# Cluster tails for critical power-law inhomogeneous random graphs

### Uittreksel

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but in finite third moment degrees was obtained. It was proved that when the degrees obey a power law with exponent $\tau \in (3,4)$, the sequence of clusters ordered in decreasing size and multiplied through by $n^{ -(\tau -2) / (\tau -1) }$ converges as $n \rightarrow \infty$ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel for the Erdös-Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.
Originele taal-2 Engels Eindhoven Eurandom 44 Gepubliceerd - 2014

### Publicatie series

Naam Report Eurandom 2014007 1389-2355

Random Graphs
Tail
Power Law
Scaling Limit
Moment
Finite Type
Weak Convergence
Large Deviations
Random variable
Exponent
Converge

### Citeer dit

Hofstad, van der, R. W., Kliem, S. M., & Leeuwaarden, van, J. S. H. (2014). Cluster tails for critical power-law inhomogeneous random graphs. (Report Eurandom; Vol. 2014007). Eindhoven: Eurandom.
@book{653a7450f3bb40e3bbf1cbc562875c8f,
title = "Cluster tails for critical power-law inhomogeneous random graphs",
abstract = "Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but in finite third moment degrees was obtained. It was proved that when the degrees obey a power law with exponent $\tau \in (3,4)$, the sequence of clusters ordered in decreasing size and multiplied through by $n^{ -(\tau -2) / (\tau -1) }$ converges as $n \rightarrow \infty$ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel for the Erd{\"o}s-R{\'e}nyi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.",
author = "{Hofstad, van der}, R.W. and S.M. Kliem and {Leeuwaarden, van}, J.S.H.",
year = "2014",
language = "English",
series = "Report Eurandom",
publisher = "Eurandom",

}

Eindhoven : Eurandom, 2014. 44 blz. (Report Eurandom; Vol. 2014007).

TY - BOOK

T1 - Cluster tails for critical power-law inhomogeneous random graphs

AU - Hofstad, van der, R.W.

AU - Kliem, S.M.

AU - Leeuwaarden, van, J.S.H.

PY - 2014

Y1 - 2014

N2 - Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but in finite third moment degrees was obtained. It was proved that when the degrees obey a power law with exponent $\tau \in (3,4)$, the sequence of clusters ordered in decreasing size and multiplied through by $n^{ -(\tau -2) / (\tau -1) }$ converges as $n \rightarrow \infty$ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel for the Erdös-Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

AB - Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but in finite third moment degrees was obtained. It was proved that when the degrees obey a power law with exponent $\tau \in (3,4)$, the sequence of clusters ordered in decreasing size and multiplied through by $n^{ -(\tau -2) / (\tau -1) }$ converges as $n \rightarrow \infty$ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel for the Erdös-Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

M3 - Report

T3 - Report Eurandom

BT - Cluster tails for critical power-law inhomogeneous random graphs

PB - Eurandom

CY - Eindhoven

ER -