Opera: Structure-Conditioned Operators for Cross-Domain Link Prediction

Paper Info
Title Opera: Structure-Conditioned Operators for Cross-Domain Link Prediction
Authors Liang Cheng, Amir Taherkordi, Martin Giese
Venue IJCKG 2026
Code GitHub: opera-ijckg2026

Motivation

Link prediction is a fundamental graph learning task — inferring missing or future interactions between nodes in social networks, recommendation systems, and knowledge graphs. GNNs have become the dominant framework, but they face a critical challenge in practice: how to generalize to entirely new domains where labeled links are scarce or unavailable?

This is the cross-domain link prediction problem. Traditional GNNs learn static weight matrices tightly coupled with the topological statistics of their training graph. A GNN trained on a dense social network learns aggregation patterns suited for high-degree hubs, yet those same weights transfer poorly when applied to a sparse citation network with a power-law degree distribution.

Recent approaches — Graph Foundation Models (GFMs) and LLM-based methods — scale pre-training to massive graph corpora or import external semantic knowledge, but they share a common limitation: once trained, they apply a fixed set of parameters to every target graph. They treat cross-domain generalization as a data-scarcity problem rather than a structural reasoning challenge.

Key Insight: Operator Transfer

We start from a simple observation: despite vast differences in global topologies, many domains exhibit recurring node-level structural roles at the local scale — hubs, bridges, peripheral nodes. A hub needs an operator that averages over dense, redundant neighborhoods; a bridge needs one that amplifies contrast between distinct communities. These computational requirements are domain-invariant.

Hypothesis: Nodes with comparable structural roles across different graphs share analogous aggregation logic. Cross-domain generalization should therefore transfer the computational operators induced by structural roles, not node identities or semantic content.

This reinterprets cross-domain link prediction as an operator transfer problem — shifting the paradigm from representation-centric to operator-centric transfer.

Method

Overall Architecture

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          ┌──────────────────────────────────────────────────┐
          │                    Opera                         │
          │                                                    │
Node text ──► [Frozen SentenceTransformer] ──► semantic emb.  │
          │                                                    │
  Graph   ──► [Structural Fingerprint (12-d)] ──► fingerprint │
          │         │                                          │
          │         ▼                                          │
          │  [Conditioned Hypernetwork]                       │
          │         │ (low-rank Z matrices)                    │
          │         ▼                                          │
          │  Dynamic Transformer (4 layers)                    │
          │    • Dynamic Fusion Attention (Q_sem · K_top)     │
          │    • Dynamic FFN                                   │
          │    • Structural top-k gating                      │
          │         │                                          │
          │         ▼                                          │
          │  Edge prediction head (BCE loss)                  │
          └──────────────────────────────────────────────────┘

Opera follows a dual-track design:

  • Semantic track: frozen language model encodes node text → fixed embeddings
  • Topological track: 12 handcrafted structural features → role descriptors
  • Hypernetwork: maps fingerprints to low-rank modulation matrices → dynamic Transformer weights
  • Prediction: edge existence via binary classifier on concatenated node embeddings

Structural Fingerprints (12-d)

Rather than learning role embeddings, we adopt interpretable, efficiently computable topology features that capture a node’s structural position:

Category Features Transfer meaning
Degree activity total / in / out degree Hubness, connectivity activity
Neighbor context mean / std neighbor degree Local role environment
Global position PageRank, hierarchy depth, two-hop count, k-core Influence, reachability
Local cohesion clustering coefficient, ego density, loop coefficient Community tightness

These are computed exclusively from the observed graph (no test-edge leakage) and standardized per-graph via z-score normalization.

Conditioned Hypernetwork

The core mechanism: a lightweight hypernetwork H maps each node’s fingerprint f_v to a low-rank modulation matrix Z_v, which combines with shared factorized bases A, B to form dynamic weights:

\begin{equation} W_v = A \cdot Z_v \cdot B + b, \quad \text{with} \quad Z_v = H(f_v) \end{equation}
  • A, B are shared across all nodes — they capture universal transformation patterns
  • Z_v is node-specific — it modulates the bases according to structural role
  • Low-rank factorization (r ≪ d) keeps the hypernetwork small and prevents overfitting

Spectral Normalization

To ensure training stability under zero-shot cross-domain conditions, we apply spectral normalization to all layers of the hypernetwork:

\begin{equation} W_{SN} = \frac{W}{\sigma_{max}(W) \cdot c} \end{equation}

This enforces a Lipschitz bound on the hypernetwork, directly linking fingerprint similarity to operator similarity:

\begin{equation} \|W_u - W_v\| \leq \|A\| \cdot L \cdot \|B\| \cdot \|f_u - f_v\| \end{equation}

This provides a formal intuition for transfer stability: nodes with similar structural roles (similar fingerprints) receive similar operators, and this mapping is smooth and well-behaved across domains.

Dynamic Fusion Attention

For each node v, Q, K, V projections are all fingerprint-conditioned:

  • Q derives from semantic embeddings (query = what you’re looking for)
  • K, V derive from topology (keys/values = what’s structurally available)
  • Each projection uses A·Z·B factorization with its own Z matrix

A structural affinity score computes pairwise similarity from topology embeddings, and for each node we retain only the top-k neighbors (k=32) — forming a sparse, structurally-relevant neighborhood.

The final attention fuses semantic similarity with structural affinity:

\begin{equation} \alpha_{uv} = \text{Softmax}_{v \in \mathcal{N}_u^{(k)}} \left( \frac{\mathbf{Q}_u^\top \mathbf{K}_v}{\sqrt{d_h}} + \lambda \cdot s_{uv} \cdot \tau \right) \end{equation}

Dynamic FFN

The FFN also applies two structure-conditioned transformations (Z₁ for expansion, Z₂ for contraction), ensuring that operator transfer influences both aggregation and downstream feature transformation:

\begin{equation} m_v = \text{GELU}\left( \left( h^{attn}_v \cdot A_1 \right) \cdot Z_{1,v} \cdot B_1 + b_1 \right) \end{equation} \begin{equation} h^{ffn}_v = \left( m_v \cdot A_2 \right) \cdot Z_{2,v} \cdot B_2 + b_2 \end{equation}

Pre-training & Zero-shot Inference

  • Objective: binary cross-entropy on edge prediction (positive + negative samples)
  • Source domain: pre-train on one graph (e.g., Arxiv or FB15K-237)
  • Zero-shot target: compute fingerprints from observed target graph → hypernetwork generates operators → single forward pass → no fine-tuning, no target labels

Validation

Experimental Setup

Item Details
Datasets 15 public benchmarks: 7 citation networks, 6 e-commerce graphs, 1 knowledge graph (FB15K-237)
Source domains Arxiv (169K nodes, 1.16M edges) and FB15K-237 (14.5K nodes, 272K edges)
Target domains 13 zero-shot target graphs (Cora, Citeseer, Pubmed, Art, Business, Geography, Sociology, Book Child, Book History, Electronic Photo, Electronic Computer, Sportsfit, Products)
Model config 4-layer dynamic Transformer, 8 heads, hidden dim 384, FFN dim 512, low-rank r=8, top-k=32
Hypernetwork 2-layer shared MLP + separate spectral-normalized heads for ZQ, ZK, ZV, Z1, Z2
Optimizer AdamW, lr=1e-3 (base), 1e-4 (hypernetwork)
Params ~3.9M total
Metric MRR (Mean Reciprocal Rank)
Baselines One4all, AnyGraph, ZeroG (graph foundation models)
Protocol 5 random repetitions, no target-domain supervision

Key Results

Opera outperforms all three GFM baselines across nearly all 13 target datasets, improving average MRR by 16.7 percentage points over the strongest baseline (ZeroG).

Notable observations:

  • Opera (Arxiv source) slightly outperforms Opera (FB15K-237 source) overall despite Arxiv being 10× larger — suggesting data scale and structural diversity are complementary
  • FB15K-237-trained model performs better on Products (e-commerce) — richer structural patterns compensate for smaller scale
  • Performance peaks on medium-scale graphs (20K–200K nodes); small graphs have unstable structural statistics, large graphs exceed the learned structural complexity capacity

Ablation Study

Variant Impact on MRR Interpretation
Full Opera baseline —
w/o Hypernet (static params) −58% to −80% (−74% avg) Dynamic parameter generation is the key to zero-shot transfer; static Transformer struggles on structurally divergent graphs
w/o Spectral Norm −40% to −45% on divergent domains Spectral normalization is a low-overhead safety net — minor effect on similar domains, critical for robust cross-domain transfer

The ablation confirms both core design choices:

  1. Structure-conditioned operator generation is essential — it’s not the Transformer architecture itself, but the dynamic parameters that enable transfer
  2. Spectral normalization is essential for stability — without it, the hypernetwork overfits to source-domain structural patterns

Key Takeaways

Dimension Finding
Main contribution Frames zero-shot cross-domain link prediction as structure-conditioned operator transfer — a new perspective distinct from representation-centric approaches
Core mechanism Lightweight spectrally-normalized hypernetwork maps 12-d structural fingerprints to low-rank Transformer modulation matrices
Why it works Structural roles are domain-invariant; Lipschitz-constrained mapping ensures similar roles → similar operators across graphs
Performance +16.7 pp MRR over strongest GFM baseline across 13 zero-shot targets
Ablation insight Dynamic operators explain ~74% of the gain; spectral normalization is essential for divergent domains
Limitations Handcrafted fingerprints (not learned), assumes observable target topology, relation-agnostic (binary edge prediction only)
Future directions Learnable role encoders (GraphWave, Role2Vec), partially observed / evolving graphs, relation-dependent operators for full KG completion

One-sentence summary: By learning to generate node-specific Transformer operators from structural fingerprints rather than memorizing fixed weights, Opera enables zero-shot cross-domain link prediction through structure-conditioned operator transfer.

BibTeX

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@inproceedings{cheng2026opera,
  title     = {Opera: Structure-Conditioned Operators for Cross-Domain Link Prediction},
  author    = {Cheng, Liang and Taherkordi, Amir and Giese, Martin},
  booktitle = {IJCKG},
  year      = {2026}
}