This is a Short Lecture Series in Mathematics (SLSM) consisting of 4 lectures of 90 minutes each. This is the third lecture in this series.
Abstract: The first half of the mini course will be an introducing to the two classical models of random graphs (a.k.a. Erdős-Rényi random graphs) and discuss the phenomenon of phase transition. We will also discuss thresholds for monotonic properties with examples including connectivity threshold and sub-graph containment threshold.
In the second half of the course we will consider other kind of random graphs. In particular, we will discuss various models for complex networks, including Albert-Barabási preferential attachment models. We will discuss "scale-freeness", asymptotic degree distribution and "small-world phenomenon". Properties of super and sub-linear preferential attachment models and some recent developments in de-preferential attachment models will also be discussed.
If time permits we will also introduce the random geometric graphs and discuss asymptotic of the connectivity threshold.