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Semicircle laws with combined variance for non-uniform Erdős-Rényi hypergraphs

Luca Avena, Elia Bisi, Eleonora Bordiga

Abstract

We consider Erdős-Rényi-type random hypergraphs that are non-uniform, in the sense that hyperedges of different sizes may coexist, and inhomogeneous, in that connection probabilities may depend on the hyperedge size. All parameters are allowed to scale with the hypergraph size. We study the random adjacency matrix whose $(u,v)$-entry counts the number of hyperedges containing both vertices $u$ and $v$, and characterize its expected limiting spectral distribution in terms of the connection probabilities and the hyperedge sizes. We provide a Pastur-type condition, in the sense of Chatterjee (2005), under which the matrix can be Gaussianized, as well as a more restrictive but simpler sufficient condition in terms of the generalized average degree of the model. As a second main result, based on such a Gaussianization, we characterize the limiting spectral distributions under non-sparse conditions as semicircle laws with an explicit parametric variance. The latter can be expressed as a convex combination of the variances arising in the uniform cases, with coefficients determined by the trade-off between the different sources of inhomogeneity.

Semicircle laws with combined variance for non-uniform Erdős-Rényi hypergraphs

Abstract

We consider Erdős-Rényi-type random hypergraphs that are non-uniform, in the sense that hyperedges of different sizes may coexist, and inhomogeneous, in that connection probabilities may depend on the hyperedge size. All parameters are allowed to scale with the hypergraph size. We study the random adjacency matrix whose -entry counts the number of hyperedges containing both vertices and , and characterize its expected limiting spectral distribution in terms of the connection probabilities and the hyperedge sizes. We provide a Pastur-type condition, in the sense of Chatterjee (2005), under which the matrix can be Gaussianized, as well as a more restrictive but simpler sufficient condition in terms of the generalized average degree of the model. As a second main result, based on such a Gaussianization, we characterize the limiting spectral distributions under non-sparse conditions as semicircle laws with an explicit parametric variance. The latter can be expressed as a convex combination of the variances arising in the uniform cases, with coefficients determined by the trade-off between the different sources of inhomogeneity.

Paper Structure

This paper contains 21 sections, 13 theorems, 123 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Content and organization of the paper
  4. Asymptotic notation
  5. Model
  6. Non-uniform hypergraphs and adjacency matrices
  7. Non-uniform inhomogeneous Erdős--Rényi hypergraphs
  8. Normalization of the adjacency matrix
  9. Main results
  10. Notation
  11. Overview of the results
  12. Convergence to the semicircle law
  13. Gaussianization
  14. Discussion
  15. Dominant and balanced regimes
  16. ...and 6 more sections

Key Result

Theorem 3.4

Consider the Erdős--Rényi hypergraph $\mathcal{H}(n,\bm{r},\bm{p})$ as in Definition our_model. For each $1 \leq i \leq k$, assume that $r_i/n \rightarrow c_i \in [0,1)$ and $1-p_i=\Theta(1)$. Suppose moreover that the non-sparsity condition D holds. Then where $\nu_{s^2}$ is the semicircle law with variance and $w_1,\dots,w_k$ are defined in weights. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Superposition of two classical Erdős--Rényi random graphs ($k=2$) on 3 vertices.

Theorems & Definitions (34)

  • Definition 2.1: $({n,}\bm{r},\bm{p})$-Erdős--Rényi hypergraph
  • Example 2.2: Homogeneous case
  • Example 2.3: Graphs and uniform hypergraphs
  • Example 2.4: Superposition of Erdős--Rényi random graphs
  • Remark 2.5
  • Remark 2.6: Complete and empty hypergraphs
  • Remark 3.3: Hyperedges of the same magnitude and non-sparsity condition
  • Theorem 3.4: Convergence to the semicircle law
  • Remark 3.5: Loss of variance of the semicircle law
  • Definition 3.6: $\bm{Z}$-Gaussianizable model
  • ...and 24 more