FastKnill - An alternative fast computation of Euler characteristics and Knill curvature from networks

Author: Danillo Barros de Souza -- ORCID: 0000-0002-7792-8862

This work is inspired by the proposed new methods for computations of cliques in Vietoris-Rips complexes [1]. We provide a set-theoretical Python coding for efficiently computing the Knill curvature and the Euler characteristics for simplicial complexes. For alternative geometric computations, see also [2]

Current version:

0.2.0

Content:

• fastknill.pyx
• compiler.py
• setup.py

Python version:

3.8.5

Package requirement:

• numpy
• networkx
• cython
• scikit-learn

Installation:

The files fastknill.pyx, compiler.py and setup.py must be placed at the same directory.

### Local installation: To this purpose, the user will need to compile the file compiler.py in cython by running the command:

python3 compiler.py build_ext --inplace

Global installation:

Run setup.py by executing the command:

pip install .

After successful compilation, the generated fastknill file can be imported as a package, for instance:

python import fastknill as fk

Functions:

• compute_knill:

Input:

- `D`, can be:
   - A `dictionary` in which the keys are integer numbers from 0 to len(D)-1 and the values are the N-dimensional points;

   - A symmetric `numpy.matrix` of float numbers;

   - A simple undirected `nx.Graph` that may include the feature `weights`on edges attributes;

   - A `string` of the `graph/graphml` file in a directory.

- `cutoff`: Float number for the threshold distance between points.
- `n_neighbors`: Interger, the number of the nearest neighbours to consider. If None, it is not applied.
- `max_dim`: Integer, the maximum simplex dimension to compute FRC.
- `metric`: string, the metric in which the pairwise distance is computed. If D is not a dictionary, this parameter is ignored.
- `mapped_nodes`: True or False - if the mapping of the original node labels is kept.

Output:

A dictionary whose keys are the noded and the values are the Knill curvature of each node.

• compute_euler:

Input:

- `D`, can be:
   - A `dictionary` in which the keys are integer numbers from 0 to len(D)-1 and the values are the N-dimensional points;

   - A symmetric `numpy.matrix` of float numbers;

   - A simple undirected `nx.Graph` that may include the feature `weights` on edges attributes;

   - A `string` of the `graph/graphml` file in a directory.

- `cutoff`: Float number for the threshold distance between points.
- `n_neighbors`: Integer, the number of the nearest neighbours to consider. If None, it is not applied.
- `metric`: string, the metric in which the pairwise distance is computed. If D is not a dictionary, this parameter is ignored.
- `max_dim`: Integer, the maximum simplex dimension to compute FRC.
- `mapped_nodes`: True or False - if the mapping of the original node labels is kept.

Output:

An integer, the Euler characteristics of the input network.

Contact and Support:

danillo.dbs16@gmail.com, dbarros@bcamath.org

References:

[1] Alternative set-theoretical algorithms for efficient computations of cliques in Vietoris-Rips complexes, Barros de Souza, Danillo; da Cunha, Jontatas, A.N. Santos, Fernando; Desroches, Mathieu & Rodigues, Serafim; link

[2] Efficient set-theoretic algorithms for computing high-order Forman-Ricci curvature on abstract simplicial complexes; Barros de Souza, Danillo; da Cunha, Jontatas, A.N. Santos, Fernando; Jost, Juergen & Rodigues, Serafim; Link