In short: snowflakes look like small connected components of random graphs; are there models using this, and what kind of model would make sense, if any?
While I was once again fascinated by falling snow and the wide variety of bigger and smaller snowflakes, a simple idea stroke me: snowflakes look very similar to small connected components that appear in an Erdős–Rényi graph, when one starts with an empty graph and adds random edges (before reaching the percolation threshold where a giant connected component appears).
In the case of snow, vertices would be $H_2O$ molecules that meet randomly according to something like Brownian motion (and wind), which creates bondings between them. Snowflakes, seen as connected components of $H_2O$ molecules, merge and grow in this way until they are too heavy to stay in the air and just fall down, for our greatest pleasure.
This made me think of the following snow model: take a large number of vertices, and iteratively add edges between random vertices in ER-like manner; stop when a connected component of more than $\tau$ vertices appears (it is the maximal size of snowflakes), and consider the connected components of the obtained graph as your snowflakes.
I have several questions:
- is there such a model of snow, based on an ER-like approach and/or connected components? where to find it? I searched the web for such models, but connected components and snow appear quite rarely in the same place.
- what is the size distribution of obtained connected components, compared to real snowflakes? what is the influece of $\tau$? I know there are many works on connected component size in random graphs; what is the most appropriate reference?
- should we push the analogy further by removing components larger than $\tau$ (falling of heavy snowflakes) when they appear and continuing the edge addition process? How to deal with bias induced by such removals? should we replace removed vertices?
- would other graph models be more appropriate?