Phillip Bonacich and Philip Lu - Introduction to Mathematical Sociology

Introduction to Mathematical Sociology

Introduction to
Mathematical Sociology

Phillip Bonacich and Philip Lu

Copyright 2012 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
press.princeton.edu

All Rights Reserved

Library of Congress Cataloging-in-Publication Data

Bonacich, Phillip, 1940

Introduction to mathematical sociology / Phillip Bonacich and Philip Lu.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-691-14549-5 (hbk.)

1. Mathematical sociology. I. Lu, Philip, 1982- II. Title.

HM529.B66 2012

302.301513dc23

2011041621

British Library Cataloging-in-Publication Data is available

This book has been composed in ITC Slimbach

Printed on acid-free paper.

Typeset by S R Nova Pvt Ltd, Bangalore, India
Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

CONTENTS

FIGURES

TABLES

PREFACE

This book is designed for the undergraduate mostly sociology students we taught at UCLA in courses on mathematical sociology and social networks. They were interested in sociological issues, and they had no background in mathematics beyond high school algebra and the statistics course they were required to take as part of the sociology major. Increasingly, they were aware of the importance of mathematical models and computer simulations in the social sciences, and, of course, over the years they were increasingly comfortable and at ease with the computer. Teaching them a usable amount of calculus would have been a major undertaking, but we knew that interesting uses of finite mathematics were well within their reach.

This book introduces all the mathematics it uses: set theory, the probability function, matrix multiplication, graphs, elementary game theory, groups and their graphs, and Markov chains. Its most distinctive aspect, and what sets it off from earlier books on mathematical sociology and most more recent books on mathematics in the social sciences, is its emphasis on social networks, an area that is clearly the most exciting and fastest growing of mathematical sociology. Moreover, students know they live in an increasingly complex, interdependent, and networked world, and they want to know more about it.

This book is also distinctive in its use of embedded computer demonstrations that are used in the homework and can be used in class. It was the availability of Mathematica Player, a free download available from Wolfram Research, that energized us to rewrite this book. Most of the demonstrations were written by us; a few we downloaded from the Mathematica web site. Player demonstrations do not require that students program. They need only move sliders and press buttons to explore models by varying their parameters. The demonstrations are available from the web and can be run either within Mathematica or with the Mathematica Player, available for free on the web. The simulations are used to increase student understanding of the material, for homework assignments, and, on occasion, to describe models that are intractable mathematically.

All the simulations are available for download at the following website: http://press.princeton.edu/titles/9741.html.

The book teaches all the mathematics that is required. We decided not to include any calculus; the student wanting to use calculus is better served by taking a standard calculus class. We decided not to teach any computer programming. Its difficult to teach both programming and substance in the same course. We decided not to focus on agent-based modeling of complex systems. Our feeling is that one can go pretty far with mathematics alone.

We have found that two different quarter-long courses can be taught from this text, one focusing on social networks and the other not. For the social-network-oriented course we use , and the Afterword.

Introduction to Mathematical Sociology

Mathematical sociology is not an oxymoron. There is a useful role for mathematics in the study of society and groups. In fact, that role is growing as social scientists and others develop better and better tools for the study of complex systems. A number of trends are converging to make the application of mathematics to society increasingly productive.

First, more and more human systems are complex, in a sense to be described soon. World economies are more and more interconnected. Our transportation and communication systems are increasingly worldwide. Social networks are less local and more global, making them more complex, producing new emergent communication patterns, a positive effect, but which also has made us increasingly vulnerable to pandemics, a negative effect. The Internet has connected us in ways that no one understands completely. Power grids are less and less local and are subject to more widespread failures than ever before. New species are increasingly introduced into local complex ecologies with unexpected effects. Our recent climate change has produced a situation in which it is more and more important to predict the future and the effects of human interventions in the complex system of the global weather. The mapping of the human genome makes available to biologists the possibility of studying the complex system of interactions between genes and proteins. All of these tendencies mean that scientists in a wide variety of areascomputer science, economics, ecology, genetics, climatology, epidemiology, and othershave developed mathematical tools to study complex systems, and these tools are available to us sociologists.

The second important trend is the growing power and ubiquity of the computer. Computer simulations and mathematics are complementary tools for the study of complex systems. They are two different ways of drawing implications for the future from what is known or assumed to be true. Mathematics can be used to draw far-reaching and sometimes unexpected conclusions using logic and mathematics. For example, many properties of networks have been proved to be true by mathematicians using traditional mathematical tools. Computer simulations use computer programs the coding of which embodies assumptions and whose conclusions are evident after the program has iterated. Computer simulations are useful in situations that are unsolved or intractable mathematically.

This text uses both mathematics and computer simulations. Sometimes the computer simulations demonstrate phenomena for which there is no exact mathematical solution. More frequently simulations are used to illustrate some model so that you, the reader of this book, will gain some understanding of how the model works and how it is affected by varying parameters even if a full mathematical treatment of the model is beyond the purpose of this book.

EPIDEMICS

At the time we are writing this chapter there is an epidemic of concern over swine fever, a variant of influenza that seems to have captured the publics attention. Both the flu and fear of this flu spread through social networks, and we want now to illustrate some of the properties of epidemics through a very simple model. The model will be illustrated both with a little simple mathematics and with a computer simulation.

Suppose that a large population consists of N individuals. Suppose that each individual in the population has small probability p of being connected to each of the others in the population and that his connection to one individual has no bearing on his connection to any other person in the population. This creates a