Attach new links in proportion to the links a node already has, and the biggest nodes pull away from the rest — the ‘rich get richer’. The share held by the top GROWS over time, all by itself, with no outside push. It’s a gap that opens rather than closes, and it’s why citations, followers, and wealth pile up on the already-large. Slide the steps and watch the leader run away.
Preferential attachment (Barabási–Albert / Yule / Simon): the probability a new link lands on node i is proportional to i’s current degree kᵢ. This rich-get-richer rule produces a scale-free, power-law degree distribution and makes the TOP node’s share of all mass GROW over time — concentration increases without any external force. It is the Matthew effect made mechanical. A fail-loud self-check throws unless the top node’s share of total degree is larger at the end than at the start. ◆ real network science, node-verified.
A small seeded simulation is the illustration; the proportional-attachment rule and the resulting concentration are exact. The instrument shows the gap WIDENS — whether that is just or should be curbed is a normative question ([[lillith]]’s), not this one’s.