Geometric random graphs with a scale-free degree distribution has been the go-to model of random graphs in the network science community to model a large class of real-life networks. In this talk, we will focus on how condensation effects arise in such models. More specifically, we look at upper tail large deviations of the total number of edges in such graphs and show that the excess number of edges leading to the large deviation event, come from a condensation effect in the underlying degree distribution. This is in sharp contrast with the condensation effect in the `classical’ random geometric graph observed by Chatterjee and Harel (https://arxiv.org/abs/1401.7577), where the condensation instead takes place in the underlying space. This difference is due to the scale-free nature of our model. Here, the randomness coming from the highly variable degrees overpower the randomness of the underlying vertex locations, which gives rise to degree condensates - vertices with high degrees responsible for the excess number of edges in the large deviation event. We will also review the local structure and the degree distribution of the conditional graph. Based on joint work with Remco van der Hofstad, Pim van der Hoorn, Céline Kerriou and Peter Mörters.

Zoom link: https://us02web.zoom.us/j/88670406480 Meeting ID: 886 7040 6480