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 [Formula: see text]-dimensional lattice at the characteristic interaction range [Formula: see text]. We find that, whilst far from the percolation threshold the random part of the giant component scales linearly with [Formula: see text] , close to criticality it extends in space until the universal length scale [Formula: see text] , for [Formula: see text] , 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 [Formula: see text] networks, and we discuss its general implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.