#### Consider an observer operating on a branchial graph

Consider an observer operating on a branchial graph

The branchial is already defining ancestral closeness

### Observer coarse-graining a generic graph

Observer coarse-graining a generic graph

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#### What is coarse-graining on a graph?

What is coarse-graining on a graph?

Connect every node directly to its 2-step as well as 1-step neighbors......

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#### Reduction method: “equate two nodes”

Reduction method: “equate two nodes”

#### Simple case: merging of triangles in triangulation

Simple case: merging of triangles in triangulation

### Another form of coarse-graining: backtrack to ancestry; or forward-track to merging

Another form of coarse-graining: backtrack to ancestry; or forward-track to merging

Eventual merging allows coarse-graining to be consistent

#### Apply graph transformation rules, and say that nodes that evolve into “each other” after n steps are equivalent

Apply graph transformation rules, and say that nodes that evolve into “each other” after n steps are equivalent

## Branchial case

Branchial case

#### Can either coarse grain by by-fiat conflating different paths at a particular time slice...

Can either coarse grain by by-fiat conflating different paths at a particular time slice...

#### Or by taking time-bins and saying things are equivalent if they evolve into other in a certain time...

Or by taking time-bins and saying things are equivalent if they evolve into other in a certain time...

But to make this meaningful, we need to project to a consistent somewhat-time-independent branchial space

#### [ Fuzzy logic analog: the observer has a Gaussian kernel in continuous branchial space ]

[ Fuzzy logic analog: the observer has a Gaussian kernel in continuous branchial space ]