This python package was adapted from the MATLAB package as first presented in Small-World Propensity and Weighted Brain Networks (2016) by Sarah Feldt Muldoon, Eric W. Bridgeford & Danielle S. Bassett. Their original MATLAB implementation can be found here.

The small-world propensity package can be installed using pip

python -m pip install small-world-propensity

small_world_propensity can be called in two ways: either with a single adjacency matrix, or with a list of adjacency matrices and a boolean list denoting whether each matrix is binary or not. In either case, small_world_propensity will return a pandas dataframe similar to the following:

Using the structural network of the cat cortex obtained from tract-tracing studies between 52 brain regions, we can visualize the process behind the calculation of the small-world propensity, $\phi$. The matrix is loaded using

cat = sio.loadmat('data/cat.mat')['CIJctx']

We can then ensure symmetry by calling

symm_cat = swp.make_symmetric(cat)

In order to get the regular version of the cat matrix, we first find the effective average radius:

r = swp.get_avg_rad_eff(symm_cat)
cat_reg = swp.regular_matrix_generator(symm_cat, r)

Finally we produce the randomized cat matrix:

cat_rand = swp.randomize_matrix(cat_symm)

The graphs visualized in a circular layout look as follows:

We can take the networks used in Muldoon et al and plot $\phi$, $\Delta_L$, $\Delta_C$, and $\delta$. Note that these networks are not the exact same as the ones used in Muldoon et al, and due to differences in how Numpy performs permutations, and the use of NetworkX and iGraph libraries, the results are not identical, but still match closely.

The adjacency matrices:

And the results:

To cite this work, please use:

@software{small-world-propensity,
  author       = {{Daniels, R. K.}},
  title        = {small-world-propensity},
  year         = 2023,
  publisher    = {Zenodo},
  version      = {v0.0.8},
  doi          = {10.5281/zenodo.10299681},
  url          = {https://github.com/rkdan/small-world-propensity}
}

Please also cite the authors of the original MATLAB implementation:

@article{Muldoon2016,
    author = "Muldoon, Sarah Feldt and Bridgeford, Eric W. and Bassett, Danielle S.",
    title = "{Small-World Propensity and Weighted Brain Networks}",
    doi = "10.1038/srep22057",
    journal = "Scientific Reports",
    volume = "6",
    number = "1",
    pages = "P07027",
    year = "2016"
}