[Paper Review] Spectra of edge-independent random graphs

Linyuan Lü, Xing Peng|arXiv (Cornell University)|Apr 27, 2012

Random Matrices and Applications16 references4 citations

TL;DR

This paper improves spectral concentration bounds for edge-independent random graphs by removing the logarithmic factor in error terms for both adjacency and normalized Laplacian matrices. Under stronger moment conditions ($\Delta \gg \ln^4 n$), it proves $\|A - \bar{A}\| \leq (2+o(1))\sqrt{\Delta}$ and $\|L - \bar{L}\| \leq \left(2 + \sqrt{\sum_{\lambda \in \Lambda}(1 - \lambda)^2} + o(1)\right)/\sqrt{\delta}$, significantly tightening prior results by Oliveira and Chung-Radcliffe.

ABSTRACT

Let $G$ be a random graph on the vertex set $\{1,2,..., n\}$ such that edges in $G$ are determined by independent random indicator variables, while the probability $p_{ij}$ for $\{i,j\}$ being an edge in $G$ is not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of $G$ are recently studied by Oliveira and Chung-Radcliffe. Let $A$ be the adjacency matrix of $G$, $\bar A=\E(A)$, and $Δ$ be the maximum expected degree of $G$. Oliveira first proved that almost surely $\|A-\bar A\|=O(\sqrt{Δ\ln n})$ provided $Δ\geq C \ln n$ for some constant $C$. Chung-Radcliffe improved the hidden constant in the error term using a new Chernoff-type inequality for random matrices. Here we prove that almost surely $\|A-\bar A\|\leq (2+o(1))\sqrtΔ$ with a slightly stronger condition $Δ\gg \ln^4 n$. For the Laplacian $L$ of $G$, Oliveira and Chung-Radcliffe proved similar results $\|L-\bar L|=O(\sqrt{\ln n}/\sqrtδ)$ provided the minimum expected degree $δ\gg \ln n$; we also improve their results by removing the $\sqrt{\ln n}$ multiplicative factor from the error term under some mild conditions. Our results naturally apply to the classic Erdős-Rényi random graphs, random graphs with given expected degree sequences, and bond percolation of general graphs.

Motivation & Objective

• To improve spectral concentration bounds for adjacency and Laplacian matrices in edge-independent random graphs.
• To remove the $\sqrt{\ln n}$ factor in error terms previously present in results by Oliveira and Chung-Radcliffe.
• To establish tighter high-probability bounds on eigenvalue deviations under stronger moment conditions.
• To extend results to general random graph models, including Erdős-Rényi, expected degree sequences, and bond percolation.

Proposed method

• Uses a refined Chernoff-type inequality for random matrices to control the spectral norm of the deviation $A - \bar{A}$.
• Applies Weyl’s theorem to relate eigenvalue deviations to the spectral norm of the matrix difference.
• Introduces a rank-$k$ approximation condition on the expected Laplacian $\bar{L}$ to handle non-uniform expected degrees.
• Employs matrix decomposition and norm estimation techniques, including $\|M_4\|$ analysis via orthogonal projections and variance bounds.
• Applies concentration inequalities with a carefully chosen threshold $\eta = \sqrt[3]{\delta}/\sqrt[3]{k}$ to control entrywise deviations.
• Uses the structure of the expected degree matrix $T$ and the diagonal matrix $D$ of actual degrees to bound $\|L - \bar{L}\|$.

Experimental results

Research questions

• RQ1 Can the $\sqrt{\ln n}$ factor in spectral deviation bounds for random graphs be removed under stronger assumptions?
• RQ2 What is the optimal spectral concentration bound for the adjacency matrix of edge-independent random graphs when $\Delta \gg \ln^4 n$?
• RQ3 How does the spectral deviation of the normalized Laplacian $L$ behave when the expected degree sequence is non-uniform?
• RQ4 Under what conditions is the expected Laplacian $\bar{L}$ well-approximated by a low-rank matrix?
• RQ5 Can the results be extended to general random graph models such as $G(\mathbf{w})$ with given expected degree sequences?

Key findings

• Almost surely, $\|A - \bar{A}\| \leq (2+o(1))\sqrt{\Delta}$ holds when $\Delta \gg \ln^4 n$, improving upon Oliveira’s $O(\sqrt{\Delta \ln n})$ bound.
• For the normalized Laplacian, $\|L - \bar{L}\| \leq \left(2 + \sqrt{\sum_{\lambda \in \Lambda}(1 - \lambda)^2} + o(1)\right)/\sqrt{\delta}$ under $\delta \gg \max\{k, \ln^4 n\}$, removing the $\sqrt{\ln n}$ factor.
• In the special case of expected degree sequences with $\text{rank}(\bar{A}) = k$, the bound simplifies to $\|L - \bar{L}\| \leq (2 + \sqrt{k} + o(1))/\sqrt{\delta}$.
• The results apply to classic models: Erdős-Rényi graphs, random graphs with given expected degree sequences, and bond percolation on general graphs.
• The proof technique relies on refined matrix concentration inequalities and a novel decomposition of the Laplacian deviation into four matrix components.
• The key innovation is the use of a threshold $\eta = \sqrt[3]{\delta}/\sqrt[3]{k}$ to control entrywise deviations in the degree matrix, enabling tighter norm bounds.