Statistical and dynamical physics of complex networks — graphs whose nodes and edges are governed by universal statistical laws rather than by a specific geometric or chemical lattice structure. Canonical ensembles: Erdős-Rényi 1959-60…
network-physics
Erdős-Rényi random graph G(N, p): giant component at ⟨k⟩ = 1
Erdős-Rényi 1959-60 (Publ. Math. Debrecen 6:290; Publ. Math. Inst. Hung. Acad. Sci. 5:17) defined the ensemble G(N, p) of undirected graphs…
Barabási-Albert 1999: preferential attachment → γ = 3
Barabási-Albert 1999 (Science 286:509) proposed that the 'scale-free' property (power-law P(k) ~ k^{-γ}) observed in the World-Wide-Web and…
Watts-Strogatz 1998 small-world: C(p) ≈ C(0)(1-p)³
Watts-Strogatz 1998 (Nature 393:440) introduced the small-world network model interpolating between regular lattices (high clustering C,…
ER giant-component threshold: ⟨k⟩_c = 1 exact (Poisson branching)
Exact derivation of the ER critical connectivity. Probability that a randomly chosen node is NOT in the giant connected component, q,…
BA exponent γ = 3 exact from P(k) = 2m(m+1)/[k(k+1)(k+2)]
Exact closed-form evaluation of the Barabási-Albert degree distribution and its large-k exponent. Master-equation solution with boundary…
WS clustering: C(0) = 1/2 at K=2, C(p=1/2)/C(0) = 1/8
Exact closed-form evaluation of the Watts-Strogatz clustering at two canonical points. Regular ring with K = 2 neighbours per side (degree…