SIGEST

Abstract

How often is it the case that not only the applications but also the methods of analysis of an important mathematical property range all the way from the deeply mathematical to the realm of the social sciences (and beyond)? Or, for that matter, that one can learn about the property under consideration in some detail on Wikipedia? The {small world phenomenon} is an instance of all of these characteristics.For the mathematical community, the small world phenomenon came alive with the 1998 paper of Watts and Strogatz that looked at the phenomenon from the perspective of a lattice network model and showed that the addition of a relatively small number of randomly chosen long—range connections to an otherwise locally connected lattice leads to a very good model of the small world phenomenon. Their work and others showed further that this type of model describes the actual connectivity and behavior of applications in areas as diverse as the brain of a small organism, the electrical power grid, and connectivity on the World Wide Web.This issue's SIGEST paper, ’’A Matrix Perturbation View of the Small World Phenomenon’’ by D. J. Higham that originally was published in the { SIAM Journal on Matrix Analysis and Applications} in 2003, makes an inspired and important contribution to the mathematical understanding of the small world phenomenon by looking at it from a new perspective. As Higham discusses in the new preamble to the paper, he was motivated to formulate a model that could display the small world phenomenon and be amenable to rigorous mathematical analysis. He succeeded by considering a Markov chain that also can be thought of as a one—dimensional periodic random walk, where the standard connections to the two nearest neighbors are augmented by low probability uniform jumps to any state, also referred to as shortcuts. Higham uses matrix perturbation theory methods to study the correspondence between the probabilities of these uniform jumps and the mean hitting time (the average number of steps to get from some random state to a given fixed state) of the random walk. In particular, the mean hitting times of a Markov chain can be computed by solving a certain system of linear equations, and by use of matrix perturbation theory, Higham analyzes how these times vary with the probability of the uniform jumps. The paper shows that this model is very effective in explaining the small world phenomenon mathematically, and that in addition, its mathematical behavior corroborates behavior that was observed experimentally by Watts and Strogatz. Higham's model also has a close connection to techniques used in Internet search engines.We hope that many SIAM readers will enjoy this interesting, accessible, and very nicely written paper that adds great insight into an important property of the natural and social worlds. And those of you with Hollywood connections will have something novel to bring up the next time the conversation turns to the ’’Six Degrees of Kevin Bacon.’’