Kuramoto model on scale-free networks: hysteresis and first-order sync
Oscillators on network (color = phase)
Order parameter r(K) — forward/backward sweep
Order parameter r = ? | Mean degree ⟨k⟩ = ? | K / K_c ≈ ?
The standard Kuramoto model on random graphs shows a continuous (second-order) phase transition at K_c = 2/π⟨ω²⟩/⟨ω⟩. But on scale-free networks where the natural frequency ω_i is correlated with the node degree k_i (ω_i ∝ k_i), the transition becomes discontinuous (first-order): explosive synchronization.
Gómez-Gardeñes et al. (2011) showed that when oscillator frequencies match hub degrees, hubs synchronize among themselves first, then absorb the rest abruptly. The result: a hysteresis loop — the forward (K increasing) and backward (K decreasing) transition points differ, a hallmark of first-order transitions.
The order parameter r = |⟨e^{iθ_j}⟩| measures global coherence: r≈0 (incoherent) to r=1 (fully synchronized). The explosive jump from r≈0 to r≈1 is abrupt and irreversible within the hysteresis window.
Applications: power grid cascades, neural synchrony, epidemic spreading on heterogeneous networks where degree-heterogeneity creates similar correlations.