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Suppose we have a distance cutoff $r_0$.

Let's say that any two stars in the galaxy with a relative distance below $r_0$ have a path between them.

Would the path-connected Milky Way look like a Web in which every star is potentially connected to all of its neighbors or would we form a type of branched Tree structure with junctions and bottlenecks?

I understand that for a high enough $r_0$, it will always look like a web. But my doubt is whether the standard-deviation of interstellar distances in the Milky way is such that there exists a value of $r_0$ for which the topology is unequivocally a Tree.

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The process where you link random points within distance $r_0$ into a graph is known as continuum percolation. As $r_0$ increases from 0 at first there are just isolated clusters (binaries, randomly very close stars) that gradually merge. At a critical distance of a few parsecs these clusters mostly merge into one major galaxy-spanning cluster, and above that distance this cluster just gets more densely connected. The big network at the critical range has a scale-free structure of local clusters, bridges, and trees: neither a proper web nor a tree.

Some illustrations from my book manuscript:

enter image description here enter image description here

It should be noted that these pictures assume a constant star density. In reality the stellar density varies, so for a given $r_0$ there is a part of the disk and bulge that are connected and parts of the outer galaxy and halo that are not connected.

enter image description here

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  • $\begingroup$ Very interesting! I did not know graph theory was used like this. What is it then useful for in understanding these systems/images? Also, the last question of the OP, "But my doubt is whether the standard-deviation of interstellar distances in the Milky way is such that there exists a value of r0 for which the topology is unequivocally a Tree." seems to perhaps require some graph theoretic techniques to answer? $\endgroup$
    – Daddy Kropotkin
    May 20, 2021 at 23:07
  • $\begingroup$ @DaddyKropotkin - The main application of continuum percolation is in nucleation of crystals, but it shows up in many domains. I have seen people apply it to the cosmic web, trying to analyse its topological properties by turning it into a graph. As for tree structure, I doubt it happens since it would require a very particular distance distribution to avoid creating self-loops. When looking at actual star distance data it is always clusters merging to a web. $\endgroup$
    – Anders Sandberg
    May 21, 2021 at 1:01

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