Spreading and shortest paths in systems with sparse long-range connections

Spreading according to simple rules (e. g. , of fire or diseases) and shortest-path distances are studied on d-dimensional systems with a small density p per site of long-range connections (``small-world'' lattices). The volume V (t) covered by the spreading quantity on an infinite system is exactly calculated in all dimensions as a function of time t. From this, the average shortest-path distance l (r) can be calculated as a function of Euclidean distance r. It is found that l (r) for rr₂. The characteristic length r₂, which governs the behavior of shortest-path lengths, diverges logarithmically with L for all p>0.