Thursday, August 31, 2017 - 15:05 , Location: Skiles 006 , Po-Ling Loh , University of Wisconsin-Madison , Organizer: Mayya Zhilova
We discuss two recent results concerning disease modeling on networks. The infection is assumed to spread via contagion (e.g., transmission over the edges of an underlying network). In the first scenario, we observe the infection status of individuals at a particular time instance and the goal is to identify a confidence set of nodes that contain the source of the infection with high probability. We show that when the underlying graph is a tree with certain regularity properties and the structure of the graph is known, confidence sets may be constructed with cardinality independent of the size of the infection set. In the scenario, the goal is to infer the network structure of the underlying graph based on knowledge of the infected individuals. We develop a hypothesis test based on permutation testing, and describe a sufficient condition for the validity of the hypothesis test based on automorphism groups of the graphs involved in the hypothesis test. This is joint work with Justin Khim (UPenn).
Thursday, April 20, 2017 - 15:05 , Location: Skiles 006 , Lutz Warnke , School of Mathematics, GaTech , Organizer: Christian Houdre
We consider rooted subgraph extension counts, such as (a) the number of triangles containinga given vertex, or (b) the number of paths of length three connecting two given vertices. In 1989 Spencer gave sufficient conditions for the event that whp all roots of the binomial random graph G(n,p) have the same asymptotic number of extensions, i.e., (1 \pm \epsilon) times their expected number. Perhaps surprisingly, the question whether these conditions are necessary has remained open. In this talk we briefly discuss our qualitative solution of this problem for the `strictly balanced' case, and mention several intriguing questions that remain open (which lie at the intersection of probability theory + discrete mathematics, and are of concentration inequality type). Based on joint work in progress with Matas Sileikis
Thursday, April 13, 2017 - 15:05 , Location: Skiles 006 , Christian Houdré , School of Mathematics, Georgia Institute of Technology , Organizer: Christian Houdre
I will revisit the classical Stein's method, for normal random variables, as well as its version for Poisson random variables and show how both (as well as many other examples) can be incorporated in a single framework.
Scalings and saturation in infinite-dimensional control problems with applications to stochastic partial differential equationsFriday, April 7, 2017 - 13:05 , Location: Skiles 270 , David Herzog , Iowa State University , email@example.com , Organizer: Michael Damron
We discuss scaling methods which can be used to solve low mode control problems for nonlinear partial differential equations. These methods lead naturally to a infinite-dimensional generalization of the notion of saturation, originally due to Jurdjevic and Kupka in the finite-dimensional setting of ODEs. The methods will be highlighted by applying them to specific equations, including reaction-diffusion equations, the 2d/3d Euler/Navier-Stokes equations and the 2d Boussinesq equations. Applications to support properties of the laws solving randomly-forced versions of each of these equations will be noted.
Thursday, April 6, 2017 - 15:05 , Location: Skiles 006 , Zhou Fan , Stanford University , Organizer: Christian Houdre
Spectral algorithms are a powerful method for detecting low-rank structure in dense random matrices and random graphs. However, in certain problems involving sparse random graphs with bounded average vertex degree, a naive spectral analysis of the graph adjacency matrix fails to detect this structure. In this talk, I will discuss a semidefinite programming (SDP) approach to address this problem, which may be viewed both as imposing a delocalization constraint on the maximum eigenvalue problem and as a natural convex relaxation of minimum graph bisection. I will discuss probabilistic results that bound the value of this SDP for sparse Erdos-Renyi random graphs with fixed average vertex degree, as well as an extension of the lower bound to the two-group stochastic block model. Our upper bound uses a dual witness construction that is related to the non-backtracking matrix of the graph. Our lower bounds analyze the behavior of local algorithms, and in particular imply that such algorithms can approximately solve the SDP in the Erdos-Renyi setting. This is joint work with Andrea Montanari.