Finally, both changes happened in a hurry. They didn't build steadily and slowly. It is instructive to look at a chart of the crime rate in New York City from, say, the mid-1960s to the late 1990s. It looks like a giant arch. In 1965, there were 200,000 crimes in the city and from that point on the number begins a sharp rise, doubling in two years and continuing almost unbroken until it hits 650,000 crimes a year in the mid-1970s. It stays steady at that level for the next two decades, before plunging downward in 1992 as sharply as it rose thirty years earlier. Crime did not taper off. It didn't gently decelerate. It hit a certain point and jammed on the brakes.

These three characteristics — one, contagiousness; two, the fact that little causes can have big effects; and three, that change happens not gradually but at one dramatic moment — are the same three principles that define how measles moves through a grade-school classroom or the flu attacks every winter. Of the three, the third trait — the idea that epidemics can rise or fall in one dramatic moment — is the most important, because it is the principle that makes sense of the first two and that permits the greatest insight into why modern change happens the way it does. The name given to that one dramatic moment in an epidemic when everything can change all at once is the Tipping Point.

A world that follows the rules of epidemics is a very different place from the world we think we live in now. Think, for a moment, about the concept of contagiousness. If I say that word to you, you think of colds and the flu or perhaps something very dangerous like HIV or Ebola. We have, in our minds, a very specific, biological notion of what contagiousness means. But if there can be epidemics of crime or epidemics of fashion, there must be all kinds of things just as contagious as viruses. Have you ever thought about yawning, for instance? Yawning is a surprisingly powerful act. Just because you read the word "yawning" in the previous two sentences — and the two additional "yawns" in this sentence — a good number of you will probably yawn within the next few minutes. Even as I'm writing this, I've yawned twice. If you're reading this in a public place, and you've just yawned, chances are that a good proportion of everyone who saw you yawn is now yawning too, and a good proportion of the people watching the people who watched you yawn are now yawning as well, and on and on, in an ever-widening, yawning circle.

Yawning is incredibly contagious. I made some of you reading this yawn simply by writing the word "yawn." The people who yawned when they saw you yawn, meanwhile, were infected by the sight of you yawning — which is a second kind of contagion. They might even have yawned if they only heard you yawn, because yawning is also aurally contagious: if you play an audiotape of a yawn to blind people, they'll yawn too. And finally, if you yawned as you read this, did the thought cross your mind — however unconsciously and fleetingly — that you might be tired? I suspect that for some of you it did, which means that yawns can also be emotionally contagious. Simply by writing the word, I can plant a feeling in your mind. Can the flu virus do that? Contagiousness, in other words, is an unexpected property of all kinds of things, and we have to remember that, if we are to recognize and diagnose epidemic change.

The second of the principles of epidemics — that little changes can somehow have big effects — is also a fairly radical notion. We are, as humans, heavily socialized to make a kind of rough approximation between cause and effect. If we want to communicate a strong emotion, if we want to convince someone that, say, we love them; we realize that we need to speak passionately and forthrightly. If we want to break bad news to someone, we lower our voices and choose our words carefully. We are trained to think that what goes into any transaction or relationship or system must be directly related, in intensity and dimension, to what comes out. Consider, for example, the following puzzle. I give you a large piece of paper, and I ask you to fold it over once, and then take that folded paper and fold it over again, and then again, and again, until you have refolded the original paper 50 times. How tall do you think the final stack is going to be? In answer to that question, most people will fold the sheet in their mind's eye, and guess that the pile would be as thick as a phone book or, if they're really courageous, they'll say that it would be as tall as a refrigerator. But the real answer is that the height of the stack would approximate the distance to the sun. And if you folded it over one more time, the stack would be as high as the distance to the sun and back. This is an example of what in mathematics is called a geometric progression. Epidemics are another example of geometric progression: when a virus spreads through a population, it doubles and doubles again, until it has (figuratively) grown from a single sheet of paper all the way to the sun m fifty steps. As human beings we have a hard time with this kind of progression, because the end result — the effect — seems far out of proportion to the cause. To appreciate the power of epidemics, we have to abandon this expectation about proportionality. We need to prepare ourselves for the possibility that sometimes big changes follow from small events, and that sometimes these changes can happen very quickly.

This possibility of sudden change is at the center of the idea of the Tipping Point and might well be the hardest of all to accept. The expression first came into popular use in the 1970s to describe the flight to the suburbs of whites living in the older cities of the American Northeast. When the number of incoming African Americans in a particular neighborhood reached a certain point — 20 percent, say — sociologists observed that the community would "tip": most of the remaining whites would leave almost immediately. The Tipping Point is the moment of critical mass, the threshold, the boiling point. There was a Tipping Point for violent crime in New York in the early 1990s, and a Tipping Point for the reemergence of Hush Puppies, just as there is a Tipping Point for the introduction of any new technology. Sharp introduced the first low-priced fax machine in 1984, and sold about 80,000 of those machines in the United States in that first year. For the next three years, businesses slowly and steadily bought more and more faxes, until, in 1987, enough people had faxes that it made sense for everyone to get a fax. Nineteen eighty-seven was the fax machine Tipping Point. A million machines were sold that year, and by 1989 two million new machines had gone into operation. Cellular phones have followed the same trajectory. Through the 1990s, they got smaller and cheaper, and service got better until 1998, when the technology hit a Tipping Point and suddenly everyone had a cell phone. (For an explanation of the mathematics of Tipping Points, see the Endnotes).

All epidemics have Tipping Points. Jonathan Crane, a sociologist at the University of Illinois, has looked at the effect the number of role models in a community — the professionals, managers, teachers whom the Census Bureau has defined as "high status" — has on the lives of teenagers in the same neighborhood. He found little difference in pregnancy rates or school drop-out rates in neighborhoods of between 40 and 5 percent of high-status workers. But when the number of professionals dropped below 5 percent, the problems exploded. For black schoolchildren, for example, as the percentage of high-status workers falls just 2.2 percentage points — from 5.6 percent to 3.4 percent — drop-out rates more than double. At the same Tipping Point, the rates of childbearing for teenaged girls — which barely move at all up to that point — nearly double. We assume, intuitively, that neighborhoods and social problems decline in some kind of steady progression. But sometimes they may not decline steadily at all; at the Tipping Point, schools can lose control of their students, and family life can disintegrate all at once.