Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits (2412.19352v2)

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 $np = n^{\Omega(1)}$, 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.

• The paper establishes CLTs for linear spectral statistics using graphon convergence in inhomogeneous random graphs.
• It identifies phase transitions in spectral fluctuations across sparse, dense, and bounded degree regimes.
• The study integrates combinatorial graph structures with rigorous random matrix methods to enhance network analysis.

Central Limit Theorems for Linear Spectral Statistics of Inhomogeneous Random Graphs with Graphon Limits

The paper "Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits" by Xiangyi Zhu and Yizhe Zhu makes significant contributions to the understanding of spectral properties of adjacency matrices derived from inhomogeneous random graphs. It establishes central limit theorems (CLTs) for the linear spectral statistics across various sparsity regimes while employing graphon theory for variance profile convergence.

Summary and Key Results

1. Graph Model and Scaling: The paper focuses on the adjacency matrix $A_n$ of sparse inhomogeneous random graphs with elements $a_{ij} \sim \text{Ber}(p s_{ij})$, where $s_{ij}$ represents the variance profile converging to a graphon $W$. The sparsity parameter $p$ distinguishes between dense, sparse, and bounded degree regimes.
2. Linear Spectral Statistics:
   • The central objects of paper are linear spectral statistics, $\mathcal{L}(f) = \sum_{i=1}^n f(\lambda_i)$, where $\lambda_i$ are the eigenvalues of $A_n$.
   • Different scaling factors are used for centered versus non-centered adjacency matrices in different sparsity regimes.
3. Phase Transition in Spectral Fluctuations:
   • The paper identifies critical phase transitions in spectral fluctuations due to centering. In regimes where $np = n^{\Omega(1)}$, centering impacts the scaling factor, whereas in $np = n^{o(1)}$, the CLTs for centered and non-centered matrices converge with the same statistics.
4. Convergence to Gaussian Processes:
   • For the centered adjacency matrix $\overline{A}_n$, the linear spectral statistics converge to Gaussian processes. In the dense regime, and using graphon limits, the covariance of these processes is expressed in terms of homomorphism densities.
   • For the adjacency matrix $A_n$, similar Gaussian process convergence is evidenced, yet the scaling differs particularly in the presence of cycles.
5. Graphon as a Limiting Object:
   • Essential to these results is the notion of graphon convergence, specifically using both the strong $\delta_1$ and weak $\delta_{\Box}$ convergences. The strength of the required convergence decreases with the graph's sparsity.
   • The graphon framework allows the authors to relate the spectral properties of infinite-size graphs characterized as limits of dense sequences.
6. Combinatorial Structures and Homomorphism Densities:
   • The paper engages with combinatorially defined graph structures, such as rooted planar trees and cycles with overlapping edges. Homomorphism densities from these structures to graphons play a crucial role in articulating the limiting spectral statistics.
   • Such detailed combinatorial and graph-theoretical treatments provide insights that build upon and extend existing random matrix theory.

Implications and Speculations

• Theoretical Significance: The convergence results and identified phase transitions enrich the mathematical framework within random matrix theory and probability theory related to graph convergence.
• Practical Relevance: By establishing CLTs for sparse regimes commonly seen in real-world networks, these findings could influence approaches in network analysis, allowing for more refined statistical tests based on spectral properties.
• Speculative Extension: Further research may extend these methods to understanding the impact of structural properties in weighted networks or those with hierarchical clustering.

Overall, the integration of graphon theory with random matrix analysis presented in this work potentially paves the way for advanced studies in both theoretical explorations and practical applications in networks and graph-based data analyses.