Exactly solvable small-world network

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 n to = 1. We find the distribution of the shortest distances P () and obtain a scaling form of (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.