Erdős & Rényi (1959) asked three questions about a graph formed by connecting n labelled points at random: What is the probability it's completely connected? How big is its largest connected component? How many connected components does it have? Answering these for growing n revealed one of the most influential results in network science: as the average number of connections per node (c) crosses 1, the graph undergoes a sudden phase transition — from many small, disconnected fragments to a single "giant component" that swallows almost every node.
Drag c below to add random edges to the network on the left, and watch where it lands on the giant-component curve on the right (computed once from a larger reference network) — the same threshold behavior the paper predicted for n → ∞.
Computing reference curve…