Giant components in random graphs

Abstract

The phase transition in random graphs was first discussed by Erdős and Renyi who showed that a random graph undergoes a drastic change in the order and structure of the largest component. In view of recent results, one can recognise two main trends that have increased our understanding of random graphs. The first trend is the study of various random graph models (e.g. random hypergraphs, random graph processes, random graphs with degree constraints, random graphs on surfaces). The second trend is finding new simple proofs (and thereby improvement) of classical results on random graphs. This article discusses developments in random graphs in the light of these trends, with focus on giant components, limit theorems, and proof techniques.