The Size of the Giant Component of a Random Graph with a Given Degree Sequence

Given a sequence of non-negative real numbers 0 1 ::: which sum to 1, we consider a random graph having approximately in vertices of degree i. In 12] the authors essentially show that ifPi(i ; 2) i > 0 then the graph a.s. has a giant component, while ifPi(i ; 2) i < 0 then a.s. all components in the graph are small. In this paper we analyze the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine 0 0 0 1 ::: such that a.s. the giant component, C, has n + o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n 0 = n;jCj vertices, and with 0 i n 0 of them of degree i. 1 

• Type: Journal
• Year: 1998
• Where: CPC
• Authors: Michael Molloy, Bruce A. Reed
• Keywords: CPC 1998, Giant Component, Non-negative Real Numbers, Random Graph