Half a link each is dust. One link each is a web.

A room of strangers. No cliques, no seating plan, nobody popular. Just links handed out between random pairs, one at a time. Count them per head and watch what the room does as that number passes one.

What you're seeing

Every dot is a person. Where they sit means nothing — there is no neighbourhood here, no distance, no reason for anyone to link to anyone. A link is drawn between two people picked at random, and the slider says how many links there are per head. Your eye will not believe that, because it reads nearness as relatedness whatever a caption says. So check it: press Reseat everyone and the same links are redrawn with everybody sitting somewhere else. The web is untouched. It was never about the seats.

At half a link each the room is grey dust. Dozens of little huddles, the biggest holding a few percent of the people. Nudge the slider up and the huddles get slightly bigger, and it looks like nothing much is happening.

Then you cross one link per person, and a single amber group appears and starts eating the room. By two links each it holds four people in five. Nothing about the rule changed. The only thing that changed is a number crossing 1.

The lower chart says why that is not a trick of the eye. The amber curve is the predicted share, worked out before any dots were drawn; the blue dots are what this particular room actually did as you dragged. They land on the curve. One room, growing: dragging the slider adds and removes links from a single fixed list rather than drawing a fresh crowd at every setting, so groups only ever merge as you go right. That is the real process, and the proof checks it gives the same answer as drawing fresh each time — but it does mean the blue dots are one room's story, not the law. Press New room to see another one land on the same curve.

Now drag the second slider. In a small room the change is a smear; in a big one it is a cliff. That is the signature of a real threshold: it gets sharper the more people you have, rather than staying equally vague. Below one link each, the biggest group does keep growing as you add people — but far slower than the room does, so its share falls away to nothing. That is exactly what “there is no giant group” means.

The rule, exactly. Take n people and m = kn/2 links, each joining a uniformly random pair (this is the Erdős–Rényi model G(n,m); k is the average degree). The share S of people in the largest connected group settles on the non-zero solution of S = 1 − e^−kS which has no non-zero solution at all until k > 1 — so the threshold is exactly one link per person. Dragging the slider reveals a prefix of one fixed shuffled list of pairs, so the room grows rather than being redrawn. The companion proof (improve/verify/107-giant-component.js, 20 checks) cross-checks the union-find against an independent breadth-first search on 40 graphs, matches simulation to the curve at five values of k, and confirms the reveal is statistically identical to drawing a fresh graph at every k. Four negative controls: at k = 0.5 the largest share falls 1.8% → 0.22% as the room grows 500 → 8000 (while its absolute size only doubles); the formula returns exactly 0 below the threshold rather than a small comforting number; exactly at k = 1 the largest group grows like n^2/3, so its share still vanishes — measured at 0.469× per 8× the crowd against a predicted 0.5; and putting the same average degree on a lattice gives a largest patch of 0.6% instead, which is why percolation is a different exhibit and not this one in a hat.