Scale-Free Networks — Barabási–Albert

Preferential attachment: new nodes connect to existing nodes with probability proportional to their degree, generating hubs and power-law degree distributions.

Network — node size ∝ degree. Drag nodes to rearrange.
Degree distribution (log-log scale)
Nodes: 0
Edges: 0
Max degree: 0
Power law γ ≈ —
2
100
5
Barabási–Albert model: begin with m₀ = m+1 fully connected nodes. Each new node attaches m edges to existing nodes with probability Π(i) ∝ k_i (degree of node i). This "rich-get-richer" rule generates a scale-free network with degree distribution P(k) ~ k^{-γ}, where γ = 3 for the BA model (independent of m). Real-world examples: World Wide Web, citation networks, protein interactions, airport connections. Hubs emerge naturally and make the network robust to random failures but vulnerable to targeted hub attacks.