Birth control for giants

13 years 4 months ago

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The standard Erd˝os-Renyi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n 2 rounds as one time unit, a phase transition occurs at t = 1 when a giant component (one of size constant time n) ﬁrst appears. Under the inﬂuence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory. We deﬁne a broad class of graph evolutions in which at each round one chooses one of two random edges {v1, v2}, {v3, v4} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a diﬀerential equation for S(t). There is a critical time tc so that S(t) → ∞ as t approaches tc from below. We show that the discrete random process asymptotically fol...

Joel H. Spencer, Nicholas C. Wormald

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