Subgraph count statistics in Erdős-Rényi random graphs

Abstract

In this chapter, we consider Erdős-Rényi model Γ(n, p _n ) of random graphs. A random graph Γ with this distribution is described as follows. Its vertex set is [n] and for each pair {i, j} ⊂ [n] with i ≠ j, there is an edge between i and j with probability p _n . Moreover, all these events are independent. We are then interested in the following random variables, called subgraph count statistics. If γ is a fixed graph of size k, then X _γ ⁽ⁿ⁾ is the number of copies of γ contained in the graph Γ(n, p _n ) (a more formal definition is given in the next paragraph). This is a classical parameter in random graph theory; see, e.g. the book of S. Janson, T. Łuczak and A. Ruciński [JŁR00].