BundleNet: Dual Neural Architectures

• Bundle Neural Networks (BuNNs) are a novel graph neural network built on flat vector bundles and diffusion PDEs to overcome key challenges like over-smoothing and over-squashing.
• The BuNN layer discretization process integrates synchronization, bundle-aware encoding, message diffusion, and local nonlinearity to enhance node-level expressivity and performance.
• BundleNet for joint advertising auctions employs bundle-based mechanisms with allocation and payment networks to achieve near-optimal revenue while enforcing incentive compatibility.

BundleNet refers to two unrelated neural architectures in contemporary research: (1) Bundle Neural Networks (BuNNs), a novel graph neural network (GNN) that leverages message diffusion over flat vector bundles to overcome structural limitations of classical message-passing neural networks; and (2) BundleNet for joint advertising, a bundle-based neural mechanism designed for automated mechanism design in multi-slot, multi-bidder auction settings, specifically for joint allocation of advertising slots. Both architectures are technically distinct and address domain-specific challenges in graph representation learning and economic mechanism design, respectively.

1. Bundle Neural Networks (BuNNs): Foundations and Mathematical Structure

Bundle Neural Networks (BuNNs) are a new class of graph neural networks constructed over diffusion-type partial differential equations (PDEs) on flat vector bundles defined on graphs. A flat vector bundle of rank $d$ on a graph $G=(V,E)$ assigns to each node $v\in V$ a $d$-dimensional real inner-product space $\mathcal{B}(v)\cong\mathbb{R}^d$ (the fiber) and to each edge $(u,v)\in E$ an orthogonal map $O_{uv}:\mathcal{B}(v)\to\mathcal{B}(u)$ with a path-independence property: the composite of the $O_{ij}$ along any path depends only on endpoints. This structure allows "synchronization" across the graph, and the set of all node-wise orthogonal matrices $O_v\in \mathrm{O}(d)$ defines a global block-diagonal matrix $O\in\mathbb{R}^{nd\times nd}$. The normalized bundle Laplacian is then $G=(V,E)$0, where $G=(V,E)$1 is the usual combinatorial Laplacian.

The continuous dynamics of node features are governed by the bundle-based heat equation:

$G=(V,E)$2

with solution

$G=(V,E)$3

where $G=(V,E)$4 is termed the bundle heat kernel. This kernel controls global feature propagation, with the nodewise update at $G=(V,E)$5 being a sum over all nodes $G=(V,E)$6:

$G=(V,E)$7

where $G=(V,E)$8 is the classical scalar heat kernel.

2. BuNN Layer Construction and Discretization

A discrete BuNN layer at depth $G=(V,E)$9 comprises four steps, each tightly coupling the vector bundle structure to feature transformation:

1. Bundle-wide synchronization: For each node $v\in V$0, compute $v\in V$1 using a small node-wise GNN or multilayer perceptron, establishing the fiber orientation.
2. Bundle-aware linear encoding: Apply $v\in V$2, where $v\in V$3 and $v\in V$4 are learnable parameters.
3. Message diffusion: Stack $v\in V$5 to $v\in V$6 and propagate by

$v\in V$7

optionally approximating the kernel using a truncated Taylor expansion for scalability.

1. Local nonlinearity: Set $v\in V$8.

This discretization recovers (with specific parameterization) a special case of Sheaf Neural Networks (SNNs), substantiating the theoretical expressivity and mitigation of classical GNN pathologies (Bamberger et al., 2024).

3. Over-Smoothing, Over-Squashing, and Expressivity in BuNNs

Classical message passing neural networks (MPNNs) on graphs face two entrenched issues:

• Over-smoothing: Feature representations of distant nodes converge under repeated aggregation, eroding node-level informativeness. With BuNNs, nontrivial bundle structure ensures that only synchronized fibers $v\in V$9 (not necessarily identical) are harmonized, with Proposition 4.1 showing the preservation of node-level features in the nontrivial bundle regime.
• Over-squashing: Gradients propagated across long-range dependencies decay exponentially, leading to vanishing signal for distant nodes. Lemma 4.2 demonstrates that in BuNN, for a linear layer,

$d$0

which remains strictly positive for all $d$1 and converges to a constant as $d$2, thereby circumventing the decay.

• Universal node-level expressivity: Theorem 5.2 proves that two-layer BuNNs, with sufficient width and injective positional encodings, can compact-uniformly approximate any continuous node-level feature transformation for potentially infinite graph families, exceeding the expressivity limitations of bounded-depth MPNNs.

4. Empirical Performance of BuNNs

BuNNs exhibit state-of-the-art accuracy on both synthetic and real-world graph learning tasks. In synthetic regression tasks that pathologically defeat classical GNNs due to over-smoothing or over-squashing (barbell graph, clique graph), BuNN achieves near-perfect mean squared error (MSE ≈ 0.01–0.03), whereas GCN, GAT, SAGE, and NSD fail (MSE ≈ 1). On heterophilic node-classification benchmarks (Roman-Empire, Amazon-Ratings, Minesweeper, Tolokers, Questions), BuNN attains average relative gains of approximately 4.4% over strong baselines, achieving state-of-the-art in all tasks. On the Long Range Graph Benchmark (Peptides-func, Peptides-struct), BuNN outperforms prior methods on peptide-function classification (AP = 71.92 vs prior 71.50) and is competitive on structure regression (MAE ≈ 24.49) (Bamberger et al., 2024).

5. BundleNet for Joint Advertising Auctions: Architecture and Theoretical Motivation

A second, entirely different BundleNet formalism appears in automated mechanism design for joint advertising auctions (Li et al., 10 Jul 2025). Here, the multi-slot joint advertising problem is modeled as a bipartite graph $d$3 between retailers $d$4 and suppliers $d$5, where bids and click-through rates (CTR) are compiled into bidder feature vectors. For each bundle (edge) $d$6:

• Divided Bids: $d$7.
• Stacked Bids: $d$8.

The BundleNet architecture consists of:

• Allocation network: Processes $d$9 through a shared MLP and generates a "doubly-stochastic" allocation matrix $\mathcal{B}(v)\cong\mathbb{R}^d$0 via row and column softmax normalization, followed by an entrywise minimum for admissibility and empty-slot handling.
• Payment network: Takes $\mathcal{B}(v)\cong\mathbb{R}^d$1 as input, feeding through a shared MLP and final sigmoid. Payments $\mathcal{B}(v)\cong\mathbb{R}^d$2 are scaled by corresponding allocation entries, enforcing ex-post individual rationality, i.e., $\mathcal{B}(v)\cong\mathbb{R}^d$3 and $\mathcal{B}(v)\cong\mathbb{R}^d$4.

6. Incentive Compatibility and Training via Regret Minimization

BundleNet is trained to maximize expected revenue:

$\mathcal{B}(v)\cong\mathbb{R}^d$5

subject to approximate dominant strategy incentive compatibility (DSIC) and individual rationality (IR). DSIC is enforced by bundle-level regret penalties. For each bundle $\mathcal{B}(v)\cong\mathbb{R}^d$6, the ex-post regret $\mathcal{B}(v)\cong\mathbb{R}^d$7 is defined by maximizing each agent's gain from unilateral misreporting, summed over both participants. Lemma 4.1 gives $\mathcal{B}(v)\cong\mathbb{R}^d$8, so zero bundle-level regret implies bidder-level IC. Optimization uses an augmented Lagrangian loss, alternating stochastic gradient updates for network parameters and multipliers, and inner maximizations for regret estimation.

7. Experimental and Practical Aspects of BundleNet in Auction Design

Extensive experiments on synthetic settings (uniform, exponential, normal valuation distributions) show that BundleNet closely matches the optimal revenue given by Myerson-like single-slot mechanisms while driving incentive violation ("IC-violat.") metrics below $\mathcal{B}(v)\cong\mathbb{R}^d$9. In multi-slot settings, BundleNet outperforms RVCG and JRegNet baselines across a range of bundle sizes and composition (Li et al., 10 Jul 2025).

Number of Bundles	Optimal Revenue	BundleNet Revenue	IC-Violation
2	0.5247	0.5286	0.0006
3	0.6705	0.6681	<0.001
4	0.7826	0.7805	<0.001
5	0.8819	0.8802	<0.001

Training time for convergence on large datasets (204,800 samples, batch size 256) is roughly 3 hours on a single GPU. Scalability is primarily determined by the number of bundles $(u,v)\in E$0 and slots $(u,v)\in E$1. Deployment at moderate scale (e.g., $(u,v)\in E$2) gives inference times below 1 ms per auction.

BundleNet's mechanism empirically recovers the allocation rules predicted by the theoretical analysis (Theorem 3.1), whereas extensions of RegretNet can show significant mis-learning. The network's design ensures compliance with revenue-optimality and incentive compatibility constraints in practical multi-agent allocation environments (Li et al., 10 Jul 2025).