A photo-sharing platform grows by the Barabasi-Albert rule throughout its history:…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Ordinary differential equations
Many Parts, One Behaviour, Ch Growing networks: preferential attachment and scale-free degree distributions
Problem
A photo-sharing platform grows by the Barabasi-Albert rule throughout its history: every new member forms m=5 mutual-follow links, chosen by linear preferential attachment, Πi=ki/2mt, where t is the number of members already on the platform (time counted in members added). The platform grows to N=2×106 members. (a) Starting from Πi=ki/2mt, derive the continuum equation for the degree ki(t) of a member who joined at time ti, and solve it (with ki(ti)=m) to obtain ki(t). (b) Find the degree, when the platform has N=2×106 members, of (i) the very first member (who joined at ti=1) and (ii) a member who joined at ti=2×104. (c) Use the continuum result to derive the natural cutoff kmax on the largest degree in the network, and check it against part (b)(i).