**Speaker:** Bruce Reed, McGill University

**Title:** How to determine if a random graph with a fixed degree sequence has a giant component

**Abstract: **

Many 21st century networks are formed via procedures, such as preferential attachment, which make some nodes much more likely to have attachments than others. For example, we observe empirically in the web that certain authoritative pages, called hubs, will have many more links entering them. Thus it is of interest to determine, for a fixed degree sequence D = { *d*1,....*d**n*}, the probability that a uniformly chosen (simple) graph on nodes {1,....,*n*} with the given degree sequence (i.e. such that node* i* is linked to *d**i* other nodes) has a giant component. A heuristic argument suggests that a giant component will almost surely exist provided the sum of the squares of the degrees is at least twice the sum of the degrees. In 1996, Molloy and Reed essentially proved this to be the case provided the degree sequence under consideration satisfied certain technical conditions. This work has attracted considerable attention and has been applied to random models of a wide range of complicated 21st century networks such as the web or biological networks operating at a sub-molecular level. Many authors, have obtained related results which prove this result under a different set of technical conditions or tie down the size of the giant component. In this paper we characterize when a random graph with a fixed degree sequence almost surely contains a giant component, essentially imposing no conditions.

This is joint work with Felix Joos , Guillem Perarnau, and Dieter Rautenbach,