Indexing the archive…
Your Universe of Digital Possibilities
Two crowds of the same 350 people, and everyone has exactly as many friends in one as in the other. One crowd grew — newcomers meeting friends of friends; the other is the same crowd with its wiring shuffled. Pick the one with a history, twelve times. Then count triangles.
For each node, the share of its pairs of neighbours that are linked — Ti the triangles through it, ki its degree — averaged over the network. Growth through introductions leaves triangles behind; a degree-preserving shuffle keeps every degree and sinks C to the floor its hubs allow by accident, about (⟨k²⟩ − ⟨k⟩)² / (N⟨k⟩³).
A new node links to an existing one with probability proportional to how many links it already has. Growth plus this one bias — the already-popular get more popular — is enough to breed hubs from nothing.
The exact central interval for K, the count of wins in n independent plays at win probability p — computed from the rules alone, before a single play. A record inside it is the rules working; a record outside it happens one time in twenty, and the band says so in advance. When the chance of each play is fixed before it is played but differs from play to play, the same band comes from the Poisson-binomial sum.
The seventh twin forges by surrogate: the original crowd with one kind of order — who was introduced to whom — destroyed and every degree kept, so the census most network statistics report cannot tell them apart. The Hub owns preferential attachment, the grower’s first step; The Contagion is where Watts and Strogatz’s small world spreads things; and The Drift runs the same move in time — the null cloud is the forgery, done again in public.