Critical Stretching of Mean-Field Regimes in Spatial Networks

We study a spatial network model with exponentially distributed link lengths on an underlying grid of points, undergoing a structural crossover from a random, Erdős-Rényi graph, to a d -dimensional lattice at the characteristic interaction range ζ . We find that, whilst far from the percolation threshold the random part of the giant component scales linearly with ζ , close to criticality it extends in space until the universal length scale ζ^{6 /(6 -d )} , for d <6 , before crossing over to the spatial one. We demonstrate the universal behavior of the spatiotemporal scales characterizing this critical stretching phenomenon of mean-field regimes in percolation and in dynamical processes on d =2 networks, and we discuss its general implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.