Complement of a hypergraph

[acombretrenouard]

I wrote a function to complement a hypergraph.
It is computationally expensive, and to core function is a bit sketchy yet : I used strings and sets to enumerate all possible hyperedges, because I think it is quicker.

Let me know what you think !

[acombretrenouard]

By the way : is it "a hypergraph" or "an hypergraph" ?
: )

[codecov]

Codecov Report

Patch coverage: 93.79% and project coverage change: +0.97% 🎉

Comparison is base (387647a) 90.90% compared to head (a637640) 91.87%.
Report is 59 commits behind head on main.

Additional details and impacted files

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Files Changed	Coverage Δ
xgi/algorithms/clustering.py	96.66% <ø> (ø)
xgi/algorithms/shortest_path.py	100.00% <ø> (ø)
xgi/linalg/hypergraph_matrix.py	100.00% <ø> (ø)
xgi/stats/edgestats.py	80.00% <ø> (ø)
xgi/stats/nodestats.py	100.00% <ø> (ø)
xgi/convert/bipartite_edges.py	44.44% <44.44%> (ø)
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xgi/stats/__init__.py	85.11% <84.00%> (+0.45%)	⬆️
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[maximelucas]

I think this should be equivalent

Suggested change

	if M == 0:
	max_edge_size = 0
	else: # there is at least one non-empty edge
	max_edge_size = H.edges.order.max() + 1
	max_edge_size = xgi.max_edge_order(H) + 1

[acombretrenouard]

On this : the function max_edge_order is defined above in the same python file (xgi/classes/functions.py). I could use it directly and write max_edge_size = max_edge_order(H) + 1 (with the xgi. prefix).

Is there a problem with this ?
The function complement might not work if it was placed above max_edge_order in the file, right ?

[maximelucas]

You're right, it's in the same file so you don't need the xgi.. The order of the functions in the file should not matter.

[leotrs]

Hi @acombretrenouard, thanks for your work on this.

I don't think we need the binary strings at all. You can use combinations from the stadard library, which does exactly the same thing (i.e. treat elements in a collection unique based on their position and not on their value). This is guaranteed to be faster.

[maximelucas]

Thanks Antoine! A few duplicate comments with Leo, we reviewed at the same time without knowing it.
I also agree with Leo that finding all possible hyperedges can be done in a simpler way:

• we already have a powerset function that does that
• or the complete_hypergraph(max_order=max_edge_size+1) function that creates a hypergraph with all possibles edges. Right now that functions creates nodes as range(N). If you want to use that, to preserve the original nodes, we should add an arguments nodes that specifies the nodes and overrides range(N) if not None.

[maximelucas]

Btw @leotrs @nwlandry did we not have a few basic hypergraph operations?
I tried H1 - H2 but it says "-" does not exist. Did we call it difference or something? In any case I couldn't find it in the docs, we should maybe make it more visible.

[leotrs]

H1.nodes - H2.nodes

[acombretrenouard]

@nwlandry I think that what we are looking for is powerset (as @maximelucas pointed out) instead of combinations (we want subsets of all sizes...).

The problem with powerset is that it has no cutoff for the maximal size of hyperedges/subsets. I could add it though, and it would be close to what complete_hypergraph does.

However, once we have the full list L of all possible hyperedges, I don't see how we can pop those that are already present in the hypergraph we want to complement.
To me, we need to access the hyperedges in L knowing only their members (nodes), and not knowing their IDs.

[acombretrenouard]

This thing with removing hyperedges from the list of all possible hyperedges to build Hc makes the use of the incidence matrix and/or a node ordering somehow necessary.
What do you think ?

[maximelucas]

The problem with powerset is that it has no cutoff for the maximal size of hyperedges/subsets. I could add it though, and it would be close to what complete_hypergraph does.

The powerset iterates over combinations: chain.from_iterable(combinations(s, r) for r in range(start, len(s) + end)). You could add a max_order argument that modifies end if specified to generate edges up to a certain order.

However, once we have the full list L of all possible hyperedges, I don't see how we can pop those that are already present in the hypergraph we want to complement. To me, we need to access the hyperedges in L knowing only their members (nodes), and not knowing their IDs.

One way would be to iterate over all those possible edges, and remove them from the list if they exist in the original HG. Otherwise if you could use the method difference for sets and but then you need to convert the list of hyperedges (all possible, and original) to sets.

[maximelucas]

This thing with removing hyperedges from the list of all possible hyperedges to build Hc makes the use of the incidence matrix and/or a node ordering somehow necessary. What do you think ?

I didn't understand what you're referring to, could you elaborate?

[acombretrenouard]

My last comment was on how to check if a possible hyperedge belong to the original hypergraph.

I didn't understand what you're referring to, could you elaborate?

I refer to the process of checking if a possible hyperedge belongs to the original HG :

One way would be to iterate over all those possible edges, and remove them from the list if they exist in the original HG.

How precisely do you think I can check this ?
Is there a way to access an hyperedge knowing only its nodes ?

[maximelucas]

How precisely do you think I can check this ?
Is there a way to access an hyperedge knowing only its nodes ?

You can do H.edges.members() to get the nodes inside each hyperedge.
Or H.edges.members(dtype=dict) (or equvalient but faster: H._edge) to have the same but keyed by hyperedge index.

Is that what you were looking for?

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[nwlandry]

I did a rebase (sorry for the mess 😄) and I fixed the outstanding issues.