INSTRUCTOR: Christian Borgs
(borgs@eecs)
TIME: Tuesday, Thursday 9:30 - 11:00
PLACE: Dwinelle 106
OFFICE HOURS: TBD
ANNOUNCEMENTS
- (12/19/24) Welcome to the course! The first class will be on Tue, Jan. 21
COURSE DESCRIPTION
Networks play a central role in our social and economic lives.
They affect our well-being by influencing the information we receive,
products we choose to buy, economic opportunities we enjoy,
and diseases we catch from others. How do these networks form?
Which network structures are likely to emerge in society?
And how does the structure of the network impact the dynamics of the
spread of an innovation or infection.
This course tries to survey the mathematical results developed in the last
few years on analyzing the structure of popular random networks, as well
as the understanding of processes on them, with particular emphasis on epidemics.
In addition it will touch on a recently very popular subject, the non-parametric modeling
of a large graph via a graphon, and the related notion of graph limits.
The course is loosely based on the text book Random Graph Dynamics by
Rick Durrett, updated to what has
happened since its publication,
including the topics of graphons and graph limits, plus a larger emphasis
on epidemics, as well as a short introduction to the topic of the spread of
information and innovation. Additional literature, including
original research papers, will be provided during the course.
Prerequisites:
The course is open to graduate students with a good level of mathematical maturity and a
strong background in probability (including some basic knowledge of
Martingales, Markov Chains, and basic notions of stochastic processes), as well as some
basic background in graph theory and differential equations.
Tentative List of Topics
Random Graph Models and Structure of Large Networks:
- Erdos-Renyi random graphs: cluster growth, formation of the giant connected component, diameter
- Models with community structure: Stochastic block model and topic models
- Scale-free graphs: random graphs with a fixed degree distribution, preferential attachment model
and Polya urns
- Non-parametric models: graphons, graph limits, estimation, differential privacy on Networks
Epidemics Models and their Behavior:
- Basic compartmental models (SIS, SIR, etc.)
- Differential Equation Approach: R0, exponential growth, size of an epidemic
- Mathematically rigorous derivation of Differential Equations from the underlying dynamic model
- Percolation and Oriented Percolation representation
- Analysis of contact inhomogeneities
Algorithmic Aspects:
- Managing epidemics:
algorithmic questions related to parameter estimation, testing, and contact tracing
- Viral spread of innovations in a social network:
models of adoption of innovation, influence maximization, formation of social norms
ATTENDANCE, ASSIGNMENTS, ETC.:
This is a small graduate class, and I hope for active participation of everyone enrolled. Therefore attendance is mandatory, and will be
(a small) part of the assessment.
Aside from class
participation, you can expect a moderate additional amount of work:
- I will periodically give problem assignments (with at least 2 weeks between assignments),
for which I encourage collaboration of 2-3 students, but expect separately written up solutions from every participant
- While some scribes will be provided by the lecturer, most will be written by participants and are part of the
class requirement. You can expect to write one or two scribes, based on handwritten notes by the lecturer
- There will be no exams, but depending on the class size, some of the homework assignments and scribes may be replaced by a final project done in groups of a few