Graph-Based Proofs of Indistinguishability of Linear Compartmental Models

Cashous Bortner; Dev Patel; John Gilliana; Zaia Tamras

Graph-Based Proofs of Indistinguishability of Linear Compartmental Models

T0 review · 0 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that two families of skeletal path models—paths with a single leak and the path with a terminal reverse edge—have identical input–output equations up to parameter relabeling, established through a graph-theoretic…

desk verdict A clean, honest reproof of two known indistinguishability results via the forest-sum formula; the explicit coefficient formulas are the real new content, and they check out. read the letter →

arxiv 2412.01135 v1 pith:QUQB7A53 submitted 2024-12-02 math.CO math.DS

Cashous Bortner , John Gilliana , Dev Patel , Zaia Tamras This is my paper · ORCID

classification math.COmath.DS MSC 05C2005C0592B05

keywords linearcompartmentalmodelsskeletalpathpermutationindistinguishabilityincomingforestselementarysymmetricpolynomialsinput-outputequationsgraph-theoreticproofsforest-sumformula

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

Linear compartmental models are used to describe how quantities move through compartments in pharmacokinetics, cell biology, and ecology. Two models are permutation indistinguishable when their input–output equations coincide after renaming parameters, meaning that experiments cannot tell them apart. This paper proves two such indistinguishability results for skeletal path models—directed paths with one leak, or with a terminal reverse edge—using a graph-based formula for the coefficients of the input–output equation. The proof shows that the coefficients are elementary symmetric polynomials minus a single correction term for the edge pairs that violate the incoming-forest condition, and that this correction term is preserved by an explicit parameter relabeling. The earlier linear-algebra proofs are thereby replaced by a combinatorial argument.

What carries the argument

The load-bearing mechanism is the forest-sum formula of Theorem 2.3: it expresses each coefficient $c_j$ as the sum of productivities of all incoming forests on the augmented graph with $n-j$ edges, and each $d_i$ as the sum over incoming forests with a path from the input to the output in the edge-reduced graph. In these path models, an incoming forest is any acyclic subgraph in which no vertex has two outgoing edges. The key observation is that the only subsets of parameters that fail this condition are precisely the two-edge sets $\{a_{0i},a_{i+1,i}\}$ for a leak at $i$, or $\{a_{n,n-1},a_{n-1,n}\}$ for the terminal reverse edge. Therefore the full elementary symmetric polynomial $\sigma_k(Q)$ decomposes into the coefficient plus a single product times $\sigma_{k-2}$ of the remaining parameters, and this decomposition is what the indistinguishability maps preserve.

What would settle it

Compute the input–output equation of $M_2=(P_4,\{1\},\{4\},\{2\})$ by direct substitution and compare the coefficient of $y'$ with the value $\sigma_3(Q)-a_{02}a_{32}\sigma_1(Q\setminus\{a_{02},a_{32}\})$; any mismatch would falsify Proposition 3.1. Alternatively, for $n=5$ and generic numerical parameters, apply the relabeling $\Phi$ of Theorem 3.5 to the coefficients of $M_2$, and check whether they equal the coefficients of $M_4$; a single generic failure would falsify the theorem.

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Extended reading notes

Core claim

The discovery is that, for skeletal path models, the input–output coefficients have a uniform graph-theoretic shape. For a path model $M_i=(P_n,\{1\},\{n\},\{i\})$ with parameter set $Q_i$, Proposition 3.1 gives $$c_j=\sigma_{n-j}(Q_i)-a_{0i}a_{i+1,i}\,\sigma_{n-j-2}(Q_i\setminus\{a_{0i},a_{i+1,i}\})$$ for the coefficient of $y^{(n-j)}$, while for the no-leak model $M_n$ with edge $n\to n-1$ and parameter set $Q_n$, Proposition 3.3 gives the same expression with the terminal pair $a_{n,n-1}a_{n-1,n}$ playing the role of the forbidden pair. The reason is that in the augmented graph $\tilde G$, the only edge subsets that violate the incoming-forest condition are exactly the subsets containing both edges that leave the leak vertex, or both edges of the two-cycle at the end of the path. The authors define bijections between parameter sets that swap the forbidden pairs, so the elementary symmetric polynomials and the correction terms match term by term. This proves Theorems 3.4 and 3.5 as purely combinatorial identities.

Load-bearing premise

The argument rests on the forest-sum formula of Theorem 2.3 and on the structural fact that, in these path models, the only edge subsets that violate the incoming-forest condition are the two outgoing edges from the leak vertex or the two-cycle edges; if either fails, the coefficient identities and the indistinguishability conclusions do not follow.

Editorial extensions

If this is right

The penultimate-leak model $M_{n-1}$ and the no-leak model $M_n$ with edge $n\to n-1$ have identical input–output equations under the parameter bijection that sends the leak parameter $a_{0,n-1}$ to the reverse-edge parameter $b_{n-1,n}$.
Any two single-leak path models $M_i$ and $M_k$ with $1\le i<k<n$ are permutation indistinguishable under the bijection that swaps $a_{i+1,i}$ with $b_{k+1,k}$ and sends $a_{0i}$ to $b_{0k}$.
Transitivity of permutation indistinguishability implies that a leak at any single non-output compartment of a path model is indistinguishable from the no-leak model with the terminal reverse edge.
Because permutation indistinguishable models are also indistinguishable in the usual sense, no input–output experiment can separate these pairs.
The coefficient formulas provide a direct way to write the input–output equation of these models from the graph, without forming the compartmental matrix.

Reading between the lines

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

The “one forbidden pair” mechanism suggests a testable sufficient condition: any two graphs whose non-incoming-forest edge subsets can be paired by an edge bijection that preserves productivity sums will be permutation indistinguishable, and this could be checked on other structured families.
For models with several leaks, the correction term would become a sum over all pairs of edges leaving the same vertex; if that sum is symmetric under some parameter bijection, the same proof pattern would apply.
One could test the boundary of the claim by adding an extra edge to a path model; the enumeration of forbidden edge subsets would have to remain a matching set of pairs for the coefficient identity to survive.
The identity may also be read as an identifiability statement: the correction term is determined by the rest of the coefficients, so the graph structure fixes how much the leak can be moved without changing the model's behavior.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This is a small but genuinely clean paper. It reproves two indistinguishability results for skeletal path models—results that Bortner and Meshkat already proved by linear algebra—using the forest-sum coefficient formula from the group's earlier work. The authors are upfront about the reproof; the stated goal is a proof-of-concept for graph-theoretic arguments.

The new content is Propositions 3.1 and 3.3, which express the input-output coefficients as an elementary symmetric polynomial minus a correction term for the single family of edge subsets that violate the incoming-forest condition. In the leak model that's the two edges leaving the leak vertex; in the return-edge model it's the two-cycles. I checked the enumerations and the bijections in Theorems 3.4 and 3.5; the maps swap exactly the forbidden pairs and preserve the path-edge product, so the proofs work. The writing is transparent about assumptions: the derivation leans on Theorem 2.3 from prior work, which is published and standard. No circularity—the target theorems are not assumed.

The soft spots are minor but real. The main theorems are not new, so the contribution is a proof technique rather than a result. The algebra is simple symmetric-polynomial manipulation; the graph theory only identifies the forbidden subsets. There's no attempt to extend the framework to other model families, so it remains a proof-of-concept. But it is exactly what it claims to be.

The stress-test note found no load-bearing flaw, and I agree. This is a correct, honest, well-scoped paper. It should go to peer review; a referee can verify it quickly and the likely outcome is acceptance with minor revisions. It would be a useful reference for anyone exploring graph-based approaches to indistinguishability, though I wouldn't cite it in my own work unless I were working directly on that technique.

Referee Report

0 major / 4 minor

Summary. The paper gives a graph-theoretic reproof of two permutation indistinguishability results for skeletal path linear compartmental models that were originally proved by Bortner and Meshkat using linear algebra. The key tool is Theorem 2.3, a forest-sum formula for the coefficients of input-output equations. The authors derive explicit coefficient formulas for a path model with a single leak (Proposition 3.1) and for a path model with a return edge from the last vertex to the penultimate vertex (Proposition 3.3). Using these formulas, they construct explicit parameter bijections that preserve the input-output equation, yielding the two indistinguishability theorems (Theorems 3.4 and 3.5). The proofs are elementary and rely only on elementary symmetric polynomial identities and a correct enumeration of the edge subsets that violate the incoming-forest condition in the associated graphs.

Significance. The central contribution is a clean proof-of-concept showing that the two known indistinguishability theorems follow directly from a graphical coefficient formula, avoiding the comparatively heavy linear algebra of Cramer's rule or differential elimination. The coefficient formulas in Propositions 3.1 and 3.3 are explicit and checkable, and I verified that the forest-violating edge subsets identified in the proofs are exactly the pairs of edges leaving the leak vertex (for the path-with-leak models) and the two-cycle edges (for the return-edge model). The permutation maps in Theorems 3.4 and 3.5 are simple bijections that send each excluded pair to the corresponding excluded pair, so the symmetric-polynomial expressions transform as claimed. The paper does not introduce new indistinguishability theorems, but it establishes a transparent combinatorial framework that is likely to be useful for future sufficient conditions based on graph structure.

minor comments (4)

[Section 3, Proposition 3.1 and Eq. (8)] The input-output equation (8) omits the c_0 y term, and the statement defines c_j only for j=1,...,n-1, even though Example 3.2 correctly computes c_0=0. Please state explicitly that c_0=0 (or include j=0 in the formula range) and similarly note that d_i=0 for i>0 in both Proposition 3.1 and Proposition 3.3.
[Section 3, proofs of Propositions 3.1 and 3.3] The sentence in the proof of Proposition 3.1 that the only non-incoming forests are those containing both a_{0i} and a_{(i+1)i} is correct, but the justification should explicitly mention that the underlying graph is acyclic and that every vertex other than the leak vertex has at most one outgoing edge; a one-sentence enumeration would make the argument fully rigorous. The analogous claim in Proposition 3.3, namely that the only forbidden subsets are those containing both edges of the two-cycle, is also correct and could be stated with the same level of explicitness.
[Section 3, Example 3.2 and Proposition 3.3] There is a small notational slip in the set complments: Example 3.2 writes Q_3 \ {a_{03}a_{(3+1)3}} instead of Q_3 \ {a_{03}, a_{43}}, and Proposition 3.3 writes Q_n \ {a_{n(n-1)}a_{(n-1)n}} instead of a set with a comma between the two excluded parameters. Please fix these to avoid ambiguity.
[Section 3, Theorem 3.4 proof] In the sentence referring to "the left-hand coefficient" of the input-output equations, the plural "coefficients" would be more accurate, since the proof compares the full sets of left-hand side coefficients. This is a wording issue only.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reproofs are a genuine derivation from the general forest-sum formula, with prior indistinguishability results used only as the stated target, not as premises.

full rationale

The paper's central claim is a graph-theoretic reproof of two previously known indistinguishability theorems (Theorems 2.5 and 2.6 from Bortner and Meshkat). The proofs of Theorems 3.4 and 3.5 never invoke Theorems 2.5 or 2.6 as premises; instead the coefficient formulas in Propositions 3.1 and 3.3 are derived from Theorem 2.3, the forest-sum formula for input-output coefficients taken from prior work [2]. This cited formula is a general, parameter-free identity with stated assumptions (single input/output, linear compartmental model) that do not include the target indistinguishability statements, so it is independent support rather than a self-citation forcing the result. The remaining work is a direct enumeration of the only forest-violating edge subsets in these path graphs (the pair of edges leaving the leak vertex, or the two-cycle edges), followed by explicit bijections that send each excluded pair to the corresponding excluded pair and induce the elementary-symmetric-polynomial identities. This is a genuine reduction rather than a renaming of the known results. The only self-citations are transparent citations to the prior formula and to the results being reproven, and none of them smuggles in the conclusion.

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

No free parameters and no invented entities: the paper is a purely mathematical proof. The main external input is the forest-sum coefficient formula from prior work, listed as an axiom here because the paper cites rather than proves it.

assumptions (4)

domain assumption Input-output coefficients equal sums of incoming forest productivities (Theorem 2.3 from [2]).
Invoked in Propositions 3.1 and 3.3 to derive explicit coefficient formulas; the paper cites [2] rather than proving it.
domain assumption Permutation indistinguishability is defined by existence of a parameter bijection mapping coefficients to coefficients (Definition 1.16).
This is the target relation under study; the paper relies on this definition to set up the bijections in Theorems 3.4 and 3.5.
standard math Elementary symmetric polynomials satisfy the standard combinatorial identities used to separate incoming forests from non-incoming edge subsets.
The subtraction of correction terms from sigma uses basic properties of symmetric polynomials; no proof given, standard background.
domain assumption A two-cycle formed by two opposite arcs between vertices n-1 and n is treated as a cycle, so including both arcs violates the incoming forest condition.
Used in Proposition 3.3 to identify the only non-forest subsets.

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Pith. "Pith review of Graph-Based Proofs of Indistinguishability of Linear Compartmental Models." pith.science (2026). https://pith.science/paper/QUQB7A53

@misc{pith2026241201135,
  author       = {Pith},
  title        = {Pith review of: Graph-Based Proofs of Indistinguishability of Linear Compartmental Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QUQB7A53}},
  note         = {Machine review of arXiv:2412.01135}
}

read the original abstract

Given experimental data, one of the main objectives of biological modeling is to construct a model which best represents the real world phenomena. In some cases, there could be multiple distinct models exhibiting the exact same dynamics, meaning from the modeling perspective it would be impossible to distinguish which model is ``correct.'' This is the study of indistinguishability of models, and in our case we focus on linear compartmental models which are often used to model pharmacokinetics, cell biology, ecology, and related fields. Specifically, we focus on a family of linear compartmental models called skeletal path models which have an underlying directed path, and have recently been shown to have the first recorded sufficient conditions for indistinguishability based on underlying graph structure. In this recent work, certain families of skeletal path models were proven to be indistinguishable, however the proofs relied heavily on linear algebra. In this work, we reprove several of these indistinguishability results instead using a graph theoretic framework.

Figures

Figures reproduced from arXiv: 2412.01135 by the authors.

**Figure 1.** On the left is an undirected graph G and on the right is a directed graph G′ , both defined in Example 1.2. From this definition of incoming forest, we can define the set of all incoming forests within a graph G with k edges to be Fk(G). We can also define the incoming forests on a graph G with k edges that also include a path from vertex i to vertex j as F i,j k (G). Example 1.4. The graph G from Example 1.2 seen i… view at source ↗

**Figure 2.** The model described in Example 1.9 families of linear compartmental models, including cycle, mammillary, catenary, and tree models [2, 5, 10, 14]. Definition 1.8. A model M = (G, In, Out, Leak) is a skeletal path model if G contains a directed path from compartment 1 to n, with In = {1} and Out = {n}. In other words, skeletal path models will have a graph that contains a “backbone” of a path along with other vertice… view at source ↗

**Figure 3.** The model described in Example 1.17 where H = P4 ∪ {4 → 3} Example 1.15. The coefficient map of the model M3 seen in Example 1.9 with corresponding input-output equation seen in Example 1.13 is given by: c : R 4 → R 4   a03 a21 a32 a43   7→   a21 + a32 + a03 + a43 a21a32 + a21a03 + a21a43 + a32a03 + a32a43 a21a32a43 + a21a32a03 a21a32a43   There are many interesting questions related to the coeffici… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: On the left is the Pe4 graph corresponding to M3 as defined in Example 1.9. On the right is the He∗ graph corresponding to M4 as defined in Example 1.17. The graph on the right of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

An introduction to compartm ental modeling for the bud- ding infectious disease modeler

Julie C Blackwood and Lauren M Childs. An introduction to compartm ental modeling for the bud- ding infectious disease modeler. Lett. Biomath., 5(1):195–221, 2018

work page 2018

[2] [2]

Identiﬁability of linear compartmental tree models and a general formula for the input-output equations

Cashous Bortner, Elizabeth Gross, Nicolette Meshkat, Anne Sh iu, and Seth Sullivant. Identiﬁability of linear compartmental tree models and a general formula for the input-output equations. Advances in Applied Mathematics , 146, May 2023

work page 2023

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Identiﬁable paths and cycles in linear compartmental mod- els

Cashous Bortner and Nicolette Meshkat. Identiﬁable paths and cycles in linear compartmental mod- els. Bulletin of mathematical biology , 84(5):53, March 2022

work page 2022

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Graph-based suﬃcien t conditions for the indistinguishabil- ity of linear compartmental models

Cashous Bortner and Nicolette Meshkat. Graph-based suﬃcien t conditions for the indistinguishabil- ity of linear compartmental models. SIAM Journal on Applied Dynamical Systems , 23(3):2179–2207, 2024

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Identifiability of Linear Compartmental Models: The Impact of Removing Leaks and Edges

Patrick Chan, Katherine Johnston, Anne Shiu, Aleksandra Sobie ska, and Clare Spinner. Identiﬁ- ability of linear compartmental models: The impact of removing leaks a nd edges. Available from arXiv:2102.04417, 2021

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Keith Godfrey. Compartmental Models and their Application . Academic Press, 1983

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