N-Hop Lifting (Graph to Combinatorial)

[martin-carrasco]

Add N-Hop lifting to combinatorial complex where the n-hop neighbourhood is a 2+k rank hyper edge. Let $G = (V, E)$ denote a graph and $N(v, k)$ denote the $k$-hop neighbours of $v$. We say that an $(2+k)$-cell $\sigma$ is the set of neighbours $u$ such that $u \in N(v, k) \cup {v}$, $rank(u) > rank(v)$ and ${v} \subset \sigma$. This definition fits with the properties of a Combinatorial Complex (CCC) as described in [1]. Similarly, this example is describe in a similar way in the Appendix.

The CCC after the lift will contain information of the underlying graph as hierarchical relations as well as set-relations in terms of the hyperedges added to represent $k$-hop neighbourhoods.

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[1] Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzmán-Sáenz, A., Ramamurthy, K. N., ... & Schaub, M. T. (2022). Topological deep learning: Going beyond graph data. arXiv preprint arXiv:2206.00606.
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Tags:
Existing lift from literature | connectivity-based | deterministic

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[gbg141]

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