Detecting illicit transactions in evolving financial networks requires models that generalize under severe class imbalance and concept drift. Existing graph neural network (GNN) approaches on the Elliptic dataset often suffer from feature collapse, over-smoothing, or poor temporal generalization, leading to suboptimal performance under strict temporal validation.

We propose SWAN (Spline-Weighted Adaptive Network), a graph learning framework that integrates
(i) a rigorously constrained sparse autoencoder with tied weights,
(ii) a novel Phoenix Mechanism that adaptively regulates sparsity based on effective feature superposition, and
(iii) a topology-aware embedding evolution inspired by E8 lattice geometry.

Under a strict temporal cross-validation protocol, SWAN achieves up to 0.99 AUPRC on the Elliptic dataset, surpassing previously reported results. Our findings demonstrate that controlling effective representation dimensionality is critical for robust temporal generalization in fraud detection.

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Illicit transaction detection in cryptocurrency networks is challenging due to extreme class imbalance, noisy relational structure, and non-stationary data distributions. The Elliptic dataset has become a standard benchmark for this task, yet many reported results rely on random splits or insufficient temporal controls, leading to overly optimistic performance estimates.

Graph neural networks (GNNs) improve relational reasoning but often exhibit representation collapse under long training horizons, especially when sparsity regularization is applied naively. This collapse manifests as a loss of effective feature diversity, harming temporal generalization.

We introduce SWAN, a model explicitly designed to resist feature collapse under concept drift by monitoring and controlling the effective number of active features during training.

• A rigorously defined sparse autoencoder with tied weights for graph representation stabilization.
• The Phoenix Mechanism, an adaptive sparsity controller driven by effective feature superposition.
• A local spline-based complexity regularizer to prevent over-fragmentation.
• State-of-the-art performance on Elliptic under strict temporal validation.

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Early approaches on Elliptic relied on classical machine learning with hand-crafted features. GNNs such as GCN and GAT later improved performance by leveraging transaction graphs. More recent work incorporates temporal modeling and autoencoder-based regularization.

However, prior methods typically apply fixed regularization schedules and do not explicitly measure representational collapse. In contrast, SWAN introduces a closed-loop, information-theoretic control mechanism over sparsity.

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We consider a temporal graph:

$$ G_t = (V_t, E_t, X_t) $$

where nodes represent transactions, edges represent fund flows, and node features
$X_t \in \mathbb{R}^d$ are observed at timestep $t$.

The task is binary classification of illicit vs. licit transactions, training on $t' < t$ and evaluating on $t' = t$.

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SWAN consists of five components:

1. Topology-Aware Embedding Layer
   A linear embedding with masked weights periodically regenerated using projections derived from an E8 lattice, promoting orthogonality and diversity.

2. Graph Backbone
   A hybrid GCN–GAT architecture combining efficient message passing with adaptive attention.

3. Rigid Sparse Autoencoder (SAE)
   A single-encoder autoencoder with tied decoder weights and no decoder bias.

4. Spline Complexity Manager
   Estimates local decision boundary complexity via near-nonlinearity intersections.

5. Classification Head
   A shallow MLP optimized with class-balanced loss.

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Given hidden representations $h \in \mathbb{R}^d$, the SAE computes:

$$ z = \mathrm{ReLU}(hW + b) $$

$$ \hat{h} = z W^\top $$

Weights are tied, ensuring identifiability and theoretical stability. Sparsity is encouraged through an $\ell_1$ penalty on $z$.

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We define the feature activation probability:

$$ p_i = \frac{\sum_s |z_{s,i}|}{\sum_{j,s} |z_{s,j}|} $$

The Shannon entropy is:

$$ H(p) = -\sum_i p_i \log p_i $$

The effective number of active features is:

$$ F_{\text{eff}} = e^{H(p)} $$

We define the superposition ratio:

$$ \Psi = \frac{F_{\text{eff}}}{d} $$

which measures how much representational capacity is effectively utilized.

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The Phoenix Mechanism adaptively regulates the sparsity coefficient $\lambda_{\ell_1}$ based on $\Psi$:

• If $\Psi$ falls below a threshold, sparsity is relaxed to allow feature revival.
• Otherwise, sparsity is gently enforced to prevent redundancy.

This feedback loop stabilizes representation dimensionality throughout training.

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The total loss is defined as:

$$ \mathcal{L} = \mathcal{L}_{\text{cls}}

• \alpha \mathcal{L}_{\text{recon}}
• \lambda_{\ell_1}(\Psi)\mathcal{L}_{\ell_1}
• \beta (\Psi - \Psi_0)^2
• \gamma \mathcal{L}_{\text{LC}}
• \delta \mathcal{L}_{\text{ortho}} $$

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We evaluate on the Elliptic dataset using strict temporal cross-validation. For each timestep $t$, models are trained on all $t' < t$ and evaluated on $t' = t$. Feature scaling is performed using training data only.

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Due to extreme class imbalance, we report Area Under the Precision–Recall Curve (AUPRC) as the primary metric.

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SWAN achieves up to 0.99 AUPRC on timestep T=08, consistently outperforming GCN, GAT, and prior hybrid models under the same protocol. Performance peaks early in training, emphasizing the importance of early stopping under temporal drift.

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Removing the Phoenix Mechanism leads to rapid collapse of $\Psi$ and a drop of 4–7 AUPRC points. Fixing the sparsity coefficient similarly degrades temporal generalization.

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Our results indicate that controlling effective representational capacity is more important than increasing nominal model size. The Phoenix Mechanism operationalizes a simple principle:

Regularization should respond to representation health, not training time.

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We presented SWAN, a graph learning framework that achieves state-of-the-art performance on temporal fraud detection by explicitly controlling feature superposition. The proposed principles extend beyond Elliptic to other non-stationary graph learning problems.

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(Omitted for brevity)