00 · Briefing

02805 · go nuts · week 2

Models & null models

Four networks. One is real.

We put the 277-character Marvel giant component next to three famous models of the same size and density, then erase the labels. Every instrument below asks the same question: which structures can a simple mechanism manufacture, and which ones survive only in the real thing?

targetshort paths targettriangles targetgiant hubs

Our wager: no single simple model matches all three at once.

Specimen M · 277 nodes · 1,421 undirected links · ⟨k⟩ = 10.3

01 · Specimen arena

select an input channel
SPECIMEN M
Drag nodes, scroll to zoom. Position is decoration; the gauges are data.

Channel readout

Marvel

02 · Tail scope

CCDF · P(K ≥ k) · no binning

The complementary cumulative distribution uses every data point and needs no bins. A heavy tail stays straight here; a Poisson tail falls off a cliff.

03 · Null model detective

5 clues · 4 suspects
clue1 / 5
score0
streak0

Clue

Press 1–4 to accuse, N for the next clue.

04 · Shuffle rig

degree-preserving edge swaps

An Erdős–Rényi baseline deletes Marvel's hubs, so it cannot tell us whether hubs cause the triangles. Edge swaps keep every character's degree exactly and destroy only who is connected to whom — the baseline that asks the right question.

0 swaps C = —
realscrambled

120 shuffled universes

Could chance reach Marvel?

distance from the null—
empirical p-value—

What surprised us

Hubs explain only half the clustering gap. Marvel falls from C = — to — once degrees are held fixed and everything else is scrambled — yet the survivor is still —σ beyond chance. Swap in a plain random graph and you would have credited the whole surplus to the hubs.

05 · Paradox draw

degree-biased sampling

Draw a random character, then one random neighbour of that character. Popular heroes sit in many neighbour lists, so the second draw is biased upward. Pull the handle enough times and the friendship paradox appears on its own.

You drew

?

— connections

Their random friend

?

— connections
0friend at least as popular
0person more popular
—paradox rate

Pull the handle. Spider-Man is waiting inside a lot of neighbour lists.

06 · Spec sheet

method & receipts
Representation
A node is a Marvel superhero Wikipedia article. An undirected, unweighted edge exists when either article links to the other. All measurements run on the 277-node giant component.
Models
Erdős–Rényi matches n and m. Watts–Strogatz uses degree 10 with rewiring q = 0.2. Barabási–Albert attaches 5 links per arriving node. One fixed seed per model, so every number on this page is one reproducible draw rather than an average.
Null model
120 samples, each applying three double-edge swaps per edge. The degree sequence is asserted unchanged. The p-value is (extreme + 1)/(samples + 1), so 120 shuffles can never report zero.
Human check
Not done by the machine: pull one shuffled snapshot, confirm by hand that Spider-Man still has 106 neighbours while the names in that list have changed. Record who checked before publishing.
Reproduce
uv run analyse_week2.py rebuilds week2/data/week2.json from the frozen snapshot of 26 August 2026.

Markus · William · Kris · source repository ↗ · all weeks