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arxiv: 2605.06288 · v1 · submitted 2026-05-07 · 📊 stat.ME · cs.AI

A Topological Sorting Criterion for Random Causal Directed Acyclic Graphs

Alexander G. Reisach , Antoine Chambaz , Gilles Blanchard , Sebastian Weichwald This is my paper

Pith reviewed 2026-05-08 07:23 UTC · model grok-4.3

classification 📊 stat.ME cs.AI
keywords causal discoverydirected acyclic graphsrandom graphstopological sortingMarkov equivalence classErdős-Rényiscale-free networkscausal order
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The pith

In standard random causal DAGs, the number of relatives increases monotonically along the causal order.

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

The paper shows that DAGs generated by first imposing a topological order and then sampling edges from Erdős-Rényi or scale-free models have a built-in pattern: the set of nodes reachable via open paths, called relatives, grows larger the further one moves along the causal order. This monotonic growth appears frequently in the random graphs used to benchmark causal discovery methods. As a result, estimating the number of relatives from data and sorting nodes by that estimate often recovers the true causal order. When the increase is strict, the graph has a unique Markov equivalence class. The authors propose time-series DAG sampling as one way to generate alternatives that may lack this pattern.

Core claim

In DAGs generated by imposing an order on Erdős-Rényi and scale-free random graphs, the set of nodes reachable via open paths, termed relatives, increases monotonically along the causal order. Sorting by the estimated number of relatives recovers the causal order. A strict increase of relatives along the causal order leads to a singular Markov equivalence class. Sampling time-series DAGs is proposed as a possible alternative generation method.

What carries the argument

The set of relatives (nodes reachable via open paths) and its monotonic increase in count along the imposed causal order in these random DAGs.

If this is right

  • Sorting nodes by estimated number of relatives provides an effective proxy for recovering the causal order in many common simulation settings.
  • A strict monotonic increase in relatives implies the Markov equivalence class is singular.
  • The pattern is prevalent under standard procedures for generating random causal DAGs.
  • Time-series DAG sampling offers an alternative that may avoid this monotonic property.

Where Pith is reading between the lines

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

  • Causal discovery algorithms tested on these DAGs may benefit from the implicit ordering information carried by relative counts, affecting how performance is interpreted.
  • Simple sorting on estimated relatives could serve as a baseline comparator for more complex causal order methods.
  • Using generation procedures without this monotonicity would create stricter tests for causal discovery algorithms.

Load-bearing premise

The monotonic increase in relatives is a reliable property of standard random DAG generation by imposing order then sampling edges, and the number of relatives can be accurately estimated from finite observational data.

What would settle it

A generated random DAG from the Erdős-Rényi or scale-free procedure where the number of relatives does not increase monotonically along the imposed order, or data showing that sorting by estimated relatives fails to recover the order accurately.

Figures

Figures reproduced from arXiv: 2605.06288 by Alexander G. Reisach, Antoine Chambaz, Gilles Blanchard, Sebastian Weichwald.

Figure 1
Figure 1. Figure 1: Rel-sortability of random ER and SF DAGs with view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of rel-sortability to other sortabilities in ER DAGs. view at source ↗
Figure 3
Figure 3. Figure 3: Comparative SID (lower is better) performance of rel-SortnRegress on ER DAGs. view at source ↗
Figure 4
Figure 4. Figure 4: A directed and cyclic summary graph and corresponding time-unrolled DAG for view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of rel-sortability to other sortabilities in SF DAGs. view at source ↗
Figure 6
Figure 6. Figure 6: Comparative SID (lower is better) performance of rel-SortnRegress on SF DAGs. view at source ↗
Figure 7
Figure 7. Figure 7: compares the different sortabilities on data from non-standardized SCMs as discussed in Section 2.2. We observe that sortability by variance is extremely high for ER and SF DAGs, corroborating the findings in Reisach et al. (2021), and suggesting information in the variance as an explanation for the performance difference of the DAGMA algorithm between the standardized settings (sSCM) in Figures 3a and 6a … view at source ↗
read the original abstract

Random directed acyclic graphs (DAGs) based on imposing an order on Erd\H{o}s-R\'enyi and scale free random graphs are widely used for evaluating causal discovery algorithms. We show that in such DAGs, the set of nodes reachable via open paths, termed relatives, increases monotonically along the causal order. We assess the prevalence of this pattern numerically, and demonstrate that it can be exploited for causal order recovery via sorting by the estimated number of relatives. We note that many simulations in the literature feature settings where this yields an excellent proxy for the causal order, and show that a strict increase of relatives along the causal order leads to a singular Markov equivalence class. We propose sampling time-series DAGs as a possible alternative and discuss implications for causal discovery algorithms and their evaluation on synthetic data.

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 / 2 minor

Summary. The paper claims that in random causal DAGs generated by imposing a topological order on Erdős-Rényi or scale-free graphs, the set of nodes reachable via open paths (termed 'relatives') increases monotonically along the causal order. It numerically assesses the prevalence of this monotonicity, shows that sorting nodes by the estimated number of relatives can recover the causal order (and is an excellent proxy in many literature simulations), proves that strict monotonicity implies a singular Markov equivalence class, proposes time-series DAG sampling as an alternative generation method, and discusses implications for evaluating causal discovery algorithms.

Significance. If the monotonicity property holds in the population and the number of relatives can be reliably estimated, the result would identify a structural bias in standard synthetic DAG generators that makes causal order recovery trivial in many simulation settings, thereby improving the design and interpretation of benchmarks for causal discovery methods. The suggestion of time-series alternatives and the Markov equivalence result are useful contributions to the literature on random graph models for causal inference.

major comments (2)
  1. [numerical assessment section] The central claim that sorting by the estimated number of relatives recovers the causal order (abstract and numerical assessment section) is load-bearing on accurate recovery of d-connections from finite observational data, but the manuscript provides no details on the conditional independence tests used, sample sizes, or robustness to test errors and faithfulness violations; this leaves the practical utility of the sorting procedure unverified even when the population monotonicity holds.
  2. [Markov equivalence section] § on Markov equivalence: While the proof that strict increase of relatives along the order yields a singular equivalence class is noted, the manuscript does not quantify how often strict monotonicity occurs under the ER/scale-free generators or discuss whether the result extends beyond the specific random graph models considered.
minor comments (2)
  1. [abstract] The abstract states a numerical assessment of prevalence but the manuscript should include a table or figure with exact simulation parameters (number of nodes, edge probabilities, number of replicates) to allow reproducibility.
  2. [introduction] Notation for 'relatives' and 'open paths' should be defined more formally with reference to d-separation in the main text before the numerical results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the constructive referee report. We address each major comment below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: The central claim that sorting by the estimated number of relatives recovers the causal order (abstract and numerical assessment section) is load-bearing on accurate recovery of d-connections from finite observational data, but the manuscript provides no details on the conditional independence tests used, sample sizes, or robustness to test errors and faithfulness violations; this leaves the practical utility of the sorting procedure unverified even when the population monotonicity holds.

    Authors: We agree that details on finite-sample estimation are needed to substantiate practical utility. The current numerical assessment demonstrates prevalence of the monotonicity property using true population d-connections, with sorting presented as a conceptual exploitation. We will revise the numerical assessment section to specify the conditional independence tests (e.g., partial correlation tests), sample sizes, and add simulations assessing robustness to test errors and faithfulness violations. revision: yes

  2. Referee: While the proof that strict increase of relatives along the order yields a singular equivalence class is noted, the manuscript does not quantify how often strict monotonicity occurs under the ER/scale-free generators or discuss whether the result extends beyond the specific random graph models considered.

    Authors: The numerical assessment already evaluates prevalence of monotonicity for ER and scale-free generators; we will revise to explicitly report the frequency of strict monotonicity. The Markov equivalence proof is general for any DAG with strict monotonicity in relatives and does not rely on the specific generators. We will add discussion clarifying this generality while noting time-series sampling as one alternative to avoid the bias in practice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from graph generation process

full rationale

The paper establishes a monotonicity property of the 'relatives' set (nodes reachable via open paths) directly from the standard random DAG generation procedure (impose topological order, then sample edges via ER or scale-free models). This is shown analytically for the population case and assessed via numerical prevalence checks on generated graphs. The proposal to sort by estimated relatives count for order recovery follows as an application, without any step where a fitted parameter or self-citation is redefined as the output. Estimation from data is discussed as a practical step but does not enter the core derivation by construction. No self-definitional loops, fitted-input predictions, or load-bearing self-citations appear in the chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis relies on standard definitions from causal graphical models and random graph theory; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Random causal DAGs are generated by first imposing a total order on nodes and then sampling edges from Erdős-Rényi or scale-free models while respecting the order.
    This is the standard construction used in the causal discovery simulation literature referenced in the abstract.

pith-pipeline@v0.9.0 · 5437 in / 1482 out tokens · 44816 ms · 2026-05-08T07:23:14.269215+00:00 · methodology

discussion (0)

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