We offer an example of a network model with a power-law degree distribution, $P\left(k\right)\mathrm{}\sim k^{-\alpha },$ for nodes, but which nevertheless has a well-defined geography and a nonzero threshold percolation probability for $\alpha >2,$ the range of real-world contact networks. This is different from $p_c=0\mathrm{}$ for $\alpha <3\mathrm{}$ results for the original well-mixed scale-free networks. In our lattice-based scale-free network, individuals link to nearby neighbors on a lattice. Even considerable additional small-world links do not change our conclusion of nonzero thresholds. When applied to disease propagation, these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering.

• Received 16 July 2002

DOI:https://doi.org/10.1103/PhysRevE.66.056105