Mapper risk tiers on breast-cancer expression that survive a held-out split, beside a discordant-survivor result defined by the outcome it predicts.

Mapper Survival Analysis on TCGA-BRCA

A review of Kibisi, Abawonse and Woukeng (2026)

Primary source: Emmanuel Kibisi, Olakunle Abawonse, and Donald Woukeng (African Institute for Mathematical Sciences, Rwanda), “Topology-Informed Survival Analysis of Breast Cancer Patients Using the Mapper Algorithm,” submitted to arXiv 16 July 2026, 25 pages, 14 figures. arXiv:2607.15022 | arXiv DOI

Access and reuse: the article declares CC BY-SA 4.0, the first share-alike source this corpus has reviewed and a licence the shared policy table carried no row for until this note needed one. The row landed upstream in @galaxy-foundry/license-policy@0.7.0 as verbatim-ok with the copyleft obligation, on the grounds that isolation answers share-alike the same way it answers the GPL. So verbatim carry is permitted here. This note is an own-words summary anyway — a note may always summarize in its own words, and nothing in this paper is worth carrying under an isolation obligation. Numbers, parameter values, and gene symbols below are facts and are reproduced as such; no upstream prose is.

Why this note exists. mapper is the one method in this corpus with implementations and no application evidence — it links kmapper-environment, giotto-tda-environment, and scikit-tda-environment, and no paper showing the technique doing biological work. This is that paper. It is also, usefully, a paper that exercises every caution mapper states.

What the paper does

It builds on Rostami, Fooshee, Carlsson and Subramaniam (IEEE OJEMB 6, 465–471, 2025), who applied Mapper to TCGA-BRCA expression in two directions at once — a sample network whose nodes are clusters of patients, and a feature network whose nodes are clusters of co-expressed genes — and found a Luminal B subgroup in it. Rostami and colleagues stopped at structure. This paper reproduces that dual construction and then asks the question they did not: does a patient’s position in the graph carry survival information?

Four steps follow: reproduce the two networks; run node-level survival analysis on the sample network; identify patients whose outcomes diverge from their PAM50 subtype’s expectation; and test whether risk tiers derived from the graph transfer to patients who were not used to build it.

The construction is a two-dimensional lens — L1 (Manhattan) eccentricity paired with the first principal component of the expression matrix — a cover of 12 intervals at 50% overlap, and DBSCAN under Euclidean distance with ε = 55, min_samples = 2 for the reproduction and 5 for the train-test graph. The cohort is described only as more than 1,000 TCGA-BRCA patients.

The result that holds

The train-test section is the paper’s real contribution and is structurally sound. The Mapper graph is built on the 80% training split alone. A univariate Cox model is fitted per node on an in-node indicator; node hazard ratios are rank-transformed into tertiles, giving High, Mid and Low risk tiers; each patient takes the tier of their highest-hazard node. Held-out patients — who are not in the graph — are placed by Pearson correlation of their expression profile against each training node’s mean profile, inheriting a tier directly above 0.65 and by a Borda count over positively correlated nodes otherwise. Clinical variables are withheld until after tiers are assigned.

That design is the right shape: the graph never sees the validation patients, and the transfer rule is stated precisely enough to implement. The risk ordering survives it, with a significant global log-rank test on held-out patients and the High-versus-Low contrast driving the separation.

A second check is more persuasive than the survival result and gets less space. Mean expression of six proliferation markers (MKI67, CCNB1, MYBL2, BIRC5, UBE2C, RRM2) is monotone across the tiers — 0.1552, −0.1595, −0.1634 from High to Low, one-way ANOVA p = 0.000158 — even though no proliferation measurement entered the graph’s construction. A partition built from expression geometry alone lands on an axis oncology already knows is prognostic. That is a real finding about what Mapper recovers, and it does not depend on any survival claim.

The reporting, though, thins out exactly where the headline lives. Section 4.2.1 states that held-out High Risk patients faced substantially elevated mortality after adjustment for age, stage, and treatment, and reports no hazard ratio, no confidence interval, no p-value, and no group sizes. The abstract’s strongest claim rests on a paragraph containing no statistics.

The result that does not

The discordant-survivor analysis is circular, and the circularity is definitional rather than subtle. Discordant Basal patients are Basal-like patients who survived longer than the median Luminal A patient. Discordant Luminal A patients are Luminal A patients who died before the median Basal-like survival time. Group membership is a function of each patient’s own observed survival.

Those groups are then plotted as Kaplan–Meier curves against their non-discordant counterparts, where they separate, and entered into Cox models with survival as the outcome, where the discordant flag is significant after adjusting for age and stage: HR = 0.223 (p = 0.0005) among Basal-like patients, HR = 65.91 (p < 0.0001) among Luminal A. A hazard ratio of 65.91 is not a finding; it is what a model returns when the predictor is a relabelling of the outcome. The separation in the survival curves is guaranteed by construction, and the paper neither notes this nor lists it among its limitations.

The differential expression built on those groups is a different matter and survives the objection. Asking which genes distinguish long-surviving Basal-like patients from short-surviving ones is ordinary outcome-stratified differential expression, and the answer reported — a cross-subtype reversal, with discordant Basal patients expressing AKAP12, TCEAL7, MYH11, KANK2, SMOC2 and other differentiation-associated genes from the Luminal-A-associated segment of the feature network — is a legitimate hypothesis. What it is not is a topological result.

What Mapper contributes, precisely

This distinction is worth stating because the paper blurs it. Mapper does not identify the discordant patients; PAM50 labels and survival times identify them, and Mapper is applied afterwards to say where they sit — near transitional and mixed nodes at subtype boundaries rather than in subtype cores. Localization is a genuine contribution and it is the weaker of the two claims the text makes for it.

One further claim outruns its evidence. Four genes read off the feature network (ERBB2, HORMAD1, plus ESR1/CA12/XBP1) are said to achieve a subtype separation that conventional classifiers need the full 50-gene PAM50 panel to approximate. No classifier is trained and no accuracy is reported; the evidence is a 3D scatter plot coloured by PAM50 label, and the genes were selected by inspecting the feature network coloured by those same labels.

The comparison that is missing

The paper’s organizing claim is that topology carries prognostic information beyond PAM50. PAM50 appears as a covariate — the global training Cox model adjusts for subtype alongside age, stage, treatment, and proliferation score, and the High Risk tier is reported to remain significant — but never as a competing model. There is no PAM50-only survival model fitted on the same split, no discrimination metric such as a concordance index, no likelihood-ratio comparison, and no numbers from the global model at all. Adjustment answers “is there residual signal”; it does not answer “is this better than what the clinic already uses”, which is the question the abstract’s phrasing invites.

Parameters, and the diagnostic that is absent

mapper states the rule this paper is the test case for: vary the parameters and show it. Lens, interval count, overlap fraction, and clustering parameters are all fixed at single values and never varied. The authors name parameter sensitivity as a limitation and run no sensitivity analysis, so a reader has no way to tell which findings survive a different cover. min_samples also changes from 2 to 5 between the reproduction and the train-test graph without comment.

ε = 55 compounds this. It is a Euclidean radius in standardized gene space, so its meaning depends entirely on how many genes are retained — and the gene count is never stated, nor is the filtering criterion, nor the exact cohort size, nor the assay or normalization beyond z-scoring. There is no data-availability statement, no code repository, and no named Mapper implementation. TCGA-BRCA is public; this analysis of it is not reconstructible from the paper.

What it gets right that is easy to miss

The landmark-node section reports a null and explains it. Distance to the highest-hazard node (HR = 2.94) separates survival by tercile at p = 0.0228, and the continuous-distance hazard ratio then moves from 0.90 (p = 0.075) to 1.01 (p = 0.90) once age and stage enter the model. Rather than dropping the result, the authors diagnose it: mean age falls from 71.3 to 54.4 years moving away from the landmark, and modal stage shifts from IIIA to IIA. The topological gradient was tracking demographics. Reporting that plainly, and listing it again under limitations, is better practice than most of what this corpus’s survey notes describe.

What a reader here should take from it

As application evidence for Mapper, this is a genuine but narrow positive: a graph built from expression alone recovers a proliferation axis it was never shown, and a partition derived from it retains prognostic ordering on held-out patients from the same cohort. That is worth knowing and it is roughly the strength of claim mapper says the method can support.

Everything stronger in the paper fails for a reason mapper already names. The discordant-survivor hazard ratios are a Mapper graph being treated as evidence for a subgroup that the outcome variable defined. The parameter choices are unvaried, so the filter is the model and no one can see how much the model chose. And the comparison to PAM50 — the claim in the title’s spirit and the abstract’s last sentence — is an adjustment, not a benchmark. A note in this Foundry making the same claim would need a matched non-topological baseline on an identical split with a stated metric, which is exactly what is absent here.

mapper is the method; this note is its first application evidence, and reads as a case study in that note’s three cautions. persistent-homology is the stable counterpart whose stability theorem is precisely what would make an unvaried-parameter analysis defensible, and which Mapper does not have.