1. Percolation

2. Percolation theory is the study of the connectivity of networks.

3. Percolation. Given a composite systems comprised of randomly distributed insulating and metallic materials: what fraction of the materials need to be metallic so that the composite system is an electrical conductor?
a, Schematic representation of a film of striped Au NPs, between two Au electrodes. b, On binding of cations (blue spheres) conductive paths (blue line) connecting the two electrodes percolate the film and connect the two electrodes. c, Sensitivity (that is, χ  =  σafter/σbefore) of HT/EG2 Au NP films on binding of K+ and Cd2+ from solutions of different concentrations.

4. Percolation. Given a porous landscape with water on the surface (or oil below), under what conditions will the water be able to drain through to the bottom (or the oil to gush through to the surface)?

6.  The plots below show the site vacancy probability p versus the percolation probability for 20-by-20 random grid (left) and 100-by-100 random grid (right).
50 open sites open sites open sites 204 open sites
The percolation threshold is 204/400 = 0.51 because the system percolates when the 204th site is opened.

7. Program 1

10. Program 2

14. Image segmentation is the partition of an image into a set of perceptually important regions (sets of contiguous pixels)

15. Estimated average degrees of separation between all people on Facebook
Estimated average degrees of separation between all people on Facebook. The average person is connected to every other person by an average of 3.57 steps. The majority of people have an average between three and four steps.
Six degrees of separation or small world phenomenon.
5.8 million users
721 million users with 69 billion friendship links
1.2 billion users

16. Catherine’s Facebook friends –dimension of clusters and their connectiviy
The most common structures consist of either two or three major clusters, all of them connected.