The Threshold Theorem is a principle in random graph theory that identifies critical probabilities at which a given property becomes likely to appear in a random graph. Specifically, it shows that as the number of edges increases in the Erdős-Rényi model, certain properties like connectivity or the presence of a giant component will emerge suddenly once a threshold probability is surpassed. This phenomenon highlights the phase transition behavior of random graphs, where small changes in edge density can lead to drastic changes in graph structure.
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