# Mini course by Sidney Resnick

**January 4, 2023**

Sidney Resnick from Cornell visiting us from January 7th to January 28th. He will be giving a series of six lectures on the topic of:

**Multivariate Power Laws and Preferential Attachment Modeling**

This is part of the lecture series funded by the NSF RTG grant DMS 2134107.

**Abstract:** In one-dimension, heavy tails or power-laws are easily understood to represent Pareto like behavior where data plotted on a log-log scale looks roughly linear. The generalization to higher dimensions is not always obvious and the infinite variety of dependence possibilities can be daunting. Multivariate regular variation of measures is a clean, flexible and clear way forward. A variety of mathematical and statistical techniques guide a user.

Network modeling of social networks using preferential attachment presents other challenges. Models can be difficult to analyze and only occasionally do simulations from these models leave a comfortable impression that simulation matches reality. One glaring discrepancy is “reciprocity”, meaning the percentage of directed edges that link to nodes (network users) in both directions. (You like me and I like you. You reference my paper and I reference yours.) Real data exhibits higher reciprocity compared to what is given by simulations from traditional preferential attachment.

In- and out-degree sequence data for many social networks marginally exhibit the expected straight line power law behavior and preferential attachment models theoretically predict this both marginally and in the two-dimensional sense. We can add reciprocity to the model by assuming something like “when I connect to you, you flip a coin to decide if you want to connect with me.” This and its generalizations corrects the empirical under-prediction of reciprocity and introduces the feature that asymptotically the limit measure of regular variation concentrates on a line. This means large values of in- and out-degree tend to always be present simultaneously, a property called “asymptotic full dependence”. Without reciprocity, preferential attachment leads to in- and out-degree having a limit measure of regular variation that concentrates on the full positive quadrant meaning that a large value of either in- or out-degree can be associated with a variety of values in the other.

**Schedule:** The lectures will be held in Hanes 125 at the following dates and times:

- TTh (Jan 10, 12): 4:15-5:30 pm
- Tue (Jan 17): 4:45-6:00 pm
- Th (Jan 19): 4:15-5:30 pm
- TTh (Jan 24, 26): 4:15-5:30 pm

STOR students may earn credit by registering for STOR 893.

**Pre-requisites:** foundation in probability and statistics at the graduate level.

**Bio:** Sidney Resnick is the Lee Teng-Hui Professor in Engineering Emeritus at Cornell. Resnick joined the Cornell faculty in 1987 after nine years at Colorado State University, six years at Stanford University, and two years at the Technion, in Haifa, Israel. He received his Ph.D. from Purdue University in 1970. His interests center in applied probability and cross the boundary into statistics. Past foci include modeling queues, storage facilities, extremes, data and social networks, risk estimation and tail estimation. Resnick is a Fellow of the Institute of Mathematical Statistics, a founding associate editor of Annals of Applied Probability, and past associate editor of Journal of Applied Probability, Stochastic Processes and their Applications, Stochastic Models, Extremes, The Mathematical Scientist. He has authored or coauthored approximately 190 papers and four books.