Four universal growth regimes in degree-dependent first passage percolation on spatial random graphs I

TL;DR

This work analyzes one-dependent first-passage percolation on three spatial scale-free graph models, showing a fourfold universal progression of transmission-time growth from explosive to polylogarithmic, polynomial, and finally linear in Euclidean distance as the penalty exponent $\mu$ increases.The authors develop a novel quenched methodology based on pseudorandom nets and a multi-round edge-exposure scheme to handle dependencies and to construct cheap-edge structures in fixed realisations, enabling sharp upper bounds for all sub-explosive regimes.Explicit growth exponents are provided (e.g., $\Delta_0$ and $\eta_0$) and depend on model parameters such as the tail exponent $\tau$, long-range parameter $\alpha$, and the near-zero behavior of the edge factor $L$ via $\beta$; the results unify long-range and finite-variance regimes and include limit cases like threshold models.The techniques yield robust, hierarchical bridging paths (the budget travel plan) and are shown to be applicable to a broad class of vertex-weighted random graphs; a companion paper complements by establishing the matching lower bounds for the polynomial and linear phases.Together, these results reveal a rich, parameter-sensitive phase space for 1-FPP on spatial inhomogeneous graphs and open avenues for applying the method to other stochastic-geometric processes.

Abstract

One-dependent first passage percolation is a spreading process on a graph where the transmission time through each edge depends on the direct surroundings of the edge. In particular, the classical iid transmission time $L_{xy}$ is multiplied by $(W_xW_y)^μ$, a polynomial of the expected degrees $W_x, W_y$ of the endpoints of the edge $xy$, which we call the penalty function. Beyond the Markov case, we also allow any distribution for $L_{xy}$ with regularly varying distribution near $0$. We then run this process on three spatial scale-free random graph models: 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 show that as the penalty-function, i.e., $μ$ increases, the transmission time between two far away vertices sweeps through four universal phases: explosive (with tight transmission times), polylogarithmic, polynomial but strictly sublinear, and linear in the Euclidean distance. The strictly polynomial growth phase here is a new phenomenon that so far was extremely rare in spatial graph models. The four growth phases are highly robust in the model parameters and are not restricted to phase boundaries. Further, the transition points between the phases depend non-trivially on the main model parameters: the tail of the degree distribution, a long-range parameter governing the presence of long edges, and the behaviour of the distribution $L$ near $0$. In this paper we develop new methods to prove the upper bounds in all sub-explosive phases. Our companion paper complements these results by providing matching lower bounds in the polynomial and linear regimes.

Figures (4)

• Figure 1: Heatmaps for the four different universality classes. The vertices are sorted by their transmission times from the origin (center vertex). The colours represent this ordering: blue infected first, red last. All four plots are generated on the same underlying graph (with parameters $\tau=2.3$ and $\alpha=5$, and edge connection probabilities $p(u,v)=1 \wedge (w_u w_v / (\mathbb{E}[W]\|u-v\|^2))^5$), where the vertices are placed on a $750 \times 750$ grid in the 2-dimensional torus. The random factors $L_{xy}$ associated to each edge are also identical in all four plots, and follow an exponential distribution (i.e. $\beta=1$). The only varying parameter is the penalty exponent $\mu$, taking values (a) $\mu=2$ for the linear regime (b) $\mu=1$ for the polynomial regime (c) $\mu=0.5$ for the polylogarithmic regime and (d) $\mu=0$ for the explosive regime. In the linear regime, the late points are -- typically -- high degree vertices carrying high penalisation. We thank Zylan Benjert for generating the simulations and the pictures.
• Figure 2: The budget travel plan with 3-edge bridging-paths: (a) first and (b) second iteration.
• Figure 3: Hyperrectangle-cover and definition of $i$-good boxes. In this figure, $d=1$, $R=3$, $r_2'/r_1' = 3$, and $r_3'/r_2' = 4$, and the requirement of (B1) for $i > 1$ is "all but at most one sub-box $B' \in \mathcal{P}_{i-1}$ of $B$ is good". The hyperrectangle-cover is denoted by coloured-boundary rectangles. The spatial dimension on the $x$ axis is covered by nested intervals, where (blue) boxes in $\mathcal{P}_2$ contain $3$ level-1 (orange) boxes and (green) boxes in $\mathcal{P}_3$ contain $4$ level-2 boxes. The weight dimension on the $y$ axis is covered by a base-2-cover $I_1,\ldots,I_6$. Hyperrectangles above $f(r_1)$ (e.g. $B_1\! \times\! I_3$) and above $f(r_2)$ (e.g. $B_2 \!\times\! I_5$), are not included in $\mathcal{P}_1, \mathcal{P}_2$, respectively, since they contain too few vertices for concentration. Good boxes are shaded and bad boxes are hatched or get no colour. Box $B_1$ is good because its two hyper-rectangles $B_1\! \times\! I_1$ and $B_1\! \times\! I_2$ (filled orange) contain the right number of vertices, making all vertices in $B_1$$2$-good, including those with weights above $I_1 \cup I_2$. Box $B_1'$ is bad (light hatching), since it contains too few vertices in $B_1\!\times\!I_1$ (cross-hatching). Box $B_2$ is good, because it only contains one bad sub-box ($B_1'$) in $\mathcal{P}_1$, and because its four hyperrectangles $B_2\! \times\! I_1,\ldots,B_2 \!\times\! I_4$ (filled blue) all contain the right number of $2$-good vertices in total. Since $B_1$ and $B_2$ are both good, vertex $v$ is $3$-good. Box $B_2'$ is "doubly" bad (filled white): it contains two level-$1$ bad sub-boxes, and the hyperrectangle $B_2' \times I_3$ contains too few $1$-good vertices. Thus no vertex in $B_2'$ is $3$-good, including $v'$. Still, $B_3$ is $3$-good: it contains enough $3$-good vertices in total, and only one bad level-$2$ sub-box ($B_2'$).
• Figure 4: A schematic representation of a $(\gamma, U, \overline{w}, 2)$-hierarchy of depth $R\!=\!3$. The horizontal axis represents the ($1$-dimensional, Euclidean) distances between the vertices, while the vertical axis shows the level of the hierarchy. The weights of all vertices except $y_0$ and $y_1$ are in the interval $[\overline{w}, 4\overline{w}]$. On level $1$, only the initial vertices $y_0, y_1$ appear. We "push down" $y_0=y_{00}, y_{1}=y_{11}$ to level $2$ (red) and we find them their respective level-$2$ sibling vertices $y_{01}$ and $y_{10}$ within Euclidean distance $2\xi^\gamma$, so that there is path of cost at most $U$ between $y_{01}, y_{10}$ (blue). Then, we "push down" to level $3$ all vertices that appeared at or before level $2$, i.e., $y_{000}, y_{011}, y_{100}, y_{111}$ (red), and find for each of them their level-$3$ siblings, i.e., $y_{001}, y_{010}, y_{101}, y_{110}$, so that each vertex is within Euclidean distance $\le 2 \xi^{\gamma^2}$ from its level-$3$ sibling, and that there is a (blue) path of cost at most $U$ between the newly appearing cousins $y_{001}, y_{010}$ and between $y_{101}, y_{110}$. An intuitive representation is in Figure \ref{['fig:intuition_hierarchy']}.

Theorems & Definitions (115)

• Definition 1.1: SFP, IGIRG, GIRG
• Definition 1.2: 1-dependent first passage percolation (1-FPP)
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Corollary 1.7: Polynomial Regime
• Remark 1.8
• Theorem 1.9
• Theorem 1.10: Extension to threshold GIRGs and $\beta=\infty$
• Definition 2.1: Net