More ways to define neighborhood relationship between cells of same rank

[ekarais]

At the moment, our codebase defines neighborhood relationships between cells as follows. Let's say we have to cells A and B with rank i and j respectively.

• If $i < j$ (w.l.o.g.): the cells are neighbors iff $A \subset B$
• If $i = j$: the cells are neighbors iff $\exists C: rank(C) = i+1 \land A \subset C \land B \subset C$

However, as @clabat9 pointed out, there are many other ways by which A and B can be considered neighbors. Namely:

• $\exists C: rank(C) = i-1 \land C \subset A \land C \subset B$ "immediate co-adjacency"
• $\exists C: rank(C) > i \land A \subset C \land B \subset C$ "general adjacency"
• $\exists C: rank(C) < i \land C \subset A \land C \subset B$ "general coadjacency"

We should add options to consider each of these other ways as well as "immediate adjacency" which the current implementation corresponds to. We should also add the option to use multiple rules simultaneously, for example immediate adjacency and coadjacency, or consider all rules.

[ekarais]

I implemented the logic to compute these different kinds of neighborhood relationships in #29 . However, all neighbors of a cell with the same rank still get processed by the same neural network. In other words, the model does not know which type of neighbors two cells A and B are, if they have the same rank.

[clabat9]

I think that, as I reported in the paper, what was really missing was the "immediate co adjacency". The other two, to me, are not necessarily something we will use. For the super cell, we just need something like: $$\exists C:\textrm{rank}(C)=\textrm{max rank and }A \subset C \textrm{ and }B\subset C$$ After the implementation of different weights, I think we will be totally able to test our model.

[ekarais]

Addressed in #36.