Mean-field theory
Assuming uncorrelated degrees and mi = m(ki, hi), the zero-temperature order parameter solves m = f(m) = ∫dk [kP(k)/k̄] G(km/Δ), with G(x) = 2∫0xp0. The shape of f near m = 0 decides everything (paper Fig. 1): an infinite slope means order at every Δ, convexity gives a jump, concavity a continuous transition. Curves here use the continuum P(k) = ck−γ for k ≥ k0, with k0 set by k̄.
Self-consistency map
Mean-field order parameter
What the theory predicts here
β(γ) from mean field (line) and the paper’s finite-size-scaling estimates (dots, Fig. 7).
Finite-size simulation
This reproduces the paper’s numerical test on static-model networks (Figs. 2–6): for each size, many independent networks and field realizations are solved exactly across a grid of Δ. The same realization is reused along Δ, as in a sweep. Larger sizes and more samples sharpen the picture but take longer; everything runs in your browser.
Order parameter ⟨m⟩ versus Δ
Paper Figs. 2 and 4.
Binder parameter U = 1 − ⟨m4⟩ / 3⟨m2⟩2
Paper Fig. 5. Curves for different N cross at a continuous transition.
Histogram H(m) near the threshold
Paper Fig. 3. Two peaks at the same Δ signal phase coexistence and a first-order jump.
Finite-size scaling collapse
Paper Fig. 6: ⟨m⟩Nβ/ν′ versus |Δc − Δ|ν′N, using points with Δ < Δc.