The quest for efficiency is ancient and universal, as time and other resources are always in shortage. Thus, the question of which tasks can be performed efficiently is central to the human experience.

A key step toward the systematic study of the aforementioned question is a rigorous definition of the notion of a task and of procedures for solving tasks. These definitions were provided by computability theory, which emerged in the 1930s. This theory focuses on computational tasks, considers automated procedures (i.e., computing devices and algorithms) that may solve such tasks, and studies the class of solvable tasks.

In focusing attention on computational tasks and algorithms, computability theory has set the stage for the study of the computational resources (like time) that are required by such algorithms. When this study focuses on the resources that are necessary for any algorithm that solves a particular task (or a task of a particular type), it is viewed as belonging to the theory of Computational Complexity (also known as Complexity Theory). In contrast, when the focus is on the design and analysis of specific algorithms (rather than on the intrinsic complexity of the task), the study is viewed as belonging to a related area that may be called Algorithmic Design and Analysis. Furthermore, Algorithmic Design and Analysis tends to be sub-divided according to the domain of mathematics, science, and engineering in which the computational tasks arise.

Diffusion in Networks

A basic issue in the preceding several chapters has been the way in which an individual's choices depend on what other people do. This has informed our use of information cascades, network effects, and rich-get-richer dynamics to model the processes by which new ideas and innovations are adopted by a population. When we perform this type of analysis, the underlying social network can be considered at two conceptually very different levels of resolution: one in which we view the network as a relatively amorphous population of individuals and look at effects in aggregate, and another in which we move closer to the fine structure of the network as a graph and look at how individuals are influenced by their particular network neighbors. Our focus in these past few chapters has been mainly on the first of these levels of resolution — capturing choices in which each individual is at least implicitly aware of the previous choices made by everyone else, and everyone takes these into account. In the next few chapters, we bring the analysis closer to the detailed network level.

What do we gain by considering this second level of resolution, oriented around network structure? To begin with, we can address a number of phenomena that can't be modeled well at the level of homogeneous populations. Many of our interactions with the rest of the world happen at a local, rather than a global, level: we often don't care as much about the full population's decisions as about the decisions made by friends and colleagues.

In the opening chapter of the book, we emphasized that the “connectedness” of a complex social, natural, or technological system really means two things: first, an underlying structure of interconnecting links, and second, an interdependence in the behaviors of the individuals who inhabit the system, so that the outcome for any one depends at least implicitly on the combined behaviors of all. The first issue – network structure – was addressed in the first part of the book using graph theory. In this second part of the book, we study interconnectedness at the level of behavior, developing basic models for this in the language of game theory.

Game theory is designed to address situations in which the outcomes of a person's decisions depend not just on how they choose among several options, but also on the choices made by the people with whom they interact. Game-theoretic ideas arise in many contexts. Some contexts are literally games; for example, choosing how to target a soccer penalty kick and choosing how to defend against it can be modeled using game theory. Other settings are not usually called games, but they can be analyzed with the same tools. Examples include the pricing of a new product when other firms have similar new products, deciding how to bid in an auction, choosing a route on the Internet or through a transportation network, deciding whether to adopt an aggressive or a passive stance in international relations, or choosing whether to use performance-enhancing drugs in a professional sport. In these examples, each decision-maker's outcome depends on the decisions made by others. This introduces a strategic element that game theory is designed to analyze.

Searching the Web: The Problem of Ranking

When you go to Google and type the word “Cornell,” the first result it shows you is www. cornelledu, the home page of Cornell University. It's certainly hard to argue with this as a first choice, but how did Google “know” that this was the best answer? Search engines determine how to rank pages using automated methods that look at the Web itself, not some external source of knowledge, so the conclusion is that there must be enough information intrinsic to the Web and its structure to figure this out.

Before discussing some of the ideas behind the ranking of pages, let's begin by considering a few basic reasons why it's a hard problem. First, search is a hard problem for computers to solve in any setting, not just on the Web. Indeed, the field of information retrieval [36, 360] dealt with this problem for decades before the creation of the Web: automated information retrieval systems starting in the 1960s were designed to search repositories of newspaper articles, scientific papers, patents, legal abstracts, and other document collections in reponse to keyword queries. Information retrieval systems have always had to deal with the problem that keywords are a very limited way to express a complex information need. In addition to the fact that a list of keywords is short and inexpressive, it suffers from the problems of synonymy (multiple ways to say the same thing, so that your search for recipes involving scallions fails because the recipe you wanted called them “green onions”) and polysemy (multiple meanings for the same term, so that your search for information about the animal called a jaguar instead produces results primarily about automobiles, football players, and an operating system for the Apple Macintosh.)

The study of epidemic disease has always been a topic that mixes biological issues with social ones. When we talk about epidemic disease, we will be thinking of contagious diseases caused by biological pathogens — things like influenza, measles, and sexually transmitted diseases — which spread from person to person. Epidemics can pass explosively through a population, or they can persist over long time periods at low levels; they can experience sudden flare-ups or even wavelike cyclic patterns of increasing and decreasing prevalence. In extreme cases, a single disease outbreak can have a significant effect on a whole civilization, as with the epidemics started by the arrival of Europeans in the Americas [130], or the outbreak of bubonic plague that killed 20% of the population of Europe over a seven-year period in the 1300s [293].

Diseases and the Networks That Transmit Them

The patterns by which epidemics spread through groups of people is determined not just by the properties of the pathogen carrying it — including its contagiousness, the length of its infectious period, and its severity — but also by network structures within the population it is affecting. The social network within a population — recording who knows whom — determines a lot about how the disease is likely to spread from one person to another. But more generally, the opportunities for a disease to spread are given by a contact network: there is a node for each person, and an edge if two people come into contact with each other in a way that makes it possible for the disease to spread from one to the other.

In Chapter 6, we developed the basic ideas of game theory, in which individual players make decisions and the payoff to each player depends on the decisions made by all. As we saw there, a key question in game theory is to reason about the behavior we should expect to see when players take part in a given game.

The discussion in Chapter 6 was based on considering how players simultaneously reason about what the other players may do. In this chapter, on the other hand, we explore the notion of evolutionary game theory, which shows that the basic ideas of game theory can be applied even to situations in which no individual is overtly reasoning or even making explicit decisions. Instead, game-theoretic analysis will be applied here to settings in which individuals can exhibit different forms of behavior (including those that may not be the result of conscious choices), and we will consider which forms of behavior have the ability to persist in the population and which forms of behavior have a tendency to be driven out by others.

As its name suggests, evolutionary game theory has been applied most widely in the area of evolutionary biology, the domain in which the idea was first articulated by John Maynard Smith and G. R. Price [375, 376]. Evolutionary biology is based on the idea that an organism's genes largely determine its observable characteristics, and hence its fitness, in a given environment. Organisms that are more fit will tend to produce more offspring, causing genes that provide greater fitness to increase their representation in the population.

In the previous chapter, we saw a first example of an institution that can synthesize information held by many people, through the ways in which markets serve to aggregate the individual beliefs of investors. We now turn to a second fundamental institution: voting.

Voting for Group Decision Making

Like markets, voting systems also serve to aggregate information across a group, and as a result, it's hard to draw a perfectly clear dividing line between these two kinds of institutions. But there are definite distinctions between the respective settings in which they are typically applied. A first important distinction is that voting is generally used in situations where a group of people is expressly trying to reach a single decision that in a sense will speak for the group. When a population votes on a set of candidates or ballot initiatives, a legislative body votes on whether to pass a bill, a jury votes on a verdict in a trial, a prize committee votes on the recipient of an award, or a group of critics votes on the top movies of the past century, the resulting decision is a single outcome that stands for the group and has some kind of binding effect going forward. In contrast, markets synthesize the opinions of a group more indirectly, as investors' beliefs are conveyed implicitly through their transactions in the market — choosing how much to invest or to bet, choosing whether to buy or not to buy, and so forth.

Six Degrees of Separation

In the previous chapter, we considered how social networks can serve as conduits by which ideas and innovations flow through groups of people. To develop this idea more fully, we now relate it to another basic structural issue — the fact that these groups can be connected by very short paths through the social network. When people try to use these short paths to reach others who are socially distant, they are engaging in a kind of “focused” search that is much more targeted than the broad spreading pattern exhibited by the diffusion of information or a new behavior. Understanding the relationship between targeted search and wide-ranging diffusion is important in thinking more generally about the way things flow through social networks.

As we saw in Chapter 2, the fact that social networks are so rich in short paths is known as the small-world phenomenon, or the “six degrees of separation,” and it has long been the subject of both anecdotal and scientific fascination. To briefly recapitulate what we discussed in that earlier chapter, the first significant empirical study of the small-world phenomenon was undertaken the social psychologist Stanley Milgram [297, 391], who asked randomly chosen “starter” individuals to each try forwarding a letter to a designated “target” person living in the town of Sharon, Massachusetts, a suburb of Boston. He provided the target's name, address, occupation, and some personal information, but stipulated that the participants could not mail the letter directly to the target; rather, each participant could only advance the letter by forwarding it to a single acquaintance that he or she knew on a first-name basis, with the goal of reaching the target as rapidly as possible.

Following the Crowd

When people are connected by a network, it becomes possible for them to influence each other's behavior and decisions. In the next several chapters, we will explore how this basic principle gives rise to a range of social processes in which networks serve to aggregate individual behavior and thus produce population-wide, collective outcomes.

There is a nearly limitless set of situations in which people are influenced by others: in the opinions they hold, the products they buy, the political positions they support, the activities they pursue, the technologies they use, and many other things. What we'd like to do here is to go beyond this observation and consider some of the reasons why such influence occurs. We'll see that there are many settings in which it may in fact be rational for an individual to imitate the choices of others even if the individual's own information suggests an alternative choice.

As a first example, suppose that you are choosing a restaurant in an unfamiliar town, and based on your own research about restaurants, you intend to go to restaurant A. However, when you arrive you see that no one is eating in restaurant A, whereas restaurant B next door is nearly full. If you believe that other diners have tastes similar to yours, and that they too have some information about where to eat, it may be rational to join the crowd at B rather than to follow your own information. To see how this is possible, suppose that each diner has obtained independent but imperfect information about which of the two restaurants is better.

Up to this point in the book, we've considered networks in which the basic units being connected were people or other social entities, like firms or organizations. The links connecting them have generally corresponded to opportunities for some kind of social or economic interaction.

In the next several chapters, we consider a different type of network, in which the basic units being connected are pieces of information, and links join pieces of information that are related to each other in some fashion. We will call such a network an information network. As we will see, the World Wide Web is arguably the most prominent current example of such a network, and while the use of information networks has a long history, it was really the growth of the Web that brought such networks to wide public awareness.

While there are basic differences between information networks and the kinds of social and economic networks that we've discussed earlier, many of the central ideas developed earlier in the book turn out to be fundamental here as well: we'll be using the same basic ideas from graph theory, including short paths and giant components; formulating notions of power in terms of the underlying graph structure; and even drawing connections to matching markets when we consider some of the ways in which search companies on the Web have designed their businesses.

Because the Web plays such a central role in the modern version of this topic, we begin with some context about the Web, and then look further back into the history of information networks that led up to the Web.

One of the powerful roles that networks play is to bridge the local and the global – to offer explanations for how simple processes at the level of individual nodes and links can have complex effects that ripple through a population as a whole. In this chapter, we consider some fundamental social network issues that illustrate this theme: how information flows through a social network, how different nodes can play structurally distinct roles in this process, and how these structural considerations shape the evolution of the network itself over time. These themes all play central roles throughout the book, adapting themselves to different contexts as they arise. Our context in this chapter begins with the famous “strength of weak ties” hypothesis from sociology [190] and explores outward from this point to more general settings.

Let's begin with some backgound and a motivating question. As part of his Ph.D. thesis research in the late 1960s, Mark Granovetter interviewed people who had recently changed employers to learn how these people had discovered their new jobs [191]. In keeping with earlier research, Granovetter found that many people learned information leading to their current jobs through personal contacts. But perhaps more strikingly, these personal contacts were often described by interview subjects as acquaintances rather than close friends. This fact is a bit surprising: your close friends presumably have the most motivation to help you when you’re between jobs, so why is it so often your more distant acquaintances who are actually to thank for crucial information leading to your new job?