Large deviations of the giant in supercritical kernel-based spatial random graphs

TL;DR

The article develops a unified framework for analyzing large deviations of the giant component in supercritical kernel-based spatial random graphs with long-range edges and heavy-tailed degrees. By formalizing kernel-based spatial random graphs (KSRG) and classifying dominant connection types, it shows that long edges and degree inhomogeneity drastically alter tail behavior: upper tails can have logarithmic speeds with a rate function tied to hub-generated connectivity, while lower tails exhibit polynomial (stretched-exponential) decay governed by the exponent ζ_star = max{ζ_long, ζ_short}. A multi-scale renormalization scheme, together with sprinkling and ER-dominance arguments, yields a sharp LLN for the giant, tight bounds on the second-largest component, and precise relationships between the origin cluster and downward-edge boundaries. These results extend to classical long-range percolation and related spatial models (GIRG, sPBM), offering a robust understanding of how long-range connections and hub structures shape the final-size distribution in complex networks with geometry. The work provides new tools for analyzing epidemics, information diffusion, and robustness in spatially embedded networks with heavy-tailed degrees.

Abstract

We study cluster sizes in supercritical $d$-dimensional inhomogeneous percolation models with long-range edges -- such as long-range percolation -- and/or heavy-tailed degree distributions -- such as geometric inhomogeneous random graphs and the age-dependent random connection model. Our focus is on large deviations of the size of the largest cluster in the graph restricted to a finite box as its volume tends to infinity. Compared to nearest-neighbor Bernoulli bond percolation on $\mathbb{Z}^d$, we show that long edges can increase the exponent of the polynomial speed of the lower tail from $(d-1)/d$ to any $ζ_\star\in\big((d-1)/d,1\big)$. We prove that this exponent $ζ_\star$ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin $\mathcal{C}(0)$. For the upper tail of large deviations, we prove that its speed is logarithmic for models with power-law degree distributions. We express the rate function via the generating function of $|\mathcal{C}(0)|$. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.

Figures (3)

• Figure 1: Simulations of 2-dimensional long-range percolation (LRP), a 2-dimensional geometric inhomogeneous random graph (GIRG), and a 1-dimensional soft Poisson--Boolean model (sPBM). In the sPBM, the $y$-axis reflects the vertex marks. The presence of long edges in these models leads to delocalized components, and affects the distributions of the sizes of three components: the speed of the lower tail of large deviations of the largest component (blue), the size of the second-largest component (red), and the distribution of the size of the cluster containing the origin (green) are all governed by the exponent $\zeta_\star$, which is defined in \ref{['eq:zeta-star']}. The upper tail of large deviations of the giant's size has linear speed in (mark)-homogeneous models as LRP, and logarithmic speed in inhomogeneous models as GIRG and sPBM.
• Figure 2: Strategy for the upper tail. The giant induced on lower-mark vertices (filled component) increases in size: hubs (vertices of mark $\Omega(n)$) connect to it by an edge, and also to smaller-size components. The resulting giant component in the entire graph is colored red. If $p<1$, the hubs do not connect by an edge to all vertices (missing edges are dashed in gray), leaving some small components or isolated vertices (blue). The probability for a component of size exactly $k$ to have no edge to one of at least $h$ many hubs is $(1-p)^{kh}$, so the giant's size increases by roughly $\sum_{k} \mathbb P(\mathcal{C}(0)=k)(1-(1-p)^{kh})$. The function $\mathrm{hubs}$ in \ref{['eq:hubs-gen']} gives the minimal number of hubs, so that the size of the giant increases above $\rho n$.
• Figure 3: Sketch of the multi-scale renormalization for the hl-regime. The level-$(i-1)$ box situated second from the left is bad because its largest connected component is too small (e.g. due to failure of Con at a previous level). The fifth box is bad because the Mark step failed at level $i-1$, even though it contains a large component. The large components in the first, third, and fourth box are connected using the connector vertices, forming the darkred component. At the end, we reveal the high-mark vertices in $[n_i^\gamma, 2n_i^\gamma)$ and their edges to the local giant in the level-i box. If there are sufficiently many high-mark vertices, the box passes the Mark-step. We will only use these vertices in the next level.

Theorems & Definitions (99)

• Definition 2.1: Kernel-based spatial random graphs (KSRG)
• Claim 2.3: Phases of $\zeta_\star$, clusterI
• Theorem 2.4: Linear speed in degree-homogeneous models
• Theorem 2.5: Logarithmic speed in degree-inhomogeneous models
• Corollary 2.6: Rate function for the upper tail
• proof
• Theorem 2.7: Speed in the lower tail of large deviations for the giant
• Corollary 2.8: Law of Large Numbers
• proof
• Theorem 2.9: Cluster-size decay