the giant component
The giant component is the single huge cluster that suddenly appears in a random graph once the average degree exceeds 1. Below this point all connected pieces are small — of size O(log n) — and roughly the same order; above it, exactly one component balloons to contain a positive fraction of all n vertices, while every other component remains O(log n). The giant is the most striking phenomenon in random graph theory: a global object emerging out of purely local, independent edge choices. The question it answers is: when does a network become 'mostly one piece', and how big is that piece?
Fix p = c/n so the average degree is c. The fundamental theorem (Erdos-Renyi) is a trichotomy. If c < 1 (subcritical), whp the largest component has only O(log n) vertices. If c > 1 (supercritical), whp there is a unique giant component of size approximately rho(c) times n, where rho(c) is the unique strictly positive solution of rho = 1 - e^(-c rho) (equivalently 1 - rho = e^(-c rho)); the second-largest component is back down to O(log n). At c = 1 (critical) the largest component is of order n^(2/3), an intermediate scale. The survival probability rho(c) is exactly the survival probability of a Galton-Watson branching process with Poisson(c) offspring, which is the heart of the heuristic for why the giant appears: exploring the cluster of a vertex looks locally like a branching process, and the process survives forever (giant) precisely when its mean offspring c exceeds 1.
The giant component is the prototype of a phase transition in a discrete probabilistic system, and the bridge between random graphs, branching processes, and percolation. It matters because it formalizes connectivity in large networks (epidemics spread to a macroscopic fraction iff a giant cluster of susceptibles exists; a communication network is robust iff a giant survives random failures). A subtlety to respect: 'unique' is a theorem, not an assumption — proving there is only one giant (no two competing macroscopic clusters) is a real part of the result, done by a sprinkling argument showing any two large clusters whp get joined by an extra edge. Also note the giant does not contain all vertices: even well above the threshold a positive fraction (those in tiny components, including isolated vertices) lie outside it until p reaches the much larger connectivity scale (log n)/n.
At average degree c = 2 the giant fraction rho solves rho = 1 - e^(-2 rho), giving rho about 0.797: roughly 80 percent of the vertices belong to the giant, the rest scattered in tiny clusters. At c = 1.5, rho solves rho = 1 - e^(-1.5 rho), about 0.583. As c decreases to 1 the giant fraction shrinks continuously to 0 — the transition is continuous (second order).
The giant fraction rho(c) = 1 - e^(-c rho) grows continuously from 0 at c = 1, the order parameter of the random-graph phase transition.
Uniqueness of the giant is a theorem proved by sprinkling, not a definition; and the giant is not the whole graph until the much higher connectivity threshold. The fraction rho(c) coincides exactly with the survival probability of a Poisson(c) Galton-Watson tree.