Large induced subgraphs of random graphs with given degree sequences

Abstract We study a random graph with given degree sequence , with the aim of characterising the degree sequence of the subgraph induced on a given set of vertices. For suitable and , we show that the degree sequence of the subgraph induced on is essentially concentrated around a sequence that we can deterministically describe in terms of and . We then give an application of this result, determining a threshold for when this induced subgraph contains a giant component. We also apply a similar analysis to the case where is chosen by randomly sampling vertices with some probability , that is, site percolation, and determine a threshold for the existence of a giant component in this model. We consider the case where the density of the subgraph is either constant or slowly going to 0 as goes to infinity, and the degree sequence of the whole graph satisfies a certain maximum degree condition. Analogously, in the percolation model we consider the cases where either is a constant or where slowly. This is similar to work of Fountoulakis in 2007 and Janson in 2009, but we work directly in the random graph model to avoid the limitations of the configuration model that they used.