EVENT DETAILS AND ABSTRACT

Title: Probabilistic Symmetry and Network Models I
Speaker: Harry Crane
Speaker Info: Rutgers University
Brief Description:
Special Note:
Abstract:

The tutorial lectures will discuss probabilistic invariance principles and structural results for network models. The two lectures will be organized roughly as follows:

Lecture 1: Basic symmetries and network sampling. The first lecture focuses on exchangeability and sampling considerations in network modeling. After introducing some well known random graph models, I compare structural results for graphon models (Aldous, 1979; Hoover, 1980; Kallenberg, 2006) and edge exchangeable models (Crane-Dempsey, 2018). I discuss qualitative similarities and differences between these cases and show how the main results extend naturally to higher-order structures and more general invariance principles (Crane-Dempsey, 2019; Crane-Towsner, 2018).

Lecture 2: Dynamic network models. The second lecture focuses on network dynamics. In this setting, I discuss graph-valued Markov process models that satisfy a certain Markov projectivity condition. These processes evolve on the state space of graphs with countable vertex set in such a way that the process induced by projecting to any subset of vertices preserves the Markov property. I discuss a general structural result for processes of this type (Crane, 2017; Crane-Towsner, 2019+) and then present a special subclass of combinatorial Levy processes, for which additional results are known (Crane, 2018).

Many questions about the material in both lectures remain open. I will highlight some such problems throughout the lectures.