Polynomial growth in degree-dependent first passage percolation on spatial random graphs

TL;DR

This work analyzes a degree-dependent first passage percolation model on spatial graphs with power-law degree distributions, where edge costs scale as $\mathcal{C}(uv)=L_{uv}(W_uW_v)^{\mu}$ and $L$ has regularly varying behaviour near $0$ with exponent $\beta$. Across IGIRG, GIRG, and SFP, the authors identify four universal growth phases for the cost-distance between distant vertices as the penalty $\mu$ increases: explosive, polylogarithmic, sublinear polynomial with exponent $\eta_0<1$, and linear. They develop a renormalisation-based lower-bound framework to handle long, expensive edges and, in a companion paper, provide matching upper bounds for the polynomial regime; they also establish a linear upper bound in the linear regime, using a dense geometric graph coupling to bond percolation. The results depend intricately on the tail of the degree distribution ($\tau$), the long-range parameter ($\alpha$), and the near-zero behaviour of $L$ ($\beta$), and they extend to limit cases like $\alpha=\infty$ or $\beta=\infty$, as well as to Hyperbolic/GIRG settings. Overall, the paper reveals a rich, phase-changing dynamics in a single underlying graph model, showing that the propagation speed can be tuned by the penalty exponent without altering the network geometry.

Abstract

In this paper we study a version of (non-Markovian) first passage percolation on graphs, where the transmission time between two connected vertices is non-iid, but increases by a penalty factor polynomial in their expected degrees. Based on the exponent of the penalty-polynomial, this makes it increasingly harder to transmit to and from high-degree vertices. This choice is motivated by awareness or time-limitations. For the iid part of the transmission times we allow any nonnegative distribution with regularly varying behaviour at $0$. For the underlying graph models we choose spatial random graphs that have power-law degree distributions, so that the effect of the penalisation becomes visible: (finite and infinite) Geometric Inhomogeneous Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees. We prove that upon increasing the penalty exponent, the transmission time between two far away vertices $x,y$ sweeps through four universal phases even for a single underlying graph: explosive (tight transmission times), polylogarithmic, polynomial but sublinear ($|x-y|^{η_0+o(1)}$ for an explicit $η_0<1$), and linear ($Θ(|x-y|)$) in their Euclidean distance. Further, none of these phases are restricted to phase boundaries, and those are non-trivial in the main model parameters: the tail of the degree-distribution, a long-range parameter, and the exponent of regular variation of the iid part of the transmission times. In this paper we present proofs of lower bounds for the latter two phases and the upper bound for the linear phase. These complement the matching upper bounds for the polynomial regime in our companion paper.

Figures (2)

• Figure 1: Illustration of the inductive step for the lower bound proof. The big box is good, hence all edges in it that are longer than one-hundredth of the smaller boxes' side-length are already too expensive. So, any cheap-enough path $\pi$ between $0$ and $x$ contained in $Q$ only uses edges shorter than that. There are only a few bad child-boxes, since $Q$ is good, and we remove the edges of $\pi$ with at least one end-point in bad child-boxes. We select from the remaining edges of $\pi$ enough sub-segments such that each sub-segment is fully contained in a good child-box of $Q$ and connects two sufficiently far vertices within that child-box. Three sub-segments are shown in bold, together with their corresponding child-boxes. By induction we have a lower bound on the costs of these sub-segments, and summing up those yields a lower bound on $\mathcal{C}(\pi)$.
• Figure 2: Example for $k=3$ cubes covering the line from $x$ to $y$. Any two adjacent cubes overlap by at least $4^{-d}\mathrm{Vol}(Q^{{(i)}})$ and contain a cube of side-length $|x-y|/(8K)$ and any two intersections have distance at least $r/2 = |x-y|/(2K)$, where $r = |x-y|/K$ is the side-length of the cubes.

Theorems & Definitions (50)

• Definition 1.1: SFP, IGIRG, GIRG
• Definition 1.2: 1-dependent first passage percolation (1-FPP)
• Theorem 1.4: Polynomial Lower Bound
• Corollary 1.5
• Theorem 1.6: Polynomial Upper Bound komjathy2022one1
• Corollary 1.7
• Theorem 1.8
• Corollary 1.9
• Theorem 1.10
• Theorem 1.11: Extension to threshold IGIRGs/GIRGs, and $\beta=\infty$