MuRiT

What MuRiT Computes

MuRiT computes pathwise persistent homology for a multifiltered clique complex.

Input model:

• finite vertex set V = {0,...,N-1},
• parameter dimension k,
• for each listed edge e = (i,j), a non-empty set of minimal generator vectors M_e ⊂ R^k (an antichain),
• a totally ordered path p_1, ..., p_T in R^k (coordinatewise non-decreasing).

Semantics:

• edge e is present at filtration value x iff exists m in M_e: m <= x coordinatewise,
• the complex at x is the clique complex on all present edges.

Pathwise reduction used by MuRiT:

• entry(e) = min{ t in {1,...,T} : exists m in M_e with m <= p_t },
• if no such t, then entry(e) = T+1 (never appears on this path),
• write lower-triangular auxiliary matrix with these integer entry indices,
• run ripser on that auxiliary matrix,
• translate interval endpoints from indices back to filtration vectors on the path.

Run

Run with the murit executable:

murit --complex <file.csv> --path <path-literal-or-path-file> [--options]

If murit is not in PATH, use the executable path directly.

--ripser behavior:

• omit --ripser: do not run ripser,
• --ripser: use ripser from PATH,
• --ripser <path>: use the provided ripser executable path. If ripser cannot be resolved, MuRiT exits with an error.

Input and Output Formats

Complex Input (--complex)

CSV with comments/blank lines allowed:

# comments are allowed
N,k
i,j,"[[a1,...,ak],[b1,...,bk],...]"

Rules:

• N >= 1, k >= 1
• 0 <= i < j < N
• one row per edge
• generator list must be valid JSON, non-empty, vectors of length k
• duplicate (i,j) rows are rejected
• dominated generators are pruned automatically to the minimal antichain
• omitted edges are treated as permanently absent

Path Input (--path)

One syntax only: a path literal = JSON array of vectors:

[[a1,a2,...],[b1,b2,...],...]

--path accepts:

• inline literal (argument starts with [),
• or a .paths file where each non-empty non-comment line is one path literal.

Validation:

• each vector has length k,
• vectors are coordinatewise non-decreasing along the path,
• empty path is invalid.

Output Files

When --path is a path file, MuRiT writes per-path files:

• <complex-stem>_path01.aux
• <complex-stem>_path01.ripser (with --ripser)
• etc.

File meanings:

• *.csv: complex input.
• *.paths: one path literal per line.
• *_pathNN.aux: lower-triangular integer entry-index matrix.
• *_pathNN.ripser: ripser output with interval endpoints translated to filtration vectors.

Ripser threshold behavior:

• MuRiT always sets ripser threshold to len(path) internally so omitted 1-simplices remain absent on the path.

Basic Usage

murit \
  --complex examples/figure8.csv \
  --path examples/figure8.paths \
  --ripser --dim 1

For the full figure-8 walkthrough (commands + interpretation), see:

• examples/README.md

Example Visualization

• examples/figure8_overview.png

Changelog

• CHANGELOG.md

Citing

Maximilian Neumann, Michael Bleher, Lukas Hahn, Samuel Braun, Holger Obermaier, Mehmet Soysal, Rene Caspart, and Andreas Ott. "MuRiT: Efficient Computation of Pathwise Persistence Barcodes in Multi-Filtered Flag Complexes via Vietoris-Rips Transformations". arXiv, July 7, 2022. https://doi.org/10.48550/arXiv.2207.03394.