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

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

1999-05-21

arxiv.org/abs/cond-mat/9905322v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

4 pages, 1 eps fig. Uses psfig

Scientific paper

10.1103/PhysRevE.60.R6263

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. We find that V(t) grows initially as t^d/d for t<< t^* = (2p \Gamma_d (d-1)!)^{-1/d} and later exponentially for $t>>t^*$, generalizing a previous result in one dimension. Using the properties of V(t), the average shortest-path distance \ell(r) can be calculated as a function of Euclidean distance r. It is found that \ell(r) = r for rr_c. The characteristic length r_c, which governs the behavior of shortest-path lengths, diverges with system size for all p>0. Therefore the mean separation s \sim p^{-1/d} between shortcut-ends is not a relevant internal length-scale for shortest-path lengths. We notice however that the globally averaged shortest-path length, divided by L, is a function of L/s only.

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