Exactly solvable analogy of small-world networks

Details Exactly solvable analogy of small-world networks Exactly solvable analogy of small-world networks

1999-07-28

arxiv.org/abs/cond-mat/9907445v1

Europhys. Lett. 50 (2000) 1

Physics

Condensed Matter

Statistical Mechanics

4 pages with 3 postscript figures

Scientific paper

10.1209/epl/i2000-00227-1

We present an exact description of a crossover between two different regimes of simple analogies of small-world networks. Each of the sites chosen with a probability $p$ from $n$ sites of an ordered system defined on a circle is connected to all other sites selected in such a way. Every link is of a unit length. Thus, while $p$ changes from 0 to 1, an averaged shortest distance between a pair of sites changes from $\bar{\ell} \sim n$ to $\bar{\ell} = 1$. We find the distribution of the shortest distances $P(\ell)$ and obtain a scaling form of $\bar{\ell}(p,n)$. In spite of the simplicity of the models under consideration, the results appear to be surprisingly close to those obtained numerically for usual small-world networks.

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