<p>The process where you link random points within distance <span class="math-container">$r_0$</span> into a graph is known as <a href="https://en.wikipedia.org/wiki/Continuum_percolation_theory" rel="noreferrer">continuum percolation</a>. As <span class="math-container">$r_0$</span> 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.</p>
<p>Some illustrations from my book manuscript:</p>
<p><a href="https://i.sstatic.net/ALb92.png" rel="noreferrer"><img src="https://i.sstatic.net/ALb92.png" alt="enter image description here" /></a>
<a href="https://i.sstatic.net/oW05J.png" rel="noreferrer"><img src="https://i.sstatic.net/oW05J.png" alt="enter image description here" /></a></p>
<p>It should be noted that these pictures assume a constant star density. In reality the stellar density varies, so for a given <span class="math-container">$r_0$</span> there is a part of the disk and bulge that are connected and parts of the outer galaxy and halo that are not connected.</p>
<p><a href="https://i.sstatic.net/O2LCR.png" rel="noreferrer"><img src="https://i.sstatic.net/O2LCR.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-05-20T23:07:26.347Z
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?
Daddy Kropotkin
CC BY-SA 4.0
2021-05-21T01:01:27.713Z
@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.
Anders Sandberg
CC BY-SA 4.0
2021-05-20T09:39:19.063Z
2021-05-20T09:39:19.063Z
Anders Sandberg
<p>Suppose we have a distance cutoff <span class="math-container">$r_0$</span>.</p>
<p>Let's say that any two stars in the galaxy with a relative distance below <span class="math-container">$r_0$</span> have a path between them.</p>
<p>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?</p>
<p>I understand that for a high enough <span class="math-container">$r_0$</span>, 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 <span class="math-container">$r_0$</span> for which the topology is unequivocally a Tree.</p>

<milky-way><interstellar>
Does the galaxy look like a Web or a Tree?
urquiza
# Does the galaxy look like a Web or a Tree?

(Asked by: username_0 on 2021-05-20)

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.

**Tags:** milky-way, interstellar

---

## 1 answers

### Answer by username_1 on 2021-05-20. Score: 11 | Accepted answer

The process where you link random points within distance $r\_0$ into a graph is known as [continuum percolation](https://en.wikipedia.org/wiki/Continuum_percolation_theory). 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](https://i.sstatic.net/ALb92.png)](https://i.sstatic.net/ALb92.png)
[![enter image description here](https://i.sstatic.net/oW05J.png)](https://i.sstatic.net/oW05J.png)

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](https://i.sstatic.net/O2LCR.png)](https://i.sstatic.net/O2LCR.png)

> **Comments:**
> **username_2**: 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?  
> **username_1**: @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.  

---

