[Paper Review] Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension

Peter Gracar, Lukas Lüchtrath+1 more|arXiv (Cornell University)|Mar 22, 2022

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes a sharp criterion for the finiteness of the percolation threshold in one-dimensional inhomogeneous long-range random graphs with heavy-tailed degrees. By introducing an effective exponent $\delta_{\text{eff}}$, the authors prove that $\delta_{\text{eff}} < 2$ guarantees a supercritical percolation phase, while $\delta_{\text{eff}} > 2$ implies no percolation, resolving a key question about giant component existence in such models.

ABSTRACT

We consider inhomogeneous spatial random graphs on the real line. Each vertex carries an i.i.d. weight and edges are drawn such that short edges and edges to vertices with large weights occur with higher probability. This allows the study of models with long-range effects and heavy-tailed degree distributions. We introduce a new coefficient $δ_ ext{eff}$ which quantifies the influence of heavy-tailed degrees on long-range connections. We show that $δ_ ext{eff}&lt;2$ is sufficient for the existence of a supercritical percolation phase in the model and that $δ_ ext{eff}&gt;2$ always implies the absence of percolation. In particular, our results complement those in Gracar et al. (Adv. Appl. Prob., 2021), where sufficient conditions were given for the soft Boolean model and the age-dependent random connection model for both the existence and the absence of a subcritical percolation phase. Our results further provide a criterion for the existence or non-existence of a giant component in large finite graphs.

Motivation & Objective

• To determine conditions under which a supercritical percolation phase exists in one-dimensional inhomogeneous long-range random graphs with heavy-tailed degrees.
• To resolve the open question of whether the percolation threshold $\beta_c$ is finite in models with finite mean degree and long-range connections.
• To introduce and analyze a new effective exponent $\delta_{\text{eff}}$ that quantifies the impact of heavy-tailed degrees on long-range connectivity.
• To establish a phase transition criterion based on $\delta_{\text{eff}}$ that distinguishes between existence and absence of an infinite cluster.
• To extend previous results on soft Boolean and age-dependent random connection models by providing a unified criterion for giant component emergence in finite graphs.

Proposed method

• Introduce a new coefficient $\delta_{\text{eff}}$ that captures the combined effect of vertex weight tails and long-range connection probabilities.
• Use a renormalization argument based on dyadic decomposition of space to analyze connectivity across scales.
• Apply moment bounds and exponential moment estimates to control the probability of long-range connections between distant vertex clusters.
• Employ Poisson thinning and scaling arguments to reduce the case $\rho(0+) = 1$ to the case $\rho(0+) < 1$, where standard estimates apply.
• Use empirical distribution functions of vertex marks and $\mu$-regularity assumptions to control local density and connection probabilities.
• Leverage the Palm distribution of a Poisson process to analyze connectivity from a typical vertex, enabling the use of invariance and scaling techniques.

Experimental results

Research questions

• RQ1 Under what conditions does the weight-dependent random connection model on $\mathbb{R}$ exhibit a supercritical percolation phase with a giant component?
• RQ2 Is the percolation threshold $\beta_c$ finite when the graph has finite mean degree but heavy-tailed degree distributions?
• RQ3 How does the interplay between heavy-tailed vertex weights and long-range connection rules affect the existence of infinite clusters?
• RQ4 Can a sharp threshold criterion be derived that distinguishes between percolation and non-percolation regimes based on a single effective exponent?
• RQ5 What is the role of the connection function's behavior at zero ($\rho(0+)$) in determining the existence of infinite clusters?

Key findings

• The existence of a supercritical percolation phase is guaranteed if $\delta_{\text{eff}} < 2$, regardless of the specific form of the connection function or weight distribution.
• If $\delta_{\text{eff}} > 2$, then no infinite cluster exists for any $\beta > 0$, implying $\beta_c = \infty$.
• The critical case $\delta_{\text{eff}} = 2$ is not fully resolved in this work, but the results suggest it may lie at the boundary of percolation.
• The criterion based on $\delta_{\text{eff}}$ is sharp and applies uniformly across a broad class of inhomogeneous long-range models, including soft Boolean and age-dependent random connection models.
• The result holds even when $\rho(0+) = 1$, by using Poisson thinning and scaling to reduce to the case $\rho(0+) < 1$, where standard estimates apply.
• The method establishes that the finiteness of $\beta_c$ depends solely on the tail behavior of the vertex weights and the decay rate of the connection function, encapsulated in $\delta_{\text{eff}}$.