Simon model
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 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 formula_1). 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 considered a network with formula_2 nodes, and each node with connectivities formula_3, formula_4. These nodes form classes formula_5 of formula_6 nodes with identical connectivity formula_1. Repeat the following steps: (i) With probability formula_8 add a new node and attach a link to it from an arbitrarily chosen node. (ii) With probability formula_9 add one link from an arbitrary node to a node formula_10 of class formula_5 chosen with a probability proportional to formula_12. For  this stochastic process, Simon found a stationary solution exhibiting power-law scaling, formula_13, with exponent formula_14 Properties. (i) Barabási-Albert (BA) model can be mapped to the subclass formula_15 of Simon's model, when using the simpler probability for a node being connected to another node formula_16 with connectivity formula_3 formula_18 (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. (ii) The only free parameter of the model formula_8 reflects the relative  growth of number of nodes versus the number of links. In general formula_8 has small values; therefore, the scaling exponents can be predicted to be formula_21. For instance, Bornholdt and Ebel studied the linking dynamics of World Wide Web, and predicted the scaling exponent as formula_22, 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, 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.