The shape of evolution: persistent homology of genetic-distance data as an observable of reticulate processes in pathogen and plant genomes

Evolutionary biology has long theorized processes—recombination, lineage divergence, drug-resistance sweeps, introgression, refugial persistence—whose signatures in genomic data are incompatible with tree structure. We argue that the shape of genetic-distance data, formalized through simplicial complexes and quantified through persistent homology, is a direct observable of these processes. The Vietoris—Rips filtration of a genetic-distance matrix yields the Betti numbers β ₀ (connected components), β ₁ (loops), and β ₂ (cavities); we read β ₁ not as a literal count of recombination events but as a quantity that is monotone in effective recombination above a sampling-dependent geometric baseline, and we organise the resulting shapes into a four-letter alphabet of topological primitives ( K ₁ clonal, K ₂ divergence, K ₃ reticulation, K ₄ higher-order reticulation). Coalescent and Wright—Fisher simulations establish the two load-bearing claims: β ₁ rises monotonically with the recombination rate over six orders of magnitude, and persistent-homology features separate reticulate from non-reticulate histories with 98—100% recall (the residual confusion falls entirely within the non-reticulate K ₁ / K ₂ pair, which β ₁ does not distinguish). We then apply the pipeline to four empirical systems. (i) On the MalariaGEN Pf7 Plasmodium falciparum dataset ( n = 20,864, 33 countries), per-population β ₁ spans two orders of magnitude and diverges significantly from a label—shuffle null (median 20.5, range 8-32); the ordering runs opposite to recombination rate-freely-recombining African populations sit lowest and clonal/swept Southeast Asian and Papuan populations highest—because at the population scale β ₁ is dominated by demographic structure rather than recombination rate, a point we reconcile explicitly with the controlled dose-response. (ii) Colombian Cauca SP-resistant samples carry β ₁ = 12 against a near-clonal SP-sensitive baseline of β ₁ = 5 (and two orders of magnitude more total persistence), the high- β ₁ , multi-origin band of K ₃ consistent with resistance carried on several genomic backgrounds. (iii) The Cambodia artemisinin sweep (2008-2018) traces a K ₃ → K ₁ trajectory, β ₁ rising to a mid-sweep peak of 45 and collapsing to 13 at fixation—to our knowledge the first direct observation of a selective-sweep transient in topological coordinates, with the caveat that the per-bin values are medians of three subsamples with wide bars. (iv) On Arabidopsis thaliana 1001 Genomes data, Iberian relict populations (Spain, β ₁ /n = 0.64) exceed post-glacial-expansion populations (Sweden 0.54; United Kingdom 0), generalising the framework beyond pathogens. A P. falciparum mitochondrial negative control recovers β ₁ = 0 across all subsamples, establishing pipeline specificity. Moving above the 1-skeleton, β ₂ is zero at the clonal/expansion limits and positive across the reticulate systems; a controlled two-vs-three-way admixture simulation confirms that β ₂ separates regimes that share a β ₁ profile, while the further suggestion that the ratio η = β ₂ /β ₁ separates microevolutionary from macroevolutionary timescales is presented, given the small number of systems and the absence of a β ₂ null, as a hypothesis for future testing. Together these results demonstrate that the topology of genetic-distance data is an evolutionary observable, with immediate implications for drug-resistance surveillance in P. falciparum .