## Charles Amick Memorial Lectures in Applied Mathematics

Charles J. Amick was an applied mathematician at the University of Chicago who died in 1991, at the age of 39. The Lecture Series was begun in 1993 as a means of honoring his painfully brief life.

### 2016 Speaker: Jennifer Chayes (Microsoft Research)

#### Lecture 1: Modeling and Estimating Massive Networks: Overview

Friday, October 28, 2016, 4:00pm–5:00pm, Ryerson 251

**Abstract**:
There are numerous examples of sparse massive networks,
including the Internet, WWW and online social networks. How do we
model and learn these networks? In contrast to conventional learning
problems, where we have many independent samples, it is often the case
for these networks that we can get only one independent sample. How
do we use a single snapshot today to learn a model for the network,
and hence predict a similar, but larger network in the future? In the
case of relatively small or moderately sized networks, it’s
appropriate to model the network parametrically, and attempt to learn
these parameters. For massive networks, a non-parametric
representation is more appropriate. I review the theory of graph
limits (graphons), developed over the last decade, to describe limits
of dense graphs and, more recently, sparse graphs of unbounded degree,
such as power-law graphs. I then show how to use these graphons to
give consistent estimators of non-parametric models of sparse
networks, and moreover how to do this in a way that protects the
privacy of individuals on the network. This first lecture gives an
overview of results, while the second two focus more on details and
methods.

#### Lecture 2: Limits and Stochastic Models for Sparse Massive Networks

Monday, November 1, 2016, 4:00pm–5:00pm, Eckhart 202

**Abstract**:
Graphons are obtained as limits of sequences graphs, and
provide non-parametric ways of modeling and generating networks. In
this lecture, I focus on analytical methods to obtain graphons as
limits of sparse networks of unbounded degree. These networks include
growing power-law networks in which vertices can disconnect from
previous friends and contacts (known as non-projective sequences of
networks), like Facebook and Google. A fundamental tool in the study
of networks and other random structures is the Szemeredi Regularity
Lemma, which has been used extensively in combinatorics and other
fields to discover apparent order within random structures. We show
how the conventional (dense network) form of the lemma breaks down for
sparse networks, and how to extend it in the sparse case.

#### Lecture 3: Exchangeability and Estimation of Sparse Massive Networks

Tuesday, November 2, 2016, 4:00pm–5:00pm, Eckhart 206

**Abstract**:
Graphons are obtained as limits of sequences graphs, and
provide non-parametric ways of modeling and generating networks. In
this final lecture, I give an alternative way to model massive sparse
networks, in this case networks which retain all previous connections
as they grow. While the approach of Lecture 2 was more analytic, this
lecture focuses on statistical considerations. I show how to extend
the classic de Finetti Theorem for infinite exchangeable sequences,
and the later Aldous-Hoover Theorem for infinite exchangeable dense
arrays, to the case of sparse arrays. I also provide proofs of how to
use graphons to consistently estimate massive sparse networks, and in
certain cases to do this in a way that preserves the privacy of the
individuals on the network.

Past Amick Lecturers include: Andrew Majda, Joseph Keller, John Ball, Martin Kruskal, Paul Roberts, David Ruelle, John Guckenheimer, Percy Deift, Keith Moffatt, Ingrid Daubechies, Yann Brenier, Felix Otto, Claude Bardos, George Papanicolaou, Michael Brenner, Ronald Coifman, Pierre-Louis Lions, Claude LeBris, L. Mahadevan, Jeffrey Rauch, Peter Constantin, Emmanuel Candes, and Amit Singer.