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- W2033468284 abstract "Let $c$ be a constant and $(e_1,f_1), (e_2,f_2), dots, (e_{cn},f_{cn})$ be a sequence of ordered pairs of edges on vertex set $[n]$ chosen uniformly and independently at random. Let $A$ be an algorithm for the on-line choice of one edge from each presented pair, and for $i= 1,hellip,cn$ let $G_A(i)$ be the graph on vertex set $[n]$ consisting of the first $i$ edges chosen by $A$. We prove that all algorithms in a certain class have a critical value $c_A$ for the emergence of a giant component in $G_A(cn) (ie$, if $c gt c_A$, then with high probability the largest component in $G_A(cn)$ has $o(n)$ vertices, and if $c > c_A$ then with high probability there is a component of size $Omega(n)$ in $G_A(cn))$. We show that a particular algorithm in this class with high probability produces a giant component before $0.385 n$ steps in the process ($ie$, we exhibit an algorithm that creates a giant component relatively quickly). The fact that another specific algorithm that is in this class has a critical value resolves a conjecture of Spencer.In addition, we establish a lower bound on the time of emergence of a giant component in any process produced by an on-line algorithm and show that there is a phase transition for the off-line version of the problem of creating a giant component." @default.
- W2033468284 created "2016-06-24" @default.
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- W2033468284 date "2006-06-07" @default.
- W2033468284 modified "2023-09-27" @default.
- W2033468284 title "Creating a Giant Component" @default.
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- W2033468284 doi "https://doi.org/10.1017/s0963548306007486" @default.
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