The rank-turbulence divergence

An attempt at implementing a python version of the rank-turbulence divergence, introduced in the recent paper:

• Dodds, P.S., Minot, J. R., Arnold, M. V., Alshaabi, T., Adams, J. L., Dewhurst, D. R., Gray, T. J., Frank, M. R., Reagan, A. J., Danforth C. M. (2020). Allotaxonometry and rank-turbulence divergence: A universal instrument for comparing complex systems. arXiv 2002.09770.

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Below is the definition of the rank-turbulence divergence. For the most detailed version of this definition, including motivation and discussion of its various features, see Section II.D (no relation) of the paper.

where r refers to the (float) ranking of element τ (for example, a team in the premier league and its rank), such that r=1.0 is the first place team. The N_{1, 2; α} term refers to a normalization factor that forces the rank-turbulence divergence to be between 0 and 1 and is expressed as follows:

(equations rendered with HTML output from https://alexanderrodin.com/github-latex-markdown/)

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Installation and Usage

In order to use this code, first clone/download the repository. Below is a simple example usage. Please feel free to reach out if you find any bugs, have any questions, or if for some reason the code does not run.

Requirements

This code is written in Python 3.x and uses the following packages / tools:

• Scipy
• Numpy
• Itertools
• Collections

Playground Notebooks

1. rank_turbulence_divergence, which uses the rtd.py script, where the actual implementation is. Note also that this is... where the bugs are going to be. If you find them, please yell at me.

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Fig. 1: Rank-turbulence divergence betweeen two simple vectors.

Fig. 2: Matlab version: Rank-turbulence divergence.

Note: Fig. 2 is a validation using this Matlab code from the authors.

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Basic usage

>>> from rtd import rank_turbulence_divergence
>>> inputA_1 = ['a', 'e', 'c', 'b', 'f', 'g', 'd']
>>> inputA_2 = ['b', 'a', 'e', 'd', 'c', 'f', 'g']
>>> print("rtd =",rank_turbulence_divergence(inputA_1, inputA_2, alpha=1.0))
rtd = 0.45924793111057804

However, there are other data-types that can be input into the rank_turbulence_divergence function:

>>> from rtd import rank_turbulence_divergence
>>> from numpy.random import shuffle
>>> inputB_1 = ['a']*20 + ['e']*14 + ['c']*8 + ['b']*7 + ['f']*4 + ['g']*2 + ['d']*1
>>> inputB_2 = ['b']*24 + ['a']*16 + ['e']*5 + ['d']*4 + ['c']*3 + ['f']*2 + ['g']*1
>>> shuffle(inputB_1)
>>> shuffle(inputB_2)

>>> print('inputB_1:',inputB_1)
inputB_1: ['a', 'a', 'a', 'c', 'g', 'f', 'c', 'a', 'a', 'c', 'e', 'e', 'a', 'e', 'e', 'a', 'b', 'a', 'a', 'a', 'a', 'a', 'g', 'c', 'e', 'b', 'c', 'a', 'e', 'b', 'e', 'e', 'e', 'b', 'e', 'f', 'c', 'f', 'e', 'b', 'b', 'a', 'a', 'a', 'c', 'f', 'a', 'a', 'a', 'a', 'c', 'e', 'd', 'e', 'e', 'b']

>>> print('inputB_2:',inputB_2)
inputB_1: ['a', 'a', 'a', 'c', 'g', 'f', 'c', 'a', 'a', 'c', 'e', 'e', 'a', 'e', 'e', 'a', 'b', 'a', 'a', 'a', 'a', 'a', 'g', 'c', 'e', 'b', 'c', 'a', 'e', 'b', 'e', 'e', 'e', 'b', 'e', 'f', 'c', 'f', 'e', 'b', 'b', 'a', 'a', 'a', 'c', 'f', 'a', 'a', 'a', 'a', 'c', 'e', 'd', 'e', 'e', 'b']

>>> print("rtd =",rank_turbulence_divergence(inputB_1, inputB_2, alpha=1.0))
rtd = 0.45924793111057804

As well as dictionaries of counts:

>>> from rtd import rank_turbulence_divergence
>>> from collections import Counter
>>> inputC_1 = dict(Counter(inputB_1))
>>> inputC_2 = dict(Counter(inputB_2))
>>> print("rtd =",rank_turbulence_divergence(inputC_1, inputC_2, alpha=1.0))
rtd = 0.45924793111057804

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Citation

Play around with this measure, it rocks! If you use it in your work, be sure to cite these folks:

Dodds, P.S., Minot, J. R., Arnold, M. V., Alshaabi, T., Adams, J. L., Dewhurst, D. R., Gray, T. J., Frank, M. R., Reagan, A. J., Danforth C. M. (2020). Allotaxonometry and rank-turbulence divergence: A universal instrument for comparing complex systems. arXiv 2002.09770.

Bibtex:

@article{dodds2020rtd,
  title = {Allotaxonometry and rank-turbulence divergence: A universal instrument for comparing complex systems},
  author = {Dodds, P.S. and Minot, J. R. and Arnold, M. V. and Alshaabi, T. and Adams, J. L. and Dewhurst, D. R. and Gray, T. J. and Frank, M. R. and Reagan, A. J. and Danforth C. M.},
  journal = {arXiv},
  year = {2020},
  url = {https://arxiv.org/abs/2002.09770}
}

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This repo uses ideas and insights about network comparison. If that interests you, also check out:

• Hartle, H., Klein, B., McCabe, S. St-Onge, G., Murphy, C., Daniels, A., & Hébert-Dufresne, L. (under review). Network comparision and the within-ensemble graph distance.
  • forthcoming work defining the within-ensemble graph distance
• McCabe, S., Torres, L., LaRock, T., Haque, S., Yang, C-H., Hartle, H., & Klein, B. (in prep.). netrd: A library for network reconstruction and graph distances}.
  • the netrd python package that, among other things, has dozens of graph distance tools implemented.