The connectivity and phase transition in inhomogeneous random graphs of finite types

A significant generalization of the Erd\"os-R\'enyi random graph model is an `inhomogeneous' random graph where the edge probabilities vary according to vertex types. We identify the threshold value for this random graph with a finite number of vertex types to be connected and examine the model's behavior near this threshold value. In particular, we show that the threshold value is $c \frac{\log n }{n}$ for some $c>0$ which is explicitly determined, where $n$ denotes the number of vertices. Furthermore, we prove that near the threshold, the graph consists of a giant component and isolated vertices. We also investigate the phase transition and provide an alternative proof of the results by Bollob\'as et al. [Random Struct. Algorithms, 31, 3-122 (2007)]. Our proofs are based on an exploration process that corresponds to the graph, and instead of relying heavily on branching processes, we employ a random walk constructed from the exploration process. We then apply a large deviations theory to show that a reasonably large component is always significantly larger, a strategy used in both connectivity and phase transition analysis.