Abstract
We consider a random subgraph _G<sub>p</sub>_ of a host graph _G_ formed by retaining each edge of _G_ with probability _p_. We address the question of determining the critical value _p_ (as a function of _G_) for which a giant component emerges. Suppose _G_ satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second-order average degree <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10390644_o_uf0001.gif"></inline-graphic> to be <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10390644_o_uf0001.gif"></inline-graphic> = Σ<sub> <em>v</em> </sub>_d_<sup>2</sup><sub> <em>v</em> </sub>/(Σ<sub> <em>v</em> </sub>_d_<sub> <em>v</em> </sub>), where _d_<sub> <em>v</em> </sub> denotes the degree of _v_. We prove that for any ∊ > 0, if _p_ > (1 + ∊)/<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10390644_o_uf0001.gif"></inline-graphic>, then asymptotically almost surely, the percolated subgraph _G<sub>p</sub>_ has a giant component. In the other direction, if _p_ > (1 − ∊)/<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10390644_o_uf0001.gif"></inline-graphic>, then almost surely, the percolated subgraph _G<sub>p</sub>_ contains no giant component. An extended abstract of this paper appeared in the WAW 2009 proceedings [Chung et al. 09]. The main theorems are strengthened with much weaker assumptions.
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