Renormalizing a network by integrating out its nodes

An interactive companion to Renormalization of complex networks with partition functions by Sungwon Jung, Sang Hoon Lee and Jaeyoon Cho (Phys. Rev. E 110, 064316, 2024; doi:10.1103/PhysRevE.110.064316). Free bosons hop across the network with couplings J_ij and local chemical potentials μ_i. Each RG step integrates one node out of the Gaussian partition function, which rewires its neighbours exactly. Under continuous observation and low density, the edge weight w_ij = J_ij² becomes the transition rate of a classical Markov process.

Network model Nodes 120 Degree exponent γ 3.0 Mean degree Initial μ₀

Hover to inspect · click a node to integrate it out

original edges couplings created by RG |μ_i| low → high

Speed Hide weights below 10⁻⁶

Current scale

G₀

Nodes remaining: –
Scale level k = log₂(N/n): –
Surviving original edges: –
Couplings created by RG: –
Shown at current threshold: –
|μ_i| range: –
Largest eigenvalue of adjacency: –

Last step

Nothing integrated out yet. The next node is chosen at random among those with the largest |μ_i|, as in the paper. Clicking a node lets you pick a different integration order.

Update rule for eliminating node k

μ_i → μ_i − J_ik² / μ_k

J_ij → J_ij − J_ik J_jk / μ_k

Because μ_k < 0, neighbours absorb population (|μ_i| shrinks) and every pair of neighbours gains a coupling, even if they were not connected before.

Partition function check

Z ∝ 1/det(−βA) with A = J + diag(μ). Integrating out node k factors out −μ_k, so these two numbers must stay equal (β = 1).

ln det(−A) of G₀: –
Σ ln(−μ_k) + ln det(−A) now: –
Difference: –

Strength distribution of this network

The strength of a node is the sum of its edge weights. With the physical map, the right definition is s_i = Σ_j J_ij², since J² is the transition rate. In G₀ it equals the degree.

After coarse-graining, s = 1 separates two populations: nodes whose strength comes mostly from newly created, weak couplings (s < 1) and nodes still holding original edges (s > 1). A single small network is noisy; the ensemble below pools many runs.

Run the experiment to fill the chart.

Ensemble experiment

This repeats the paper's analysis at a smaller size: generate many networks, renormalize each down to G₁ … G₄ (N/2^k nodes left), and pool the strength distributions. For scale-free inputs, both the s < 1 and s > 1 branches keep a power-law shape across scales. For Erdős–Rényi and Watts–Strogatz inputs, the shape changes.

Model Nodes Samples μ₀

The paper uses about 10⁴ nodes and 100 samples; expect noisier tails here. Switching to the false strength reproduces the paper's Fig. 3 check: the clean power-law branches blur, because J itself is an oscillation frequency, not a rate.