Random Geometry in Spatial Networks
Universality, where the same sort of probabilistic scaling occurs in seemingly different systems, occurs across physics, including in avalanches in piles of sand, the rate of grain pouring out of a silo, cracks in rocks, random matrices, complex networks, quantum gravity and beyond. We give an example of universality as it occurs in random spatial networks, such as road networks, or the brain. This is based on first passage percolation. In the classic model, concerning shortest paths between points of a square lattice with random edge weights, for pairs of vertices separated by a Euclidean distance $L$, geodesics exhibit deviations from their mean length $L$ that are of order $L^\chi$, while the transversal fluctuations, known as wandering, grow as $L^\xi$. In this talk, we show that when weighting edges directly with their Euclidean span in various spatial network models, now in a continuum, we have two distinct universality classes defined by different exponents $\xi=3/5$ and $\chi = 1/5$, or $\xi=7/10$ and $\chi = 2/5$, depending only on coarse details of the specific connectivity laws used. The first class contains proximity graphs such as the hard and soft random geometric graph, and the $k$-nearest neighbor random geometric graphs, where via Monte Carlo simulations we find $\xi=0.60\pm 0.01$ and $\chi = 0.20\pm 0.01$, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as $\beta$-skeletons and the Delaunay triangulation and are characterized by the values $\xi=0.70\pm 0.01$ and $\chi = 0.40\pm 0.01$, with a nearly theoretically maximal wandering exponent. We also show numerically that the Kadar-Parisi-Zhang (KPZ) relation $\chi = 2\xi -1$ is satisfied for all these models. These results shed some light on the how spatial networks, important physical systems that underpin much of complexity, interact with important probabilistic ideas such as universality and random geometry.