Exactly solvable analogy of small-world networks

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 $\bar{\mathrm{\ell }}\sim n$ to $\bar{\mathrm{\ell }}=1$. We find the distribution of the shortest distances $P(\mathrm{\ell })$ and obtain a scaling form of $\bar{\mathrm{\ell }}(p,n)$. 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.