Graph Theory

A graph begins as a bare list of vertices: cities, people, neurons, web pages. On their own they tell you almost nothing.

Join two vertices whenever something passes between them: a road, a friendship, a synapse, a link. The moment you do, you have an object with properties you can measure and prove things about.

Now the classical questions open up. What is the cheapest way from A to B? Dijkstra's algorithm answers it in milliseconds, and that single answer underpins navigation, logistics, and how a fleet of driverless cars organises itself.

Connections are never spread evenly. A few vertices do most of the work, and the busiest one is where information piles up, where an epidemic spreads fastest, and where the network is most fragile if it fails.