Probability Graphons and Large Deviations

• Probability graphons are measurable, symmetric functions mapping vertex pairs to probability measures, thus generalizing classical graphon models with edge decorations.
• They enable rigorous large deviation analysis of dense weighted networks by quantifying rare events using an integrated KL divergence rate function.
• This framework supports statistical inference and model validation through weak regularity and variational methods, extending applications beyond the Bernoulli case.

Probability graphons generalize the concept of classical graphons by encoding, for every pair of “vertex types” in the unit square, a probability measure valued in a general, typically compact Polish, space. This extension enables rigorous analysis of dense weighted (or decorated) networks and supplies a flexible analytic framework to paper the limit behavior, fluctuations, convergence, and rare events in large networks with edge distributions beyond the Bernoulli case.

1. Definition, Framework, and Motivation

A probability graphon is a measurable, symmetric function

$W : [0,1]^2 \to \mathcal{P}(Z)$

where $\mathcal{P}(Z)$ denotes the set of Borel probability measures over a compact Polish space $Z$ (the “edge state” or “decoration” space). For each $(x, y)$, $W(x,y)$ is a probability measure representing the law of the edge-weight (or decoration) between types $x$ and $y$. This formulation strictly generalizes the classical real-valued graphon $w : [0,1]^2 \to [0,1]$, which prescribes a (Bernoulli) edge probability; for example, in the binary case $Z = \{0,1\}$, the classical graphon $w$ is embedded as

$$W(x, y; dz) = w(x, y) \, \delta_1(dz) + (1 - w(x, y)) \, \delta_0(dz).$$

The construction enables modeling and asymptotic analysis for large dense weighted or edge-decorated graphs, where edges can carry general features (e.g., weights, spatial distances, colors), by interpreting observed (random) networks via their empirical probability graphon representations. This extension is both structurally and analytically essential for realistic weighted network models and for understanding rare events in them (Dionigi et al., 17 Sep 2025).

2. Large Deviation Principle for Probability Graphons

The primary result is a Large Deviation Principle (LDP) for random, dense weighted graphs encoded as random probability graphons. Consider the model:

• Fix a reference probability measure $\nu \in \mathcal{P}(Z)$.
• Generate a random graph on $n$ vertices by independently sampling each edge weight from $\nu$.
• Represent this graph by its empirical probability graphon—typically a step function $W_n$ such that for each block $(x, y)$, $W_n(x, y)$ gives the empirical law of the observed edge.

The law of the resulting random probability graphon $W_n$ (modulo measure-preserving relabelings, i.e., in the quotient space) is a random element in a Polish space under the unlabeled cut metric $\delta_{\square}$ (see (Abraham et al., 2023)). The main theorem proves that, as $n \to \infty$, these measures satisfy an LDP with rate function

$$I_\nu(W) = \int_{[0,1]^2} \mathcal{H}\Bigl( W(x, y) \;\|\; \nu \Bigr) dx dy$$

where $\mathcal{H}(\mu \| \nu)$ is the Kullback-Leibler (KL) divergence (relative entropy) of the measure $\mu$ with respect to $\nu$. The LDP takes the form: for open $\mathcal{O}$ and closed $\mathcal{C}$ in the probability graphon space,

$$-\inf_{W \in \mathcal{O}} I_\nu(W) \le \liminf_{n \to \infty} \tfrac{2}{n^2} \log \Pr(W_n \in \mathcal{O}) \le \limsup_{n \to \infty} \tfrac{2}{n^2} \log \Pr(W_n \in \mathcal{C}) \le -\inf_{W \in \mathcal{C}} I_\nu(W).$$

This is a Sanov-type theorem for random weighted graphs, which strictly generalizes the binary/Bernoulli case of Chatterjee and Varadhan (Dionigi et al., 17 Sep 2025).

3. Mathematical Formulation: Rate Function and Metric Structure

The rate function $I_\nu(W)$ is well-defined and lower semicontinuous on the quotient space of probability graphons (modulo measure-preserving bijections and weak topology). This requires:

• The weak regularity lemma for probability graphons: step-function (empirical) graphons are dense under $\delta_{\square}$ (Abraham et al., 2023).
• The cut metric is defined as

$$d_{\square}(U, W) = \sup_{S, T \subseteq [0,1]} d_m\big(U(S \times T; \cdot), W(S \times T; \cdot)\big)$$

where $d_m$ metrizes the weak topology on $\mathcal{P}(Z)$, such as the Prokhorov or Kantorovich–Rubinstein distance.

• The invariant version takes the infimum over all relabelings (measure-preserving bijections).

These properties ensure that $I_\nu(\cdot)$ is a good rate function: its sublevel sets are compact, and the space of probability graphons is Polish with respect to the quotient cut metric (Abraham et al., 2023).

4. Applications: Rare Event Analysis and Statistical Inference

The LDP provides an explicit tool to estimate the likelihood of rare events in large networks with distributionally decorated edges, in the sense that for “atypical” macroscopic events (e.g., unusually high clustering or edge-weight concentration),

$$\Pr(W_n \text{ in } A) \asymp \exp\Big( -\tfrac{n^2}{2} \inf_{W \in A} I_\nu(W) \Big).$$

This has several direct consequences:

• Determination of the most likely limit objects (via empirical minimizers of $I_\nu$) when observing rare configurations, yielding conditional laws for weighted networks.
• Statistical inference in random graph models under distributional edge uncertainty—estimating $\nu$ or testing for structural anomalies by comparing empirical minimizers with theoretical predictions (Dionigi et al., 17 Sep 2025).
• Model validation and hypothesis testing via entropy minimization, supporting variational approaches for fitting probability graphon models to observed weighted networks.

5. Connection to and Generalization of Classical Results

This theory extends previous foundational work on dense graph limits and large deviations, in particular:

• Chatterjee and Varadhan’s LDP for Erdős–Rényi (binary) graphs is recovered as the special case $Z = \{0,1\}$ and illustrates the unified nature of the approach (Dionigi et al., 17 Sep 2025).
• The extension from $[0,1]$-valued graphons to $\mathcal{P}(Z)$-valued kernels enables modeling networks with edge-dependent randomness or decorations in arbitrary compact spaces, crucial for real-world applications involving weights, capacities, or continuous attributes.
• Technical advances include the use of the weak regularity lemma for probability graphons, extensions of the KL divergence to graphon-valued observables, and a projective limit (Dawson–Gärtner) approach to LDPs in infinite-dimensional settings.

6. Technical Challenges and Open Directions

The proof of the LDP involves several challenges:

• Handling infinite-dimensionality: The space of probability graphons is infinite-dimensional and requires controlling convergence, approximation, and tightness using metric and measure-theoretic tools adapted from empirical process theory and weak regularity.
• Rate function minimization often involves high-dimensional optimization over the space of measurable $\mathcal{P}(Z)$-valued kernels, requiring convex analytic and variational methods.
• A plausible implication is that extensions to sparse regimes—or expanding beyond independent edge-weight models—will require new combinatorial and measure-concentration tools.

7. Summary

Probability graphons $W : [0,1]^2 \to \mathcal{P}(Z)$ provide a unifying continuum representation for the limit behavior of large dense weighted or decorated graphs. The established large deviation principle, with the integrated KL divergence

$$I_\nu(W) = \int_{[0,1]^2} \mathcal{H}\big(W(x, y) \,\|\, \nu\big)\, dx\,dy,$$

as rate function, quantifies the exponential rates for rare events in random weighted networks, extending the core results from the classical (binary) setting to networks with arbitrary edge-distributional structure (Dionigi et al., 17 Sep 2025). This framework rigorously supports inference, model selection, and conditional sampling in modern network data analysis, facilitating a broad range of applications in mathematical statistics, statistical physics, and data science with distributional edge uncertainty.