Neighborhood/Dowker Lifting (Graph2Simplicial)

[Snopoff]

This lifting constructs a neighborhood simplicial complex as it is usually defined in the field of topological combinatorics. In this lifting, for each vertex in the original graph, its neighborhood is the subset of adjacent vertices, and the simplices are these subsets.

That is, if $G=(V,E)$ is a graph, then its neighborhood complex $N(G)$ is a simplicial complex with the vertex set $V$ and simplices given by subsets $A\subseteq V$ such, that $\forall a\in A; \exists v: (a,v)\in E$. That is, say, 3 vertices form a simplex iff there's another vertex which is adjacent to each of these 3 vertices.

This construction differs from another lifting with the similar naming (I've started working on it a little earlier and indeed should have been create this PR earlier). The difference is, for example, that in this construction the edges of an original graph doesn't present as the edges in the simplicial complex.

This lifting is really a Dowker construction since an edge between two vertices in the graph can be considered as a symmetric binary relation between these vertices.

[review-notebook-app]

Check out this pull request on 

See visual diffs & provide feedback on Jupyter Notebooks.

---

Powered by ReviewNB

[levtelyatnikov]

@Snopoff, please update the linting so your submission passes the tests and qualifies for the competition.

[gbg141]

Hello, @Snopoff! 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:

• your first and last name (as well as any other team members)
• the title of the method you implemented
• the input domain of the method you implemented
• the output domain of the method you implemented
• your pull request number (Neighborhood/Dowker Lifting (Graph2Simplicial) #6)

Before July 12, make sure that your submission respects all Submission Requirements laid out on the challenge page. Any submission that fails to meet this criteria will be automatically disqualified.