new project_to_dag algorithm

[blengerich]

Prompted by #205 , here is a simplified project_to_dag algorithm. Instead of searching over edge weights to threshold out, it does a topological sort of the nodes and then returns only the forward edges. The topological sort is just a simple ordering by sum of edge weight magnitudes (more and stronger edges => higher priority to keep the edges in this node).

A quick test of performance is included in the Jupyter notebook. The topological sort function could (possibly should?) be improved, but the sort-based projection appears to get similar performance to the search-based algorithm:

with much reduced runtime:

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[cnellington]

The speedup is definitely worth the cost here, but there's a few edge cases to consider.
Here's one failure case for topological sort.

X -2-> Y 
Y -4-> Z

Where it's already a dag but topological sort by outgoing edge magnitudes will remove the first edge. Both algorithms often ignore root nodes with few/small outgoing edges. Just to emphasize that neither are perfect, both sort and score fail here.

X -2-> Y
Y -4-> Z
Z -3-> Y

I wonder if it's even possible to come up with an edge-removal function $f$ such that for any $G \in R^{n \times n}$, we maximize
$max_f \sum_{i,j}|f(G)_{i,j}|$ s.t. $f(G)$ is a dag?

[cnellington]

LGTM, feel free to merge.

[cnellington]

Re: #205 (comment)
Is it possible that the binary search implementation is causing runtime issues? Python is very slow at recursion. Seems like it might not be the is_dag check that's the issue, and I don't think binary search should incur serious runtime problems? It might be worthwhile reimplementing the binary search as a while loop to see if we can get the search algorithm to work quickly.

[CheeseLee888]

Hi,
I am Peter Li, new member in Lengerich Lab and I am working on this issue.