  Power lines, roadways, pipelines and other physical infrastructure are
critical to modern society. These structures may be viewed as spatial networks
where geographic distances play a role in the functionality and construction
cost of links. Traditionally, studies of network robustness have primarily
considered the connectedness of large, random networks. Yet for spatial
infrastructure physical distances must also play a role in network robustness.
Understanding the robustness of small spatial networks is particularly
important with the increasing interest in microgrids, small-area distributed
power grids that are well suited to using renewable energy resources. We study
the random failures of links in small networks where functionality depends on
both spatial distance and topological connectedness. By introducing a
percolation model where the failure of each link is proportional to its spatial
length, we find that, when failures depend on spatial distances, networks are
more fragile than expected. Accounting for spatial effects in both construction
and robustness is important for designing efficient microgrids and other
network infrastructure.
