Georgia Institute of Technology College of Sciences

We consider the modeling and computations of random dynamical processes of viral signals propagating over time in social networks. The viral signals of interests can be popular tweets on trendy topics in social media, or computer malware on the Internet, or infectious diseases spreading between human or animal hosts. The viral signal propagations can be modeled as diffusion processes with various dynamical properties on graphs or networks, which are essentially different from the classical diffusions carried out in continuous spaces. We address a critical computational problem in predicting influences of such signal propagations, and develop a discrete Fokker-Planck equation method to solve this problem in an efficient and effective manner. We show that the solution can be integrated to search for the optimal source node set that maximizes the influences in any prescribed time period. This is a joint work with Profs. Shui-Nee Chow (GT-MATH), Hongyuan Zha (GT-CSE), and Haomin Zhou (GT-MATH).

A large variety of Constraint Satisfactoin Problems can be classified as "computationally hard". In recent years researchers from statistical mechanics have investigated such problems via non-rigorous methods. The aim of this talk is to give a brief overview of this work, and of the extent to which the physics ideas can be turned into rigorous mathematics. I'm also going to point out various open problems.

I will talk about two model problem concerning a diffusion with a cellular drift (a.k.a array of opposing vortices). The first concerns the expected exit time from a domain as both the flow amplitude $A$ (or more precisely the Peclet number) goes to infinity, AND the cell size (or vortex seperation) $\epsilon$ approaches $0$ simultaneously. When one of the parameters is fixed, the problem has been extensively studied and the limiting behaviour is that of an effective "homogenized" or "averaged" problem. When both vary simultaneously one sees an interesting transition at $A \approx \eps^{-4}$. While the behaviour in the averaged regime ($A \gg \eps^{-4}$) is well understood, the behaviour in the homogenized regime ($A \ll \eps^{-4}$) is poorly understood, and the critical transition regime is not understood at all. The second problem concerns an anomalous diffusive behaviour observed in "intermediate" time scales. It is well known that a passive tracer diffusing in the presence of a strong cellular flows "homogenizes" and behaves like an effective Brownian motion on large time scales. On intermediate time scales, however, an anomalous diffusive behaviour was numerically observed recently. I will show a few preliminary rigorous results indicating that the stable "anomalous" behaviour at intermediate time scales is better modelled through Levy flights, and show how this can be used to recover the homogenized Brownian behaviour on long time scales.

Piecewise linear Fermi-Ulam pingpongs. We consider a particle moving freely between two periodically moving infinitely heavy walls. We assume that one wall is fixed and the second one moves with piecewise linear velocities. We study the question about existence and abundance of accelerating orbits for that model. This is a joint work with Jacopo de Simoi