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 a network of judicial opinions, each new opinion cites a mean of c=15 earlier opinions (its out-degree), and every opinion, however new, has an additional fixed offset a=3 added to its in-degree q before new citations are distributed by preference (so an opinion with no citations yet is still citable). The Price model's rate equation for this network reduces, in the mean-field continuum limit, to the recursion pq=q+a+1+a/cq−1+apq−1,q≥1, with p0 fixed by normalisation. (a) Solve the recursion as a product, write the product using Gamma functions, and identify it as proportional to the Beta function B(q+a,2+a/c). (b) Using the quoted large-x asymptotic B(x,y)∼Γ(y)x−y, derive the tail exponent γ=2+a/c and evaluate it for a=3, c=15. (c) Check the special case a=c against the Barabasi-Albert exponent derived from pure linear preferential attachment.