What does this research mean for the field?

The Ordinal Locality Filtration method enables the highly efficient computation of Betti numbers for large Vietoris-Rips complexes of hypercube graphs on standard consumer hardware, drastically reducing the required memory and time compared to traditional cluster-based approaches. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.SUPPORTS_CONSENSUS.

What question did this study set out to answer?

Compute the Betti numbers of large Vietoris-Rips complexes of hypercube graphs efficiently on consumer-grade hardware using Ordinal Locality Filtration.

September 6, 2026Open Access

Betti Numbers of Vietoris-Rips Complexes of Hypercube Graphs in Degrees 7 and 15 on a Single Laptop

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Compute the Betti numbers of large Vietoris-Rips complexes of hypercube graphs efficiently on consumer-grade hardware using Ordinal Locality Filtration.
Implemented Ordinal Locality Filtration, a topological algorithm that bypasses metric consultation and threshold filtering.
Evaluated computational runtime and memory usage on a single laptop for the 64-vertex complex X(6,4) with 2,932,100,732 simplices and top dimension 21, as well as the 256-vertex complex X(8,3).
Computed Betti numbers β7 = 239 and β15 = 14 for X(6,4) in 14 minutes with 17.5 GB peak memory, compared to ~18 hours and ≥800 GB on a university cluster in prior work.
Directly calculated Betti numbers β4 = 351 and β7 = 1,120 for the 256-vertex complex X(8,3) in under 90 seconds each.
Verified all 11 computations against published theorems, confirmed by an Euler characteristic of −252 = 1 − 239 − 14.

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Implication

Computational study demonstrates rapid Betti number calculation for hypercube Vietoris-Rips complexes on consumer hardware, indicating dramatic scalability gains over cluster-based approaches.

Cite This Study

Timothy Gleason (2026) studied this question.

synapsesocial.com/papers/6a9d1eb628139818eab21cd2 https://doi.org/10.5281/zenodo.22308416

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Implication

Computational study demonstrates rapid Betti number calculation for hypercube Vietoris-Rips complexes on consumer hardware, indicating dramatic scalability gains over cluster-based approaches.

Betti Numbers of Vietoris-Rips Complexes of Hypercube Graphs in Degrees 7 and 15 on a Single Laptop

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Implication

Cite This Study

Betti Numbers of Vietoris-Rips Complexes of Hypercube Graphs in Degrees 7 and 15 on a Single Laptop

Key Points

Discussion

Implication

Cite This Study

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