arxiv: 2605.06466 · v1 · submitted 2026-05-07 · 💻 cs.LG

Recognition: unknown

Diversity Curves for Graph Representation Learning

Katharina Limbeck , Nadja H\"ausermann , Martin Carrasco , Guy Wolf , Bastian Rieck

Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:38 UTC · model grok-4.3

classification 💻 cs.LG

keywords diversity curvesgraph representation learningedge contractiongraph coarseningisometry invariantsstructural diversitygraph embeddingsunsupervised graph analysis

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The pith

Diversity curves track the spread of graphs under edge contraction coarsening to produce size-aware and more expressive graph representations.

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

The paper seeks to solve the problem of comparing graphs that have different numbers of nodes even when they come from similar distributions, especially for unsupervised tasks that need clear and scalable representations. It does this by following how a graph's structure changes as edges are contracted to create coarser versions, recording a quantity called the spread of graphs at each step to form diversity curves. These curves are built to be directly comparable across graphs of varying sizes and to carry more information about geometry and metric properties than fixed structural descriptors can provide. A reader would care because the approach promises a practical way to cluster, visualise, and differentiate graphs from simulations, single cells, molecules, and shapes without requiring size normalisation or supervision.

Core claim

By defining the spread of graphs as an isometry invariant that encodes metric diversity and geometry, then applying edge contraction coarsening to compute diversity curves across multiple levels, the resulting embeddings improve expressivity and outperform structural descriptors alone when used for graph-level tasks.

What carries the argument

Diversity curves generated by tracking the spread of graphs (a novel isometry invariant) during successive edge contractions, which coarsen the graph while preserving key structural features.

If this is right

Diversity curves cluster and visualise simulated graphs of different sizes more effectively than static descriptors.
The curves distinguish geometric properties in single-cell graphs drawn from biological data.
They enable direct structural comparisons between molecular graph datasets of varying cardinalities.
The same construction characterises and separates geometric shapes represented as graphs.

Where Pith is reading between the lines

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

The size-invariance of the curves could support transfer of learned representations between graph collections that differ only in scale.
Because the method is unsupervised and interpretable, it might serve as a preprocessing step before applying supervised graph neural networks to improve their robustness to node-count variation.
The geometric focus of the spread invariant suggests possible connections to other metric-based graph tools such as persistent homology summaries.

Load-bearing premise

The spread of graphs must act as a reliable isometry invariant that captures metric diversity and geometry, and edge contraction coarsening must retain the structural features needed for the claimed gain in expressivity.

What would settle it

A controlled test in which two graphs with provably different geometries or metric properties produce nearly identical diversity curves across the same coarsening sequence.

Figures

Figures reproduced from arXiv: 2605.06466 by Bastian Rieck, Guy Wolf, Katharina Limbeck, Martin Carrasco, Nadja H\"ausermann.

**Figure 1.** Figure 1: Diversity curves for three example graphs. Our framework first view at source ↗

**Figure 2.** Figure 2: Spearman correlation between the degree of perturbation and the change in diversity curve per dataset. As a first correctness check, we investigate how diversity curves behave under perturbations of the input graphs [63]. We fix a degree of perturbation p ∈ [0, 1] and apply edge perturbations by consecutively adding, removing, swapping, or rewiring edges, resulting in edits to all graphs in the SBM, PLANA… view at source ↗

**Figure 3.** Figure 3: PCA of diversity curves for ER, RP, SBM, and RG models. Second, we test the ability of our method to characterise and distinguish between graphs generated from the same underlying random graph model using three different parameter choices. For each graph distribution, parameter setting as described in Appendix C.2, and graph sizes from 10 to 29 nodes we generate three graphs. As before, we use the Euclidea… view at source ↗

**Figure 4.** Figure 4: against our baselines, where we achieve comparable silhouette scores to the best kernel baselines for the ER and RP distributions and obtain the highest scores for the RG and SBM graphs. This confirms that diversity curves outperform alternative unsupervised methods at distinguishing between parameters used in a random graph model, while being robust to size differences. Additional clustering quality metri… view at source ↗

**Figure 5.** Figure 5: PCA visualisation of the diversity curves for all 169 single-cell graphs (left). Examples of two trajectory-like graphs and two cluster-like single-cell graphs (right). 4.4 Diversity curves characterise molecular graph datasets ENZYMES PROTEINS enzyme graphs view at source ↗

**Figure 6.** Figure 6: Two sample permutation testing results (p-values) for testing the equality in mean diversity curves per class in the PROTEINS and ENZYMES datasets using the L 2 -norm between curves. Characterising the coarse geometry of graphs, diversity curves can be used to compare and find structural similarities between graph datasets. Specifically, we explore common trends across classes of molecular graphs from the … view at source ↗

**Figure 7.** Figure 7: PCA visualisation of view at source ↗

read the original abstract

Graph-level representations are crucial tools for characterising structural differences between graphs. However, comparing graphs with different cardinalities, even when sampled from the same underlying distribution, remains challenging. Unsupervised tasks in particular require interpretable, scalable, and reliable size-aware graph representations. Our work addresses these issues by tracking the structural diversity of a graph across coarsening levels. The resulting graph embeddings, which we denote diversity curves, are interpretable by construction, efficient, and directly comparable across coarsening hierarchies. Specifically, we track the spread of graphs, a novel isometry invariant that is inherently well-suited for encoding the metric diversity and geometry of graphs. We utilise edge contraction coarsening and prove that this improves expressivity, thus leading to more powerful graph-level representations than structural descriptors alone. Demonstrating their utility over a range of baseline methods in practice, we use diversity curves to (i) cluster and visualise simulated graphs across varying sizes, (ii) distinguish the geometry of single-cell graphs, (iii) compare the structure of molecular graph datasets, and (iv) characterise geometric shapes.

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.

Desk Editor's Note private letter to a colleague

Diversity curves give a practical size-aware graph embedding by tracking spread under coarsening, but the expressivity proof is not visible enough to judge its strength.

read the letter

The main thing here is the introduction of diversity curves, which track a novel spread isometry invariant across edge contraction coarsening to create size-aware graph embeddings. This directly tackles comparing graphs of different sizes in unsupervised tasks. They do well by making the embeddings interpretable and efficient, and by testing on relevant domains like single-cell graphs and molecular datasets. The results show it can cluster simulated graphs by size and distinguish geometries better than some structural descriptors. The applications to cheminformatics and single-cell biology are a good fit for the method. The soft spots are around the central claim. They say they prove that edge contraction improves expressivity, but without seeing the actual derivation or theorem statement, it's hard to assess if the proof is rigorous or if it really leads to more powerful representations than existing methods. The experiments seem targeted, but more controls on whether the coarsening preserves the necessary features would help. This paper is for researchers in graph ML applied to biology and chemistry. Readers looking for new descriptors that handle varying cardinalities would get value from it. It has enough going for it to deserve a serious referee, mainly to verify the theoretical improvement and see if the results hold up under closer scrutiny. Recommendation: I would send this to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces diversity curves as graph-level embeddings that track a novel isometry invariant called the spread of graphs across successive levels of edge-contraction coarsening. The central claim is that this coarsening provably increases expressivity relative to static structural descriptors, yielding size-aware, interpretable representations. The authors demonstrate the approach on simulated graphs of varying cardinalities, single-cell graphs, molecular datasets, and geometric shapes.

Significance. If the claimed proof of expressivity gain is rigorous and the spread is a genuine isometry invariant, the method would supply a practical, theoretically grounded tool for unsupervised tasks that require comparing graphs of unequal size. The interpretability-by-construction and direct comparability across hierarchies are genuine strengths; the attempt at a formal proof distinguishes the work from purely empirical coarsening studies.

major comments (2)

[Abstract / Theory] Abstract and theory section: the manuscript asserts that edge contraction coarsening 'improves expressivity' and supplies a proof, yet the abstract contains neither a theorem statement nor any derivation or assumptions. Because this expressivity claim is load-bearing for the superiority over structural descriptors, the full manuscript must contain an explicit theorem (with hypotheses on the spread and the contraction operator) together with its proof.
[Definition of spread] Definition of spread (§2 or §3, depending on numbering): the spread is introduced as a new isometry invariant without reduction to previously established quantities. To underwrite the claim that it reliably encodes metric diversity and geometry, the manuscript must supply (i) a precise mathematical definition and (ii) a short proof that the quantity is unchanged under isometries of the underlying metric space.

minor comments (2)

[Experiments] Experiments section: the four demonstration tasks are listed, but the manuscript should report quantitative controls (e.g., how size variation is isolated from structural variation, choice of baselines, and statistical significance) so that the claimed superiority can be assessed.
[Notation] Notation: ensure that the symbols for diversity curves, coarsening levels, and the spread operator are introduced once and used consistently; a small notation table would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We have revised the manuscript to make the expressivity result and the spread invariant fully explicit with theorem statements, definitions, and proofs.

read point-by-point responses

Referee: [Abstract / Theory] Abstract and theory section: the manuscript asserts that edge contraction coarsening 'improves expressivity' and supplies a proof, yet the abstract contains neither a theorem statement nor any derivation or assumptions. Because this expressivity claim is load-bearing for the superiority over structural descriptors, the full manuscript must contain an explicit theorem (with hypotheses on the spread and the contraction operator) together with its proof.

Authors: We agree that the expressivity claim benefits from an explicit theorem. In the revised manuscript we have extracted the existing argument as Theorem 3.1 (Section 3), stating that, under the hypotheses that spread is an isometry invariant and that each edge-contraction step preserves the metric up to uniform scaling, the resulting diversity curve is strictly more expressive than any static structural descriptor for distinguishing non-isometric graphs of unequal size. The complete proof, including all hypotheses on the contraction operator, now appears immediately after the theorem statement. The abstract has been updated to reference Theorem 3.1 and its key assumptions. revision: yes
Referee: [Definition of spread] Definition of spread (§2 or §3, depending on numbering): the spread is introduced as a new isometry invariant without reduction to previously established quantities. To underwrite the claim that it reliably encodes metric diversity and geometry, the manuscript must supply (i) a precise mathematical definition and (ii) a short proof that the quantity is unchanged under isometries of the underlying metric space.

Authors: We have revised Section 2 to contain a self-contained Definition 2.1 of spread (the normalized standard deviation of the multiset of all pairwise distances in the graph metric). We have also added Lemma 2.2, whose short proof shows that spread is unchanged under isometries: any isometry induces a distance-preserving bijection on vertices, so the multiset of distances (and hence its standard deviation) remains identical. This formalizes the invariance claim that was previously implicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract introduces 'diversity curves' and 'spread of graphs' explicitly as novel definitions and constructions rather than derivations from prior fitted quantities or self-referential parameters. The claimed proof that edge contraction coarsening improves expressivity is presented as an independent result, with no equations or steps shown that reduce by construction to the inputs (e.g., no fitted parameter renamed as a prediction, no uniqueness theorem imported from self-citation, and no ansatz smuggled via prior work). The central claims rest on new isometry-invariant definitions and empirical demonstrations across tasks, which are self-contained against external benchmarks and do not exhibit any of the enumerated circularity patterns. No load-bearing self-citations or reductions are visible, consistent with the reader's assessment of score 2.0 as a minor non-circularity flag at most.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim rests on the unproven assertion that edge contraction improves expressivity and on the utility of the newly introduced spread measure.

axioms (1)

domain assumption Edge contraction coarsening improves expressivity of graph representations beyond structural descriptors
Stated as a proven fact in the abstract without supporting derivation visible here.

invented entities (1)

spread of graphs no independent evidence
purpose: isometry invariant encoding metric diversity and geometry of graphs
Presented as a novel quantity introduced by the authors.

pith-pipeline@v0.9.0 · 5493 in / 1341 out tokens · 32707 ms · 2026-05-08T12:38:21.566352+00:00 · methodology

discussion (0)

Reference graph

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