Source-linked AI summary
REGAL: Representation Learning-based Graph Alignment
Mark Heimann, Haoming Shen, Tara Safavi, Danai Koutra
TL;DR
Network alignment seeks corresponding nodes across multiple graphs, but many existing methods struggle with computational difficulty and representations that are not directly comparable across networks. REGAL learns and matches cross-network node representations using xNetMF, which captures structural and optional attribute similarity through efficient implicit matrix factorization. The authors report up to 30× faster representation learning and up to 30% greater accuracy than baselines, while identifying limitations for proximity-based embeddings on disconnected graphs and noise-sensitive settings.
Problem
Network alignment requires efficiently inferring node correspondences across graphs from directly comparable representations, a challenge for existing optimization and feature-design approaches.
Method
REGAL jointly learns and matches node embeddings across graphs, using xNetMF to factorize structural and optional attribute similarities without explicitly constructing the full similarity matrix.
Results
30× faster representation learning and up to 30% greater accuracy than baselines are reported for REGAL and xNetMF.
Takeaways & Limitations
Unsupervised representation learning-based alignment can compare nodes across multiple networks using structural and attribute identity without requiring prior alignment information.
Takeaways & Limitations
Proximity-based embeddings perform poorly for corresponding nodes in disconnected graphs, while REGAL's accuracy can trade off against speed under higher structural noise.
Abstract
from arXiv · showhide
Problems involving multiple networks are prevalent in many scientific and other domains. In particular, network alignment, or the task of identifying corresponding nodes in different networks, has applications across the social and natural sciences. Motivated by recent advancements in node representation learning for single-graph tasks, we propose REGAL (REpresentation learning-based Graph ALignment), a framework that leverages the power of automatically-learned node representations to match nodes across different graphs. Within REGAL we devise xNetMF, an elegant and principled node embedding formulation that uniquely generalizes to multi-network problems. Our results demonstrate the utility and promise of unsupervised representation learning-based network alignment in terms of both speed and accuracy. REGAL runs up to 30x faster in the representation learning stage than comparable methods, outperforms existing network alignment methods by 20 to 30% accuracy on average, and scales to networks with millions of nodes each.
1 INTRODUCTION
Network alignment identifies corresponding nodes across multiple graphs, but existing approaches face difficult optimization and cross-network comparability challenges. REGAL addresses these challenges by matching jointly learned representations, with xNetMF designed for structural and attribute-based comparison across networks.
- Motivation: Network alignment finds corresponding nodes across different networks and supports applications in social, natural, and other sciences.Examples include matching users, analyzing chemical compounds, and studying protein-protein interactions.
- Motivation: Existing methods face a computationally hard optimization problem and difficulty designing node features directly comparable across networks.Recent representation learning advances motivate learning latent node features automatically instead.
- xNetMF: xNetMF factorizes a similarity matrix incorporating structural similarity and available attribute agreement across nodes in disjoint graphs.It uses an extended Nyström low-rank approximation to avoid explicitly constructing the full similarity matrix.
- REGAL: REGAL learns node alignments by jointly embedding multiple graphs and matching the most similar embeddings without performing all pairwise comparisons.The framework is intended to efficiently infer node correspondences from learned representations.
- Results: 30× faster representation learning and 20-30% higher accuracy summarize the reported advantages of xNetMF and REGAL over comparison methods.The speed result concerns existing network embedding techniques, while the accuracy result concerns traditional network alignment methods.
2 RELATED WORK
Prior network alignment and node embedding methods often rely on specialized formulations, proximity-based similarity, or procedures that do not transfer consistently across networks. REGAL instead combines structural identity, cross-network comparability, matrix factorization, and scalable low-rank approximation.
- Network Alignment: Traditional network alignment is commonly formulated as minimizing adjacency-matrix mismatch under a permutation or relaxed matching matrix.Proposed solutions include genetic, spectral, clustering, probabilistic, and distributed belief-propagation methods.
- Network Alignment: Prior alignment methods often require tailoring to graph formats, attributes, or known seed alignments, whereas REGAL supports attributed and unattributed graphs without prior alignment information.The supplied passage contrasts these requirements with REGAL's unsupervised formulation.
- Node Representation Learning: Most node embedding methods use proximity or neighborhood overlap, which can produce inconsistent embeddings across networks.These methods include shallow and deep architectures, random-walk approaches, and methods using first- and second-order connections.
- Node Representation Learning: Struc2vec preserves structural similarity regardless of node proximity and produces embeddings reported as more visually comparable for structurally similar nodes.Earlier structural role-discovery methods mainly relied on hand-engineered features.
- Matrix Approximation: REGAL's embedding module is distinguished by casting structural-identity node embedding with SGNS in a matrix-factorization framework.The authors state this is the first such formulation for structural identity.
- Matrix Approximation: The paper applies the Nyström method to node embedding, extending a low-rank approximation previously used for large dense similarity matrices in kernel machines.The supplied related-work passage identifies this as a new application context for Nyström approximation.
3 REGAL: REPRESENTATION LEARNING-BASED GRAPH ALIGNMENT
REGAL aligns nodes across graphs by learning comparable representations from structural identity and optional attributes. xNetMF uses implicit Nyström matrix factorization to avoid quadratic similarity construction, enabling efficient embedding and alignment.
- REGAL framework: REGAL processes graphs through node identity extraction, similarity-based representation learning, and fast embedding alignment.The first two steps comprise xNetMF, while the final step greedily matches embeddings across graphs.
- 3.1 Step 1: Node Identity Extraction: xNetMF defines node identity using structural roles and attribute-based identity rather than cross-graph node proximity.Structural identity incorporates neighborhood degree information across multiple hop distances; attributes are included when available.
- 3.1 Step 1: Node Identity Extraction: Logarithmic degree binning shortens neighborhood degree vectors to ⌈log2 D⌉ dimensions and makes them more robust to small degree noise.Higher-order neighborhoods are aggregated with a discount factor δ across up to K hops.
- 3.1 Step 1: Node Identity Extraction: The combined similarity function weights structural and attribute identity, aggregates discounted neighborhood degree vectors, and ignores attributes when unavailable.The structural aggregation uses δ^(k−1) across K hop distances.
- 3.2 Step 2: Efficient Similarity-based Representation: xNetMF implicitly factorizes a low-rank similarity approximation using n×p node-to-landmark similarities and an SVD of the small landmark matrix.This avoids constructing the n×n matrix and provides embeddings with runtime and storage savings.
- 3.2 Step 2: Efficient Similarity-based Representation: Explicit factorization of the full similarity matrix has quadratic computation and storage costs, limiting scalability to multiple large networks.Heuristic sparse similarities can also miss important comparisons and be fragile under noise.
- 3.4 Complexity Analysis: O(n′ log n′) average-case alignment finds top matches through a k-d tree, while the total method can achieve sub-quadratic time and space.Small logarithmic choices of K, p, and b support this complexity in practice.
4 EXPERIMENTS
The experiments evaluate REGAL against alignment baselines and embedding variants under structural and attribute noise, runtime, scalability, and parameter settings. REGAL’s structural embeddings are consistently accurate, while xNetMF provides substantial speed and sub-quadratic scaling, with trade-offs under some noise and parameter regimes.
- 4.1 Accuracy: REGAL and REGAL-struc2vec are significantly more accurate than non-representation-learning baselines across noise levels and datasets.
- 4.1 Runtime: 6-8× faster than REGAL-node2vec and 22-31× faster than REGAL-struc2vec, REGAL scales better with xNetMF.The speed advantage reflects avoiding random-walk context sampling, dynamic time warping, and SGNS training costs.
- 4.1 Baseline comparison: REGAL is competitive in runtime and significantly more accurate than non-representation-learning baselines, while remaining faster than FINAL and Klau.NetAlign can be faster on small datasets, but its message passing becomes expensive on larger datasets; IsoRank is fastest but among the least accurate.
- 4.1.2 Effects of attribute-based noise: REGAL mostly outperforms FINAL under attribute noise or limited attribute information, while FINAL can be slightly more accurate with abundant, clean attributes.FINAL’s runtime also increases significantly as it uses more attribute information.
- 4.2 Q2: Scalability: REGAL’s total runtime is sub-quadratic, supporting scalability to very large networks; similarity computation and alignment become costlier as embedding dimensionality grows.The dominant representation step computes O(n log n) similarities to landmarks, while larger p increasingly affects lookup times.
- 4.3 Q3: Sensitivity Analysis: Higher-order structure helps through K = 2, δ between 0.01–0.1 performs best, and increasing landmarks generally improves accuracy at greater runtime cost.The recommended settings γs = γa = 1 are stable, while larger landmark counts increase computation and embedding dimensionality.
5 CONCLUSION
REGAL aligns nodes across multiple networks through learned embeddings, with xNetMF producing representations comparable across networks. The framework combines improved accuracy with substantially faster representation learning.
- REGAL aligns nodes via learned embeddings that are comparable across multiple networks.xNetMF is introduced within REGAL to learn these representations.
- REGAL was presented as the first unsupervised representation learning-based network alignment method, to the authors’ knowledge.
- 30% greater accuracy and 30× faster representation learning were reported relative to baselines.The speed figure specifically concerns the representation learning stage.
- xNetMF captures structural and attribute-based node similarities for cross-network analysis.The formulation is designed to preserve relevant similarities across disjoint networks.
A CONNECTIONS: xNetMF AND SGNS
The appendix connects xNetMF to SGNS by showing that similarity-graph context sampling leads to an equivalent matrix-factorization view under stated assumptions. This connection clarifies why xNetMF can replace extensive random-walk sampling with an efficient factorization procedure.
- SGNS and similarity-graph context: struc2vec samples context through random walks on a similarity graph, producing a node-context co-occurrence matrix.The context distribution is governed by structural similarity between nodes.
- SGNS and similarity-graph context: SGNS optimizes embeddings for sampled node-context pairs and, under assumptions, is equivalent to factorizing their co-occurrence matrix.The equivalence is described as MF(D, Y⊤C).
- Connection to xNetMF: xNetMF connects similarity-graph SGNS optimization with graph factorization under the stated assumptions.
- Connection to xNetMF: With infinitely many length-1 random walks, the co-occurrence matrix converges to a scaled similarity matrix, yielding MF(S, Y⊤C).The convergence follows from the Law of Large Numbers, and matrix-factorization objectives are invariant to scaling.
- Implication for xNetMF: xNetMF corresponds to maximal context sampling and minimal walk length, while longer walks sample structurally more distant context.Increasing the number of walks reduces variance; increasing walk length changes the sampled context.