Shankar Bhamidi (University of North Carolina) (Two talks)
Two Themes in Dynamic Network Modles: Critical Scaling Limits and Reinforcement Processes, I & II
Abstract: The last few years have witnessed an explosion in the amount of empirical data on real networks motivating an array of mathematical models for the evolution of such networks. Examples range from biological networks (brain networks of interact- ing neurons), information transmission (Internet), transportation, social networks and swarm intelligence and the evolution of self-organized behavior through the interactions of simple agents. This has stimulated vigorous activity in many fields, including biology, statistical physics, statistics, mathematics and computer science to understand these models and quantify their predictions and relevance to real systems. The aim of this course is to introduce junior researchers to one corner of this vast field, Dynamic networks: systems that evolve over time through probabilistic rules. The following two main themes will be pursued: * Emergence of macroscopic connectivity: The first lecture will delve into techniques for understanding how macroscopic connectivity in the network arises via microscopic interactions between agents in the network. We will consider various random graph models and study the nature of emergence of the giant component, in particular establishing sufficient conditions for these objects to belong to the same universality class as governed by Aldous’s multiplicative coalescent. We will show how these techniques can be used to study not just sizes of maximal components in the critical regime but also show that the maximal components appropriately scaled converge to limiting random metric spaces. * Evolving networks and continuous time branching: The second lecture will study the so-called preferential attachment family of network models em- phasizing one particular technical tool: continuous time branching processes. We will show how this technique leads to rigorous asymptotic descriptions to a number of problems in the statistical modeling of real world systems ranging from Twitter event networks to change point detection in evolving networks.
Ruth Williams (University of California, San Diego) (Two talks)
Reflected Diffusions, Stochastic Networks and Biology, I & II
Abstract: Stochastic dynamic models of complex networks are used in a variety of applications in science and engineering, and increasingly so in the biological sciences. The analysis of such models present many challenging mathematical problems. These two talks will feature problems motivated by biology. The first talk will describe how reflected diffusion processes can be used to approximate the most commonly used Markov chain model for (bio)chemical reaction networks. The second talk will illustrate how a critical correlation phenomenon in enzymatic networks was discovered via stochastic analysis.
Wei-Kuo Chen (University of Minnesota)
Energy Landscape of Mean Filed Spin Glasses Abstract: The Sherrington-Kirkpatirck (SK) model is a mean-field spin glass introduced by theoretical physicists in order to explain the strange behavior of certain alloys, such as CuMn. Despite of its seemingly simple formulation, it was conjectured to possess a number of profound properties. This talk will be focused on the energy landscapes of the SK model and the mixed p-spin model with both Ising and spherical configuration spaces. We will present Parisi formule for their maximal energies followed by descriptions of the energy andscapes near the maximum energy. Based on joint works with A. Auffinger, M. Handschy, G. Lerman, and A. Sen.
Xin Guo (University of California, Berkeley)
How Much Aggregation is Too Much: Paly Mean Field Game Through Several Examples
Nike Sun (University of California, Berkeley)
Phase Transitions in Random Constraint Satisfaction ProblemsAbstract: I will discuss a class of random constraint satisfaction problems (CSPs), including the