Georgia Institute of Technology College of Sciences

This talk will focus on tree automata, which are tools to analyze existential monadic second order properties of rooted trees. A tree automaton A consists of a finite set \Sigma of colours, and a map \Gamma: \mathbb{N}^\Sigma \rightarrow \Sigma. Given a rooted tree T and a colouring \omega: V(T) \rightarrow \Sigma, we call \omega compatible with automaton A if for every v \in V(T), we have \omega(v) = \Gamma(\vec{n}), where \vec{n} = (n_\sigma: \sigma \in \Sigma) and n_\sigma is the number of children of v with colour \sigma. Under the Galton-Watson branching process set-up, if p_\sigma denotes the probability that a node is coloured \sigma, then \vec{p} = (p_\sigma: \sigma \in \Sigma) is obtained as a fixed point of a system of equations. But this system need not have a unique fixed point. Our question attempts to answer whether a fixed point of such a system simply arises out of analytic reasons, or if it admits of a probabilistic interpretation. I shall formally defined interpretation, and provide a nearly complete description of necessary and sufficient conditions for a fixed point to not admit an interpretation, in which case it is called rogue.Joint work with Tobias Johnson and Fiona Skerman.

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).

Introduction of the new Faculty, Postdocs, Academic Professionals and Staff.