[Paper Review] Merging percolation on $Z^d$ and classical random graphs: Phase transition

Tatyana S. Turova, Thomas Vallier|ArXiv.org|Dec 21, 2006

Complex Network Analysis Techniques11 references4 citations

TL;DR

This paper studies a random graph model on $\mathbb{Z}^d$ that superimposes bond percolation with long-range random edges, showing a second-order phase transition analogous to classical random graphs. The key result identifies a critical threshold $c^{\text{cr}}(p)$ such that for $c < c^{\text{cr}}(p)$, the largest component is $O(\log |B(N)|)$, while for $c \geq c^{\text{cr}}(p)$, a giant component of size $\Theta(|B(N)|)$ emerges whp as $N \to \infty$. The analysis leverages inhomogeneous random graph theory and generates explicit formulas for the component size scaling.

ABSTRACT

We study a random graph model which is a superposition of the bond percolation model on $Z^d$ with probability $p$ of an edge, and a classical random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous} random graph, has a natural relation to the so-called "rank 1 case" of {\it inhomogeneous} random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters $c\geq 0$ and $0 \leq p

Motivation & Objective

• To analyze the connectivity phase transition in a random graph model combining short-range bond percolation on $\mathbb{Z}^d$ and long-range random edges from $G(n,c/n)$.
• To determine the critical threshold $c^{\text{cr}}(p)$ for the emergence of a giant connected component in the superposition model.
• To characterize the asymptotic size of the largest connected component above and below the phase transition using tools from inhomogeneous random graph theory.
• To establish a duality between $c^{\text{cr}}(p)$ and $p^{\text{cr}}(c)$, showing that for any $c < 1$, there exists a unique $p^{\text{cr}}(c) < p_c(d)$ such that the giant component appears.

Proposed method

• The model $G_N(p,c)$ is defined as a superposition of bond percolation on $B(N) = \{-N,\dots,N\}^d$ with edge probability $p$, and a classical random graph $G(n,c/n)$ with edge probability $c/|B(N)|$.
• The analysis uses the theory of inhomogeneous random graphs, particularly the 'rank 1' case, to relate the model to known results on component size distributions.
• Key equations include $c^{\text{cr}}(p) = 1/\mathbb{E}|C|$, where $|C|$ is the size of a cluster in the bond percolation model, and $\alpha(p,c)$ defined via the solution to $\mathbb{E}[c|C|e^{c|C|y}] = 1$.
• The asymptotic behavior of the largest component is derived using convergence of cluster size distributions: $\nu_N(k) \to \rho(k)\mu(k)$, where $\rho(k)$ solves a fixed-point equation.
• Concentration inequalities and exponential tail bounds (e.g., $\mathbb{P}\{|C|=k\} \sim e^{-\zeta(p)k}$) are used to control the tail of the component size distribution.
• The proof establishes $\mathbb{P}\left\{ \left| \frac{C_1(G_N(p,c))}{|B(N)|} - \beta \right| > \varepsilon \right\} \to 0$ as $N \to \infty$, with $\beta$ solving $\beta = 1 - \mathbb{E}[e^{-c\beta|C|}]$.

Experimental results

Research questions

• RQ1 What is the critical threshold $c^{\text{cr}}(p)$ for the emergence of a giant component in the superposition of $\mathbb{Z}^d$ bond percolation and $G(n,c/n)$ random graphs?
• RQ2 How does the size of the largest connected component scale with $N$ when $c < c^{\text{cr}}(p)$?
• RQ3 What is the asymptotic size of the largest component when $c \geq c^{\text{cr}}(p)$, and how does it depend on $p$ and $c$?
• RQ4 Is there a duality between $c^{\text{cr}}(p)$ and $p^{\text{cr}}(c)$, such that for each $c < 1$, there exists a unique $p^{\text{cr}}(c) < p_c(d)$ where the phase transition occurs?
• RQ5 Does the largest component remain $O(\log |B(N)|)$ for all $c < c^{\text{cr}}(p)$, and is the bound $\alpha(p,c)$ sharp?

Key findings

• For $c < c^{\text{cr}}(p)$, the largest connected component in $G_N(p,c)$ is $O(\log |B(N)|)$ with high probability as $N \to \infty$, and the threshold $\alpha(p,c)$ is defined via the solution to $\mathbb{E}[c|C|e^{c|C|y}] = 1$.
• For $c \geq c^{\text{cr}}(p)$, the largest component grows linearly with $|B(N)|$, and its asymptotic fraction $\beta(p,c)$ is the maximal solution to $\beta = 1 - \mathbb{E}[e^{-c\beta|C|}]$.
• The critical threshold $c^{\text{cr}}(p) = 1/\mathbb{E}|C|$ is continuous and strictly decreasing in $p$ on $[0, p_c)$, with $c^{\text{cr}}(0) = 1$ and $c^{\text{cr}}(p_c) = 0$.
• For $d=1$, the model is exactly solvable: $c^{\text{cr}}(p) = (1-p)/(1+p)$, and $p^{\text{cr}}(c) = (1-c)/(1+c)$, showing a perfect duality between $c$ and $p$.
• The phase transition is of second order, similar to classical random graphs, with a continuous emergence of the giant component as $c$ crosses $c^{\text{cr}}(p)$.
• The paper leaves open whether the upper bound $\alpha(p,c)$ is sharp, i.e., whether $\mathbb{P}\{C_1 > \alpha_1 \log |B(N)|\} \to 0$ for all $\alpha_1 < \alpha(p,c)$ when $c < c^{\text{cr}}(p)$.