Characterize variety of histories in the history DAG

[willdumm]

Given a history DAG with many histories, how do we know how similar these histories are? Do they just vary by a few ancestral sequences but remain otherwise the same? Are histories found by the DAG fundamentally different than those used to seed it?

Ideas:

• (implemented) comparing the number of input trees and expressed trees, we get an idea of the level of tree rearrangement that's happening in the DAG. Also internal_avg_parents.
• (implemented, slow) count_topologies gives an idea of how many histories are the same with different ancestral sequences
• How big are the clades which are being swapped?
• Is there some natural way to cluster histories in the hDAG?
• Is variation all in certain clades, with others really constant? A start at this could be counting how many histories each hDAG node takes part in.

[willdumm]

Node counting in #19

[willdumm]

Some ideas about how we can use the number of trees each node takes part in, where for purposes of this comment, a node's uncertainty is inversely proportional to the number of trees in the dag that it takes part in.

• How does uncertainty vary across the DAG?
• Is uncertainty concentrated above certain leaves? Are nodes above a subset of leaves very certain?
• ^ maybe an equivalent question; where in the DAG are the uncertain nodes? Close to leaves, or higher up, near the UA node? (how big are their clade unions [union of their child clade sets]?)
• Can we count the number of trees in the DAG that each edge takes part in (something like edge support / edge certainty)
• ^ If we can, then we can evaluate how frequently a given parent/child label pair shows up in trees
• ^^ We can use this to characterize the path uncertainty (in label space) to each of the leaf node labels in a useful way?

See #20 for path DAG methods