Graphon-Level Bayesian Predictive Synthesis

• The paper introduces a graphon-level Bayesian predictive synthesis method that optimally aggregates network models through an L2 projection, achieving minimax optimality and oracle inequalities.
• It employs a least-squares projection to combine multiple agent graphon models, transferring estimation error explicitly to network properties like edge and triangle densities.
• The approach demonstrates a 'combination beats components' phenomenon while preserving key characteristics in heavy-tailed degree distributions and ERGM settings.

Graphon-level Bayesian Predictive Synthesis (BPS) formalizes the combination of predictive distributions from multiple agents at the level of random graph limit objects known as graphons, offering a decision-theoretic method for aggregating network models. Framed as an $L^2$ projection, graphon-level BPS achieves minimax optimality, oracle inequalities, and precise transfer of estimation error to network structural properties while also encompassing behaviors relevant for heavy-tailed degree distributions and exponential random graph model (ERGM) families (Papamichalis et al., 21 Dec 2025).

1. Foundational Framework: Graphons, Agent Models, and Synthesis

A graphon $G_0:[0,1]^2\to [0,1]$ is a symmetric, measurable function defining an infinite exchangeable random network via the Aldous–Hoover representation: $U_i\sim$ Unif$[0,1]$, edges $(i,j)$ included independently with probability $G_0(U_i,U_j)$. Each agent model $w_i:[0,1]^2\to [0,1]$, $i=1,\dots, K$, specifies an alternative random graph law converging (in cut-distance) to $w_i$ as $n\to\infty$.

The synthesis objective is to construct a single synthesized graphon in the affine span of agents,

$$w(\cdot;\theta) = \theta_0 + \sum_{i=1}^K \theta_i w_i,$$

with $\theta\in \mathbb{R}^{K+1}$, optimal in $L^2$ distance to the unknown true graphon $w_0$:

$$\theta^* \in \operatorname{argmin}_{\theta\in\mathbb{R}^{K+1}} \| w_0 - [\theta_0 + \sum_{i=1}^K \theta_i w_i] \|_{L^2}^2,$$

where $\|f\|_{L^2}^2 = \int_0^1\int_0^1 f(u,v)^2\,du\,dv$ and defining $$\mathcal{H} = \operatorname{span}\{1, w_1, \ldots, w_K\} \subset L^2([0,1]^2)$$.

2. Least-Squares Projection and Synthesis Algorithm

Define the $(K+1)$-dimensional feature vector

$F(u,v) = (1, w_1(u,v), \ldots, w_K(u,v))^T.$

Let $U_1, U_2\sim$ Unif$[0,1]$; then the population Gram matrix $G$ and vector $c$ are

$$G = E[F(U_1, U_2) F(U_1, U_2)^T], \quad c = E[w_0(U_1,U_2) F(U_1,U_2)].$$

The risk of a linear combination $\beta$ is

$$R(\beta) = E\left\{ \left[w_0(U_1,U_2) - \beta^T F(U_1,U_2)\right]^2 \right\}.$$

The unique risk minimizer is given by:

$\beta^* = G^{-1} c,$

yielding the synthesized graphon $w_{BPS}(u,v) = (\beta^*)^T F(u,v)$ as an orthogonal $L^2$ projection of $w_0$ onto $\mathcal{H}$. In empirical settings, sampled edges yield empirical Gram matrices and vectors for finite-sample least squares estimation:

$$\widehat{G}_m = \frac{1}{m}\sum_{s = 1}^m F(X_s)F(X_s)^T, \quad \widehat{c}_m = \frac{1}{m}\sum_{s=1}^m F(X_s) Y_s,$$

where $X_s \sim$ Unif$[0,1]^2$, $Y_s \mid X_s \sim \mathrm{Bernoulli}(w_0(X_s))$.

3. Nonasymptotic Guarantees and Minimax Optimality

Sampling $m$ i.i.d. edges, the empirical least-squares estimator $$\widehat{\beta}_m = \widehat{G}_m^{-1} \widehat{c}_m$$ achieves the following oracle inequality:

$$E \left[ \| \widehat{w}_m - w_0 \|_{L^2}^2 \right] \le 2 \inf_{\beta\in\mathcal{H}} \| \beta^T F - w_0 \|^2 + C \frac{d}{m},$$

where $d = K+1$ and $C$ is a constant under mild conditions. This separates intrinsic approximation error from the sample-size-dependent estimation term.

For $w_0$ in a ball of radius $R$ in $\mathcal{H}$, the minimax rate holds:

$$\inf_{\widehat{w}}\,\sup_{w_0\in\mathcal{H}(R)} E \| \widehat{w} - w_0 \|^2 \asymp \frac{d}{m},$$

showing minimax-rate optimality for least-squares BPS.

4. Combination-Beats-Components Phenomenon

For $w_0$ lying in the convex hull $$\mathcal{W}_{\text{conv}} = \operatorname{conv}\{w_1, ..., w_K\}$$, any estimator that selects a single $w_j$ (single-agent selection) incurs a uniform L$^2$ error lower-bounded by some $\delta > 0$, independent of $m$, and thus does not converge. In contrast, least-squares graphon-BPS achieves error $O(d/m)\to 0$, strictly outperforming all individual agent models. This result formalizes a combination beats components effect intrinsic to the BPS framework.

5. Lipschitz Transfer of Graphon Error to Network Properties

Key graphon functionals include:

• Edge density: $e(w)=\int w$
• Degree: $d_w(x)=\int w(x,y)\,dy$
• Triangle density: $t(w) = \int w(x,y)w(y,z)w(x,z)\,dx\,dy\,dz$
• Wedge density: $s(w) = \int d_w(x)^2\,dx$
• Clustering: $C(w)= t(w)/s(w)$ (if $s(w)>0$)

The $L^2$-Lipschitz theorem provides for any two graphons $w, w'$:

• $|e(w)-e(w')| \le \|w-w'\|_2$
• $\|d_w - d_{w'}\|_2 \le \|w-w'\|_2$
• $|t(w)-t(w')| \le 3\|w-w'\|_2$
• $|s(w)-s(w')| \le 2\|w-w'\|_2$
• For $s(w), s(w') \ge s_0 > 0$, $|C(w) - C(w')| \le (3/s_0 + 2/s_0^2)\|w-w'\|_2$

A direct corollary is that $L^2$-risk control at the graphon level yields explicit, quantitative control of errors in network summaries such as edge density, degree distributions, clustering coefficients, and phase transition points for giant components.

6. Heavy-Tailed Degree Distributions and Entropic Tilting

Suppose some agents in the mixture possess heavy-tailed degree distributions $P(D\ge k) \sim c_j k^{-\gamma_j}$ with $\gamma_j>1$. If the BPS mixture assigns weight to at least one $\gamma_j$ of minimal value, then the combined degree distribution follows

$$P(D \ge k) \sim C_{\text{mix}}\, k^{-\gamma_{\min}},$$

i.e., the slowest-decaying power law dominates.

For a degree-tilted edge law $dP_\ell/dP_0 \propto \ell(D)$:

• If $\ell$ is slowly varying (index $0$), the tail exponent $\gamma$ is preserved.
• If $\ell(k) \sim k^\rho$, the tail exponent shifts to $\gamma - \rho$.
• For polynomially controlled tilts, the degree tail is sandwiched between $k^{-(\gamma+\beta_-)}$ and $k^{-(\gamma-\beta_+)}$.
• Entropic tilts using bounded graph statistics $s(G)\in [-B,B]$ via $\exp(\lambda^T s(G))$ leave the exponent unchanged: $P_{P_\lambda}(D\ge k) \sim k^{-\gamma}$.

7. Closure Under Log-Linear Pooling: Exponential Random Graph Models

If agent models are ERGMs with sufficient statistics $T^{(j)}$ and natural parameters $\theta_j$, a log-linear BPS pool

$$f(A) \propto \prod_j p_j(A)^{\omega_j} \cdot \exp\{\tau^T T_{\text{stack}}(A)\}, \quad \sum_j \omega_j = 1,\,\omega_j \ge 0$$

preserves the ERGM form, now acting on the stacked statistic $T_{\text{stack}} = (T^{(1)}, \ldots, T^{(J)})$ with new parameter blocks $\omega_j \theta_j + \tau^{(j)}$. This ensures compatibility and representational coherence within the log-linear Bayesian pooling inventory.

Summary Table: Synthesis Properties and Guarantees

Property	BPS Result	Comparative Statement
Risk minimization	$L^2$ projection, least-squares solution	Minimax optimal in class
Oracle inequality	Yes: separation of approximation and estimation error	Matches parametric rate
Combination vs. components	Combination strictly beats any single component	Single-agent is inconsistent
Structural error transfer	Explicit Lipschitz bounds linking $\|w-w_0\|_2$ to error in network summaries	Not available for selectors
Heavy-tail preservation	Mixture tail matches slowest-decaying agent	Dominance vs. components
Closure for ERGMs under pooling	Log-linear pooling preserves ERGM form	Ensures model class integrity

All content is summarized from "Graphon-Level Bayesian Predictive Synthesis for Random Network" (Papamichalis et al., 21 Dec 2025).