(ii) With probability

add one link from an arbitrary node to a node

of class

chosen with a probability proportional to

For this stochastic process, Simon found a stationary solution exhibiting power-law scaling,

, with exponent

Properties

(i) Barabási-Albert (BA) model can be mapped to the subclass

of Simon's model, when using the simpler probability for a node being

connected to another node

with connectivity

(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 Pareto and Zipf's laws.

growth of number of nodes versus the number of links. In general

has small values; therefore, the scaling exponents can be predicted to be

. For instance, Bornholdt and Ebel^[2] studied the linking dynamics of World Wide Web, and predicted the scaling exponent as

, which was consistent with observation.

(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, [https://web.archive.org/web/20070128124135/http://austria.phys.nd.edu/netwiki/index.php/Graph_Spectra spectral properties], and clustering coefficient, cannot be obtained from this mapping.

The Simon model is related to generalized scale-free models with growth and preferential attachment properties. For more reference, see.^[3]^[4]

References

1. ^Simon, H. A., 1955, Biometrika 42, 425.
2. ^¹Bornholdt, S. and H. Ebel, Phys. Rev. E 64 (2001) 035104(R).
3. ^Barabási, A.-L., and R. Albert, Statistical mechanics of complex networks, Reviews of Modern Physics, Vol 74, page 47-97, 2002.
4. ^Amaral, L. A. N., A. Scala, M. Barthelemy, and H. E. Stanley, 2000, Proc. Natl. Acad. Sci. U.S.A. 97, 11149.