Katz centrality

[acombretrenouard]

New Katz centrality algorithms, computed with a power-series over the adjacency matrix.
Another option would be a direct matrix inversion (computationally harder ?).

The usual contribution pipeline runs without errors (pytest, isort ., etc...).

[codecov]

Codecov Report

Patch coverage: 95.23% and project coverage change: +0.15 🎉

Comparison is base (5cff847) 90.74% compared to head (1d4c046) 90.90%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #370      +/-   ##
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+ Coverage   90.74%   90.90%   +0.15%     
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  Files          42       42              
  Lines        3113     3178      +65     
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+ Hits         2825     2889      +64     
- Misses        288      289       +1     

Impacted Files	Coverage Δ
xgi/algorithms/centrality.py	97.39% <95.00%> (-0.53%)	⬇️
xgi/algorithms/shortest_path.py	100.00% <100.00%> (ø)

... and 2 files with indirect coverage changes

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[leotrs]

Hey @acombretrenouard thank you so much for your work on this! This is almost ready, just left some minor things :)

[leotrs]

Check out this link to see how to markup LaTeX within a docstring.

Also, the section title should be Notes, note Note (sphinx is very fussy about this...)

[acombretrenouard]

I put variables in between ``, is that ok ?

[maximelucas]

Thanks a lot Antoine, great work.
I left a few additional comments mainly to make sure we give enough details in the docs.

Just a question: what values can this take?
I remember we had to 1-normalise other centralities to avoid problems (like negative values), and maybe it would be better to make this one consistent with the others in that respect. @nwlandry I think you did this?

[nwlandry]

Thanks a lot Antoine, great work. I left a few additional comments mainly to make sure we give enough details in the docs.

Just a question: what values can this take? I remember we had to 1-normalise other centralities to avoid problems (like negative values), and maybe it would be better to make this one consistent with the others in that respect. @nwlandry I think you did this?

Yeah, I did this in in the clique eigenvector centrality because the eigenvector is not guaranteed to be in the correct direction (it can have all positive entries or all negative entries).

[acombretrenouard]

Thank you for your comments ! I tried to take them into account. The current version should be much better.

[acombretrenouard]

Hello again !

I added the minor fixes you mentioned.

On the normalization : the Katz-centrality counts the number of walks starting from a single node. This is a local feature (as there is the exponential attenuation of walks). In this sense, the Katz centrality is a sort of 'clustering coefficient'.
For the moment the code does not normalize the vector.

Conversely, normalizing would add the global contribution of every other node. Maybe this is what we want to reflect ?
To me, the problem with that would be that a node with a given neighborhood would have different centrality values depending on the more distant parts of the graph...

[nwlandry]

Nice work! The only thing is that there is a duplicate import but I will fix that once it's merged in.

[maximelucas]

Thanks Antoine !