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The agreement distance of unrooted phylogenetic networks

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Pith's one-line read The paper proves that the agreement distance between two unrooted phylogenetic networks is within a factor of two of their TBR distance and within a factor of four of their prune-and-regraft distance.

desk verdict A solid unrooted-network extension of agreement forests with useful constant-factor bounds on TBR and PR distances; mostly right, but the endpoint ordered-embedding proof is a genuine gap. read the letter →

arxiv 1908.08183 v4 pith:H6MUVJTF submitted 2019-08-22 math.CO cs.DMq-bio.PE

Jonathan Klawitter This is my paper · ORCID
classification math.COcs.DMq-bio.PE MSC 05C0505C9092D15
keywords phylogeneticnetworkrearrangementoperationagreementdistancemaximumgraphendpointTBRPR
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 extends the classical maximum agreement forest idea to unrooted phylogenetic networks, defining a maximum agreement graph whose minimal number of disagreement edges forms the agreement distance. It proves this distance is a metric, agrees with the TBR distance on trees and on tree-network pairs, and bounds the TBR distance of any two proper binary networks within a factor of two. A second construction, maximum endpoint agreement graphs, gives an endpoint agreement distance that bounds the prune-and-regraft distance within a factor of three. This matters because rearrangement distances are NP-hard to compute and previously lacked a graph-based certificate for networks; a single shared structure that approximates these distances is a step toward algorithms and heuristic searches.

What carries the argument

The maximum agreement graph (MAG): a graph whose connected components split into agreement subgraphs—parts that embed identically into both networks—and disagreement edges, single edges on unlabelled vertices that account for edges moved or added when passing from one network to the other. A MAG is an agreement graph with the fewest disagreement edges, and the count of those edges defines $d_{\mathrm{AD}}$. For endpoint agreement, a maximum endpoint agreement graph is the same idea but with one-ended sprouts that model prune-and-regraft moves; the endpoint agreement distance counts sprouts on agreement subgraphs plus disagreement edges. The argument that these counts are metrics and that they sandwich the rearrangement distances runs through ordered agreement embeddings, which fix a systematic ordering of disagreement edges, and through explicit constructions that turn a rearrangement sequence into an agreement graph and vice versa.

What would settle it

Run a brute-force check over small proper binary networks, testing whether every maximum endpoint agreement graph admits an ordered agreement embedding; a single failure would supply a counterexample to the proof of the upper bound $d_{\mathrm{PR}} \le 3\,d_{\mathrm{EAD}}$ as written.

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Extended reading notes

Core claim

The central claim is Theorem 5.7: for any two proper unrooted binary phylogenetic networks on the same taxa, $d_{\mathrm{AD}}(N,N') \le d_{\mathrm{TBR}}(N,N') \le 2\,d_{\mathrm{AD}}(N,N')$. The agreement distance is therefore not just a lower bound; it is a constant-factor approximation of the TBR distance, and through the known two-sided comparison between PR and TBR it also satisfies $d_{\mathrm{AD}}(N,N') \le d_{\mathrm{PR}}(N,N') \le 4\,d_{\mathrm{AD}}(N,N')$. The paper also proves that the endpoint agreement distance is a metric and satisfies $d_{\mathrm{EAD}}(N,N') \le d_{\mathrm{PR}}(N,N') \le 3\,d_{\mathrm{EAD}}(N,N')$. For trees, and whenever one network is displayed by the other, the agreement distance coincides with the TBR distance; the gap between the distances is a genuinely network phenomenon, exhibited by a pair of tier-seven networks with agreement distance $2$ and TBR distance $3$.

Load-bearing premise

The metric property of the endpoint agreement distance and the factor-three PR bound both depend on the claim that every maximum endpoint agreement graph admits an ordered agreement embedding; that existence is asserted by analogy to an earlier result rather than proven in this paper.

Editorial extensions

If this is right

  • Because $d_{\mathrm{AD}}(N,N') \le d_{\mathrm{TBR}}(N,N') \le 2\,d_{\mathrm{AD}}(N,N')$, any algorithm that computes $d_{\mathrm{AD}}$ gives a 2-approximation of the TBR distance and, through Corollary 5.8, a 4-approximation of the PR distance.
  • The agreement distance equals the TBR distance on trees and on tree-network pairs, so in those settings there is no approximation loss at all.
  • Both $d_{\mathrm{AD}}$ and $d_{\mathrm{EAD}}$ are metrics, so they give network comparison scores that obey the triangle inequality, a property useful for clustering or outlier detection among inferred networks.
  • Computing the agreement distance is NP-hard, so the new constant-factor bounds are structural certificates rather than immediate practical algorithms.

Reading between the lines

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

  • A practical reading of Theorem 5.7 is that an exact or approximate solver for maximum agreement graphs would immediately yield approximation algorithms for TBR and PR distances; the paper stops short of proposing such an algorithm.
  • The gap between $d_{\mathrm{AD}}$ and $d_{\mathrm{TBR}}$ is not known to be tight, since the only exhibited separation is $2$ versus $3$; tightening the factor-two bound would likely require understanding when ordered agreement embeddings force extra rearrangement moves.
  • For search heuristics, the endpoint agreement distance looks more natural than the ordinary agreement distance for prune-and-regraft moves because its sprouts model one-ended pruning; that analogical advantage is not tested in this paper.
  • The tier-seven separating example leaves open whether agreement distance and TBR distance coincide on low-tier networks; checking tiers one through six directly would be a small computational experiment.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper introduces maximum agreement graphs (MAGs) and maximum endpoint agreement graphs (MEAGs) for unrooted, binary, proper phylogenetic networks, generalizing maximum agreement forests from trees. It defines the agreement distance dAD and the endpoint agreement distance dEAD, proves that both are metrics, and studies their relations to the TBR and PR distances. The main results are dAD(N,N') ≤ dTBR(N,N') ≤ 2 dAD(N,N') (Theorem 5.7), the resulting bound dAD ≤ dPR ≤ 4 dAD (Corollary 5.8), and dEAD ≤ dPR ≤ 3 dEAD (Theorem 5.10), together with dAD ≤ dEAD ≤ 2 dAD (Proposition 5.9). The paper also shows that dAD coincides with the TBR-distance on unrooted trees and on tree/network pairs, and gives an example where the two distances differ for networks. The overall contribution is a constant-factor approximation framework for rearrangement distances on unrooted phylogenetic networks.

Significance. If the results hold, this is a meaningful step: it provides the first constant-factor bounds relating agreement-type distances to TBR and PR distances on unrooted networks, and the chain dAD ≤ dPR ≤ 4 dAD gives a single proxy for two rearrangement metrics. The constructions in Lemma 5.5 and Lemma 5.6 are detailed and nontrivial, and the paper is generally careful in building on published results rather than re-deriving them. However, the endpoint version of ordered agreement embeddings is only asserted by analogy with prior work and a short outline, and this lemma is load-bearing both for the metric property of dEAD and for the upper bound in Theorem 5.10. The main ideas are credible, but the manuscript is not fully vetted until that gap is closed.

major comments (2)
  1. [Section 4, ordered agreement embedding; used in Proposition 4.1 and Theorem 5.10] The existence of ordered agreement embeddings for endpoint agreement graphs is asserted without a complete proof. The text says the proof works analogously to Theorem 3.1 and Lemma 3.2 of Klawitter (2019) and gives a two-sentence outline. This existence is load-bearing: Corollary 4.2 obtains the metric property of dEAD from Proposition 4.1, whose reverse inequality fixes ordered embeddings of a MEAG into N and N′, and Theorem 5.10 fixes such embeddings and uses the ordering of disagreement edges when adding edges by PR+. The outlined argument does not show that the two conditions—no sprout of an agreement subgraph attached to a disagreement edge, and the ordering condition Ej may be attached to Ei only for i ≤ j—can be enforced simultaneously in the unrooted endpoint setting, where agreement subgraphs may themselves carry sprouts. An embedding change that detaches a sprout from a disagreement edge can alter which disagreement edges are attached to which, and the interaction with the ordering condition is not addressed. I request a full proof of the existence of ordered agreement embeddings for endpoint agreement graphs, or an explicit reduction to the rooted lemma that accounts for sprouts in agreement subgraphs.
  2. [Section 5, Lemma 5.3] The converse direction of Lemma 5.3 contains the displayed inequality dAD(T,N) ≥ dAD(T,T′) + dAD(T,N) = dTBR(T,N). The middle term dAD(T,N) makes the inequality self-referential and cannot be correct; the intended second summand is presumably dAD(T′,N), which equals r when T′ is displayed by N. As printed, the proof of the lower bound dAD(T,N) ≥ dTBR(T,N) is invalid, although the surrounding argument suggests a straightforward repair by replacing the erroneous term and justifying why the ordered embedding of G into N yields an agreement graph of T and T′ with k−r disagreement edges. Please correct the equation and expand this step.
minor comments (5)
  1. [Throughout] Several cross-references use the wrong kind of label: 'Theorem 3.1' should be Lemma 3.1, 'Theorem 2.1' should be Observation 2.1, 'Theorem 3.3' should be Proposition 3.3, 'Theorem 5.2' should be Lemma 5.2, and 'Theorem 4.1' should be Proposition 4.1. Please harmonize the references.
  2. [Section 4, proof sketch after the definition of ordered agreement embedding] The outline begins 'Starting with an agreement embedding of G into N′', but the object being embedded is the endpoint agreement graph, which has been called H; the symbol G is not defined in this paragraph.
  3. [Concluding remarks] The text says 'maximum endpoint agreement graphs (MAEGs)', whereas the body of the paper consistently uses the abbreviation MEAG. Please make the abbreviation uniform.
  4. [Section 5, proof of Proposition 5.9] The sentence 'Since ¯v is either a degree vertex or a labelled leaf' should state the degree explicitly, presumably 'degree-three vertex'; as written it is ambiguous.
  5. [Abstract] The phrase 'maximum agreement graphs as a generalisations' should be 'as a generalization' or 'as generalizations'.

Circularity Check

1 steps flagged · score 4.0 of 10

dAD/dTBR bounds are self-contained, but the EAD metric and dPR≤3·dEAD depend on ordered endpoint agreement embeddings whose existence is deferred by analogy to the author's own prior work.

  1. self citation load bearing [Section 4, definition of ordered agreement embedding for endpoint agreement graphs]
    "A proof that ordered agreement embeddings exists works analogously to the proof of Theorem 3.1 and the proof of Lemma 3.2 (Klawitter, 2019), yet we outline the proof idea here."

    The existence of ordered agreement embeddings for endpoint agreement graphs is asserted by analogy to the author's own prior rooted-network proof and not proved in this paper. This assertion is load-bearing: Proposition 4.1 uses fixed ordered embeddings for the inequality dEAD ≥ dR, Corollary 4.2 (dEAD is a metric) inherits this, and Theorem 5.10 fixes ordered endpoint embeddings to construct the PR-sequence for dPR ≤ 3 dEAD. Unlike the rooted case, endpoint agreement subgraphs may themselves contain sprouts, so the cited analogy does real work. The EAD metric and the PR upper bound therefore rest on a self-citation rather than on a self-contained derivation.

full rationale

The main agreement-distance chain is not circular. dAD is defined from MAGs and compared to dTBR via explicit construction: Lemma 5.5 turns any TBR-sequence into an agreement graph, and Lemma 5.6 turns a MAG into a TBR-sequence using Lemma 3.1, whose ordered-embedding existence is proved in this paper. Proposition 3.3 and Lemma 5.2 use Allen–Steel and Janssen–Klawitter as ordinary published, parameter-free results; Corollary 5.8 is a straightforward composition. No fitted quantity is renamed as a prediction. The endpoint side is less self-contained: Section 4 states rather than proves the existence of ordered agreement embeddings for MEAGs, saying the proof works analogously to the author's rooted-network paper and giving a two-sentence outline. Corollary 4.2 and Theorem 5.10 fix such embeddings, so if the analogy fails—endpoint agreement subgraphs may carry sprouts—dEAD could fail to be a metric and the dPR ≤ 3 dEAD construction would lack support. This is a load-bearing reliance on a self-citation, although not a definitional or fitted circularity. One line in Lemma 5.3, 'dAD(T,N) ≥ dAD(T,T′) + dAD(T,N)', is evidently a typo; the intended dAD(T′,N) term makes the displayed inequality correct. Net: no reduction of the AD results to their inputs, but one central EAD claim rests on an unproved self-cited analogy, so the score is moderate rather than zero.

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

The paper introduces no free parameters and no empirical entities. The definitions of maximum agreement graphs and maximum endpoint agreement graphs are new mathematical constructions, not fitted quantities. The main axioms are the prior definitions and metricity results for TBR and PR on unrooted networks from Janssen and Klawitter (2019), taken as domain assumptions.

assumptions (4)
  • domain assumption The TBR and PR rearrangement operations and their induced distances on unrooted proper binary phylogenetic networks are well-defined metrics, as established by Janssen and Klawitter (2019, Corollary 4.4).
    The paper builds on these prior definitions and metric properties (Section 2, Distances). This is a domain assumption from the author's cited prior work, not proven here.
  • domain assumption For a tree T and a network N, the tree T' displayed by N that minimises dTBR(T,N) = dTBR(T,T') + dTBR(T',N) exists (Janssen and Klawitter 2019, Theorem 4.13).
    Used in the proof of Lemma 5.3, which itself contains a typo; the result is cited from a self-authored prior paper.
  • domain assumption The network space is restricted to binary, proper networks; parallel edges are allowed, but networks are always proper unless otherwise stated.
    The definitions and proofs rely on degree-three non-leaf vertices and on cut-edges separating leaves (Section 2).
  • standard math Standard graph-theoretic operations: vertex suppression, edge subdivision, cyclomatic number, and the existence of DFS-based path searches in connected graphs.
    Used throughout the proofs, e.g., Lemma 3.1.

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Pith. "Pith review of The agreement distance of unrooted phylogenetic networks." pith.science (2026). https://pith.science/paper/H6MUVJTF

@misc{pith2026190808183,
  author       = {Pith},
  title        = {Pith review of: The agreement distance of unrooted phylogenetic networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6MUVJTF}},
  note         = {Machine review of arXiv:1908.08183}
}
abstract

A rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune and regraft (PR) (called subtree prune and regraft (SPR) on trees). Each of these operations induces a metric on the sets of phylogenetic trees and networks. The TBR-distance between two unrooted phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest, that is, a forest with a minimum number of components that covers both $T$ and $T'$ in a certain way. This characterisation has facilitated the development of fixed-parameter tractable algorithms and approximation algorithms. Here, we introduce maximum agreement graphs as a generalisations of maximum agreement forests for phylogenetic networks. While the agreement distance -- the metric induced by maximum agreement graphs -- does not characterise the TBR-distance of two networks, we show that it still provides constant-factor bounds on the TBR-distance. We find similar results for PR in terms of maximum endpoint agreement graphs.

Figures

Figures reproduced from arXiv: 1908.08183 by the authors.

Figure 1
Figure 1. An SPR prunes the edge e in T1 and regrafts it to the edge incident to leaf 1 to obtain T2. A TBR moves the edge e 0 to obtain T3 from T2. On unrooted phylogenetic trees, two popular rearrangement operations are subtree prune and regraft (SPR), which cuts (prunes) an edge at one side and then reattaches it, and tree bisection and reconnection (TBR), which removes an edge and then reconnects the two resulting smaller… view at source ↗
Figure 2
Figure 2. Compared to a sequence of trees that describe a TBR-distance, MAFs provide a single structure [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 2
Figure 2. The graph F is a maximum agreement forest for T1 and T3 of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figures from the paper (11 more)
Figure 3
Figure 3. Figure 3: A PR prunes the edge {u, v} at u in N1 and regrafts it to the edge incident to leaf 3 to obtain N2. A TBR moves the edge {u, v} to obtain N3 from N2. The graph G is a maximum agreement graph for N1 and N3 shown with an embedding into N3. Similar to the TBR-distance of …
Figure 4
Figure 4. Figure 4: An unrooted, binary phylogenetic tree T ∈ uT6 and an unrooted, binary proper phylogenetic network N ∈ uN6. The unrooted, binary phylogenetic network M is improper since the cut-edge e does not lie on a path that connects two leaves. A network N has reticulation number …
Figure 5
Figure 5. Figure 5: Illustration of the TBR operation. The network N2 can be obtained from N1 by a TBR0 that moves the edge {u, v} and the network N3 can be obtained from N2 by a TBR+ that adds the edge {u 0 , v0 }. Each operation has its corresponding TBR0 and TBR− operation, respectivel…
Figure 6
Figure 6. Figure 6: A maximum agreement graph G for N, N0 ∈ uNn. On the right, how G embeds into N 0 . Note that the disagreement edge E2 is only needed for an agreement embedding into N 0 . Note that if G contains m agreement subgraphs, then it also contains at least m−1 disagreement edg…
Figure 7
Figure 7. Figure 7: A maximum agreement graph G for N, N0 ∈ uNn. On the right, how G embeds into N 0 . Note that G contains an agreement subgraph without labelled vertices. forest. Note that the commonly used definition of agreement forests disregards disagreement edges (Allen and Steel, …
Figure 8
Figure 8. Figure 8: An embedding change with respect to u¯ and v¯. Agreement distance. Let N, N0 ∈ uNn. Let G be a MAG of N and N0 with k disagreement edges. We define the agreement distance dAD of N and N0 as dAD(N, N0 ) = k. Note that the agreement distance also equals half the number o…
Figure 9
Figure 9. Figure 9: How to obtain Gi (right) from Gi−1 (left) with respect to the agreement embedding of Gi−1 into Mi−1 when an edge of an agreement subgraph is mapped precisely to e (a) or to a path containing e (b); or when an disagreement edge is mapped to a path containing e (c). It i…
Figure 10
Figure 10. Figure 10: An example for the constructions of the sequences (N 0 = M0, M1, . . . , Mk00 = G 00) and (G 0 = G0, G1, . . . , Gk00 = G). Here, the networks N and N 0 have agreement distance 2 with maximum agreement graph G 0 ; the networks N 0 and N 00 have agreement distance 3 wi…
Figure 11
Figure 11. Figure 11: A maximum endpoint agreement graph H for N, N0 ∈ uNn. On the right, how H embeds into N 0 . is attached to a disagreement edge. This way the first property can be enforced. For the second property, apply embedding changes if Ei that is attached to a disagreement edge …
Figure 12
Figure 12. Figure 12: Two networks N, N0 ∈ uNn with dAD(N, N0 ) = 2, but with dTBR(N, N0 ) = 3 as proven in Theorem 5.4 (for example with a TBR-sequence via M). The graph G illustrated with an agreement embedding into N is a MAG of N and N 0 . Lemma 5.4. The networks N and N0 in [PITH_FUL…
Figure 13
Figure 13. Figure 13: Construction of a TBR-sequence from N to N 0 based on a MAG G of N and N 0 . Let G0 be G without l disagreement edges. Therefore, G0 has k disagreement edges and an agreement embedding into N0 = N without spare disagreement edges. For i ∈ {1, . . . , d} let Gi be Gi−1…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.