In this first part of the book we develop some of the basic ideas behind graph theory – the study of network structure. This approach allows us to formulate basic network properties in a unifying language. The central definitions discussed here are simple enough that we can describe them relatively quickly at the outset; after this, we consider some fundamental applications of the definitions.

Basic Definitions

Graphs: Nodes and Edges. A graph is a way of specifying relationships among a collection of items. A graph consists of a set of objects, called nodes, with certain pairs of these objects connected by links called edges. For example, the graph in Figure 2.1(a) consists of four nodes labeled A, B, C, and D; node B is connected to each of the other three nodes by edges, and nodes C and D are also connected by an edge. We say that two nodes are neighbors if they are connected by an edge. Figure 2.1 shows the typical way to draw a graph: a small circle represents each node, and a line connects each pair of nodes that are linked by an edge.

When looking at Figure 2.1(a), think of the relationship between the two ends of an edge as being symmetric; the edge simply connects them to each other. In many settings, however, we want to express asymmetric relationships – for example, that A points to B but not vice versa.

The past decade has seen a growing public fascination with the complex “connectedness” of modern society. At the heart of this fascination is the idea of a network – a pattern of interconnections among a set of things – and one finds networks appearing in discussion and commentary on an enormous range of topics. The diversity of contexts in which networks are invoked is in fact so vast that it's worth deferring precise definitions for a moment while we first recount a few of the more salient examples.

To begin with, the social networks we inhabit – the collections of social ties among friends – have grown steadily in complexity over the course of human history, due to technological advances facilitating distant travel, global communication, and digital interaction. The past half-century has seen these social networks depart even more radically from their geographic underpinnings – an effect that has weakened the traditionally local nature of such structures but enriched them in other dimensions.

The information we consume has a similarly networked structure: these structures too have grown in complexity, as a landscape with a few purveyors of high-quality information (publishers, news organizations, the academy) has become crowded with an array of information sources of wildly varying perspectives, reliabilities, and motivating intentions. Understanding any one piece of information in this environment depends on understanding the way it is endorsed by and refers to other pieces of information within a large network of links.

Our technological and economic systems have also become dependent on networks of enormous complexity. This has made the behavior of these systems increasingly difficult to reason about and increasingly risky to tinker with.

Over the past decade, there has been a growing public fascination with the complex “connectedness” of modern society. This connectedness is found in many incarnations: in the rapid growth of the Internet and the Web, in the ease with which global communication now takes place, and in the ability of news and information as well as epidemics and financial crises to spread around the world with surprising speed and intensity. These are phenomena that involve networks, incentives, and the aggregate behavior of groups of people; they are based on the links that connect us and the ways in which each of our decisions can have subtle consequences for the outcomes of everyone else.

Motivated by these developments in the world, there has been a coming-together of multiple scientific disciplines in an effort to understand how highly connected systems operate. Each discipline has contributed techniques and perspectives that are characteristically its own, and the resulting research effort exhibits an intriguing blend of these different flavors. From computer science and applied mathematics has come a framework for reasoning about how complexity arises, often unexpectedly, in systems that we design; from economics has come a perspective on how people's behavior is affected by incentives and by their expectations about the behavior of others; and from sociology and the social sciences have come insights into the characteristic structures and interactions that arise within groups and populations. The resulting synthesis of ideas suggests the beginnings of a new area of study, focusing on the phenomena that take place within complex social, economic, and technological systems.

Advertising Tied to Search Behavior

The problem of Web search, as traditionally formulated, has a very “pure” motivation: it seeks to take the content people produce on the Web and find the pages that are most relevant, useful, or authoritative for any given query. However, it soon became clear that a lucrative market existed within this framework for combining search with advertising, targeted to the queries that users were issuing.

The basic idea behind this is simple. Early Web advertising was sold on the basis of “impressions,” by analogy with the print ads one sees in newspapers or magazines: a company like Yahoo! would negotiate a rate with an advertiser, agreeing on a price for showing its ad a fixed number of times. But if the ad you're showing a user isn't tied in some intrinsic way to their behavior, then you're missing one of the main benefits of the Internet as an advertising venue, compared to print or television. Suppose, for example, that you're a very small retailer who's trying to sell a specialized product; say, for example, that you run a business that sells calligraphy pens over the Web. Then paying to display ads to the full Internet-using population seems like a very inefficient way to find customers; instead, you might want to work out an agreement with a search engine that said, “Show my ad to any user who enters the query ‘calligraphy pens'.”

The final broad class of social institutions we consider is concerned with the allocation of resources in a society via property rights. Property rights give the holder of the right the ability to use a resource, the ability to exclude others from using it, and usually the right to sell or transfer the resource to another person. Property can take many forms, ranging from physical property such as a plot of land or a can of Diet Coke, to intellectual property such as a song or a manufacturing process. In this chapter we examine how the existence and form of property rights, or the lack of property rights, can affect social outcomes for each of these types of property. The central message of this chapter is that the property rights a society chooses to establish will affect the allocations that occur, and some property rights are more likely than others to result in socially optimal allocations.

Externalities and the Coase Theorem

In Chapter 17 we argued that the allocation of goods that arises in a market equilibrium (for an economy without network effects) is socially optimal. In a market equilibrium, the goods that are produced are assigned to the consumers who value them the most, and any unit of a good that is produced costs society less to produce than it is worth to the consumer who receives the good. This results in maximum total social surplus.

In Chapter 8, we considered a first extended application of game-theoretic ideas in our analysis of traffic flow through a network. Here we consider a second major application – the behavior of buyers and sellers in an auction.

An auction is a kind of economic activity that has been brought into many people's everyday lives by the Internet, through sites such as eBay. But auctions also have a long history that spans many different domains. For example, the U.S. government uses auctions to sell Treasury bills and timber and oil leases; Christie's and Sotheby's use them to sell art; and Morrell & Company and the Chicago Wine Company use them to sell wine.

Auctions will also play an important and recurring role in this book, since the simplified form of buyer–seller interaction they embody is closely related to more complex forms of economic interaction as well. In particular, in the next part of the book, when we discuss markets in which multiple buyers and sellers are connected by an underlying network structure, we'll use ideas initially developed in this chapter for understanding simpler auction formats. Similarly, in Chapter 15, we'll study a more complex kind of auction in the context of a Web search application, analyzing the ways in which search companies like Google, Yahoo!, and Microsoft use an auction format to sell advertising rights for keywords.

Types of Auctions

In this chapter we focus on different simple types of auctions and how they promote different kinds of behavior among bidders. We consider the case of a seller auctioning one item to a set of buyers.

In Chapter 3 we considered some of the typical structures that characterize social networks, and some of the typical processes that affect the formation of links in the network. Our discussion in Chapter 3 focused primarily on the network as an object of study in itself, relatively independent of the broader world in which it exists.

However, the contexts in which a social network is embedded will generally have significant effects on its structure. Each individual in a social network has a distinctive set of personal characteristics, and similarities and compatibilities between two people's characteristics can strongly influence whether a link forms between them. Each individual also engages in a set of behaviors and activities that can shape the formation of links within the network. These considerations suggest what we mean by a network's surrounding contexts: factors that exist outside the nodes and edges of a networks, but which nonetheless affect how the network's structure evolves.

In this chapter we consider how such effects operate and what they imply about the structure of social networks. Among other observations, we will find that the surrounding contexts affecting a network's formation can, to some extent, be viewed in network terms. By expanding the network to represent the contexts together with the individuals, we will see in fact that several different processes of network formation can be described in a common framework.

Homophily

One of the most basic notions governing the structure of social networks is homophily – the principle that we tend to be similar to our friends. Typically, your friends don't look like a random sample of the underlying population.

Popularity as a Network Phenomenon

For the past two chapters, we have been studying situations in which a person's behavior or decisions depend on the choices made by other people — either because the person's rewards are dependent on what other people do or because the choices of other people convey information that is useful in the decision-making process. We've seen that these types of coupled decisions, where behavior is correlated across a population, can lead to outcomes very different from what we find in cases where individuals make independent decisions.

Here we apply this network approach to analyze the general notion of popularity. Popularity is a phenomenon characterized by extreme imbalances: while almost everyone goes through life known only to people in their immediate social circles, a few people achieve wider visibility, and a very, very few attain global name recognition. Analogous things could be said of books, movies, or almost anything that commands an audience. How can we quantify these imbalances? Why do they arise? Are they somehow intrinsic to the whole idea of popularity?

We will see that some basic models of network behavior can provide significant insight into these questions. To begin the discussion, we focus on the Web as a concrete domain in which it is possible to measure popularity very accurately. While it may be difficult to estimate the number of people worldwide who have heard of famous individuals such as Barack Obama or Bill Gates, it is easy to take a snapshot of the full Web and simply count the number of links to high-profile Web sites such as Google, Amazon, or Wikipedia.

In our analysis of economic transactions on networks, particularly the model in Chapter 11, we considered how a node's position in a network affects its power in the market. In some cases, we were able to come up with precise predictions about prices and power, but in others the analysis left open a range of possibilities. For example, in the case of perfect competition between traders, we could conclude that the traders would make no profit, but it was not possible to say whether the resulting situation would favor particular buyers or sellers – different divisions of the available surplus were possible. This is an instance of a broader phenomenon that we discussed earlier, in Chapter 6: when there are multiple equilibria, some of which favor one player and some of which favor another, we may need to look for additional sources of information to predict how things will turn out.

In this chapter, we formulate a perspective on power in networks that can help us further refine our predictions for the outcomes of different participants. This perspective arises dominantly from research in sociology, and it addresses not just economic transactions but also a range of social interactions more generally that are mediated by networks. We will develop a set of formal principles that aim to capture some subtle distinctions in how a node's network position affects its power. The goal will be to create a succinct mathematical framework enabling predictions of which nodes have power, and how much power they have, for arbitrary networks.

Price Setting in Markets

In Chapter 10 we developed an analysis of trade and prices on a bipartite graph consisting of buyers, sellers, and the edges connecting them. Most importantly, we showed that market-clearing prices exist and that trade at these prices results in maximal total valuation among the buyers and sellers, and we found a procedure that allowed us to construct market-clearing prices. This analysis shows in a striking way how prices have the power to direct the allocation of goods in a desirable way. What it doesn't do is provide a clear picture of where prices in real markets tend to come from. That is, who sets the prices in real markets, and why do they choose the particular prices they do?

Auctions, which we discussed in Chapter 9, provide a concrete example of price determination in a controlled setting. In our discussion of auctions, we found that if a seller with a single object runs a second-price sealed-bid auction – or equivalently an ascending-bid auction – then buyers bid their true values for the seller's object. In that discussion, the buyers were choosing prices (via their bids) in a procedure selected by the seller. We could also consider a procurement auction in which the roles of buyers and sellers are reversed, and a single buyer is interested in purchasing an object from one of several sellers. Here, our auction results imply that if the buyer runs a second-price sealed-bid auction (buying from the lowest bidder at the second-lowest price), or equivalently a descending-offer auction, then the sellers will offer to sell at their true costs.

In this final part of the book, we build on the principles developed thus far to consider the design of institutions and how different institutions can produce different forms of aggregate behavior. By an institution, we mean something very general — any set of rules, conventions, or mechanisms that synthesizes individual behavior across a population into an overall outcome. In the next three chapters, we will focus on three fundamental classes of institutions: markets, voting, and property rights.

We begin by discussing markets, and specifically their role in aggregating and conveying information across a population. Each individual participant in the market arrives with certain beliefs and expectations — about the value of assets or products, and about the likelihood of events that may affect these values. The markets we study will be structured so as to combine this set of beliefs into an overall outcome — generally in the form of market prices — that represents a kind of synthesis of the underlying information.

This is part of a broad issue we have seen several times so far: the fact that individuals' expectations affect their behavior. For example, we saw this in Chapter 8 with Braess's Paradox, where the optimal route depends on which routes others are expected to choose; in Chapter 16 on information cascades, where people draw inferences about the unknown desirability of alternatives (restaurants or fashions) from the behavior of others; and in Chapter 17 on network effects, where the unknown value of a product (a fax machine or a social networking site) depends on how many others are also expected to use the product.

At the beginning of Chapter 16, we discussed two fundamentally different reasons why individuals might imitate the behavior of others. One reason was based on informational effects: since the behavior of other people conveys information about what they know, observing this behavior and copying it (even against the evidence of one's own private information) can sometimes be a rational decision. This was our focus in Chapter 16. The other reason was based on direct-benefit effects, also called network effects: for some kinds of decisions, you incur an explicit benefit when you align your behavior with the behavior of others. This is what we will consider in this chapter.

A natural setting where network effects arise is in the adoption of technologies for which interaction or compatibility with others is important. For example, when the fax machine was first introduced as a product, its value to a potential consumer depended on how many others were also using the same technology. The value of a social networking or media-sharing site exhibits the same properties: it's valuable to the extent that other people are using it as well. Similarly, a computer operating system can be more useful if many other people are using it: even if the primary purpose of the operating system itself is not to interact with others, an operating system with more users will tend to have a larger amount of software written for it and will use file formats (e.g., for documents, images, and movies) that more people can easily read.

We have now seen a number of ways of thinking both about network structure and about the behavior of agents as they interact with each other. A few of our examples have brought these together directly – such as the issue of traffic in a network, including Braess's Paradox – and in the next few chapters we explore this convergence of network structure and strategic interaction more fully, and in a range of different settings.

First, we think about markets as a prime example of network-structured interaction between many agents. When we consider markets creating opportunities for interaction among buyers and sellers, there is an implicit network that encodes the access between these buyers and sellers. In fact, there are a number of ways of using networks to model interactions among market participants, and we will discuss several of these models. Later, in Chapter 12 on network exchange theory, we will discuss how market-style interactions become a metaphor for the broad notion of social exchange, in which the social dynamics within a group can be modeled by the power imbalances of the interactions within the group's social network.

Bipartite Graphs and Perfect Matchings

Matching markets form the first class of models we consider, as the focus of the current chapter. Matching markets have a long history of study in economics, operations research, and other areas because they embody, in a very clean and stylized way, a number of basic principles: the way in which people may have different preferences for different kinds of goods, the way in which prices can decentralize the allocation of goods to people, and the way in which such prices can in fact lead to allocations that are socially optimal.

Among the initial examples in our discussion of game theory in Chapter 6, we noted that traveling through a transportation network, or sending packets through the Internet, involves fundamentally game-theoretic reasoning: rather than simply choosing a route in isolation, individuals must evaluate routes in the presence of the congestion resulting from the decisions made by themselves and everyone else. In this chapter, we develop models for network traffic using the game-theoretic ideas developed thus far. In the process, we will discover a rather unexpected result – known as Braess's Paradox [76] – which shows that adding capacity to a network can sometimes actually slow down the traffic.

Traffic at Equilibrium

Let's begin by developing a model of a transportation network and how it responds to traffic congestion; with this model in place, we can then introduce the game-theoretic aspects of the problem.

We represent a transportation network by a directed graph: we consider the edges to be highways, and the nodes to be exits where you can get on or off a particular highway. There are two particular nodes, which we call A and B, and we assume everyone wants to drive from A to B. For example, we can imagine that A is an exit in the suburbs, B is an exit downtown, and we're looking at a large collection of morning commuters. Finally, each edge has a designated travel time that depends on the amount of traffic it contains.