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Linear relations of colored Gaussian cycles

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Pith's one-line read Linear binomials in colored Gaussian cycles correspond to graph symmetries for 3, 5, and 7-cycles, but this fails for all other cycle lengths, with a revised theorem for uniform edge colorings of odd cycles.

desk verdict Genuinely new counterexamples and a positive result for 3,5,7-cycles, but the main theorem's scope outruns the construction and two proof steps look wrong as written. read the letter →

arxiv 2506.23936 v1 pith:7EFSV7LB submitted 2025-06-30 math.CO math.AGmath.STstat.TH

Hannah Göbel , Pratik Misra This is my paper · ORCID
classification math.COmath.AGmath.STstat.TH
keywords coloredconjecturecycleslinearconcentrationgaussiangraphmodel
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

Colored Gaussian graphical models are statistical models that describe how random variables depend on each other using a graph whose edges and vertices carry colors. Two edges with the same color mean the corresponding correlations are forced to be equal. A natural question is whether the equations defining such a model are always explained by the graph's symmetries, meaning rotations or reflections that preserve all colors. For cycles, Davies and Marigliano conjectured this is true. This paper shows the conjecture holds for cycles with 3, 5, or 7 vertices, but fails for every cycle of length 4 or larger, and even for many longer odd cycles. The key idea is to study paths inside the cycle. The covariance between two vertices can be written using the two paths that connect them in the cycle. For a symmetry to exist, certain subpaths must have equal determinants, which usually forces the subpaths to be identical or mirror images. The authors found colorings of longer paths that have equal determinants without being identical or mirror images, and these "fake equalities" produce linear equations in the model that have no corresponding symmetry. They also prove a repaired version of the conjecture for odd cycles where all edges share one color: in that setting the original statement becomes true again.
Extended reading notes

Core claim

Theorem 1.1: Let G be a colored n-cycle. If n is 3, 5, or 7, then every linear binomial in the vanishing ideal I_G corresponds to a graph symmetry. However, if n is 4, 6, 8 or larger, then there can exist colored cycles whose vanishing ideal contains linear binomials that do not correspond to any graph symmetry.

Load-bearing premise

The theorem's assertion that counterexamples exist for every n equal to 4, 6, 8 or larger rests on the unproven assumption that the construction patterns of Examples 4.1 and 4.2 extend to every even and every odd n beyond the explicitly drawn cases (4, 6, 8, 9, 10). The paper states a general technique for even cycles but gives no explicit recipe for odd n of size 11 or more, and no proof that the required colorings can be realized without accidentally introducing a graph symmetry.

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Referee Report

5 major / 7 minor

Summary. The paper studies linear binomials in the vanishing ideal of colored Gaussian graphical models whose underlying graph is an n-cycle. It claims to prove a conjecture of Marigliano and Davies for cycles of length 3, 5, and 7, and to disprove it for every other length by constructing colored cycles with linear binomials not induced by graph symmetries. The construction is based on comparing determinants of concentration matrices of colored paths, and the paper also proposes and proves a revised conjecture for odd cycles with uniform edge coloring. The main positive tool is a formula expressing path determinants via disjoint edge sets (Lemma 3.3), and the counterexamples use two non-trivial path color configurations (Theorems 3.6 and 3.8) together with explicit 4-, 8-, 9-, and 10-cycles.

Significance. If fully established, the paper would settle a conjecture in coloured graphical models and give a clear structural explanation of when linear binomials arise from graph symmetries. The paper contains useful ingredients: a concrete determinant formula for path concentration matrices, explicit path colorings with equal determinants, and small counterexamples that are easy to verify computationally. The computational observation about reducing ideal computations via symmetries is also a nice practical contribution. However, the main theorem as stated is not supported by the provided constructions, and several intermediate proofs contain algebraic errors or gaps. The paper's central idea is plausible and the small examples are valuable, but the current version does not substantiate the full range of claims.

major comments (5)
  1. [Theorem 1.1; Section 4, Examples 4.1 and 4.2] Theorem 1.1 claims counterexamples for n = 4, 6, 8 or larger, but the paper explicitly constructs counterexamples only for n = 4, 8, 9, 10 (and n = 6 in Example 4.5). The sentence 'Applying the technique illustrated in Example 4.1 always yields cycles of even length' is not a proof: no explicit recipe or induction is given for all even n, and no odd counterexample beyond n = 9 is provided. In particular, no construction is given for any odd n ≥ 11. The universal 'or larger' clause of Theorem 1.1 is therefore not established; at present the paper disproves Conjecture 2.7 only for finitely many cycle lengths.
  2. [Lemma 4.9] The recurrence displayed in the proof of Lemma 4.9 is algebraically incorrect. Expanding det(KPj+1) = p_{j+1} det(KPj) - e^2 det(KPj-1) with det(KPj) = p_j det(KPj-1) - e^2 det(KPj-2), one obtains det(KPj+2) = (p_{j+2}p_{j+1}p_j - p_{j+2}e^2 - e^2 p_j) det(KPj-1) + (-e^2 p_{j+2}p_{j+1} + e^4) det(KPj-2). The paper instead writes -e^2(p_{j+2}p_{j+1}+1) det(KPj-2), omitting the e^4 term and changing the coefficient of det(KPj-2). Since the claim that det(KPj+2) ≠ det(KQj+2) relies on this expansion, the proof of Lemma 4.9 is incomplete; this lemma is the key step in the proof of Theorem 4.6.
  3. [Theorem 3.8] The displayed determinant expansion for the m = 5 case of Theorem 3.8 is incorrect. A direct cofactor expansion of the shown 5x5 matrix yields det(KP) = A^3BC - A^2Ba^2 - A^2b^2C - A^2a^2C - A^2Bb^2 + Ab^4 + Aa^2b^2 + Aa^4 (with A = k11, B = k22, C = k44, a = k12, b = k23), whereas the paper's expression is roughly the negative of this and has the wrong sign on the k23^2 det(KP-2) term. The same issue affects the expansion for KQ. The conclusion may still be true, but the proof as written is not a valid derivation; this theorem is one of the two path configurations used to build the counterexamples in Section 4.
  4. [Examples 4.1 and 4.2] The claims that the exhibited cycles have no graph symmetry are not proved. In Example 4.1 the assertion that 'the only potential symmetry in this 8-cycle is the reflection along the axis passing through the edges {8,1} and {4,5}' and in Example 4.2 the assertion that the 9-cycle has no symmetry are plausible but are not demonstrated; graph symmetries of an n-cycle form the dihedral group, so one must check all rotations and all reflections (or give an argument that the coloring distinguishes every nontrivial dihedral element). Because the absence of symmetry is essential to the counterexample, this needs to be made explicit.
  5. [Lemma 3.4] The proof of Lemma 3.4 compares monomials in the expanded determinants as if the variables were all distinct. Under a coloring, however, different monomials can become equal after identifying variables (for example, in a length-two path with all vertices one color and both edges another color, the two subtracted terms merge). The conclusion of the lemma may still be true, but the monomial-comparison argument needs to be made robust under color identifications; this matters because Lemma 3.4 is used in the proof of Theorem 3.5 for 3, 5 and 7 cycles.
minor comments (7)
  1. [Abstract and Section 1] There are typographical issues: 'a undirected colored cycle' should be 'an undirected colored cycle', and the phrase 'for 3, 5, and 7 cycles' reads awkwardly.
  2. [Lemma 3.3] The condition 'S⊆EP disjoint' should be stated more precisely as 'S is a set of pairwise disjoint edges'; as written it is ambiguous.
  3. [Theorems 3.6 and 3.8] In Theorem 3.6 condition (1), the indexing 'i,j∈{1,2,...,m-1}' is unclear because j is not defined in that condition; similarly in Theorem 3.8 condition (3), 'for all odd i' and 'all even j' should specify the ranges explicitly.
  4. [Example 4.2] The notation 'det(K1↔3\{1,3})' is confusing; previously the notation K\P meant the matrix with rows and columns corresponding to the vertices on path P deleted, and writing '\{1,3}' instead of the path indicator is inconsistent.
  5. [Lemma 5.2] There is a typo: 'muliset' should be 'multiset'. Also, the definition of 'odd edges' is only given parenthetically and would be clearer if spelled out in the statement.
  6. [Example 2.9] The computational timings are reported without specifying the hardware, software, or version used; adding this information would improve reproducibility.
  7. [Theorem 3.5] The proof for n = 7 ends with 'A similar argument can be repeated when the complementary path is of length four' without giving the details; if this case is not literally identical, the proof should be completed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the colored-cycle construction is derived from first principles and the cited conjecture is used only as the target.

full rationale

The paper's central claims are derived from first principles rather than from its own conclusions. Lemma 3.3 gives an induction proof of a determinant expansion for tridiagonal path concentration matrices. Theorem 3.5 proves the 3/5/7 case by monomial vertex/edge degree comparison, using only the path determinant structure. Theorems 3.6 and 3.8 construct non-identical, non-reflection colored paths with equal determinants by explicit Leibniz expansion and induction, respectively. Section 4 then assembles these path configurations into cycles and verifies the sufficient conditions of Corollary 3.1, with the resulting linear binomials confirmed by direct computation of the vanishing ideal. No fitted parameter is renamed as a prediction, and the Davies-Marigliano conjecture is the target to be proved or disproved, not an input to the derivation. The self-citations to [2], [8], and [9] are background remarks about toric ideals and do not support any later load-bearing step. Theorem 2.10 is an external path-weight formula. The proof of the revised conjecture, Theorem 4.6, is supported by its own Lemmas 4.7 and 4.9. Possible weaknesses, such as the assertion that the Example 4.1 technique yields even cycles without an explicit recipe for every even n, or the informal 'only potential symmetry' checks, are completeness or correctness concerns rather than circular reductions. Therefore no significant circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. Its load-bearing external input is the Jones-West path formula for covariance entries; the rest uses standard polynomial-ring arguments. The color variables are generic formal indeterminates, not fitted constants.

assumptions (4)
  • domain assumption Theorem 2.10 (Jones-West): for a cycle, sigma_ij can be written as a sum over the two paths between i and j of signed products of edge parameters times a complementary determinant.
    Invoked in Corollary 3.1 and throughout Sections 3 and 4; this is a cited external result, not proven in the paper.
  • domain assumption Concentration matrices are positive definite, so every principal submatrix has nonzero determinant.
    Standard for Gaussian graphical models and used in Lemma 4.9 and elsewhere to justify cancellation-free comparisons.
  • standard math Color variables are distinct formal indeterminates, so equality of monomials in the polynomial ring forces equality of the corresponding color labels.
    Used throughout the 'degree argument' to conclude that two products are equal only when the color variables coincide.
  • standard math A linear binomial sigma_ij - sigma_xy lies in I_G if and only if the corresponding rational function in the concentration parameters is identically zero.
    This is the definition of the vanishing ideal as the kernel of the rational map rho_G, used throughout the paper.

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Pith. "Pith review of Linear relations of colored Gaussian cycles." pith.science (2026). https://pith.science/paper/7EFSV7LB

@misc{pith2026250623936,
  author       = {Pith},
  title        = {Pith review of: Linear relations of colored Gaussian cycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7EFSV7LB}},
  note         = {Machine review of arXiv:2506.23936}
}
read the original abstract

A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. Marigliano and Davies conjectured that every linear binomial that appears in the vanishing ideal of an undirected colored cycle corresponds to a graph symmetry. We prove this conjecture for 3,5, and 7 cycles and disprove it for colored cycles of any other length. We construct the counterexamples by proving the fact that the determinant of the concentration matrices of two colored paths can be equal even when they are not identical or reflection of each other. We also explore the potential strengthening of the conjecture and prove a revised version of the conjecture.

Figures

Figures reproduced from arXiv: 2506.23936 by the authors.

Figure 1
Figure 1. 4-cycle △ Graph coloring and symmetry: Additional symmetries can be introduced in the model when certain partial correlations interact in the similar way. This was first introduced in [5] where the authors used colored graphs to represent these symmetries. For any given graph G, we assign colors to the vertices and edges of G with the condition that the sets of vertex and edge colors are disjoint. Let λ(i) and λ({i,… view at source ↗
Figure 2
Figure 2. Colored 4-cycle △ Observe that in the previous example, we obtained some linear binomials in IG which were not present in IG. In general, the only time we get linear relations in IG is when G is a disconnected graph, and the linear relations are of the form σij , where i and j are disconnected in G (Proposition 2.5 [3]). This follows from the fact that all the kij s are independent variables in uncolored graphs. How… view at source ↗
Figure 3
Figure 3. Colored 6-cycle △ In order to explore the necessary condition of the conjecture, it is important to first understand the image of each σij under the ρG map. In [6], the authors developed a combinatorial connection between the image of σij and the structure of the graph. Theorem 2.10. [6, Theorem 1] Consider an n-dimensional multivariate normal distribution with a finite and non-singular covariance matrix Σ, and conc… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: G1: Colored 8- cycle counterexample 1 2 3 4 5 6 7 8 9 10 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 6
Figure 6. Figure 6: G1: Colored 4- cycle counterexample 1 2 3 4 5 6 7 8 9 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 8
Figure 8. Figure 8: G1: Uniform vertex colored even cycle 1 2 3 4 5 6 7 8 9 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 10
Figure 10. Figure 10: Uniform edge colored even cycle △ Note: The above example is also the first example where σij − σxy ∈ IG but the sets {λ(i), λ(j)} and {λ(x), λ(y)} are different. In particular, det(K1↔5\{1,5}) ̸= det(K2↔4\{2,4}) and det(K1 c↔5\{1,5} ) ̸= det(K2 c↔4\{2,4} ) even thoug…
Figure 13
Figure 13. Figure 13: G1 and G2 1 2 3 6 5 4 1 2 3 6 5 4 [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]
Figure 14
Figure 14. Figure 14: G3 and G4 To the best of our knowledge, this problem has not been studied before. The only related study done in this direction is in [7], where the authors study the conditions on when the generalized characteristic polynomial of two graphs are equal. For a given gra…

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Cited by 1 Pith paper

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  1. Computing the continuous symmetries of a parametrized variety

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    The symmetry Lie algebra of a unirational variety equals the linear maps sending each point into its tangent space and can be recovered from the Jacobian of a parametrization in polynomial-time Monte Carlo fashion.

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