Propagation in a two-dimensional weighted local small-world network.

Propagation properties in a two-dimensional network with local small-world effects corresponding to the influence zone of each active site are studied. Two different weights based on characteristic times are introduced. The propagation of the front (here a forest fire front) in such a network exhibits two thresholds: the first one is geometrical corresponding to the percolation threshold and the second one is dynamical and results from the weighting procedure. The geometrical threshold is found to be a second-order phase transition as for regular networks. Further results are provided on the fractal dimension of the area covered during the propagation below the percolation threshold.