INSTRUCTOR: Christian Borgs
TIME: Tuesday, Thursday 2:30-4:00
PLACE: Online (Zoom)
OFFICE HOURS: Online, TBD
- (8/26/20) Welcome to the course! The Zoom Link for the class has been posted
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.
The course is open to graduate students with a good level of mathematical maturity and a
strong background in probability (including some 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:
Epidemics Models and their Behavior:
- 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
- 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
- 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.:
The course will be online, and I hope for active participation of everyone enrolled.
Therefore attendance is mandatory, and will be part of the assessment.
I will periodically give problem assignments, for which I encourage collaboration of
2-3 students, but expect separately written up solutions from every participant.
Depending on the size of the class, part of the course may be a final project done in groups of a few.
These projects would involve the reading and deep understanding of a research paper,
writing up what you learned, plus something “extra”, which can range from
a simulation demonstrating or visualizing some of the results of the paper(s),
to working out some part which in the paper was “left to the reader”, to solving
an unsolved research problem.
- Introduction, Branching Process (Lectures 1-2)
- Erdos-Renyi random graphs (Lectures 3-5)
- Small Worlds (Lecture 6)
- Stochastic Block Model (Lecture 7)
- Inhomogneous Random Graph, Graphons and Graph Limits (Lectures 8-10)
- Configuration Model, Weak Local Limit (Lecture 11)
- Preferential Attachment (Lecture 12)
- Graphons II: Sparse Graph Limits (Lecture 13)
- Graphons III: Estimation, Differential Privacy and Recommendation Systems (Lecture 14)
- Polya Urn and Weak Local Limit for Preferential Attachment (Lecture 15)
- Introduction to Epidemics (Lecture 16)
- Guest Lecture on Epidemics (Lecture 17)
- Infection Digraph, Bow Tie Structure (Lecture 18)
- Contact Inhomogeneities (Lecture 19)
- Guest Lecture on Information Diffusion (Lecture 20)
- Time Evolution for SIR on the Configuration Model (Lecture 21)
- Guest Lecture on Economic Impact of Covid 19 (Lecture 22)
- SIS model, innoculation strategies (Lecture 23)
- Summary (Lecture 24)