# Critical stretching of mean-field regimes in spatial networks

## TL;DR

This paper investigates how spatial networks transition from random to lattice structures, revealing a critical stretching phenomenon near percolation thresholds with implications for various real-world systems.

## Contribution

It introduces the concept of critical stretching in spatial networks, showing how incipient clusters extend over a universal length scale near criticality.

## Key findings

- Linear scaling of incipient cluster far from criticality
- Universal length scale of ^{3/2} near criticality
- Critical stretching observed in percolation and dynamical processes

## Abstract

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 $\zeta^{3/2}$ 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.