The Largest Component of Near-Critical Random Intersection Graph with Tunable Clustering

Shiying Huang,Bin Wang,Barbara Martinucci

doi:10.1155/2021/2284300

Shiying Huang, Bin Wang + Show 1 more

Open Access

https://doi.org/10.1155/2021/2284300

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Abstract

In this paper, we study the largest component of the near-critical random intersection graph G n , m , p with n nodes and m elements, where m = Θ n which leads to the fact that the clustering is tunable. We prove that with high probability the size of the largest component in the weakly supercritical random intersection graph with tunable clustering on n vertices is of order n ϵ n , and it is of order ϵ − 2 n log n ϵ 3 n in the weakly subcritical one, where ϵ n ⟶ 0 and n 1 / 3 ϵ n ⟶ ∞ as n ⟶ ∞ .

Highlights

Introduction and Main ResultOne of the important issues in random graphs is to determine the size of the largest component
Wang et al [13,14] studied√t h e component evolution in G(n, m, p) when p 1 ± ε(n)/ nm, where m nα(α > 1) and ε(n) ⟶ 0 as n ⟶ ∞. ey obtained that the order of |C1| and the width of the scaling window around the critical probability depend on whether α ≥ (5/3)
If α ≥ (5/3), they are n2/3 and n− (1/3), respectively, which are independent of α, and the evolution is similar to that in the critical Erdos-Renyi random graph; if 1 < α < 5/3, they are n(3− α)/2 and n− (α− 1/2), respectively

Summary

Introduction and Main Result

One of the important issues in random graphs is to determine the size of the largest component. For the upper bound of the size of largest component in random graphs, one of standard methods is to use a branching process to dominate the corresponding exploration process from above. To bound |C1| from below in G(n, 􏼄βn􏼅, p), it does not work by the comparison of G(n, m, p) and ErdosRenyi random graph G(n, p􏽢) for some suitable p􏽢 through the result in [16], as it does not cover the case when m and n are of the same order. We get the upper bound for the weakly supercritical regime in Section 3.1 and that for the weakly subcritical one in Section 3.2 through the branching process method and the coupling technique, respectively

The Proof of the Lower Bound

The Upper Bound