Recent years have seen a proliferation of large datasets possessing some type of network structure. Various generative models have been derived to reflect mathematical properties of real-world networks, including degree distributions and levels of connectivity and clustering. In the first part of the talk, we will discuss recent results in community detection and estimation. This is a very active area of research in electrical engineering and statistics, with diverse scientific applications in physics, biology, and social and behavioral sciences. However, the case where the observed adjacency matrix is "weighted" or "colored" has received relatively little attention. We will present sharp thresholds for when community detection is possible in weighted stochastic block models (SBMs), stated in terms of an information-theoretic expression known as the Renyi divergence. We will then discuss algorithms for recovering communities in an efficient manner in the regime when recovery is possible. In the second part of the talk, we will shift our focus to the problem of studying node centrality in growing random graphs. Many notions of graph centrality have been formulated in network science to assign numerical measures of importance to nodes in a graph. As the graph evolves dynamically over time, however, the central node(s) may also change. We will show that for particular classes of growing random graphs, namely preferential and uniform attachment trees, a particular notion of centrality persists -- meaning the most central node(s) settle down after finitely many steps. Furthermore, we will show how to guarantee this property for the first node if it seeds the graph with sufficiently many neighbors. We will conclude by discussing interesting challenges and open questions in network science and engineering.
Varun Jog received the B.Tech. degree in Electrical Engineering from IIT Bombay in 2010, and the Ph.D. in Electrical Engineering and Computer Sciences (EECS) from the University of California at Berkeley (UCB) in 2015. Since 2016, he is an Assistant Professor at the Electrical and Computer Engineering Department and a fellow at the Grainger Institute for Engineering at the University of Wisconsin - Madison. His research interests include information theory, network science, energy harvesting, convex geometry, and optimal transport. He is a recipient of the Eliahu Jury award from the EECS Department at UCB (2015) and the Jack Keil Wolf student paper award at the IEEE International Symposium on Information Theory (2015).