Statistical and Machine Learning Approaches for Network Analysis

7.1 Introduction

In the late 1960s of the last century, the theory of random graphs was developed by Erdös and Rényi [1, 2]. Most commonly studied in the literature are the and the model. The first consists of all simple graphs possessing n vertices, such that each of the n2 possible edges is chosen independently with probability p. In contrast, by using the model, a member of the set of all simple graphs consisting of n nodes and M edges is selected, such that each graph is chosen with the same probability. Despite this different definition, these models are closely related. A lot of analysis has been done to address this topic, see, for example, Refs. [1] or [2] for further information.

In this chapter, we consider generalizations of the model. More precisely, we admit the occurrence of multiple edges and loops. Furthermore, we define , a similar model of bipartite random graphs. To be precise, we deal with graphs possessing two kinds of labeled vertex sets, say V₁ and V₂, where | ...