Mean-field solution of the small-world network model

The small-world network model is a simple model of the structure of social networks, which possesses characteristics of both regular lattices and rand om graphs. The model consists of a one-dimensional lattice with a low densi ty of shortcuts added between randomly selected pairs of points. These shor tcuts greatly reduce the typical path length between any two points on the lattice. We present a mean-field solution for the average path length and f or the distribution of path lengths in the model. This solution is exact in the limit of large system size and either a large or small number of short cuts.