Graphon-Limit Law for Polynomial Averages

• Graphon-limit law is a principle describing the deterministic convergence of polynomial observables in large graph-based systems.
• It integrates spectral analysis, homomorphism densities, and mean-field techniques to quantify convergence in random graphs and particle systems.
• Its operator-theoretic formulation enables rigorous spectral filtering and effective modeling in high-dimensional and heterogeneous networks.

A graphon-limit law for polynomial averages is a mathematical principle describing the asymptotic behavior of observables—constructed as polynomial functions of eigenvalues, homomorphism densities, or particle states—in large networks, random graphs, or mean-field systems whose structure is governed by graphons. These laws unify approaches in ergodic theory, spectral graph theory, random graph models, and stochastic particle systems. In such settings, polynomial averages (e.g., moments, filter outputs, recurrence statistics) computed on a finite system converge to deterministic limits characterized by integration against a graphon kernel or associated degree profile.

1. Graphon Frameworks and Polynomial Averages

A graphon is a symmetric, measurable (often bounded, sometimes Lipschitz) function $W: [0,1]^2 \to [0,1]$, representing the limit of a sequence of dense graphs as the number of nodes $n$ goes to infinity. In signal processing and spectral applications, graphons induce self-adjoint integral operators on $L^2([0,1])$, with the operator $(Wf)(x) = \int_0^1 W(x, y) f(y)\,dy$ admitting a countable spectrum.

The concept of polynomial averages covers:

• Spectral averages: expressions like $(1/n)\sum_{i=1}^n P(\lambda_i^{(n)})$, where $\lambda_i^{(n)}$ are graph eigenvalues and $P$ is a polynomial.
• Homomorphism densities in random graphs: $t(F, G_n(W))$, the density of motifs $F$ in $W$-random graphs, often involving moments and cumulants of such counts (Méliot, 2021).
• Mean-field particle averages: normalized sums or integrals of particle states influenced by graphon-weighted interaction (Bayraktar et al., 2020, Coppini et al., 13 Feb 2024, Chen et al., 27 May 2024).

The graphon-limit law states that as $n \to \infty$, polynomial averages computed with respect to a finite structure converge to deterministic functionals (most often integrals or operator traces) derived from the graphon, with randomness and local fluctuations vanishing in the limit.

2. Mathematical Formulation and Spectral Convergence

Spectral Graphon Limit Laws

For a sequence of graphs with adjacency matrices $A^{(n)}$, let $W_n$ be the induced step graphon approximating $W$. The spectrum $\{\lambda_i^{(n)}\}$ of the rescaled Laplacian converges, and for any polynomial $P$:

$$\lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n P(\lambda_i^{(n)}) = \int_0^1 P(\delta(x)) \, dx$$

where $\delta(x) = \int_0^1 W(x,y) \, dy$ is the degree profile (Wang et al., 27 Sep 2025). For spectral filtering, the graphon-based filter output is:

$$h_G = \frac{1}{n}\sum_{i=1}^{n} \lambda_i f(\lambda_i)^2 \longrightarrow \int_0^1 \delta(x) f(\delta(x))^2 dx$$

as both graph and feature dimension grow. This convergence justifies designing spectral procedures, filters, or heterophily metrics in the graphon domain and guarantees their limiting output (Ruiz et al., 2020, Maskey et al., 2021, Wang et al., 27 Sep 2025).

Polynomial Limits in Mean Field and Particle Systems

In mean-field models defined over increasingly large networks, interactions may be linear or nonlinear polynomial functions (see (Coppini et al., 13 Feb 2024, Chen et al., 27 May 2024)):

• The limiting behavior is governed by the graphon-weighted empirical measure or polynomial average, evaluated as an integral over the graphon kernel.
• DDynamic particle systems under graphon interaction admit limiting equations for macroscopic polynomial averages, with LLN and uniform-in-time convergence (Bayraktar et al., 2020).

3. Random Graph Polynomial Averages and Central Limit Theorems

A fundamental class of observables in random graph theory are subgraph counts (homomorphism densities), which are polynomial in the adjacency matrix entries:

• For singular graphons (i.e., constant graphons corresponding to Erdős–Rényi models), rescaled polynomial averages $n(t(F, G_n(\gamma))-\mathbb{E}[t(F, G_n(\gamma))])$ converge in joint distribution, generalizing classical CLTs (Méliot, 2021).
• In general, the variance and cumulant scaling of polynomial averages, as captured by combinatorial expansions (dependency graphs, contractions, or matrix-tree theorems), encode fine-grained information about the graphon's structure (Méliot, 2021).
• The graphon-limit law characterizes when the limiting distribution is Gaussian (constant graphon) or more exotic, with spectral criteria (Fredholm determinants, vanishing fourth cumulants) providing explicit characterizations.

4. Operator-Theoretic and Homomorphism Density Limit Laws

Central limit theorems for sums of $\epsilon$-independent random variables are governed by graphons (Cébron et al., 20 Nov 2024):

• The moment formula for the graphon-based limit law is

$$\int x^{2p} d\mu_w = \sum_{\pi \in P_2(2p)} \rho(f_\pi, w)$$

where $f_\pi$ is the intersection graph of a pair partition, and $\rho(f_\pi, w)$ is the homomorphism density weighted by the graphon. The limit law $\mu_w$ thus interpolates between Gaussian, semicircular, and mixed (e.g., free convolution) cases depending on $w$.

• Extensions to matrix-valued ("decorated") graphons and multivariate systems generalize the graphon-limit law for polynomial averages beyond the scalar setting.

5. Graphon Particle Systems, Law of Large Numbers, and High-Dimensional Algorithms

Graphon particle systems (Bayraktar et al., 2020, Coppini et al., 13 Feb 2024, Chen et al., 27 May 2024) model large collections of interacting particles (or SGD iterates in distributed optimization) with coupling through a graphon kernel:

• Polynomial averages (empirical means, variances, higher moments) of the system states converge as both particle number and temporal discretization parameters vanish.
• Convergence is established using two-level approximation schemes—spatial graphon discretization and time-step Euler–Maruyama schemes—supported by Wasserstein distance estimates (often W₂), Gronwall-type bounds, and propagation-of-chaos arguments.
• The law of large numbers and uniform-in-time convergence results guarantee that all macroscopic polynomial averages (as well as nonlinear functionals) have deterministic graphon-driven limits.

6. Nonlinear and Multivariate Extensions, Limit Laws for Random Polynomials

Recent extensions allow for nonlinear dependence (polynomial in local averages):

• For polynomials $P_n$ in independent (possibly non-Gaussian) variables $(X_i)_{i \ge 1}$, any limiting distribution of $P_n((X_i))$ can be represented as a law of $P_\infty((X_i, G_i))$ for independent Gaussians $(G_i)_{i \ge 1}$, with rank and degree bounds characterized in terms of maximal directional influence (Herry et al., 9 Dec 2024).

This generalization underscores that the graphon-limit law for polynomial averages extends from linear to nonlinear (polynomial in arbitrary degree) regimes, with explicit control over limiting structures and dimensionality reduction.

7. Applications, Research Directions, and Theoretical Consequences

Graphon-limit laws for polynomial averages underpin:

• Spectral analysis and signal processing on large networks (Ruiz et al., 2020, Wang et al., 27 Sep 2025)
• Design and transferability of graph neural networks under consistent phenomena (Maskey et al., 2021)
• Stochastic modeling and control in large-scale games, mean-field systems, and distributed computation (Gao et al., 2020, Gao et al., 2020)
• Rigorous validation and design of synthetic benchmarks with controlled feature heterophily (Wang et al., 27 Sep 2025)

Open directions include:

• Quantitative rates and error bounds for convergence of polynomial averages under more general graphon structures;
• Investigation of non-ergodic or non-commuting transformation settings, where polynomial averages may not converge (Kosloff et al., 8 Sep 2024);
• Connections to combinatorial limit theory, especially for systems with complex dependency structures.

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In summary, the graphon-limit law for polynomial averages formalizes the principle that as networks or systems grow large with structure encoded by a graphon, all polynomial functionals of interest—spectral, combinatorial, mean-field, or stochastic—converge to deterministic limits computable as integrals, traces, or functionals of the underlying graphon kernel. This law enables rigorous analysis, algorithmic design, and synthetic modeling in high-dimensional, heterogeneous graph-based systems.