Power-law or Poisson? Removing nodes from a scale-free network

An interactive version of the sequential removal process in Mi Jin Lee, Jung-Ho Kim, Kwang-Il Goh, Sang Hoon Lee, Seung-Woo Son, and Deok-Sun Lee, “Degree distributions under general node removal: Power-law or Poisson?”, Phys. Rev. E 106, 064309 (2022). Each step removes one node with probability ∝ (k+1)−θ, using the current degrees.

closer to scale-free (SF)closer to Poisson (PO)
f = 0.000removed
closer to SF

Degree distribution Pf(k)

nowSF ref.PO ref.f = 0

Relative entropies

S(SF)S(PO)

Mean degree mf

this runm₀(1−f), random removal

Giant component G/Nf

this run

Where does the power law survive?

The paper measures the distance from Pf(k) to two reference distributions with the same mean degree, the Poisson distribution of an Erdős–Rényi graph and the degree distribution of the static scale-free model, using the Kullback–Leibler divergence:

Sf = Σk Pf(k) ln [ Pf(k) / P(ref)(k) ]

Whichever entropy is smaller wins. Both references keep their form under uniform random removal, which is why they act like fixed points. The paper's finding: for θ ≥ 0 the distribution stays closer to SF all the way to f → 1, and only hub-targeting removal (θ < 0) opens a Poisson regime above a crossover f*(θ).

The map on the right sweeps θ with the current network settings, averaging a few runs per θ, and marks where your current run sits.

Takes a few seconds at N₀ = 1000. The sweep uses up to 2000 nodes.

SF regimePO regimeyour run