Giant component

In network theory, a giant component is a connected component of a given random graph that contains a constant fraction of the entire graph's vertices.

Giant components are a prominent feature of the Erdős–Rényi model of random graphs, in which each possible edge connecting pairs of a given set of n vertices is present, independently of the other edges, with probability p. In this model, if ${\displaystyle p\leq {\frac {1-\epsilon }{n}}}$ for any constant ${\displaystyle \epsilon >0}$, then with high probability all connected components of the graph have size O(log n), and there is no giant component. However, for ${\displaystyle p\geq {\frac {1+\epsilon }{n}}}$ there is with high probability a single giant component, with all other components having size O(log n). For ${\displaystyle p={\frac {1}{n}}}$, intermediate between these two possibilities, the number of vertices in the largest component of the graph is with high probability proportional to ${\displaystyle n^{2/3}}$.

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