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Complex Networks

Posted April 2004.
Feature Column Archive

1. Introduction

Family Ties was the name of a popular situation comedy. The title captures succinctly one notion of the complex web of relationships that bind humans to other humans. John Donne suggested that "No man is an island." Human beings are, in fact, rarely totally isolated. Each individual is part of a "community" with blood, friendship, or economic ties or connections. The connectivity of people in different ways is captured by a variety of well-known phrases: instant messenger buddy list, network of friends, social connections, etc. However, it's not only people who have ties. Humans create ties between phones and web pages. Human societies create electrical and telephone networks, networks that deliver water and natural gas to us, and networks that carry away sewage. We create arrays of processors on chips that communicate with each other in complex ways. Within our bodies there are networks of neurons and our cells are connected via various "messenger" molecules.

The types of networks mentioned, computer networks, electrical networks, and the increasingly relied-upon World Wide Web, have similarities and differences. The theme of this year's Mathematics Awareness Month (April 2004) is "The Mathematics of Networks." Given mathematics' amazing ability to generalize, unify, abstract, and compare and contrast, why not use it as a tool to study complex network phenomena?

Joseph Malkevitch
York College (CUNY)


  1. Introduction
  2. Some history and a network primer
  3. Random networks
  4. Networks and epidemics
  5. Insights from probability and statistics
  6. The Erdős graph
  7. References