Scale-Free Network

Barabási-Albert preferential attachment: each new node connects to m existing nodes with probability proportional to k^α (where k is the existing degree). When α=1 this is the BA model, producing a power-law P(k) ~ k^(-3) degree distribution — the "rich get richer" mechanism. Hubs with disproportionately many connections emerge naturally. When α=0, attachment is uniform (Erdős-Rényi-like, Poisson degree distribution). When α=2, attachment is superlinear — one winner-takes-all hub dominates. The log-log degree histogram (right) reveals the power-law slope γ fitted by linear regression. Real networks — the Web, protein interactions, citation graphs — sit near γ≈2-3.