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, Erd\H{o}s–R'enyi graph to a $2D$ lattice at the characteristic interaction range $\zeta$. We find that, whilst far from the percolation threshold the random part of the incipient cluster scales linearly with $\zeta$, close to criticality it extends in space until the universal length scale ${\zeta }^{3\mathrm{/}2}$ before crossing over to the spatial one. We demonstrate this {\em critical stretching} phenomenon in percolation and in dynamical processes, and we discuss its implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.