# John Lewis Lecture Series organised jointly by
HMI & DIAS:

### THE MATHEMATICS OF DYNAMIC RANDOM NETWORKS

Professor Jennifer Tours Chayes

Microsoft

Redmond, Washington

**Tuesday 26 February 2008 11 am, DIAS:**

**Models of the Internet and the World Wide Web**

**Wednesday 27 February 2008**
11 am, Salmon Lecture Theatre**, Hamilton Building, TCD:**

**Mathematical Behavior of Random Scale-Invariant Networks **

During the past decade, dynamic random networks have
become increasingly important in communication and information
technology. Vast, self-engineered networks, like the Internet, the
World Wide Web, and Instant Messaging Networks, have facilitated the
flow of information, and served as media for social and economic
interaction. I will present simple mathematical models that allow us to
explain many observed properties of these networks, e.g., the
scale-free nature of their degree distribution, the ease of information
transmission (including transmission of viruses), and the
first-to-market advantage for early nodes on these networks.

LECTURE 1: Models of the Internet
and the World Wide Web

Although the Internet and the World Wide Web have
many distinct features, both have a self-organized structure, rather
than the engineered architecture of previous networks, such as the
phone or transportation systems. As a consequence of this
self-organization, the Internet and the World Wide Web have a host of
properties which differ from those encountered in engineered
structures: a broad "power-law" distribution of connections (so-called
"scale-invariance"), short paths between two given points (so-called
"small world phenomena" like "six degrees of separation"), strong
clustering (leading to so-called "communities and subcultures"),
robustness to random errors, but vulnerability to malicious attack,
etc. During this lecture, I will first review some of the
distinguishing observed features of these networks, and then review the
recent models which have been devised to explain these features.
The basic models have their origins in graph theory and statistics.

LECTURE 2: Mathematical Behavior of
Random Scale-Invariant Networks

This lecture will be devoted to a mathematical
analysis of some of the standard models of random scale-invariant
networks, including models of the Internet, the World Wide Web, and
social networks. I will show how these models can be rewritten in
terms of a Polya urn representation, which will allow us to prove that
the models exhibit some of the observed properties of real-world
networks, including scale-invariance and vulnerability to attacks by
viruses. Using these models, I will also examine various
strategies for containment of viruses and epidemics on technological
and social networks.