Elisabetta Candellero, University of Birmingham
Clustering in random geometric graphs on hyperbolic spaces

In this talk we introduce the concept of random geometric graphs on hyperbolic spaces and discuss its applicability as a model for social networks. In particular, we will discuss issues that are related to clustering, which is a phenomenon that often occurs in social networks: two individuals that have a common friend are somehow more likely to be friends of each other. We give a mathematical expression of this phenomenon and explore how this depends on the parameters of our model. (Joint work with Nikolaos Fountoulakis).

Wenqing Hu, University of Maryland at College Park
On diffusion and wave front propagation in narrow random channels

We consider a solvable model for the motion of molecular motors. Based on the averaging principle, we reduce the problem to a diffusion process on a graph. We then calculate the effective speed of transportation of these motors. We also consider a reaction-diffusion equation in narrow random channels. We approximate the generalized solution to this equation by the corresponding one on a random graph. By making use of large deviation analysis we study the asymptotic wave front propagation.

This talk is based on the following papers: [1] Freidlin, M., Hu, W., On diffusion in narrow random channels, Journal of Statistical Physics 152 (2013), 136–158. [2] Freidlin, M., Hu, W., Wave front propagation for a reaction-diffusion equation in narrow random channels, Nonlinearity (accepted).

Miklos Racz, University of California at Berkeley
Coexistence in preferential attachment networks

We introduce a new model of type adoption on a dynamic network. The key property of our model is that type choices evolve simultaneously with the network itself. When a new node joins the network, it chooses neighbors according to preferential attachment, and then chooses its type based on the number of initial neighbors of each type. This can model a new cell-phone user choosing a cell-phone provider, a new student choosing a laptop, or a new athletic team member choosing a gear provider. We are able to provide a detailed analysis of the new model; in particular, we are able to determine the limiting proportions of the various types. The main qualitative feature of our model is that, unlike other current theoretical models, often several competing types will coexist, which matches empirical observations in many current markets. (This is joint work with Tonci Antunovic and Elchanan Mossel.)

Patrick Rebeschini, Princeton University
Filtering in high dimension

A problem that arises in many applications is to compute the conditional distributions of stochastic models given observed data. While exact computations are rarely possible, particle filtering algorithms have proved to be very useful for approximating such conditional distributions. Unfortunately, the approximation error of particle filters grows exponentially with dimension. This phenomenon has rendered particle filters of limited use in complex data assimilation problems that arise, for example, in weather forecasting or oceanography. In this short talk I will argue that it is possible to develop “local” particle filtering algorithms whose approximation error is dimension-free by exploiting conditional decay of correlations properties of high-dimensional models. As a proof of concept, we prove for the simplest possible algorithm of this type an error bound that is uniform both in time and in the model dimension. (Joint work with R. van Handel.)