Abstract
We present an exact description of a crossover between two different regimes of simple
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.
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(l) and obtain a
scaling form of (p,n). In spite of the simplicity of the models
under consideration, the results
are close to those obtained numerically for usual small-world networks.