This is a Short Lecture Series in Mathematics (SLSM) consisting of 4 lectures of 90 minutes each. This lecture schedule in this series is the following:
1st Lecture: Monday, October 09, 2017 - 11:30 to 13:00
2nd Lecture: Tuesday, October 10, 2017 - 11:30 to 13:00
3rd Lecture: Wednesday, October 11, 2017 - 16:30 to 18:00
4th Lecture: Thursday, October 12, 2017 - 15:30 to 17:00
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.
1. Random Graphs by Svante Janson, Tomasz Łuczak and Andrzej Rucinski;
2. Random Graphs by Béla Bollobás;
3. Random Graphs and Complex Networks by Remco van den Hofstad;
4. Random Geometric Graphs by Mathew Penrose.