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REVIEW 2 major objections 2 minor 44 references

Directed Graph Topology Inference via Graph Filter Identification

T0 review · 2 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read The diffusion filter is identifiable from output measurements and input covariance priors under spectral diversity assumptions on the inputs, enabling recovery of the directed graph topology via a sparse commuting shift.

desk verdict The paper extends filter-based topology inference to directed graphs with correlated inputs via quadratic identification plus a commuting sparse shift, plus a joint alternating algorithm. read the letter →

arxiv 2606.27455 v1 pith:SIZF7M3A submitted 2026-06-25 stat.ML cs.LGcs.SIeess.SP

Rasoul Shafipour , Andrei Buciulea , Santiago Segarra , Antonio G. Marques , Gonzalo Mateos This is my paper · ORCID
classification stat.MLcs.LGcs.SIeess.SP
keywords directedgraphinferencefilteridentificationtopologyrecoverydiffusiondynamicsquadraticequationscommutingoperatorssparseshiftsStiefelmanifold
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that observations from linear diffusion on a directed graph can be used to identify the underlying graph convolutional filter when prior knowledge of input signal covariances is available. This identification reduces to solving a system of quadratic matrix equations that becomes solvable when the input covariances have diverse spectra. Once the filter is recovered, the graph topology is inferred by finding a sparse graph-shift operator that commutes with the filter and satisfies structural admissibility conditions. A joint algorithm that alternates between filter identification and topology recovery is also developed to improve performance with limited samples. This approach extends previous methods to directed graphs and correlated inputs.

What carries the argument

The graph convolutional filter as a polynomial of the graph-shift operator, identified via quadratic equations under Stiefel manifold constraints, followed by finding a sparse commuting shift for topology inference.

What would settle it

A counterexample where two different graph filters produce identical output covariance matrices for the same set of input covariances would show the identifiability claim does not hold.

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Extended reading notes

Core claim

We show that the diffusion filter, defined as a polynomial in the graph-shift operator, can be identified by solving quadratic matrix equations derived from output covariances and known input statistics, provided the input covariances satisfy spectral diversity. The network topology is then recovered by identifying a sparse admissible shift operator that commutes with the estimated filter.

Load-bearing premise

The input signal covariances must have sufficiently diverse spectra to ensure the quadratic equations uniquely determine the filter coefficients.

Editorial extensions

If this is right

  • The topology of directed graphs can be inferred from nodal measurements without requiring white input excitations or joint diagonalizability of shift and covariance.
  • A joint alternating algorithm between filter and topology steps achieves better sample complexity than separate identification.
  • The method applies to real-world problems such as urban mobility analysis and portfolio optimization.
  • Recovery succeeds when the estimated filter admits a sparse polynomial representation in the graph shift.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If input covariances lack spectral diversity, multiple filters may fit the data, leading to ambiguity in topology recovery.
  • The commuting condition could be relaxed or combined with other priors for graphs with additional structure.
  • Tests on larger real networks could reveal scalability limits of the Stiefel-constrained optimization.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper addresses inferring directed graph topologies from nodal measurements generated by linear diffusion dynamics, modeled as a polynomial graph convolutional filter (with unknown coefficients) applied to a graph-shift operator, excited by an ensemble of independent but arbitrarily correlated input signals. It first identifies the diffusion filter from output measurements and input covariance priors by solving a system of quadratic matrix equations, shown to be identifiable under spectral-diversity assumptions on the input covariances; this is recast as smooth quadratic minimization on the Stiefel manifold. Topology recovery then reduces to finding a sparse, admissible shift operator that commutes with the estimated filter. A joint alternating algorithm is proposed to improve sample complexity, with numerical validation on synthetic digraphs and real-data applications including urban mobility and portfolio optimization.

Significance. If the identifiability result and commuting-shift recovery hold, the work extends graph topology inference to the directed case with correlated excitations, moving beyond the undirected/white-input settings common in prior literature. The spectral-diversity condition and manifold-constrained formulation provide a concrete route to filter identification, while the joint alternating scheme offers a practical improvement in sample efficiency. These elements, combined with the real-data case studies, position the contribution as relevant for network inference tasks in signal processing and machine learning.

major comments (2)
  1. [identifiability section] § on identifiability (around the quadratic matrix equations): the claim that spectral diversity on input covariances renders the system identifiable requires an explicit uniqueness argument showing that distinct filter coefficient vectors cannot produce the same output covariances under the given assumptions; without this step-by-step derivation or a supporting lemma, the transition from the quadratic equations to unique recovery remains unverified in the provided text.
  2. [topology recovery] Topology recovery paragraph (commuting shift step): the assertion that a sparse admissible shift commuting with the estimated filter uniquely recovers the graph-shift operator needs a supporting argument or counter-example exclusion showing that no other admissible sparse operators satisfy the commutativity condition; this step is load-bearing for the end-to-end claim but currently lacks explicit verification beyond the standard polynomial-in-shift property.
minor comments (2)
  1. [abstract] The abstract and introduction would benefit from a brief statement of the precise spectral-diversity condition (e.g., distinct eigenvalues or rank conditions on the input covariance matrices) to make the identifiability claim immediately checkable.
  2. [numerical tests] Numerical experiments section: reporting the exact number of Monte Carlo trials, the range of sample sizes tested, and quantitative metrics (e.g., edge recovery F1 or normalized error) for both the filter identification and topology recovery stages would strengthen reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and detailed review of our manuscript. We address each major comment point by point below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [identifiability section] § on identifiability (around the quadratic matrix equations): the claim that spectral diversity on input covariances renders the system identifiable requires an explicit uniqueness argument showing that distinct filter coefficient vectors cannot produce the same output covariances under the given assumptions; without this step-by-step derivation or a supporting lemma, the transition from the quadratic equations to unique recovery remains unverified in the provided text.

    Authors: We appreciate the referee's observation. The manuscript derives identifiability by showing that the system of quadratic matrix equations admits a unique solution for the filter coefficients when the input covariances satisfy the spectral-diversity condition, as the distinct eigenvalues prevent different coefficient vectors from yielding identical output covariances. However, we agree that an explicit step-by-step uniqueness argument would improve clarity and verifiability. In the revised version we will insert a supporting lemma immediately after the quadratic equations that formally proves uniqueness under the stated assumptions, including the necessary algebraic steps. revision: yes

  2. Referee: [topology recovery] Topology recovery paragraph (commuting shift step): the assertion that a sparse admissible shift commuting with the estimated filter uniquely recovers the graph-shift operator needs a supporting argument or counter-example exclusion showing that no other admissible sparse operators satisfy the commutativity condition; this step is load-bearing for the end-to-end claim but currently lacks explicit verification beyond the standard polynomial-in-shift property.

    Authors: We thank the referee for this comment. The topology recovery step relies on the fact that any admissible sparse shift commuting with the estimated filter must be a polynomial in the true graph-shift operator, combined with the structural constraints (sparsity and admissibility) to enforce uniqueness. While the polynomial-in-shift property is standard, we acknowledge that an explicit argument excluding other candidate sparse commuting operators would make the claim more rigorous. We will add a short proposition in the topology-recovery section that shows, under the problem assumptions, no other admissible sparse operator satisfies the commutativity condition. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The derivation proceeds from external output measurements and input covariance priors to identify the diffusion filter via a system of quadratic equations shown identifiable under spectral-diversity assumptions on those priors; topology recovery then applies the independent structural constraint that the shift must commute with the recovered filter (forcing it to be a polynomial in the shift). No step reduces by construction to a fitted parameter renamed as prediction, a self-citation chain, or a self-definitional loop; the commuting condition is a standard algebraic consequence of the polynomial filter model rather than an ansatz or fitted input. The paper is therefore self-contained against its stated external data and assumptions.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The approach rests on modeling assumptions about linear diffusion dynamics, polynomial filter representation, and the existence of a sparse admissible graph shift; no new entities are postulated.

assumptions (2)
  • domain assumption Observations are outputs of a graph convolutional filter (polynomial in the graph-shift operator) excited by independent graph signals with arbitrarily correlated nodal components.
    Stated in the abstract as the observation model.
  • domain assumption Spectral-diversity assumptions on the input covariances render the quadratic matrix equations identifiable.
    Invoked to guarantee unique recovery of the filter.

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Cite this review

Pith. "Pith review of Directed Graph Topology Inference via Graph Filter Identification." pith.science (2026). https://pith.science/paper/SIZF7M3A

@misc{pith2026260627455,
  author       = {Pith},
  title        = {Pith review of: Directed Graph Topology Inference via Graph Filter Identification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SIZF7M3A}},
  note         = {Machine review of arXiv:2606.27455}
}
read the original abstract

We address the problem of inferring a directed network from nodal measurements generated by linear diffusion dynamics on the sought graph. Observations are modeled as the outputs of a graph convolutional filter, i.e., a polynomial (with unknown coefficients) of a local diffusion graph-shift operator encoding the latent graph topology, excited with an ensemble of independent graph signals with arbitrarily-correlated nodal components. Unlike prior efforts that considered undirected graphs and white signal excitations, here the graph-shift operator and the observations' covariance matrix are not simultaneously diagonalizable. In this challenging context, we first rely on measurements of the output signals along with prior statistical information on the inputs to identify the diffusion filter. Such system identification problem involves solving a system of quadratic matrix equations, which we show is identifiable under spectral-diversity assumptions on the input covariances. For algorithmic purposes we recast it as a smooth quadratic minimization subject to Stiefel manifold constraints. Subsequent identification of the network topology given the graph filter estimate boils down to finding a sparse and structurally admissible shift that commutes with the given filter, thus, forcing the latter to be a polynomial in the sought graph-shift operator. A joint graph filter and topology identification algorithm is also proposed, which alternates between the aforementioned steps in a mutually reinforcing fashion to offer improved sample complexity. Numerical tests corroborate the effectiveness of the proposed algorithms in recovering synthetic digraphs and real-data case studies, and illustrate their potential utility on urban mobility analyses as well as portfolio optimization.

Figures

Figures reproduced from arXiv: 2606.27455 by the authors.

Figure 1
Figure 1. Schematic view of the digraph identification methods [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Performance of the proposed open loop (OL) and closed loop (CL) approaches, in terms of (a) estimation error and (b) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Estimation error and F-score under different experimental conditions: (a) varying the number of samples [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The columns show the estimated mobility links of three different approaches: TITD [16], GFI [8], and Algorithm 3. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Portfolio performance obtained by considering four [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

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Reference graph

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