The game. N agents each keep a list of names (an inventory) for one object. At each step a random speaker meets a random neighbour (the hearer). If the speaker's inventory is empty it invents a new word. The speaker utters one random word from its list. If the hearer already knows the word, it is a success and both agents delete all their other words. If not, it is a failure and the hearer adds the word. No one ever sees the whole population and there is no leader, yet a single shared word emerges. This is the minimal naming game of Baronchelli, Felici, Loreto, Caglioti and Steels (2006).
The S-curve. The number of words in all heads first grows (everyone is inventing), peaks, and then collapses to N as one word wins; the success rate jumps from near 0 to 1 at the same time. On a complete graph the peak memory scales as N^1.5 and the convergence time grows with a similar power law (about N^1.5 at large N; runs at 100 to 3200 agents give a slightly smaller fitted value for the time, which the page reports honestly).
Topology. Who can talk to whom changes the dynamics. On a ring with 4 neighbours consensus spreads only locally (clusters of one word, like dialects, merge slowly). Rewiring a few links into long-range shortcuts (Watts-Strogatz small world) shortens convergence (the more shortcuts, the closer to the complete graph; with 5 percent at N = 100 the gain is modest), and scale-free hubs act as amplifiers.
Committed minority. If a fraction p of agents never change their word (commitment), an established convention can still be overturned. For a complete graph and large N the critical fraction is p_c = 0.098 (Xie, Zhang, Sreenivasan, Szymanski, Korniss et al. 2011): below it the old word survives or is destroyed only after a very long time, above it the minority word wins quickly, and the time diverges at the threshold. With a finite population like N = 100 the transition is smeared, so the sweep shows an approximate tipping point, not an exact one.
Checks on this page. (1) The incremental word counters used for speed are compared with a full recount of every inventory in three topologies, including committed agents and mishearing; (2) the fitted exponent of peak memory against N compared with the published 1.5; (3) the tipping-point behaviour: 5 percent never wins and 14 percent always wins for N = 200. Experiment 1 reports both fitted exponents so that you can compare.
Honest simplifications. One object, one word per utterance, no meaning space, no learning bias, no grammar, no generations, no mutation beyond the optional mishearing rate, and uniform choice of speaker and hearer. Real language change is far richer; this model shows how conventions can arise from purely local negotiation, and why a stubborn minority matters. Results are stochastic and depend on the random seed (New run changes it). Literature values are quoted from memory and flagged for a human spot-check.