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Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits

Xiangyi Zhu, Yizhe Zhu

TL;DR

This work develops central limit theorems for the linear spectral statistics of adjacency matrices of inhomogeneous random graphs across all sparsity regimes, linking variance-profile convergence to graphon limits. It introduces a unified framework that yields explicit covariance formulas in terms of graphon homomorphism densities, capturing how centering affects global eigenvalue fluctuations and revealing phase transitions between centered and non-centered regimes. The results cover dense, sparse, and bounded-degree regimes, and they hold under progressively weaker graphon convergence assumptions as sparsity increases. By connecting graph limit theory with random matrix fluctuations, the paper provides a blueprint for understanding spectral statistics of network data and informs graphon-based inference in random graphs.

Abstract

We establish central limit theorems (CLTs) for the linear spectral statistics of the adjacency matrix of inhomogeneous random graphs across all sparsity regimes, providing explicit covariance formulas under the assumption that the variance profile of the random graphs converges to a graphon limit. Two types of CLTs are derived for the (non-centered) adjacency matrix and the centered adjacency matrix, with different scaling factors when the sparsity parameter $p$ satisfies $np = n^{Ω(1)}$, and with the same scaling factor when $np = n^{o(1)}$. In both cases, the limiting covariance is expressed in terms of homomorphism densities from certain types of finite graphs to a graphon. These results highlight a phase transition in the centering effect for global eigenvalue fluctuations. For the non-centered adjacency matrix, we also identify new phase transitions for the CLTs in the sparse regime when $n^{1/m} \ll np \ll n^{1/(m-1)}$ for $m \geq 2$. Furthermore, weaker conditions for the graphon convergence of the variance profile are sufficient as $p$ decreases from being constant to $np \to c\in (0,\infty)$. These findings reveal a novel connection between graphon limits and linear spectral statistics in random matrix theory.

Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits

TL;DR

This work develops central limit theorems for the linear spectral statistics of adjacency matrices of inhomogeneous random graphs across all sparsity regimes, linking variance-profile convergence to graphon limits. It introduces a unified framework that yields explicit covariance formulas in terms of graphon homomorphism densities, capturing how centering affects global eigenvalue fluctuations and revealing phase transitions between centered and non-centered regimes. The results cover dense, sparse, and bounded-degree regimes, and they hold under progressively weaker graphon convergence assumptions as sparsity increases. By connecting graph limit theory with random matrix fluctuations, the paper provides a blueprint for understanding spectral statistics of network data and informs graphon-based inference in random graphs.

Abstract

We establish central limit theorems (CLTs) for the linear spectral statistics of the adjacency matrix of inhomogeneous random graphs across all sparsity regimes, providing explicit covariance formulas under the assumption that the variance profile of the random graphs converges to a graphon limit. Two types of CLTs are derived for the (non-centered) adjacency matrix and the centered adjacency matrix, with different scaling factors when the sparsity parameter satisfies , and with the same scaling factor when . In both cases, the limiting covariance is expressed in terms of homomorphism densities from certain types of finite graphs to a graphon. These results highlight a phase transition in the centering effect for global eigenvalue fluctuations. For the non-centered adjacency matrix, we also identify new phase transitions for the CLTs in the sparse regime when for . Furthermore, weaker conditions for the graphon convergence of the variance profile are sufficient as decreases from being constant to . These findings reveal a novel connection between graphon limits and linear spectral statistics in random matrix theory.
Paper Structure (28 sections, 14 theorems, 114 equations, 5 figures, 1 table)

This paper contains 28 sections, 14 theorems, 114 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Inhomogeneous random matrices and graph limits
  3. Global eigenvalue fluctuations
  4. Contribution of this paper
  5. Linear spectral statistics of Wigner-type matrices
  6. The centering effect
  7. Three types of graphon convergence
  8. Graphon theory
  9. Main results
  10. Centered adjacency matrices
  11. Adjacency matrices
  12. Proof for the centered adjacency matrices
  13. Proof of Theorem \ref{['thm: centered_CLT_inhomo_dense']}
  14. Proof of Theorem \ref{['thm: centered_CLT_inhomo_sparse1']}
  15. Proof of Theorem \ref{['thm: inhomo_bounded_c']}
  16. ...and 13 more sections

Key Result

Lemma 2.4

Let $\{W_n\}$ be a sequence of graphons in $\mathcal{W}_0$ and let $W\in \mathcal{W}_0$. Then $t(F,W_n)\to t(F,W)$ for all finite simple graphs if and only if $\delta_{\Box}(W_n,W)\to 0$.

Figures (5)

  • Figure 1: We give an example of $k=8$ and $h = 6$. The left figure is an element in $\mathcal{T}^{k,h}_1$ constructed by a rooted tree one of length 4 with edges $\{1,2\}, \{2,3\}, \{3,4\}, \{4,5\}$ and the root vertex $3$, and another rooted tree of length 3 with edges $\{4,5\}, \{5,6\}, \{6,7\}$ and a root vertex $6$, and one overlapping edge $\{4,5\}$. The right figure is a corresponding element in $\mathcal{T}^{k,h}_2$ constructed by two rooted planar trees, one of length 4 with edges $\{1,2\}, \{2,3\}, \{3,4\}, \{4,5\}$ and another one of length 3 with edges $\{4,5\}, \{5,6\}, \{6,7\}$ having two shared vertices $4$ and $5$. Vertices $4$ and $5$ are connected by two distinct edges.
  • Figure 2: We given an example of $k = 12$, $h = 10$ and $r = 4$. The left figure are two graphs $G_1 \in \mathcal{TC}^{k}_{r}$ and $G_2 \in \mathcal{TC}^{h}_{r}$. The right graph shows two possible graphs in $\mathcal{TC}^{k,h}$.
  • Figure 3: Given the rooted trees $T_1$ and $T_2$ in the left graph, the right graph shows two examples in $P_{\#}(T_1, T_2)$, where $T_1$ and $T_2$ overlap in the blue part for each case.
  • Figure 4: We give an example of $k=4$ and $h=5$. The left figure is an example of $F^{k,h}_1$ constructed by cycle one of length 4 with $E = \{\{1,2\}, \{2,3\},\{3,4\}, \{4,1\} \}$ and cycle two of length 5 with $E = \{ \{2,3\}, \{2,7\}, \{7,6\}, \{6,5\}, \{5,3\}\}$ having one overlapping edge $\{2,3\}$. The right figure is an example of $F^{k,h}_2$ constructed by cycle one of length 4 with $E = \{ \{1,2\}, \{2,3\},\{3,4\}, \{4,1\} \}$ and cycle two of length 5 with $E = \{ \{2,3\}', \{2,7\}, \{7,6\}, \{6,5\}, \{5,3\} \}$ having two shared vertices 2 and 3. And vertices 2 and 3 are connected by two distinct edges
  • Figure 5: We give an example of $h=5$. The left figure is an example of $C_h$ and the right figure is an example of $C_{2,h}$. In particular, the blue part in the left figure is an example of $K_2$ and the blue part in the right figure is an example of $C_2$.

Theorems & Definitions (36)

  • Definition 2.1: cut norm
  • Definition 2.2: homomorphism density
  • Definition 2.3: left convergence
  • Lemma 2.4: Theorem 11.5 in lovasz2012large
  • Lemma 2.5: Lemma C.2, janson2010graphons
  • Lemma 2.6: Lemma C.4, janson2010graphons
  • Lemma 2.7
  • Remark 3.5: Hierarchy of graphon convergence assumptions
  • Definition 3.6: Two rooted trees with overlapping edges
  • Definition 3.7: Unicyclic graphs
  • ...and 26 more