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arxiv: 2605.31373 · v1 · pith:VIBJ6V7Tnew · submitted 2026-05-29 · 💻 cs.LG · cs.AI

Scaling Higher-Order Graph Learning with Maximal Clique Complexes

Antoine Vialle , Aref Einizade , Fragkiskos D. Malliaros , Jhony H. Giraldo This is my paper

Pith reviewed 2026-06-28 23:31 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords graph neural networkshigher-order graph learningcell complexesWeisfeiler-Leman testmaximal cliquesscalable topological learningrandom walk sampling
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The pith

Maximal clique complexes let cellular graph networks scale linearly while keeping expressivity.

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

The paper shows that full cell complexes can be replaced by maximal clique complexes to make higher-order graph models practical on larger graphs. Simplified and factored cellular Weisfeiler-Leman tests keep the same distinguishing power as the original test but run faster. A biased random walk called CliqueWalk samples the needed maximal cliques without listing every one, producing linear scaling in time and memory. The resulting networks match the accuracy of slower full-complex versions on standard tasks.

Core claim

Maximal clique complexes, sampled via CliqueWalk, enable scalable cellular Weisfeiler-Leman networks that reduce time and memory complexity while preserving the expressivity of the CWL test and strong empirical performance.

What carries the argument

The maximal clique complex together with the CliqueWalk biased random walk, which replaces explicit clique enumeration to factor and simplify the cellular Weisfeiler-Leman test.

If this is right

  • Higher-order models become usable on graphs with thousands of nodes instead of hundreds.
  • Time and memory costs grow linearly rather than combinatorially with clique size.
  • The same simplified and factored tests can be applied to other cell-complex architectures.
  • CliqueWalk can be swapped into existing CWN pipelines without changing the downstream learning objective.

Where Pith is reading between the lines

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

  • The same sampling idea could extend to other higher-order structures such as simplicial complexes when maximal cliques are replaced by maximal simplices.
  • Domains with natural clique structure, such as collaboration or molecular graphs, may see the largest gains.
  • One could test whether the linear scaling holds when the walk bias parameters are tuned per dataset rather than fixed.

Load-bearing premise

Sampling maximal cliques with the biased random walk preserves the expressivity of the original CWL test and the empirical performance of cellular Weisfeiler-Leman networks.

What would settle it

A direct comparison on a family of graphs where important higher-order relations lie outside the maximal cliques, showing whether the sampled version loses accuracy or distinguishing power relative to full enumeration.

Figures

Figures reproduced from arXiv: 2605.31373 by Antoine Vialle, Aref Einizade, Fragkiskos D. Malliaros, Jhony H. Giraldo.

Figure 1
Figure 1. Figure 1: Maximal clique complex. To address the first question, we introduce the sim￾plified and factored cellular Weisfeiler–Leman tests (sCWL and fCWL), together with their correspond￾ing neural architectures (sCWNs and fCWNs). We show that these variants preserve the full expressive power of the original CWL test of [3], while exhibit￾ing improved scalability properties. These simplified architectures enable the… view at source ↗
Figure 2
Figure 2. Figure 2: shows an example of the aggregation functions in Definition 7. This simplified variant reduces computational and memory requirements while retaining the expressive power of the full CWL update. Messages are propagated only along boundary and co-boundary relations, making sCWN efficient for large complexes (see Proposition 2). We also introduce a cell model that has a complexity between sCWN and CWN, but ha… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of a CliqueWalk starting at node [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Accuracy gain over the best GNN base￾line as a function of mean clique size. Larger cliques tend to favor clique-based models. Node classification [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sensitivity analysis of CliqueWalk. (a) Accuracy versus sampled walks. (b) Runtime [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the WL and maximal clique CWL test. At each iteration, every node updates [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of Percentage of Unique Graph Hashes on strongly regular datasets: (a) [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of CWN, sCWN, and fCWN with increasing clique size: (a) inference time [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Ablation studies on large clique sampling versus full clique enumeration. [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Ablation studies on model depth and clique sampling. [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Projected datasets: (a) contact-primary-school and (b) contact-high-school. [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
read the original abstract

Graph neural networks (GNNs) are limited to modeling pairwise interactions, while higher-order models based on cell complexes achieve greater expressivity but often suffer from poor scalability. We introduce simplified and factored cellular Weisfeiler Leman tests (sCWL and fCWL), which preserve the expressivity of the CWL test while improving computational efficiency. We further introduce the maximal clique complex, enabling scalable CWNs with reduced time and memory complexity while retaining strong empirical performance. To avoid explicit clique enumeration, we propose CliqueWalk, a biased random walk that samples maximal cliques and scales linearly with graph size. These contributions yield a scalable topological learning framework for higher-order graph representation.

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

2 major / 1 minor

Summary. The manuscript introduces simplified (sCWL) and factored (fCWL) cellular Weisfeiler-Leman tests claimed to preserve the expressivity of the original CWL test while improving efficiency. It further proposes the maximal clique complex together with CliqueWalk, a biased random walk for sampling maximal cliques without explicit enumeration, asserting that this construction yields scalable cellular Weisfeiler networks (CWNs) with linear time and memory complexity that retain strong empirical performance.

Significance. If the preservation claims are rigorously established, the work would address a central scalability barrier in higher-order topological graph learning, enabling cell-complex methods on graphs too large for full clique enumeration while maintaining the distinguishing power of CWL-based models.

major comments (2)
  1. [Abstract] Abstract: the claim that sCWL and fCWL preserve the expressivity of the CWL test is stated without any derivation, invariance argument, or reduction showing that the simplifications do not reduce distinguishing power on non-isomorphic graphs.
  2. [Abstract] Abstract: the assertion that CliqueWalk sampling produces a maximal clique complex whose induced sCWL/fCWL tests retain the original expressivity lacks any concentration bound, bias analysis, or topological invariance proof; a biased walk can systematically under-sample low-probability but topologically critical maximal cliques.
minor comments (1)
  1. The abstract refers to 'strong empirical performance' and 'reduced time and memory complexity' but supplies no datasets, baselines, or quantitative results to support these statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that sCWL and fCWL preserve the expressivity of the CWL test is stated without any derivation, invariance argument, or reduction showing that the simplifications do not reduce distinguishing power on non-isomorphic graphs.

    Authors: The preservation of expressivity is established rigorously in the body of the paper (Theorems 3.1 and 3.2), which contain the required invariance arguments and reductions to the original CWL test. The abstract is necessarily concise and therefore omits the derivations; we will revise the abstract to include a short pointer to these theorems. revision: yes

  2. Referee: [Abstract] Abstract: the assertion that CliqueWalk sampling produces a maximal clique complex whose induced sCWL/fCWL tests retain the original expressivity lacks any concentration bound, bias analysis, or topological invariance proof; a biased walk can systematically under-sample low-probability but topologically critical maximal cliques.

    Authors: Section 4.3 supplies a topological argument that the sampled complex retains the higher-order structures needed for the expressivity claims, together with extensive empirical verification. We do not supply concentration bounds or a complete bias analysis of CliqueWalk. We will add an explicit limitations paragraph discussing the possible effect of biased sampling on expressivity. revision: partial

Circularity Check

0 steps flagged

No circularity; new tests and sampling method introduced as independent constructions

full rationale

The abstract and description present sCWL/fCWL as simplified variants that preserve CWL expressivity by design, the maximal clique complex as a new structure for scalability, and CliqueWalk as a separate biased random walk sampler. No equations or steps reduce a claimed prediction or expressivity result to a fitted parameter, self-definition, or self-citation chain. The central claims rest on explicit construction and empirical retention rather than tautological equivalence to inputs. This is the expected non-finding for a methods paper whose innovations are stated as additive rather than derived from prior fitted results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 4 invented entities

Ledger populated from abstract only; full paper would likely reveal additional domain assumptions around expressivity preservation and complexity analysis.

axioms (1)
  • domain assumption Graph neural networks are limited to modeling pairwise interactions while higher-order models based on cell complexes achieve greater expressivity.
    Stated as background motivation in the abstract.
invented entities (4)
  • simplified cellular Weisfeiler-Leman test (sCWL) no independent evidence
    purpose: Preserve expressivity of CWL while improving computational efficiency
    Introduced as a new test in the abstract
  • factored cellular Weisfeiler-Leman test (fCWL) no independent evidence
    purpose: Preserve expressivity of CWL while improving computational efficiency
    Introduced as a new test in the abstract
  • maximal clique complex no independent evidence
    purpose: Enable scalable CWNs with reduced time and memory complexity
    Introduced as a new complex construction in the abstract
  • CliqueWalk no independent evidence
    purpose: Sample maximal cliques via biased random walk to avoid explicit enumeration
    Proposed as a new sampling method in the abstract

pith-pipeline@v0.9.1-grok · 5650 in / 1416 out tokens · 14741 ms · 2026-06-28T23:31:31.966654+00:00 · methodology

discussion (0)

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