Georgia Institute of Technology College of Sciences

On the two-dimensional square grid, remove each nearest-neighbor edge independently with probability 1/2 and consider the graph induced by the remaining edges. What is the structure of its connected components? It is a famous theorem of Kesten that 1/2 is the ``critical value.'' In other words, if we remove edges with probability p \in [0,1], then for p < 1/2, there is an infinite component remaining, and for p > 1/2, there is no infinite component remaining. We will describe some of the differences in these phases in terms of crossings of large boxes: for p < 1/2, there are relatively straight crossings of large boxes, for p = 1/2, there are crossings, but they are very circuitous, and for p > 1/2, there are no crossings.

Network data analysis has wide applications in computational social science, computational biology, online social media, and data visualization. For many of these network inference questions, the brute-force (yet statistically optimal) methods involve combinatorial optimization, which is computationally prohibitive when faced with large scale networks. Therefore, it is important to understand the effect on statistical inference when focusing on computationally tractable methods. In this talk, we will discuss three closely related statistical models for different network inference problems. These models answer inference questions on cliques, communities, and ties, respectively. For each particular model, we will describe the statistical model, propose new computationally efficient algorithms, and study the theoretical properties and numerical performance of the algorithms. Further, we will quantify the computational optimality through describing the intrinsic barrier for certain efficient algorithm classes, and investigate the computational-to-statistical gap theoretically. A key feature shared by our studies is that, as the parameters of the model changes, the problems exhibit different phases of computational difficulty.

In this talk we outline the core group theoretic routine, the "Local Certificates" algorithm, underlying the new Graph Isomorphism test. The basic strategy follows Luks's group-theoretic divide-and-conquer approach (1980). We address the bottleneck of Luks's technique via local-global interaction based on a new group theoretic lemma. Undergraduate-level familiarity with the basic concept of group theory (homomorphism, kernel, quotient group, permutation groups) will be assumed.