Randomized trial illustrates the behavior of spreading and shortest-path lengths in systems with long-range connections, suggesting key patterns in distance metrics.
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)~r for r<rc=[2pΓd(d-1)!]^-1/dlog(2pΓdLᵈ) and l(r)~rc for r>rc. The characteristic length rc, which governs the behavior of shortest-path lengths, diverges logarithmically with L for all $p>0.$
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Cristian F. Moukarzel (1999) studied this question.
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