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math, statistical physics, graphs, connected components, network science
- Percolation theory relates to how networks/graphs behave when edges or vertices areremoved from the system
- Representative question:
- "given that a liquid is poured onto some porous material, will the liquid be able to make its way from hole to hole and reach the bottom?"
- often modeled as a 3-dimensional graph of size
n x n x n, where edges (bonds) between vertices (sites) allow the liquid to pass through probabilistically
- edges will independently allow liquid between two given vertices with probability
p, or not with probability
1 - p
- so the likelihood of percolation depends on the "site vacancy probability"
- "therefore, for a given
p, what is the probability that an open path exists for the liquid?"
- particularly when
n (the dimension of the porous material) is large
- a system "percolates" if and only if (iff) there is an open path from top to bottom
- reflected in many real-world systems: