[Paper Review] The spectrum of a random geometric graph is concentrated

Sanatan Rai|ArXiv.org|Aug 9, 2004

Random Matrices and Applications7 references4 citations

TL;DR

This paper establishes that the spectral measure of the transition matrix for simple random walk on a random geometric graph in $[0,1]^d$ is asymptotically equidistributed with that of a deterministic grid graph, under appropriate edge radius scaling. Using Hilbert-Schmidt norm concentration and Chernoff-Höffding bounds, it proves that the spectral distance between the random and deterministic graphs converges to zero with high probability, implying spectral concentration.

ABSTRACT

Consider $n$ points distributed uniformly in $[0,1]^d$. Form a graph by connecting two points if their mutual distance is no greater than $r(n)$. This gives a random geometric graph, $\gnrn$, which is connected for appropriate $r(n)$. We show that the spectral measure of the transition matrix of the simple random walk (\abbr{srw}) on $\gnrn$ is concentrated, and in fact converges to that of the graph on the deterministic grid.

Motivation & Objective

• To establish that the spectral measure of the transition matrix for simple random walk on a random geometric graph converges to that of a deterministic grid graph.
• To quantify the rate at which the spectral measures of random and deterministic geometric graphs become asymptotically equidistributed.
• To prove that the spectral measure of the random geometric graph is concentrated around that of the deterministic grid, using probabilistic concentration inequalities.
• To bridge the gap between random geometric graphs and deterministic lattice approximations in spectral graph theory.

Proposed method

• Define the random geometric graph $G(oldsymbol{ u}_n; r(n))$ by connecting points in $[0,1]^d$ within distance $r(n)$, with $r(n) \to 0$ but ensuring connectivity whp.
• Define the deterministic grid $\mathcal{D}_n$ as the $n$-point lattice with spacing $n^{-1/d}$, and consider the transition matrix $P(\mathcal{D}_n)$ for simple random walk on it.
• Use the Hilbert-Schmidt norm $\|P(\mathcal{X}_n) - P(\mathcal{D}_n)\|_{\mathrm{HS}}$ as a proxy for spectral distance, leveraging Theorem 2.1 to relate matrix difference to spectral discrepancy.
• Apply Chernoff-Höffding bounds to control the tail behavior of the reciprocal of neighborhood degrees, ensuring concentration of the transition matrix entries.
• Use the Wasserstein distance $\|\mu(\mathcal{X}_n) - \mu(\mathcal{D}_n)\|_{\mathrm{WS}}$ to measure spectral measure concentration, and derive high-probability bounds on its deviation.
• Introduce the minimum bottleneck matching distance $M_n$ between $\mathcal{X}_n$ and $\mathcal{D}_n$ to control geometric displacement and link it to matrix norm concentration.

Experimental results

Research questions

• RQ1 Does the spectral measure of the transition matrix for simple random walk on a random geometric graph converge to that of a deterministic grid graph?
• RQ2 What is the rate of convergence of the spectral measures of random and deterministic geometric graphs?
• RQ3 Can the spectral concentration of the random geometric graph be quantified using probabilistic concentration inequalities?
• RQ4 How does the geometry of point placement (random vs. lattice) affect the spectrum of the transition matrix?

Key findings

• The spectral measures of the random geometric graph $G(\mathcal{X}_n; r(n))$ and the deterministic grid graph $G(\mathcal{D}_n; r(n))$ are asymptotically equidistributed with high probability.
• The Wasserstein distance between the spectral measures satisfies the high-probability bound: $\mathbb{P}\left\{\|\mu(\mathcal{X}_n) - \mu(\mathcal{D}_n)\|_{\mathrm{WS}} > \frac{t}{a(n)^{1/4}}\right\} \leq \frac{16n a(n)^{1/4}}{t} \left[2\exp\left(-\frac{1}{2}\left(\frac{t^4}{8t^4 + 4096}\right)^2 a(n)\right) + \exp\left(-c_d \frac{t^8}{512} a(n)\right)\right]$, where $a(n) = n \pi_d r(n)^d$.
• The spectral concentration holds under the condition that the minimum bottleneck matching distance $M_n = o(r_n)$ almost surely and with high probability.
• The convergence is established via elementary tools: Hilbert-Schmidt norm control and Chernoff-Höffding bounds, without relying on advanced spectral theory or Stieltjes transforms.
• The result implies that the eigenvalue distribution of the random geometric graph's transition matrix is tightly concentrated around that of the deterministic grid, even though the graph is random.
• The paper conjectures that the eigenvalues of the two graphs are asymptotically equally distributed in $L^2$, though symmetry is not assumed, and a Wielandt-Hoffman-type bound would be needed to confirm this.