From the weighted Hamiltonian to a free energy in m̃
The model is Hf = −K̃Σi<j(kikj/Nz1)sisj = −(K̃/2Nz1)(Σkisi)² + const (Eqs. 8–9). The paper motivates it as the average of the Ising model over random graphs with link probability pij = kikj/Nz1, which needs pij < 1 and so restricts it to λ > 3 (its note 14). Taken as a weighted, globally coupled network, the model needs no such condition, and everything below also covers 2 < λ ≤ 3. A saddle point in the auxiliary field u then gives the free-energy density f(m̃) = −z1K̃m̃²/2 + z1u0m̃ − ⟨ln 2cosh(u0k)⟩ with m̃ = ⟨k tanh(u0k)⟩/z1 (Eqs. 13–15). Every curve below integrates e−Nf numerically for the actual finite degree sequence, so it is the paper’s theory at finite N, not just its asymptotes.
Degree sequence
Deterministic rule of Eq. (54) at the N chosen below; the dashed line is k−λ.
Free-energy density
What the theory predicts here
Order parameter against coupling
Paper Figs. 1–3, computed for N = 10³ to 10⁷.
Critical finite-size scaling
Paper Figs. 4(a) and 5(a): ⟨m̃⟩ at the pseudo-critical point ε = 0 and at the bulk critical point εb = 0. Dashed lines have the predicted slopes.
Monte Carlo test
As in the paper’s Sec. IV, Metropolis dynamics runs on the globally coupled model of Eq. (8) with deterministic degree sequences. Each size is swept from the ordered side down in ε, reusing the equilibrated state, and also run at the bulk critical point (for λ ≤ 3, where that point is zero, at a fixed coupling instead). Lines are the saddle-point theory for the same sequence. The paper went to N = 32 768 × 10³ with 100 samples; this runs in your browser, so the sizes are smaller and the data noisier.
⟨m̃⟩ against ε
Markers: Monte Carlo. Lines: theory for the same N.
⟨m̃⟩ at ε = 0 and εb = 0
Markers: Monte Carlo. Lines: theory, extended to larger N.
Scaling collapse
Sample-to-sample fluctuations
Drawing the N degrees independently from P(k) instead of deterministically makes the degree moments fluctuate between samples (Sec. V). The critical amplitude depends on b = z14z4/z24, and z4 is not self-averaging for λ < 5 at the natural cutoff; for λ < 3 neither is z2, so even K̃c(N) = z1/z2 fluctuates. Here each sample’s ⟨m̃⟩c is the saddle-point thermal average at its own K̃c, so the spread shown comes purely from the sampled degrees.
Histogram of ⟨m̃⟩c/[⟨m̃⟩c]
Paper Fig. 6. A self-averaging quantity sharpens toward a spike at 1 as N grows; a non-self-averaging one keeps its width.