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REVIEW 2 major objections 4 minor 46 references

A graph vertex's importance can be measured by how much (Eulerian) magnitude homology drops when the vertex is deleted, and the drop at scale l depends only on the vertex's l-hop neighborhood.

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2026-08-01 21:27 UTC pith:U4RR3CBP

load-bearing objection New local centrality construction with a correct-looking locality theorem, but the weak size axiom claim is contradicted by the paper's own formulas and must be fixed. the 2 major comments →

arxiv 2607.16377 v1 pith:U4RR3CBP submitted 2026-07-17 math.AT cs.SI

A Centrality Measure Using Magnitude Homology

Nadja H\"ausermann , Bastian Rieck This is my paper
classification math.AT cs.SI
keywords magnitude homologyEulerian magnitude homologycentralityBetti numberslocalitygraph homologyrelative homologyvertex importance
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The authors introduce a family of centrality measures for undirected graphs built from magnitude homology, a bigraded homological invariant of a graph. For a fixed scale l, the centrality of a vertex is the summed drop in Betti numbers of magnitude homology, or its Eulerian variant, when that vertex is removed. They prove the central property of locality: at scale l, the measure depends only on the induced subgraph of the vertex's l-hop neighborhood. This makes l a tunable range parameter, recovers degree centrality at l=1, and gives a new ranking of vertices that agrees partly with standard measures while distinguishing graphs that classic isomorphism tests do not.

Core claim

The central claim is that subtracting the homology of the graph after deleting a vertex from the homology of the whole graph yields a meaningful, local measure of that vertex's role. Writing beta for the Betti numbers of magnitude homology, and beta^E for Eulerian magnitude homology, the measure is C_MH(G,v,l) = sum over l'=0..l, k'=0..l' of (beta_{k',l'}(G) - beta_{k',l'}(G without v)), with the Eulerian version defined analogously. The load-bearing result, Proposition 3.16, states that C_MH(G,v,l) = C_MH(G_v^l,v,l) and likewise for the Eulerian version, so the centrality is local: the global graph can be replaced by the subgraph induced by the l-hop neighborhood of v. From this the paper d

What carries the argument

The machinery is the (Eulerian) magnitude homology of a graph over F2, whose Betti numbers beta_{k,l} count independent length-l chains that pass through k+1 landmarks, modulo boundaries. The centrality is simply the summed Betti-number drop after deleting the vertex. The proof of locality rests on a support lemma: any minimal cycle contributing to the drop has all its vertices inside the l-hop neighborhood of the deleted vertex, because two length-l walks with the same endpoints form a closed walk of length 2l, forcing any two support vertices to be within distance l. This lemma, together with the map induced in homology by inclusion, yields Proposition 3.16.

Load-bearing premise

The Eulerian version's locality is backed by a one-line assertion that the proof is the same as the standard case, without a detailed check that the minimal-cycle and support arguments survive the no-repeated-vertices restriction; if that check fails, C_EMH loses its local character and its main theoretical guarantee.

What would settle it

Enumerate all connected simple graphs up to eight vertices; for every vertex v and l=2,3 compute C_EMH(G,v,l) and compare it with C_EMH(G_v^l,v,l). Any mismatch would refute the Eulerian locality claim. The same exhaustive comparison for the standard version would independently confirm or refute Proposition 3.16 on small graphs.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Because the measure is local, computing the centrality of a vertex at scale l only requires the induced subgraph of its l-hop neighborhood, making l an explicit range-versus-cost tradeoff.
  • At l=1 the measure reduces to degree (up to the constant 1+2*deg), so the construction is a graded extension of a classical centrality.
  • The measure can separate graphs that are hard for standard isomorphism heuristics: two strongly regular graphs with the same parameters have equal l=2 values but differ at l=3.
  • In experiments on small connected graphs, the new measures correlate most strongly with subgraph centrality and least with betweenness centrality, indicating they capture local subgraph participation rather than global path brokerage.
  • The standard variant with l=2,3 satisfies the density axiom and the weak size axiom; empirical checks support the top-node axiom for all tested variants and the endpoint-increase axiom for standard magnitude centrality on small graphs.

Where Pith is reading between the lines

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

  • One could treat the whole family {C(G,v,l)}_l as a multiscale importance profile for each vertex; vertices with equal degree or equal l=2 rank can still be separated by how their centrality changes across scales.
  • The negative Betti-number differences observed for some regular graphs suggest splitting the measure into a positive part (homology destroyed by deletion) and a negative part (homology created by deletion); the negative part may flag vertices whose removal makes the graph topologically more complex.
  • Given the strong empirical correlation with subgraph centrality and the known counting interpretation of Eulerian magnitude homology, a closed-form or spectral approximation of this centrality may be possible, which would remove the current computational bottleneck and allow application to large networks.

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 / 4 minor

Summary. The paper defines a new family of vertex centrality measures using (Eulerian) magnitude homology. For a vertex v and a degree bound l, the centrality is the total change in (Eulerian) magnitude homology Betti numbers up to bidegree (l,l) when v is removed. The main theoretical result is locality: the centrality of v depends only on the l-hop neighbourhood of v (Prop. 3.16). The paper also studies a list of centrality axioms, reports computational experiments comparing the measure with degree, betweenness, closeness, subgraph, and PageRank centrality, and shows that the measure distinguishes the Rook and Shrikhande graphs for l=3.

Significance. The idea of deriving a local vertex centrality from magnitude homology is novel and well motivated, and the locality theorem for the standard magnitude-homology version appears plausible and would be a useful contribution. The paper also provides reproducible code and extensive empirical comparisons, which are strengths. However, the axiomatic section contains a concrete numerical contradiction: the claimed weak size axiom is false for the standard version, and the corresponding discussion for the Eulerian version is also incorrect. The Eulerian locality claim is deferred with 'the proof is the same' and is not actually verified. These issues affect the paper's central claim that the measure satisfies several expected properties, so the current version needs substantive revision.

major comments (2)
  1. [Section 4.1, Definition 4.1, Table 7] The claim that CMH satisfies the weak size axiom for l∈{2,3} is directly contradicted by the formulas derived in the same section. Definition 4.1(i) requires that for every n there exists P_n such that for all p≥P_n the centrality of a cycle node in C_p⊔K_n exceeds that of a complete-graph node. The paper computes CMH(C_p,v,2)=9 for p>5 and CMH(K_n,v,2)=3n^2−5n+3. For n=3, the complete-graph value is 15, already larger than 9, and the gap grows with n; hence no such P_n exists. The l=3 case is identical: CMH(C_p,v,3)=13 for p>7, while CMH(K_n,v,3)=4n^3−12n^2+14n−5 equals 37 for n=3 and grows with n. Consequently Table 7's 'weak ✓' for CMH is incorrect. The Eulerian discussion in the same section is also backwards: CEMH(K_4,v,2)=25 vs. CEMH(C_p,v,2)=5 shows that weak size axiom fails for n=4 (the complete-graph value is already larger, not smaller), so the assertion that this 'is enough t
  2. [Section 3.3, after Definition 3.7; Prop. 3.16] For the Eulerian version, the paper states 'For the Eulerian version the proof is the same' and gives no details. This is a load-bearing gap: the Eulerian chain complex forbids repeated vertices, so the supporting objects in the proof of Prop. 3.16 — minimal-cycle decompositions (Lemma 3.12), support containment (Lemma 3.14), and the two-type analyses of compl(im f_*) and ker(f_*) — all need to be re-verified for EMC. If any step fails, CEMH would not be local, and the Eulerian variant's main theoretical property would be unsupported. I found no obvious obstruction, but the manuscript as written does not provide the proof. Please supply the Eulerian argument explicitly or give a precise reduction to the standard case.
minor comments (4)
  1. [Section 3.1 and Section 3.3] Internal references are inconsistent: 'Theorem 2.8' should be Proposition 2.8, and in the proof of Prop. 3.16 'Theorem 2.13' is actually Remark 2.13, while 'Theorem 3.14' and 'Theorem 3.15' should be Lemma 3.14 and Remark 3.15. Please correct.
  2. [Section 4.1] The sentence 'The full size axiom cannot hold because magnitude homology is a local centrality measure' is not by itself an argument; locality does not automatically preclude the full size axiom. The concrete formulas in the same section are what show the failure. Consider replacing this sentence with a direct derivation.
  3. [Section 4.1] Typo: 'The magnitude homology for for trees' should read 'for trees'. There are also several similar small typos throughout (e.g., 'homlogy').
  4. [Section 5.1, table after centrality values] The table header mixes CMH and CEMH columns and only one vertex name appears in the header. Since both graphs are vertex-transitive this is not mathematically harmful, but the table should be relabelled to clarify which columns refer to which graph and which homology variant.

Circularity Check

0 steps flagged

No significant circularity: the centrality measure is defined directly from magnitude homology Betti numbers, and the paper's support comes from external, non-self-cited results.

full rationale

The derivation chain is self-contained rather than circular. The centrality measure is explicitly defined as a difference of Betti numbers (Definition 3.3), with no fitted parameters and no quantity that is reused as its own prediction. The locality theorem (Prop. 3.16) is proved from the definition of minimal cycles, the support lemma (Lemma 3.14), and the induced map in homology, resting on Hepworth–Willerton's and Gu's external results rather than on a self-citation. The paper's known l=1 recovery of degree is acknowledged as such, not presented as a prediction. Self-citations involving Rieck are limited to background references on Weisfeiler–Leman tests and magnitude in machine learning, and they are not load-bearing for the main claims. The empirically tested axioms are external benchmarks and separate from the construction. The suspicious passages noted by the reviewer—the claimed weak size axiom in Section 4.1 and the Eulerian 'proof is the same' in Section 3.3—are correctness or proof-gap concerns, not circularity: they do not reduce a derived output to an input by definition or by self-citation. Accordingly, the honest circularity finding is a score of 0.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical or graph-theoretic entities. Its free parameter is the locality degree l. The main external inputs are the established theory of magnitude homology and known Betti-number formulas, plus the assumption that the Eulerian locality proof transfers without modification.

free parameters (1)
  • l = 2 or 3 (in experiments)
    The degree (locality scale) of the centrality is user-chosen. It is a hyperparameter of the measure, not fitted to data, but it changes the value and locality radius.
axioms (4)
  • standard math Standard magnitude homology definitions and properties from Hepworth-Willerton [22] and Leinster [28] are assumed.
    The construction of Magnitude chain complexes, differentials, and the k≤l vanishing are cited from prior work as background.
  • domain assumption Betti number formulas for complete graphs, trees, and cycles (for the axiom verification) are taken from [22] and [18].
    Section 4.1 uses β_{k,l}(K_n)=n(n-1)^l and β_{k,l}(T_n) formulas; Section 4.2 uses cycle Betti numbers from [18]. These are external results, not re-proven here.
  • standard math The inclusion map G\{v}→G is a graph map and induces the chain maps used in Prop. 3.8.
    Definition 2.12/2.14 guarantee induced maps in homology; this is standard category-of-graphs machinery.
  • ad hoc to paper For the Eulerian version, the locality proof 'is the same' as the standard case.
    Section 3.3 after Def. 3.7: the paper asserts the Eulerian proof is identical, but does not provide it. The Eulerian chain complex has stricter distinctness conditions, so the minimal-cycle and support arguments may require separate verification.

pith-pipeline@v1.3.0-alltime-deepseek · 23692 in / 24986 out tokens · 205845 ms · 2026-08-01T21:27:32.346285+00:00 · methodology

0 comments
read the original abstract

The magnitude of a metric space constitutes an expressive invariant that subsumes numerous different geometrical-topological invariants. Building on recent advances in magnitude homology, i.e., a bigraded homology theory that recovers the magnitude, we develop a novel local measure of the centrality or importance of nodes in a graph. Our measure is inspired by the concept of relative homology as it considers the change in magnitude homology when removing a vertex. We show that our proposed measure satisfies several properties a centrality measure is reasonably expected to respect and demonstrate that we introduce a new perspective on centrality by comparing to several established centrality measures.

discussion (0)

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