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**Maximizing the size of the giant.**
*(English)*
Zbl 1257.05158

Summary: Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring ‘on’ the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs.

First, the Poissonian random graph in which each node \(i\) is given an independent and identically distributed (i.i.d.) random weight \(X_{i}\) with \(E(X_{i})=\mu \), and where there is an edge between \(i\) and \(j\) with probability \(1-e^{-X_{i}X_{j}/(\mu n)}\), independently of other edges.

The second model is the thinned configuration model in which the \(n\) vertices of the ground graph have i.i.d. ground degrees, distributed as \(D\), with \(E(D) = \mu \). The graph of interest is obtained by deleting edges independently with probability \(1-p\). In both models the fraction of vertices in the largest connected component converges in probability to a constant \(1-q\), where \(q\) depends on \(X\) or \(D\) and \(p\).

We investigate for which distributions \(X\) and \(D\) with given \(\mu \) and \(p, 1-q\) is maximized. We show that in the class of Poissonian random graphs, \(X\) should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, \(D\) should have all its mass at 0 and two subsequent positive integers.

First, the Poissonian random graph in which each node \(i\) is given an independent and identically distributed (i.i.d.) random weight \(X_{i}\) with \(E(X_{i})=\mu \), and where there is an edge between \(i\) and \(j\) with probability \(1-e^{-X_{i}X_{j}/(\mu n)}\), independently of other edges.

The second model is the thinned configuration model in which the \(n\) vertices of the ground graph have i.i.d. ground degrees, distributed as \(D\), with \(E(D) = \mu \). The graph of interest is obtained by deleting edges independently with probability \(1-p\). In both models the fraction of vertices in the largest connected component converges in probability to a constant \(1-q\), where \(q\) depends on \(X\) or \(D\) and \(p\).

We investigate for which distributions \(X\) and \(D\) with given \(\mu \) and \(p, 1-q\) is maximized. We show that in the class of Poissonian random graphs, \(X\) should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, \(D\) should have all its mass at 0 and two subsequent positive integers.

### MSC:

05C80 | Random graphs (graph-theoretic aspects) |

05C90 | Applications of graph theory |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

92D30 | Epidemiology |

82B43 | Percolation |

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\textit{T. Britton} and \textit{P. Trapman}, J. Appl. Probab. 49, No. 4, 1156--1165 (2012; Zbl 1257.05158)

### References:

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