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Spreading and shortest paths in systems with sparse long-range connections. (English) Zbl 1062.82507
Summary: 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 $l(r)\sim r$ for $$r<r\sb c=[2p\Gamma\sb d(d-1)!]\sp {-1/d}\log(2p\Gamma\sb dL\sp d)$$ and $l(r)\sim r\sb c$ for $r>r\sb c$. The characteristic length $r\sb c$, which governs the behavior of shortest-path lengths, diverges logarithmically with $L$ for all $p>0$.

82C20Dynamic lattice systems and systems on graphs
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