PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Growing networks: preferential attachment and scale-free degree distributions
Problem
In an online strategy game, each new clan forms m=3 alliance links. For each link, with probability ρ the ally is chosen uniformly at random among existing clans, and with probability 1−ρ the ally is copied from the corresponding alliance of one sponsor clan, chosen uniformly at random. This is the vertex-copying mechanism, with attachment probability Πi=ρ/t+(1−ρ)qi/(mt) and exponent γ(ρ)=(2−ρ)/(1−ρ). (a) Show that as ρ→1, Πi reduces exactly to the uniform-random-attachment rule Πi=1/t, independent of qi. (b) Quote (without rederiving) the stationary uniform-random-attachment distribution for m=3, and note its functional form. (c) Evaluate γ(ρ) at ρ=0.9, 0.99 and 0.999. (d) Explain in one or two sentences why a diverging γ as ρ→1 is consistent with, rather than contradicting, the geometric tail in part (b).