Probabilistic Clique Lifting (Graph to Combinatorial)

[alvarolmartinez]

In this PR we introduce a (novel) probabilistic version of the clique lifting. This lifting is appropriate under the assumption that some edges have been randomly and uniformly erased from the graph.
The algorithm does not introduce extra edges to the complex, and the probabilistic part is only involved in adding higher-order cells.

The algorithm finds the cliques $C_{i,j}$ of size $i$, and considers nodes which are highly connected to the clique. Given a probability parameter $p=$probability representing the erasure probability (assuming independence of erasures), it considers a node $x$ as part of a clique $C_{i,j}$ if

$$|N(x)\cap C_{i,j}| \ge (1-p)|C_{i,j}|$$

When $p=0$, it recovers the clique lifting, see the notebook for a custom example for various values of $p$.

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In comparison with the clique lifting, the Probabilistic Clique Lifting produces fewer, larger maximal cliques (or cells) which overlap less with each other.

The lifting leverages the property of combinatorial complexes (as opposed to simplicial complexes, see [1]) that not all subcells of a cell must be in the complex.

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[gbg141]

Hello @alvarolmartinez! Thank you for your submission. As we near the end of the challenge, I am collecting participant info for the purpose of selecting and announcing winners. Please email me (or have one member of your team email me) at guillermo_bernardez@ucsb.edu so I can share access to the voting form. In your email, please include:

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