Abstract
In this paper, by the branching process and the martingale method, we prove that the size of the largest component in the critical random intersection graphGn,n5/3,pis asymptotically of ordern2/3and the width of scaling window isn−1/3.
Highlights
It is natural to ask what is the order of |C1| and what is the width of scaling window in critical G(n, ⌊n5/3⌋, p), which is proposed in [5]. e aim of the present paper is to study the critical G(n, ⌊n5/3⌋, p)
By eorem 2, to give a lower bound for |C1|, the following theorem on the size of the largest component in the critical Erdos–Renyi random graph is useful
Let |C1′| denote the size of the largest component in G(n, p′)
Summary
(1) Weakly subcritical case: if mp (1 − ε(n))/n, there are two positive constants C1 and C2 such that w.h.p. C1ε− 2(n)log nε3(n) ≤ C1 ≤ C2ε− 2(n)log nε3(n). (2) Critical case: if mp (1 + λn(− 1/3))/n for some constant λ, there are a positive function. Mathematical Problems in Engineering ω1(n)( < log n), which tends to infinity as n ⟶ ∞, and a constant C3 C3(λ) such that w.h.p. ω−11(n)n2/3 ≤ C1 ≤ C3n2/3log n. (3) Weakly supercritical case: if mp (1 + ε(n))/n, there are two positive constants C4 and C5 such that w.h.p. C4nε(n) ≤ C1 ≤ C5nε(n). By eorem 2, to give a lower bound for |C1|, the following theorem on the size of the largest component in the critical Erdos–Renyi random graph is useful
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