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More than fifty years ago, Paul Erd˝os and Alfred R´enyi discovered that the random graph G(n, p) underwent a phase transition (in modern language) near p = 1/n. For p a bit smaller (e.g., 0.99/n) all of the components were very small and had simple structures. But for p a bit bigger (e.g., 1.01/n) a “giant component” had emerged with a complex behavior. We now understand how to slow down this process so as to see the incipient giant component (the dominant component) at an early stage and to define a Critical Window through which the process moves from subcriticality to supercriticality.