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arxiv: 2604.08492 · v1 · submitted 2026-04-09 · 💻 cs.LG

The Impact of Dimensionality on the Stability of Node Embeddings

Tobias Schumacher , Simon Reichelt , Markus Strohmaier This is my paper

Pith reviewed 2026-05-10 17:53 UTC · model grok-4.3

classification 💻 cs.LG
keywords node embeddingsdimensionalitystabilitygraph representation learningnode2vecGraphSAGEdownstream performance
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The pith

Varying the dimensionality of node embeddings changes their stability in method-specific ways, and the dimension with highest stability does not always give the best downstream performance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Previous work showed that node embeddings can differ across training seeds even with fixed parameters. This paper tests whether and how the choice of embedding dimension influences that instability. It runs five standard methods across multiple datasets while tracking both how much the embedding vectors themselves shift and how much the outputs on tasks like node classification shift. Some methods gain stability as dimensions increase, but others show flat or inconsistent patterns, and the dimension that maximizes stability rarely matches the one that maximizes task accuracy. The results point to a practical need to weigh dimension choices against both reliability and accuracy rather than treating dimension as a simple performance knob.

Core claim

Embedding stability varies significantly with dimensionality, but different methods follow different patterns: node2vec and ASNE tend to become more stable at higher dimensions while DGI, GraphSAGE, and VERSE do not exhibit the same trend. Across all methods, the dimensionality that produces maximum stability does not align with the dimensionality that produces optimal performance on downstream tasks.

What carries the argument

Representational stability (similarity of embedding vectors across random seeds) and functional stability (consistency of downstream task outputs across seeds) measured as functions of embedding dimension for the five algorithms ASNE, DGI, GraphSAGE, node2vec, and VERSE.

If this is right

  • Higher embedding dimensions can increase stability for some methods such as node2vec and ASNE.
  • Stability and task performance must be optimized separately when selecting embedding dimension.
  • Computational cost of higher dimensions is not always offset by gains in either stability or accuracy.
  • Different embedding algorithms respond differently to changes in dimension, so method-specific tuning is needed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practitioners may need to run stability checks across a range of dimensions rather than relying on performance alone to pick a fixed size.
  • The observed trade-offs could motivate new training procedures that explicitly regularize for stability at moderate dimensions.
  • The method-dependent patterns suggest that algorithm choice itself can be informed by stability requirements of the target application.

Load-bearing premise

That the five chosen embedding methods, the stability metrics, and the selected datasets are representative enough for the observed patterns to hold beyond this experimental setup.

What would settle it

A replication that uses the same five methods but additional datasets or different random seeds and finds that every method becomes steadily more stable with higher dimensionality and that stability and task performance peak at the same dimension would contradict the reported variation in trends and the misalignment result.

Figures

Figures reproduced from arXiv: 2604.08492 by Markus Strohmaier, Simon Reichelt, Tobias Schumacher.

Figure 1
Figure 1. Figure 1: Impact of Dimension on Downstream Performance. We present the average accuracy of each embedding method as aggregated across 30 embeddings, using a logistic regressor as downstream classifier. We observe that in most cases, performance improves with increasing dimension, up to a plateau. Yet, specifically for VERSE and GraphSAGE, very high dimensions also tend to yield decreasing performance. Local Stabili… view at source ↗
Figure 2
Figure 2. Figure 2: Impact of dimension on representational stability in terms of k-NN Jaccard Similarity. We depict average similarity scores aggregated across all pairs of the 30 embeddings we computed; 435 pairs in total. Higher values indicate higher stability. Stars indicate the dimension at which the best downstream accuracy was achieved (cf [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of dimension on representational stability in terms of aligned cosine similarity. We depict average similarity scores aggregated across all pairs of the 30 embeddings we computed; 435 pairs in total. Higher values indicate higher stability. Stars indicate the dimension at which the best downstream accuracy was achieved (cf [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Impact of dimension on functional stability in terms of stable core. The stable core is derived from the total number of instances at the predictions from all 30 embeddings agree. High values indicate highly consistent predictions. Stars indicate the dimension at which the best downstream accuracy was achieved (cf [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Impact of dimension on functional stability in terms of Jensen-Shannon divergence. We depict average similarity scores aggregated across all pairs of the 30 embeddings we computed; 435 pairs in total. Lower values indicate higher similarity. We observe that for node2vec and ASNE, predictions get increasingly similar with higher dimensions. DGI follows a similar trend, with a notable exception on the Wikipe… view at source ↗
Figure 10
Figure 10. Figure 10: 16 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 6
Figure 6. Figure 6: Impact of dimension on downstream performance, using a MLP as downstream classifier. Results are overall very similar to those of logistic regression, with slightly higher scores at the optimum. Exceptions are present for higher dimensions of DGI embedings—here, MLP performance collapses way below logistic regression performance, indicating that our hyperparameter grid did not include the optimal trainng r… view at source ↗
Figure 7
Figure 7. Figure 7: Impact of dimension on representational stability in terms of second-order cosine similarity. Stars indicate the dimension at which the best downstream accuracy was achieved (cf [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Impact of dimension on representational stability in terms of distance correlation. Stars indicate the dimension at which the best downstream accuracy was achieved (cf [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Impact of dimension on functional stability in terms of disagreement. Stars indicate the dimension at which the best downstream accuracy was achieved (cf [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Impact of dimension on functional stability in terms of min-max normalized disagreement. Stars indicate the dimension at which the best downstream accuracy was achieved (cf [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
read the original abstract

Previous work has established that neural network-based node embeddings return different outcomes when trained with identical parameters on the same dataset, just from using different training seeds. Yet, it has not been thoroughly analyzed how key hyperparameters such as embedding dimension could impact this instability. In this work, we investigate how varying the dimensionality of node embeddings influences both their stability and downstream performance. We systematically evaluate five widely used methods -- ASNE, DGI, GraphSAGE, node2vec, and VERSE -- across multiple datasets and embedding dimensions. We assess stability from both a representational perspective and a functional perspective, alongside performance evaluation. Our results show that embedding stability varies significantly with dimensionality, but we observe different patterns across the methods we consider: while some approaches, such as node2vec and ASNE, tend to become more stable with higher dimensionality, other methods do not exhibit the same trend. Moreover, we find that maximum stability does not necessarily align with optimal task performance. These findings highlight the importance of carefully selecting embedding dimension, and provide new insights into the trade-offs between stability, performance, and computational effectiveness in graph representation learning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper conducts an empirical investigation into the effect of embedding dimensionality on the stability of node embeddings generated by five methods (ASNE, DGI, GraphSAGE, node2vec, VERSE) across multiple datasets. Stability is measured from both representational (e.g., similarity across seeds) and functional (e.g., downstream task consistency) perspectives, with comparisons to task performance. Key findings are that stability trends with increasing dimension are method-dependent (improving for node2vec and ASNE but not uniformly for others) and that the dimensionality yielding maximum stability does not necessarily coincide with optimal downstream performance.

Significance. If the reported patterns hold under the described experimental controls, the work supplies actionable guidance for hyperparameter tuning in graph representation learning by quantifying stability-performance trade-offs as a function of dimension. It extends prior observations of seed-induced variability with a systematic dimension sweep, which is a practically relevant axis, and underscores that stability and accuracy may need independent optimization.

minor comments (3)
  1. The abstract refers to 'representational perspective and a functional perspective' for stability; the methods section should provide explicit mathematical definitions or pseudocode for these two metrics (including any distance functions or correlation measures used) to allow exact reproduction.
  2. Results should include per-method, per-dataset tables or plots with error bars or statistical significance tests (e.g., paired t-tests across seeds) rather than qualitative trend descriptions alone, so readers can assess the magnitude and reliability of the reported differences.
  3. The manuscript should state the exact range of dimensions tested, the number of random seeds per configuration, and the data-split protocol (train/validation/test) to clarify whether any post-hoc selection of dimensions or seeds occurred.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We are grateful to the referee for the positive and insightful summary of our manuscript. The recommendation for minor revision is welcome, and we appreciate the acknowledgment of the practical relevance of our findings on stability-performance trade-offs in node embeddings. Given that no major comments were specified, we have no point-by-point rebuttals to provide. We stand ready to make any necessary revisions to the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a purely empirical study that evaluates five node embedding methods (ASNE, DGI, GraphSAGE, node2vec, VERSE) across datasets and dimensions, measuring representational and functional stability plus downstream task performance. No derivation chain, first-principles prediction, or mathematical result is claimed; all findings are direct experimental observations scoped to the tested setup. No self-definitional quantities, fitted inputs renamed as predictions, or load-bearing self-citations appear. The work is self-contained against external benchmarks and does not reduce any claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an empirical evaluation study. No new mathematical axioms, free parameters, or invented entities are introduced; the work relies on standard definitions of node embeddings and stability already present in the cited literature.

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