Growth of components in random graphs

Abstract

The creation and growth of components of a given complexity in a random graph process are studied. In particular, the expected number and total size of all such components is found. It follows that the largest '-component during the process is Op(n 2/3 ) for any given '. The results also yield a new proof of the asymptotic behaviour of Wright's coecients. between the two processes is that in {G(n,m)}, the edges are added at the fixed times 1,2,..., so at time m we have the random graph G(n,m) with m edges, while in {G(n,t)} the edges are added at random times in such a way that at time t = p, we have the random graph G(n,p). ({G(n,t)} may be constructed by letting each edge e in the complete graph Kn appear at a random time Te, with Te independent and uniformly distributed on (0,1), and letting G(n,t) contain the edges that have appeared before t.) In this paper, we will study random variables that depend on the order the edges appear in the process, but not on the time scale. All such variables will thus have the same distribution for both processes (not only asymptotically, but also for each finite n). Hence all results below are valid for both processes. We define the complexity of a connected graph to be its number of edges minus its number of vertices. A component of a graph of complexity ' is called an '-component. Here ' 1; a ( 1)-component is a tree, a 0-component is unicyclic, and '-components with ' 1 are known as complex components. In the beginning of the random graph process, there are no edges at all, and thus n components of order 1 and complexity 1; at the end, we have the complete graph, with a single component of complexity n 2 n. Several authors have studied what happens in between, see e.g. (4, 9, 6, 7, 11, 8). We will here add some results obtained by studying, as in (6), the ways '-components are created. Each time a new edge is added, there are two possibilities: