Introduction to Networks: The Maths Behind Everything Connected

A network (or graph, in the mathematical sense) consists of vertices (nodes) and edges (connections between them). Edges can be directed (one-way, like a one-way street) or undirected (two-way, like a friendship). The degree of a vertex is the number of edges meeting at it. A path is a sequence of edges connecting two vertices. A cycle is a path that returns to its starting point. A tree is a connected graph with no cycles. Networks model any situation involving discrete objects and connections between them.

Seven bridges and one brilliant mathematician

Graph theory was effectively founded by Leonhard Euler in 1736, solving the Königsberg Bridge Problem. The city of Königsberg (now Kaliningrad, Russia) had seven bridges connecting its islands and riverbanks. The question: is it possible to walk through the city crossing each bridge exactly once? Euler proved it was not, by showing that such a walk (now called an Eulerian path) requires all but at most two vertices to have an even degree — and Königsberg's graph doesn't satisfy this condition. His proof was the first to treat the problem as a pure graph structure, ignoring the actual geography — a founding act of abstract mathematics. Hamilton paths (visiting each vertex exactly once) and the Travelling Salesman Problem followed later, still active research areas today.

Networks in the modern world

The internet is a network: websites are vertices, hyperlinks are directed edges. Google's original PageRank algorithm ranked web pages by their position in this network — pages linked to by many high-ranking pages were themselves highly ranked. Social networks (Facebook, LinkedIn) are graphs where people are vertices and friendships or connections are edges. GPS navigation finds the shortest path between two points in the road network. Epidemiologists model disease spread through contact networks to identify super-spreaders and optimal vaccination targets. Airline route planning optimises the network of flights to minimise costs and delays. Electrical power grids are networks whose failure modes are studied to prevent cascading blackouts. Network theory is one of the most practically powerful branches of mathematics in the modern world.