Critical Percolation in Geometric Inhomogeneous Random Graphs

Abstract

Geometric inhomogeneous random graphs (GIRGs) are a class of random graphs where each vertex has an assigned weight coming from a power law distribution, and a spatial location according to a Poisson point process on the underlying geometric space. We consider GIRGs on the d-dimensional torus $\mathbb T^d := [-n^{1/d}/2,n^{1/d}/2)^d$ and perform percolation with edge-retention probability $\pi_n = \lambda n^{(3-\tau)/2}$ for some $\lambda > 0$. We prove that there exists a $\lambda_c$, such that for all $\lambda > \lambda_c$ a largest component unique in its order of magnitude exists within the highest $\ceil{an^{(\tau-3)/2}}$ weight vertices (the core) for some $a > 0$. Furthermore, we also prove novel results on the spatial distribution of the highest weight vertices. Starting from some fixed distance $r_0$, we sweep through the powers of 2 by defining $r_i := 2^ir_0$ and show that the amount of vertex pairs between $r_i$ and $r_{i+1}$ concentrates within its own expectation. We then consider the slightly supercritical regime, where we have already established the existence of a giant within the core. Extending the core to the entire graph, we establish that the largest component consists of mostly the emerging giant inside the core, together with the vertices reached outside the core, which we call its span. We then show a law of large numbers result on the size of the span of the giant component inside the core, where we show that for any $\varepsilon > 0$, we have that $\zeta^\lambda - \varepsilon \le \frac{\Span(\mathscr C_{(1)}^a)|}{\sqrt n}\leq \zeta^\lambda + \varepsilon$.