A toy gene-regulatory network with genes is examined through three separate idealised…
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Random graphs, the giant component and percolation on networks
Problem
A toy gene-regulatory network with N=10000 genes is examined through three separate idealised pictures, one for each result below.
((a)) Treat the network as an Erdos-Renyi random graph with mean degree c=7 regulatory links per gene. Starting from the binomial degree distribution and taking the sparse limit N→∞ at fixed c, obtain the Poisson degree distribution, and use it to find the expected number of genes with no regulatory links at all.
((b)) To picture the local clustering this random graph ignores, imagine instead that every gene is linked only to its k=14 nearest neighbours on a fixed reference ring (a Watts-Strogatz ring with no rewiring). By counting the triangles formed by a node and its neighbours, derive C(0)=3(k−2)/[4(k−1)] and evaluate it for k=14.
((c)) Return to the Erdos-Renyi picture of part (a). Using G0(x)=G1(x)=ec(x−1) for a Poisson degree distribution, derive the site-percolation threshold ϕc=1/G1′(1) and evaluate it for c=7. Compare with the square-lattice site threshold 0.5927.