A music-streaming service builds new playlists automatically
PHYSICS · Multi-part · 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
A music-streaming service builds new playlists automatically. Each new playlist links to m=6 earlier playlists it borrows from: it picks one existing playlist uniformly at random as its prototype, and for each of the m links, with probability ρ it links to an existing playlist drawn uniformly at random (a random link), and with probability 1−ρ it copies the corresponding link of the prototype (a copied link). Time t is counted in playlists created.
((a)) Starting from the attachment probability under copying, Πi=ρ/t+(1−ρ)qi/(mt), where qi is the number of later playlists already linking to playlist i, show that this rule is a Price model in disguise, and identify the effective offset a in terms of ρ and m (with mean out-degree c=m).
((b)) Derive the resulting exponent γ=(2−ρ)/(1−ρ).
((c)) A maximum-likelihood fit to real playlist data gives γ=3.5. Find ρ and the copying probability 1−ρ.