Logo The Hamilton Mathematics Institute, TCD


PUBLIC EVENT

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.