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Advances in communication and transportation technologies have the poten- tial to bring people closer together and create a “global village.” However, they also allow heterogeneous agents to segregate along special interests, which gives rise to communities fragmented by type rather than by geography. We show that lower communication costs should always decrease separation between individual agents even as group-based separation increases. Each measure of separation is pertinent for distinct types of social interaction. A group-based measure captures the diversity of group preferences that can have an impact on the provision of public goods. While an individual measure correlates with the speed of informa- tion transmission through the social network that affects, for example, learning about job opportunities and new technologies. We test the model by looking at coauthoring between academic economists before and during the rise of the Internet in the 1990s.


I. INTRODUCTION
Do new communication technologies, on balance, bring us closer together, or do they push us apart? Observers greeted the introduction of new transportation technologies such as the railroad and the automobile, on the one hand, and the spread of electronic communication such as the telephone and electronic mail, on the other hand, with the expectation that they would help overcome geographic boundaries and therefore draw communities closer together. 1 However, advanced communication technologies can create new divisions by making heterogeneous agents more selective. If agents prefer to communicate with agents of their own type, communities will fragment along types rather than geographic location. The automobile and the telephone strengthened social interactions based on common inter-ests and along generational lines. 2 Furthermore, the very development of the technological underpinnings of the Internet, emailing and the World Wide Web, were driven by the desire to facilitate cooperation between scattered groups of specialized researchers across the globe. 3 We construct a simple theoretical model to explain how decreasing communication costs can simultaneously decrease separation between agents in one dimension and increase separation in another dimension. Each type of agents in our model belongs to some group such as a political party, an ethnic community, or an academic specialization or subfields. Agents prefer to collaborate with own-type neighbors, but they face a trade-off between conducting costly collaboration with distant own-type neighbors or starting less profitable projects with close distinct-type neighbors. We then study two measures of separation between agents. Group separation captures the separation between types of agents by looking at the share of messages that are exchanged within groups rather than between groups. In our model group separation always increases as communication becomes less expensive because agents become selective: the desire to segregate into type-based groups is the very reason agents communicate more with distant agents as costs fall. In contrast, individual separation describes the distance between two randomly chosen individual agents measured by the time it takes for news to travel between these agents. A priori, lower communication costs have an ambiguous effect on individual separation. While the distance between agents of the same type is always reduced, the increase in group separation makes it more difficult for news to reach members of a different group. However, we show that for sufficiently large social networks this latter effect is small and indi-2. Sproull and Kiesler [1991] describe how the spread of the telephone strengthened affiliation among teenage peer groups. Lynd and Lynd [1929, Chapter XIX, footnote 8] report in their Middletown study the tendency of young people to mingle with peers in neighboring cities: "The young people go miles away, but fail to get well acquainted with those near by." Social life in the town became increasingly fragmented and centered around shared interests: club groups became prominent and an increasing number of friends were recruited in these organized environments. Lynd and Lynd interviewed a group of working class and business class wives. In the first group, ten out of 173 friends were recruited in clubs, compared with two out of 116 friends of their mothers. In the business class group 26 out of 75 friends were first met in clubs, compared with 6 out of 71 friends of their mothers. A similar trend holds for the husbands.
3. Emailing was a by-product of the ARPANET program which was funded by the U. S. Department of Defense. The HTML markup language of the World Wide Web was invented by Tim Berners-Lee at CERN in an effort to make the information sharing between particle physicists easier [Hafner and Lyon 1996]. vidual separation always decreases. This result holds both for lattice social networks such as the circle and for small-world networks that were popularized by Watts and Strogatz [1998]. Small-world networks resemble real-world social networks much better than lattice graphs and can be easily constructed from lattice graphs by adding a small number of "shortcuts." We prove that if individual separation in the underlying lattice graph decreases by a factor a then it decreases in the corresponding small-world graph by a factor ͌ a.
Our concepts of individual and group separation allow us to decompose the welfare effects of lower communication costs. Intuitively, our results on individual separation imply that each agent who communicates more with distant neighbors exerts a positive externality on every other agent by speeding up information transmission. This allows agents to learn more quickly, for example, about job opportunities or technological innovations [Granovetter 1973;Udry and Conley 2002]. 4 In contrast, the welfare effects of increased group separation are ambiguous. On the one hand, the increase in within-group communication allows groups to more easily form complementary institutions such as political organizations (i.e., the civil rights movement, antiglobalization protest groups, or environmental campaigns) or new field journals in the case of academic specializations. These institutions amplify the private benefits of increased communication with own-type neighbors. At the same time, greater group separation gives rise to divergent group preferences because agents spend more time talking to like-minded neighbors. Increased preference heterogeneity reduces mutual understanding between groups and makes coordination across groups more difficult because of divergent social norms. Taste heterogeneity has been associated with a decrease in public goods provision and increased conflict between groups [Alesina, Baqir, and Easterly 1999]. Interventions that reduce group separation have been 4. Granovetter [1973] was the first to emphasize the importance of friends and relatives as sources of employment information. Montgomery [1991] reviews the case study evidence on job-finding methods used by workers which suggests that approximately 50 percent of all workers currently employed found their jobs through friends and relatives. Topa [2001] estimates a careful structural model of the interaction effects in the Chicago labor market. Udry and Conley [2002] illustrate the role of social networks in the spread of pineapple farming in Ghana. shown to align agents' preferences and promote empathy and cooperation. 5 We demonstrate the differential impact of decreasing communication costs on group and individual separation empirically by looking at changes in patterns of coauthoring between academic economists. We believe that the results are interesting in their own right and have implications for academic knowledge production in a more connected world. Our data include all coauthored papers in top economic journals between the years 1969 and 1999. This time period covers the rise of the Internet after the invention of the World Wide Web in 1991. It is a well-documented fact that coauthoring, in particular coauthoring with distant collaborators, increased strongly during this time period. We find that the relative probability of realizing a potential project with a distant U. S. collaborator increased by 30 percent in the 1990s compared with the 1980s. We also show that the increased attractiveness of long-distance collaborations made researchers more selective just as our model predicts: they were 20 percent less likely to realize a project with a dissimilar collaborator in the 1990s.
The paper most related to our work is van Alstyne and Brynjolfsson [1997] who introduce the possibility of greater group separation as communication costs decrease but do not formally analyze the relationship between group and individual separation. Another related literature analyzes the impact of lower communication costs on the relative use of specific communication technologies such as face-to-face communication versus electronic communication [Gasper and Glaeser 1998]. This approach is complementary to ours: while we assume a single communication technology but heterogeneous agents, Gasper and Glaeser look at changes in the relative use of different communication technologies among homogeneous agents.
The remainder of the paper is organized as follows. Section II introduces a simple formal model. Section III defines our two distinct measures of separation that capture the social distance between groups of people and between individuals. Section IV 5. Duncan, Boisjoly, Levy, Kremer, and Eccles [2003] show that white students with randomly assigned African-American roommates are more likely to support redistribution to the poor and affirmative action. Gurin, Peng, Lopez, and Nagda [1999] also find a positive correlation between the degree of interaction and declining racial stereotypes. Experiments in social psychology demonstrate that cooperative activities between members of distinct groups tend to promote tolerance [Sherif, Harvey, White, Hood, and Sherif 1961;Aronson 1975]. introduces our main result for lattice graphs which we extend to small-world networks in Section V. In Section VI we measure separation of researchers in academia using coauthoring and confirm the usefulness of our two measures. Section VII concludes.

II. THE BASIC MODEL
We build a very stylized model of communication with two different types of agents who exhibit a preference for communication with their own type. There are 2n agents (n Ͼ 3) who are located along a circle (see Figure I). One half of all agents are of type A, and the other half are of type B. Agents' types alternate along the circle: every type A agent has precisely two type B agents as direct neighbors and vice versa.
We will refer to agent's four neighbors who are located at FIGURE I Society with n ϭ 5 Type A and 5 Type B Agents most a distance two away from her as her close neighbors and the remaining four neighbors who are located at most a distance four away from her as her distant neighbors. Therefore, each agent has a total of eight neighbors, and all other agents are nonneighbors.

II.A. Projects and Communication
Time is discrete, and in each time period every agent can initiate projects with her neighbors. A project can be, for example, coauthoring a research paper, a lunch or dinner engagement, or merely a conversation. For simplicity, we assume that the benefits of a project accrue only to the initiator of a project. However, this assumption can be easily relaxed.
Each agent can start exactly four projects in each time period and can do at most one project with each of her neighbors. Collaborating on a project requires communication between both agents. We assume that the initiator has to send precisely one message to his partner. Moreover, communication with a close neighbor is costless, while sending a message to a distant neighbor has an (additive) cost C.
By choosing a lattice graph, we rely on the Euclidean notion of distance. Therefore, the types of communication technologies that best fit our basic model are those for which usage cost increases steeply with distance. Examples include the automobile and telephony before the dramatic decrease in long-distance rates during the second half of the twentieth century.
An alternative notion of "close" and "distant" neighbors labels any agent who is not close to be distant. This notion of distance better fits communication technologies such as modern long-distance telephony, the World Wide Web, and emailing with usage costs depending only weakly or not at all on geographical distance. We will be able to analyze both types of communication technologies together when we introduce small-world networks in Section V.

II.B. Preferences
Collaborating on a project with a distinct type neighbor gives utility U គ while a partnership with an own type gives utility Ũ which is distributed over [U គ ,ϱ) with cumulative distribution function F(Ũ ). The utility that can be derived from each potential project is observable by agents before they initiate collaboration.
Because our model is symmetric in both types, we can restrict attention to the decision-making process of a type A agent. Clearly, a type A agent will always collaborate with her two close own-type neighbors. The only trade-off she faces is whether to work on the remaining two projects with her two close type B neighbors at zero cost, or start a more profitable project with her two distant own-type neighbors and pay a communication cost C. A type A agent will pay for costly communication with a distant type A neighbor if the project has sufficiently high potential: This will be the case with probability ␥(C) ϭ 1 Ϫ F(U គ ϩ C). Note that the probability ␥(C) is decreasing in C: new means of communication that decrease the cost C of sending messages make more projects with distant neighbors profitable.
In each time period our type A agent will make one of three decisions.
1. With probability (1 Ϫ ␥(C)) 2 communicating with her two distant type A neighbors is not profitable enough to justify the higher cost of communication, and she will instead collaborate only with her four close neighbors. Hence our type A agent will send half of her messages to own-type neighbors. 2. With probability 2␥(C)(1 Ϫ ␥(C)) exactly one of the two projects with distant type A neighbors is sufficiently promising to drop collaboration with a close type B neighbor. She will send 75 percent of her messages to own-type neighbors. 3. With probability ␥(C) 2 collaboration with both distant type A neighbors is valuable enough to drop projects with both close type B neighbors. In this case type A agent will communicate only with own-type neighbors. In case (2) there is a small indeterminacy because the type A agent can stop collaborating with either of her two close type B neighbors in favor of the more profitable project with a distant own-type neighbor. As a tie-breaking rule we assume that every agent of type A drops the project with her left (right) type B neighbor if she wants to work instead with her distant left (right) type A neighbor.
Note that in our model the total number of projects started by an agent (and hence the amount of communication she conducts in each period) is the same for all communication costs C. In a richer environment the effect of lower communication costs on the total volume of communication is ambiguous. On the one hand, agents substitute away from local projects toward less expensive long-distance projects (the substitution effect). On the other hand, the lower overall cost of communicating allows agents to start more projects (the income effect). The total amount of communication might therefore increase or decrease as a result. We choose to abstract away from these effects.
We also do not consider endogenous location choice of agents. If agents could costlessly pick their location before playing the communication game, we would expect agents of similar types to move together and maximize utility through local communication alone. We acknowledge that the desire to live close to similar agents gives rise to some degree of clustering, and that complete mixing is an analytically convenient rather than a realistic assumption. However, complete segregation is unlikely because the choice of location is influenced by many factors other than the desire to be close to friends, such as career concerns, choice of school for children, or idiosyncratic preferences for a certain location or apartment to name a few.

III. MEASURES OF SEPARATION
In this section we formally define group and individual separation and discuss the welfare implications of changes in each measure. Our measures of group and individual separation are closely related to the indices of "balkanized affiliation" and "balkanized communication" introduced by van Alstyne andBrynjolfsson [1996, 1997].

III.A. Group Separation
Assume that agent i sends an expected number x ij of messages to agent j i in every time period. We can then define the degree of group separation ⌸ between type A and type B as the share of total messages that are exchanged between agents of the same type: The indicator function J(i, j) is defined as J͑i, j͒ ϭ ͭ 1 if i and j are of the same type 0 otherwise.
Since our model is symmetric in types and agents, this group measure collapses to the share of messages some single agent i sends to agents of the same type. Larger values of ⌸ indicate a greater degree of group separation. Society is completely segregated into noncommunicating communities if ⌸ ϭ 1. This case is excluded so long as the cost of communication does not become zero.

III.B. Individual Separation
Our second measure describes the degree of individual separation between two random agents in our society. We define it with the following simple model of information diffusion in mind. Assume that at time t ϭ 0 agent 1 has some brilliant idea about a new technology, and she starts to share it with all of her four collaborators. We assume that the agent derives no utility from other agents using the technology and also cannot demand payment from any other agent for relaying the information.
Note that the share of messages sent to neighbors of her own type is exactly ⌸(C), the degree of group separation. At the end of the first period, five agents will know about the news: herself, two neighbors to her right, and two neighbors to her left. In the second period, each of these five agents will send two more messages to her right and left neighbors; e.g., agents who already know about the superior technology will continue to transmit to their neighbors. As long as communication with distant neighbors is at least somewhat costly, (i.e., ⌸ Ͻ 1) every agent j will hear almost surely about the new technology.
This will take a random number of time periods T j . We then define the degree of individual separation S j between agent 1 (the originator of the idea) and some agent j 1 as the expected time it takes to communicate the news between those two agents: The degree of separation Ŝ is defined as the average expected waiting time to reach a random agent j: 6 6. Due to the symmetry of our model, the initial agent 1 (including her type) can be chosen randomly on the network.
Our definition of individual separation is closely related to the game called Six Degrees of Separation that was popular on American campuses in the 1980s. The aim of the game is to find the shortest path of acquaintances that connects two randomly chosen players. 7 In the context of our model, we can provide a more realistic measure of individual separation that takes into account the strength of links along the path.

III.C. Separation Measures and Welfare
In order to understand the welfare effects of changes in the cost of communication, we decompose the total expected utility U Total of an agent in each period as follows: The first term is simply the sum of utilities from the four projects that the agent conducts with close or distant neighbors in each period. The next three terms capture the communication externality that results from the collaboration decisions of all other agents. It consists of three components: the benefit T of transmitted ideas from other agents, the cost v(⌬E) generated by differences ⌬E in group opinions, and the benefit w(I) of institutions serving the needs of specific groups.
We define the benefit T of transmitted ideas as follows. In each time period every agent has an idea that can benefit exactly one other agent in the society. This idea saves that agent a cost c in each time period. The average loss from waiting for this idea to reach its recipient is therefore T ϭ ϪŜ c which is equal to the expected steady state loss. A decrease in individual separation therefore always improves welfare by giving agents quicker access to cost-saving ideas. 8 Specific examples of ideas are innovative technologies or information about job opportunities: in the 7. The game was originally invented by a group of mathematicians who defined two agents to be linked if they had a coauthored paper. The aim of the game was to find the shortest path which linked the agent to the famous graph theorist and mathematician Paul Erdös.
8. Note, that the total number of messages received by agents in the steady state in each time period is constant across all communication costs because the flow of received messages has to equal the flow of produced messages. Changes in latter case we can interpret T, respectively, as the opportunity costs of using an inferior technology and being matched to an inferior job (or no job at all).
In contrast, differences in group tastes and beliefs are affected by group separation. Greater group separation gives rise to more diverse group preferences making collective decisionmaking more difficult. The public finance literature has identified several possible channels. First, group separation can affect the tastes of the median voter and, more generally, will increase the median distance from the median voter. Alesina, Baqir, and Easterly [1999] show how such an increase in the heterogeneity of preferences can reduce the provision of public goods in a community. Second, Alesina and la Ferrara [2000] build a model of group formation to explain the empirical fact that participation in social activities and hence social capital is lower in more heterogeneous communities.
To illustrate the connection between group separation and group opinions, we present the following simple model. Each agent has a preference i for the type of public good that is provided in their community. A type A agent at time t has taste i t ϭ ␣ A t ϩ i t with real support, and a type B agent has taste i t ϭ ␣ B t ϩ i t . It consists of two components: a type dependent component ␣ A t (␣ B t ) and an idiosyncratic component i t . We assume that the idiosyncratic component is identically and independently distributed among agents and has mean 0. The type dependent component captures the idea that the median preferences of voters in each group differ a priori. For example, the young might prefer to spend money on bicycle lanes and playgrounds, while the old prefer to spend money on public transportation and making buildings accessible for the disabled. Another example would be the preferences of researchers in different subfields for what type of research to fund: labor economists might prefer funding of large-scale natural experiments, while industrial economists prefer to collect better industry data. Similarly, astronomers would like NASA to build bigger and better space telescopes, while particle physicists prefer to invest in accelerators.
Communication with neighbors affects preferences: we simply assume that the final preference of an agent is a weighted communication costs only affect the distribution of vintages of ideas. In particular, decreasing communication costs do not produce "spam" in our model. average of her own taste i and those of her neighbors with communication shares as the weights on her neighbors' preferences. This captures the idea that an agent will be more heavily influenced by the preferences of neighbors with whom she communicates more frequently. We can calculate the final preference i t of a type A agent as follows: The random variable i is a weighted average of the agent's idiosyncratic component and those of her neighbors. We can calculate the "average" or median preference for each type by summing over all individuals of the same type. 9 If society is sufficiently large, the idiosyncratic components cancel out by the law of large numbers, and we obtain the group preferences E A t : Analogously, we obtain an expression for the group preference of type B agents after they update their initial preferences: The difference in median group preferences can be calculated as This formula illustrates how group separation can preserve initial group-specific differences. The more separated agents are, the less they take the opinions of other types into account, which tends to increase the "median distance" to the median voter. Differences in the preferences of group members can be further amplified by a phenomenon known as group polarization documented in the experimental social psychology literature [Brown 1986]. Agents tend to weigh the views of more strongly opinionated peers more heavily than those of less opinionated ones when forming their own preferences. Polarization appears to be particularly prevalent if communication is computer-mediated [Hightower and Sayeed 1995], as content on the internet can be 9. This would be the opinion observed by a Gallup poll over a large sample of individuals. easily searched for Web sites and newsgroups that support one's opinion [Sunstein 2001].
Finally, higher group separation also promotes the formation of group-specific institutions I such as political parties in the political context or specialized field journals in academia. These institutions tend to amplify the private benefits agents derive from communicating more with own-type agents.
To summarize, lower individual separation promotes welfare, while an increase in individual separation has ambiguous welfare consequences. On the one hand, higher group differences decrease welfare, because they make coordination on public good provision more difficult. On the other hand, group separation gives rise to institutions complementing private gains.

IV. THE EFFECTS OF LOWER COMMUNICATION COSTS ON GROUP AND INDIVIDUAL SEPARATION
In this section we analyze how a decrease in the cost of communication affects group and individual separation, respectively.

IV.A. Group Separation
A decrease in communication costs will always increase group separation. Lower communication costs make agents more selective in their choice of collaborators, and allow them to collaborate on more projects with own-type agents. Formally, we can derive the degree of group separation as follows: This expression is the weighted sum of the following terms: with probability (1 Ϫ ␥(C)) 2 half of the messages go to same type neighbors, with probability 2␥(C)(1 Ϫ ␥(C)) three-quarters of the messages go to same types, and with probability ␥(C) 2 all messages go to same type neighbors. Note that group separation ⌸(C) is decreasing in the cost of communication C; i.e., communication increasingly focuses on own-type neighbors. In particular, we have ⌸(C) ϭ 1 ⁄2 for C ϭ ϱ and lim C30 ⌸(C) ϭ 1.

IV.B. Individual Separation
The effect of a decrease in the cost of communication from C H to C L Ͻ C H on individual separation is a priori ambiguous.
Agents will send more messages to distant own-type neighbors which will tend to increase the speed of within-type diffusion. This connectivity effect on its own would work to decrease the degree of individual separation. However, higher within-type communication increases group separation and makes it harder for messages to travel between types. In particular, if communication costs become very small (C L 3 0), news from a type A agent will almost never reach type B agents. This would make the degree of individual separation infinite. Therefore, the net effect of lower communication costs on the degree of individual separation is ambiguous.
However, for large networks sizes n we can show that the first effect dominates the group separation effect. The intuition is that connectivity is a global property while group separation is a local one. As the size of agents' neighborhoods increases due to better communication technologies, the number of time periods necessary for news to travel the distance between two randomly drawn same-type agents decreases proportionally. Nevertheless, the expected number of time periods to bridge this distance will be of order n. In contrast, the waiting time for news to travel between some agent and her close distinct-type neighbor is 1/F(U គ ϩ C): 10 news will at some point "cross over" the type barrier as they spread through the (large) social network. Therefore, the presence of distinct types does not greatly affect average individual separation.
The next theorem formalizes this first result: while group separation increases due to lower communication costs, individual separation always decreases for sufficiently large societies. We normalize individual separation Ŝ by dividing through by n to compare separation for different communication costs.
It is increasing in C.
10. The probability of sending a message to this neighbor is F(U គ ϩ C).
A heuristic proof of Theorem 1 proceeds as follows. Two random agents live, on average, a distance n/ 2 apart (since the circle has length 2n). The cluster of agents who heard about the news expands on both ends an expected distance of ⌬d(C) ϭ 1/a(C) Ͼ 2(1 Ϫ ␥(C)) ϩ 4␥(C). 11 Hence, it will take on average (n/ 2)/⌬d(C) time periods for news to travel between two randomly selected agents.
Theorem 1 has the following immediate corollary.
COROLLARY 1. The relative degree a(C H ,C L ) of individual separation in a regime with a high cost C H of communication versus a regime with a low cost C L Ͻ C H of communication satisfies Theorem 1 and Corollary 1 can be extended to more general lattice graphs. In particular, if we look at circular graphs with a larger radius of interaction, both results will hold but the function a(C) will change. On two-and more-dimensional graphs we have to additionally normalize by the degree of individual separation by n d , where d is the dimension of the lattice.
Our results differ from van Alstyne and Brynjolfsson [1997] who focus on group separation and argue that their measures of group and individual separation comove. However, they derive this conclusion from simulations on small networks where lowering communication costs has an ambiguous effect on individual separation.

V. SMALL-WORLD NETWORKS
Recently, various researchers have observed that real-world social networks exhibit small-world features (see, for example, Watts and Strogatz [1998]). Small-world networks are characterized by (a) a high degree of clustering and (b) small characteristic path length. The coefficient of clustering C(G) of some graph G measures the degree to which neighboring agents' individual neighborhoods overlap, and therefore captures the degree of 11. The outermost agent on the right boundary of a cluster can expand the cluster by a length 2 or 4. However, if the agent to the direct left of him sent a message to a distant agent, the cluster expands by a length of 3. "cliquishness" of the network. 12 Regular lattice graphs such as our circle are the prototypes of highly clustered networks.
The characteristic path length L(G) of a graph G measures the average "length" of the shortest chain connecting two random agents. The length of a chain is defined as the time it takes to transmit a message along the chain. 13 The average path length is closely related to our measure of individual separation. 14 In particular, as we increase the size n of a circle graph both individual separation and the average path length increase at rate n.

V.A. A Simple Small-World Network
We adapt a model of small worlds developed by Watts and Strogatz [1998]: the "skeletal" network of our small world is the same circular network we introduced in Section II. However, we allow for additional shortcuts between agents on the circle. Each type A (type B) agent on the circle has a random link with some other own-type agent in the network. 15 For simplicity, we assume that each agent has precisely one of those "weak" links.
We call a shortcut between two agents i and j a weak link because their individual neighborhoods do not overlap. On the other hand, agents have strong links to neighbors on the circle because they share many common neighbors. This distinction 12. Formally, assume that each agent has m neighbors and starts p projects in each time period (in our model we have m ϭ 8 and p ϭ m/ 2 ϭ 4). She starts a project with some neighbor j with probability x ij in each time period. Agents i and j conduct an expected number Y ij Յ min ( p Ϫ x ij ,m ) of projects with the same set of agents in each time period where m Յ m Ϫ 1 is the number of common neighbors of both agents which can be at most m Ϫ 1. Then the coefficient of clustering C(G) is defined as the weighted average of the share of messages neighboring agents send to the same destinations: The coefficient of clustering always lies between 0 and 1. It tends to zero for large random graphs and is 1 for complete graphs (where all agents communicate equally with each other). 13. Formally, consider the chain of agents C ϭ (A 0 ,A 1 , . . . ,A m ,A mϩ1 , . . . ,A m ) that connects agents i and j (i.e., A 0 ϭ i and A m ϭ j) and denote the volume of messages that are sent between agents A m and A mϩ1 with x m . The length of the chain is then defined as L C ϭ ¥ mϭ0 m Ϫ1 1/x m . 14. However, it is not the same. When constructing our measure Ŝ , we allowed agents to continue sending messages even after they have heard about news in order to model the spread of news through society. Each agent j can therefore be reached by agent i through more than one path. Hence agent j will not always find out about the news through the shortest possible path.
15. Our results do not depend on agents having links only to own-type agents. In fact, they stay unchanged for any kind of type-correlation along weak links. between weak and strong links was first made by Granovetter [1973Granovetter [ , 1995 in his analysis of job search.
As in Granovetter [1973], agents communicate more frequently along weak links than along strong links: in each period there is a small probability ␦ that the agent can conduct one additional project. This project can only be conducted either (a) with one of the two direct neighbors to the right or left which yields utility U គ and involves costless communication; or (b) with the weak-link neighbor which yields random utility Û distributed according to the distribution function F over [U គ ,ϱ). 16 In the latter case the cost of communication is assumed to be Ĉ . Therefore, the probability of conducting this additional project with the weaklink neighbor is Note that the function A(Ĉ ) is the inverse probability of conducting the additional project with the weak-link neighbor and is hence increasing. While the previously defined function a(C) captures the rate of expansion of news along strong links, the new function A(Ĉ ) describes the rate of expansion of news along weak links. An improvement in communication and transportation technology weakly decreases both the short-range communication cost C and our new long-range communication cost Ĉ . However, they can improve at different rates: early telephony and automobiles decreased short-range communication costs but had little effect on the long-range cost. Vice versa, modern advances in electronic communication decreased the long-range cost at a faster rate than the short-range cost.
Our definitions for group separation ⌸ S and individual separation Ŝ S carry over to small-world networks with one caveat. Our definition of a small-world network does not define a unique network but a probability distribution over a class of small networks because the identity of the weak-link neighbor is chosen randomly. With a probability that is exponentially declining in the network size n, some of these networks look just like lattice 16. Collaboration with distant weak-link neighbors seems contrived. A more natural extension of the model would have agents choose collaborators among their three distant own-type neighbors and their two close distinct-type neighbors. However, the advantage of extending the model as suggested here is that it is easier to compare the new model with the original setup.
networks. 17 To focus attention on "typical" small-world networks, we calculate the expected time S j for news to travel from agent 1 to agent j by taking the expectation over all possible small-world networks.

V.B. Group and Individual Separation in Small Worlds
Unsurprisingly, the presence of weak links does not greatly affect group separation. In fact, the degree of group separation ⌸ S (C,C ) of the small-world model is simply a linear transformation of the degree of group separation ⌸(C) of the original model: Therefore, group separation increases as communication costs decrease just as it did in the original model. The effects of weak links on individual separation are dramatic. Weak links provide shortcuts through which distant parts of the circle graph can get "infected" by news. Therefore, the individual degree of separation no longer increases linearly in the size n of the circle but only at the rate ln (n) as the next theorem shows. It is in this sense that weak links make the world "small." THEOREM 2. In the small world average individual separation satisfies Proof. See Appendix 2.
In fact, when we compare the relative degree of separation in the high and low cost regime, we find that our insights from the basic model continue to hold. Interestingly, improvements in short-range and long-range communication technologies affect the rate of diffusion in exactly the same way.
Note that in small worlds a doubling of the local rate of diffusion a(C) decreases the individual degree of separation only by a factor ͌ 2 instead of a factor 2 as in the basic model. An interesting special case is homogeneous improvements in communication technologies which increase the rate of shortrange and long-range diffusion (i.e., the parameters a(C) and A(C) by the same factor f. In this case, the degree of individual separation will also decrease by the same factor f.
Our small-world results easily generalize to circular graphs with larger individual neighborhoods. If we consider different skeletal networks (i.e., a square lattice rather than a circle), our proofs can be adapted to derive precise results on a case-by-case basis. However, qualitatively, the results remain the same: lower communication costs decrease individual separation in both small worlds and on lattice graphs, but the relative decrease is less pronounced for small worlds.

VI. COLLABORATION OF ACADEMIC ECONOMISTS BETWEEN 1969 AND 1999
We test our model by looking at the evolution of academic coauthoring between 1969 and 1999. Several new technologies decreased the cost of communication substantially starting around 1980. First, fax technology became ubiquitous in the 1980s: by 1985 already more than 100,000 machines were shipped annually and by 1990 this number had increased twentyfold [Economides and Himmelberg 1995]. Second, emailing and file transfer through FTP was common by the beginning of the 1990s at U. S. universities [Arfman and Roden 1992;Walsh 1997]. 18 Third and perhaps most importantly, the rise of the Internet in the 1990s made it dramatically easier to publish and search for working papers using the HTML markup language and browser software. Moreover, deregulation of the U. S. airline and telephone industries in the 1980s drastically decreased the cost of traveling and making long distance telephone calls. The calling 18. In the United States 24 percent of physicists and 34 percent of mathematicians had email addresses in 1991 [Walsh 1997]. rates for state-to-state calls, for example, fell by almost half between 1984 and 1989 during the price wars that followed the breakup of AT&T in 1984 [FCC 1999].
The period 1980 -1999 therefore provides a natural testing ground for our theory. We would also expect the effects of decreasing communication costs on group and individual separation to be particularly strong within the academic community. Research departments were early adopters of fax machines, and academics were the first users of both email and the Internet because the original Arpanet was specifically designed as a research tool.
Our model predicts that decreasing communication costs should lead to more collaboration between "similar" researchers but at the same time decrease individual separation of all researchers. Increased group separation can have undesirable welfare consequences: if subfields develop divergent methodologies such as "natural experimentalists" versus "empirical labor economists," it can complicate resource allocation procedures and affect teaching of the discipline. On the other hand, we would expect lower individual separation to be unambiguously positive because it accelerates the transmission of useful and yet unpublished word-of-mouth information such as the availability of new data sources or preliminary results of other researchers. 19 We use a data set that contains all articles published between 1969 and 1999 in eight top economics journals. 20 We measure collaboration and communication between researchers by looking at their coauthored publications. Our data set contains 8838 authors of whom 6201 authors published at least one coauthored paper.

VI.A. Changes in Group Separation
In order to measure changes in group separation over time, we first have to define metrics for measuring geographic distance between coauthors and for measuring the similarity of their types which can be easily mapped into our model.
We measure the type similarity of coauthors by the field overlap of their publication records prior to publication of their coauthored article. The distance between coauthors is coded in 19. Such word-of-mouth transmission is particularly important in economics where publication of new results typically takes two years and more. 20. Glenn Ellison generously shared his data with us. The data have been collected from the CD version of EconLit. two different ways. First of all, we simply distinguish between coauthor relationships where both coauthors are affiliated with U. S. institutions (U. S./U. S. coauthors) and U. S./foreign collaborators. Second, we restrict attention to U. S./U. S. coauthors and distinguish between coauthors who work less than 200 kilometers (125 miles) apart and those who live farther apart. Although the precise cutoff distance is somewhat arbitrary, we had several reasons to choose 200 kilometers. First of all, it is close to the median distance between U. S. coauthors. Second, it implies a total commute time by car of about 4 -5 hours for one coauthor to visit her collaborator. We consider this close to the maximum distance that would allow regular face-to-face contact between two collaborators without having to travel for more than one day or use an airplane.
We test our predictions on group separation by embedding our model into a simple discrete choice framework. There is a stream of potential projects y i with characteristics (D i ,S i ,X i ), where D i and S i are dummy variables that are set to 1 if both coauthors are distant and similar, respectively. The vector X i captures other attributes of the potential project such as the field of study and other coauthor attributes such as their degree of specialization and the number of previously published papers. The probability that a potential project y i will be realized is where g is an increasing function. With probability 1 Ϫ g(D i ,S i ,X i ) the project will not be realized. We estimate the empirical model separately for the periods 1980 -1989 and 1990 -1998 and make the following predictions. H1: Improved means of communication decrease the cost of coauthoring with a distant author such that H2: The opportunity cost of coauthoring with a distinct type coauthor increases because it becomes more profitable to wait for a project with an own-type coauthor. Agents become more selective which implies that

VI.B. Description of the Data
We extract all coauthored papers between 1980 and 1999. To simplify our analysis, we use just the first two coauthors of each paper-more than 80 percent of all coauthored papers during this period have exactly two coauthors. We only include papers where at least one coauthor is affiliated with a U. S. research institution and where each coauthor has at least one prior publication during the preceding ten years. 21 The latter restriction is necessary because we use an author's publication record to determine her type and to measure the degree of type similarity between two coauthors. The resulting subsample contains 1772 coauthored articles. Summary statistics are provided in Table I.
We observe only publication dates rather than the dates on which collaboration between two coauthors started. This introduces a potentially troublesome source of measurement error into our analysis especially since mean submit-accept times increased in four out of the five top economics journals from less than 12 months in the early 1970s to 18 -30 months in the 1990s [Ellison 2001]. However, our analysis focuses on comparing decades rather than particular years. Therefore, any measurement error will only misclassify articles at the beginning and the end of each of the two decades. 22 We measure type similarity of coauthors by the degree of overlap of their publication records. Thus, we do not define an author's type directly but only relative to her coauthor: they are of more similar type if their publication records overlap to a greater degree. Formally, for each paper i and author j, we construct a vector v ij (c) of size 17 which summarizes the share of publications in field c. We then define our basic type similarity measure AUSIMIL as follows: This index also takes values between 0 and 1: larger values 21. Econlit provides affiliation only after 1988. For 1969For -1988 were manually added to the data set by searching through paper copies of the eight journals in our sample.
22. Another potential source of measurement error results from a possible change in affiliations from the start of the project until the publication date. Since we do not have good data on working papers, we are not able to observe whether the projects were started while coauthors were at the same institution and then published when they had distinct affiliations or the other way around. indicate greater similarity. A value of 1 implies that both authors allocated their research time equally across the same fields. Note that AUSIMIL always takes the value 0 if both authors work in distinct fields. To map the data more closely into our model, we construct a discrete measure of type similarity. Two authors are similar (S ϭ 1) if AUSIMIL is above its median and dissimilar (S ϭ 0) otherwise.
NATDIFF is an indicator variable which is 1 if one of the The data set comprises all coauthored papers in eight economics journals between 1980 and 1998 with at least one U. S. author: Journal of Political Economy, American Economic Review, Quarterly Journal of Economics, Econometrica, Review of Economic Studies, Review of Economics and Statistics, Rand Journal of Economics, and Brookings Papers on Economic Activity. The variable AUSIMIL measures the similarity of coauthors based on their publication records up to ten years prior to publication of their joint paper. AUSIMIL50 is an indicator variable and set to 1 if AUSIMIL is greater than its median value. AUSPEC indicates how specialized authors are. NATDIFF is an indicator variable which is 1 if one of the coauthors lives outside the United States. DISTANCE is distance between U. S. coauthors locations in kilometers, and LONGDIST is 1 if the distance exceeds 200 kilometers (125 miles). YEAR indicates the year of publication, and calendar year 1980 is set to 0. Each paper falls into one of seventeen field categories, which are labor, econometrics, productivity, experimental, micro theory, industrial organization, finance, macro, international, development, history, public finance, environmental economics, political economy, law and economics, and other fields. coauthors is foreign. DISTANCE measures the geographic distance between academic institutions of both coauthors in kilometers for U. S./U. S. coauthors. LONGDIST is an indicator variable equal to 1 iff the distance between two U. S.-based coauthors is more than 200 kilometers.
We collect information about the ten-year prior publication record of each coauthor j by counting her total number of publications AUPREVj ( j ϭ 1,2) and her degree of specialization AUSPECj. We use a simple Herfindahl-type index of specialization defined as follows: This index of specialization is a real number between 0 and 1 and takes the value 1 if the author is completely specialized; i.e., all her publications are in a single field.

VI.C. Analysis
The patterns of coauthoring with foreign authors (NATDIFF) and coauthoring with long-distance U. S. authors (LONGDIST) are consistent with hypothesis H1. Between 1969Between and 1979Between and 1980 to 1989 the share of U. S./foreign papers was about 16 percent and increased to 19 percent thereafter. Among U. S./U. S. coauthor relationships long-distance collaborations increased from 43 percent before 1980 to 50 percent between 1980 and 1989 and 55 percent thereafter. These results are consistent with those found in Gasper and Glaeser [1998].
When we regress both distance measures on a year trend and sixteen field controls an interesting trend emerges: U. S./foreign coauthoring increased mainly in the 1990s while U. S./U. S. long-distance coauthoring already accelerated in the 1980s. U. S./ foreign coauthoring increased at an annualized rate of 1.4 percent in the 1990s after decreasing slightly in the 1980s (see Table II). In contrast, long-distance collaborations within the United States increased at an annualized rate of 1.4 percent in the 1980s (see Table III). This is consistent with the fact that the United States deregulated their airline and telecommunications markets earlier than most other countries and was also a leader in introducing electronic means of communication.
In Figure II we decompose the changes in the patterns of coau-thoring between the 1980s and the 1990s. Two related trends emerge from "eyeballing" the U. S. data: (1) coauthoring between distinct-type and close coauthors has declined strongly; while (2) coauthoring with distant own-type collaborators has increased by roughly the same amount. These trends also show up in the U. S./ foreign coauthoring data but less strongly so. Both phenomena are exactly consistent with the predictions of our model: agents become more selective when communication costs decrease and substitute Significance levels: †: 10 percent; *: 5 percent; **: 1 percent. The dependent variable is NATDIFF; standard errors are shown in parentheses. The field controls include experimental economics, micro theory, industrial organization, finance, macro, international, development, urban economics, history, public finance, labor economics, econometrics, productivity, environmental economics, political economy, and law and economics. The first column includes all coauthored papers published between 1980 and 1998 while the next two columns restrict attention to the 1980s (1980 -1989) and 1990s (1990 -1998  Significance levels: †: 10 percent; *: 5 percent; **: 1 percent. The dependent variable is LONGDIST; standard errors are shown in parentheses. The field controls include experimental economics, micro theory, industrial organization, finance, macro, international, development, urban economics, history, public finance, labor economics, econometrics, productivity, environmental economics, political economy, and law and economics. The first column includes all coauthored papers published between 1980 and 1998 while the next two columns restrict attention to the 1980s (1980 -1989) and 1990s (1990 -1998). low-value projects with close but dissimilar coauthors with highvalue projects with distant but similar collaborators.
Ideally, we would like to formally test our joint hypotheses H1 and H2 by separately estimating the probability p i that a potential project y i with characteristics (D i ,S i ,X i ) is implemented in the 1980s and 1990s. Unfortunately, we lack the data to fully estimate such a discrete model because we only observe successful projects ( y i ϭ 1). However, under some additional assumptions we can estimate the change in coefficients between the periods 1980 -1989 and 1990 -1998.  1980 -1989 and 1990 -1998 The two diagrams on the left illustrate changes in coauthoring between U. S. coauthors, while the two diagrams on the right show changes in coauthoring between U. S. and foreign coauthors. In each case the top diagram shows coauthoring patterns in the 1980s, and the bottom diagram shows patterns in the 1990s. In each diagram we classify all coauthored papers along two dimensions: coauthoring with close (D ϭ 0) and distant (D ϭ 1) authors and coauthoring with similar (S ϭ 1) and dissimilar (S ϭ 0) authors.
We choose the following functional form for estimating our discrete choice model: Note that we can interpret ␣ D (and similarly ␣ S ) as the relative percentage increase in probability that a potential project will be realized if both coauthors are distant (or of similar type). Using Bayes' rule, we obtain The additional project characteristics X i include field controls and dummy variables capturing the degree of specialization of each coauthor and the experience of each coauthor measured by the number of articles which he or she has published previously. The cutoff values for our two specialization dummies (one for each coauthor) and our two experience dummies are simply the median values of AUSPEC1, AUSPEC2, AUPREV1, and AUPREV2. The joint project characteristics (D i ,S i ,X i ) therefore divide the data set into discrete cells. To simplify notation, we will use the subindex i both to denote an individual observation and a cell with characteristics (D i ,S i ,X i ).
In order to ferret out the effect of distance on coauthoring, we can simply compare two cells with the same characteristics except distance: (23) Using our functional form assumption, the left-hand side of this equation simplifies to exp(␣ D ). Term I on the right-hand side can be easily estimated from the data. Only Term II presents a problem because we do not know the distribution of coauthor characteristics in the universe of potential (as opposed to actual) projects. However, if we assume that this distribution did not change between the periods 1980 -1989 and 1990 -1998, then we can obtain a formula for the change in the distance coefficient ␣ D when estimated separately for both periods: For each pair of cells with characteristics (D i ϭ 1,S i ,X i ) and (D i ϭ 0,S i ,X i ), we thus get a different estimate ⌬␣ ͑Si,Xi͒ with precision h (S i ,X i ) . 23 By summing over all cell pairs, we can thus get an improved estimate of ⌬␣ D and its standard error 2 : We derive an estimator for the change ⌬␣ Ŝ in the preference for coauthoring with a similar author in an exactly analogous way.
The assumption that the distribution of coauthor characteristics for the universe of potential projects did not change between 1980-1989 and 1990-1998 is important for the derivation of this estimator. It implies that the geographic distribution of economists across U. S. universities according to fields and degree of specialization has not changed very much during the last twenty years. We do not have data to verify this assumption: but to the extent that economics departments tend to replicate themselves when replacing vacant positions with new researchers in order to preserve the balance of the various subfields within the department we believe that the assumption can be justified. which gives us four distinct cells. In the second column of both tables we add controls for 17 fields giving us 4 ϫ 17 ϭ 68 cells. In the third column we also control for each coauthor's degree of Significance levels: †: 10 percent; *: 5 percent; **: 1 percent. Standard errors are shown in parentheses. Distance is measured by NATDIFF. The field controls create seventeen cells and include experimental economics, micro theory, industrial organization, finance, macro, international, development, urban economics, history, public finance, labor economics, econometrics, productivity, environmental economics, political economy, and law and economics. The second column adds controls for coauthor specialization using the median values of AUSPEC1 and AUSPEC2 as cutoffs to distinguish between nonspecialized and specialized authors. These controls subdivide each field cell into four subcells. The third column adds controls for the number of previously published papers by each coauthor using the median values of AUPREV1 and AUPREV2 as cutoffs. Significance levels: †: 10 percent; *: 5 percent; **: 1 percent. Standard errors are shown in parentheses. Distance is measured by LONGDIST. The field controls create seventeen cells and include experimental economics, micro theory, industrial organization, finance, macro, international, development, urban economics, history, public finance, labor economics, econometrics, productivity, environmental economics, political economy, and law and economics. The third column adds controls for coauthor specialization using the median values of AUSPEC1 and AUSPEC2 as cutoffs to distinguish between nonspecialized and specialized authors. These controls subdivide each field cell into four subcells. The fourth column adds controls for the number of previously published papers by each coauthor using the median values of AUPREV1 and AUPREV2 as cutoffs.
specialization giving us 4 ϫ 68 ϭ 272 cells, and in the fourth column we control for each coauthor's experience. Increasing the number of controls any further is problematic given our sample size: while each new control dummy allows us to better control for project heterogeneity, it also doubles the number of cell pairs and cuts the observations per cell on average by half. Eventually, an increasing number of cells contain no observations. For our U. S./U. S. coauthoring data our estimates of both ⌬␣ D and ⌬␣ S are positive and significant which confirms hypotheses H1 and H2. The effects are quite large: in the 1990s a potential project with a distant author is 30 percent more likely to be realized than in the 1980s. The increased choice set makes researchers about 20 percent less likely to realize a project with a dissimilar author compared with the 1980s. The estimated coefficients are remarkably stable across all four specifications.
For U. S./foreign coauthoring we estimate a slightly bigger increased preference for coauthoring with own-type coauthors compared with the estimated coefficients for U. S./U. S. data in Table IV. However, we do not obtain estimates of ⌬␣ D which are significantly different from zero. Unfortunately, the share of U. S./foreign coauthored papers lies around only 17 percent from 1980 -1998. This makes it hard to apply our estimation technique while at the same time controlling for sources of heterogeneity such as fields and experience.

VI.D. Changes in Individual Separation
To demonstrate changes in individual separation, we simply calculate the average number of coauthors who separate two randomly chosen researchers i and j. We calculate this distance using 1989 and 1999 as base years and by considering all papers published during a twenty-or fifteen-year time frame prior to those base years. We say that two researchers are linked if they have coauthored a paper during the respective twenty-or fifteenyear time frame.
We then compare the network distance between two random researchers in the 1969 -1989 network with the same measure in the 1979 -1999 network. We repeat the same exercise with a comparison based on twenty-year time frames (i.e., 1974 -1989 compared with 1984 -1999).
One problem with this simple approach is that the resulting graphs are not always connected (and our measure of individual separation is hence not well defined): some researchers never coauthor or coauthor with an exclusive clique of colleagues. Fortunately, almost all researchers belong to the same giant connected cluster. 24 For example, the coauthoring graph of all economists who have published at least one paper between 1969 and 1998 consists of a giant component with 3443 distinct authors, while the next largest component of the graph contains only 15 authors. The same giant component exists within all the subgraphs defined for the pre-1989 and pre-1999 networks. To keep our analysis simple, we ignore the small number of author-nodes who do not belong to the giant component.
Another complication arises due to the fact that coauthoring has become increasingly common. Between 1969 and 1989 every author in the giant cluster coauthored on average C 1 ϭ 2.54 papers. Between 1979 and 1999 that number increased to about C 2 ϭ 2.73 papers, an increase of about 7.5 percent. In recent work, Goyal, van der Leijy, and Moraga-Gonzalez [2004] argue that this effect can explain most of the observed decline in individual separation.
We want to abstract away from this increase in the density of the network when comparing average individual separation. We achieve this by deleting links in the pre-1999 coauthoring network with probability 1 Ϫ C1/C2. This "pruning" of links preserves the structure of the network in terms of the share of "close" and "distant" links and makes the pre-1989 networks comparable to the pre-1999 networks.
The left two columns of Table VI show the evolution of average individual separation over time for twenty-and fifteen-year time frames. In both cases individual separation decreases by 7 and 16 percent, respectively, between 1989 and 1999. These declines are amplified if we restrict attention to authors with more than two publications (see right two columns in Table VI). Such a restriction excludes many "peripheral authors" who have only weak connections to the giant cluster and tend to drive up the degree of separation. 25 Now average individual separation decreases by 16 and 18.5 percent, respectively. 24. The presence of a giant connected cluster is typical for real world social networks. Watts and Strogatz [1998] analyze the network of film actors who are linked if they acted in a film together and find that about 90 percent of all actors belong to the giant connected cluster.
25. This might also help to reconcile our results with those of Goyal van der Leijy, and Moraga-Gonzalez [2004].
VII. CONCLUSION Our model shows how advances in communication technology have the potential to simultaneously bring us together and push us apart. We test this hypothesis by looking at collaboration patterns between academic economists and find support for both lower average individual separation and greater group separation.
There are a number of possible directions for extending our empirical analysis. First of all, it would be interesting to see whether our dual observations of lower individual separation and greater group separation can be replicated for other data sets. Second, the link between different measures of separation and economic outcomes should be explored carefully. This would mean, for example, to carefully map a process of technological diffusion through different types of social networks.
APPENDIX 1: PROOF OF THEOREM 1 We start by introducing some notation. We denote the cluster of agents who "hear about" news sent by some agent i by time t The size of the giant connected cluster is n, and the average number of links of every author in this cluster is L. The average degree of separation is Ŝ . Each column corresponds to a different time frame-the twenty-year frame for example compares the period 1969 -1988 (pre-1989) with 1979 -1998 (pre-1999). The two right columns only include authors who had more than two publications during the time frame. The adjusted pre-1999 network is obtained by (a) calculating the average number of links in the pre-1989 network (C1) and the pre-1999 network (C2); and (b) deleting links in the pre-1999 network randomly with probability 1 Ϫ (C1/C2). For this table the values of (C1/C2) were 0.930, 0.924, 0.896, and 0.911 (left to right). This link pruning to obtain the adjusted network is replicated 50 times for each of the four columns, and we report the average cluster size n, link number L and degree of separation Ŝ from these simulations. Sample standard errors are shown in parentheses.
with H i t . We adopt the convention H i 0 ϭ {i}. Since a cluster has two boundaries on a circular graph, we focus on the expansion of the right boundary without loss of generality.
The leading agent h i t is the member of the set H i t who is the farthest away from i on the right. We define the distance between agents i and h i t as d i t : The expansion of the right boundary of the cluster is determined by the leading agent and the agent next to him if she has heard the news already. Therefore, the right boundary can be in one of two states: in state 01 the agent next to the leading agent has not heard about the news. In state 11 the agent next to the leading agent has heard about it too. From state 01 the process transits to state 11 with probability 1 Ϫ ␥(C) (if the leading agent collaborates with his close neighbors only). From state 11 the process transits to state 01 with probability (1 Ϫ ␥(C))␥(C) (if the leading agent collaborates with his two own-type neighbors and the agent next to him only collaborates with his close neighbors). Since the probability flow between both states has to be the same in steady state we can deduce that the probability that the right boundary of the cluster is in state 01 converges to ␥/(1 ϩ ␥).
In state 01 the leading agent sends a message with probability 1 Ϫ ␥(C) to her two close neighbors and with probability ␥(C) to her two own-type neighbors (one close and one distant). We can therefore describe the evolution of d i t through the following transition matrix: In state 11 both the leading agent and the agent next to her can send messages with probability 1 Ϫ ␥(C) to their two close neighbors and with probability ␥(C) to their two own-type neighbors. The boundary of the cluster can therefore expand by either a distance of 2, 3, or 4. The transition matrix becomes 1. Each agent within the convex hull of an infected cluster can start a new cluster with probability ␦/A(C ) through his weak link in each period. 2. Clusters evolve without overlap. Both assumptions speed up diffusion. To derive the rate of diffusion of this simpler process, we introduce some notation. At each point in time t the process generates a new (stochastic) set of clusters ⌶ t . The superset of all these clusters is denoted by ⌸ t ϭ ഫ sϽt ⌶ t . Each cluster ʦ ⌶ is said to have a vintage . It grows over time at a stochastic rate, and we call the size of its convex hull at time t the span D(,t). We use the convention D(,t) ϭ 0 if t Ͻ . The number of clusters formed at time t is denoted by X t ϭ ͉⌶ t ͉, and the total number of agents inside the span on all clusters which have formed up to time t is Y t . Note that for the coupled process we have (32) Y t ϭ ʦ⌸t D͑,t͒.
The number of infected agents Y t increases over time because there are new infections and because existing clusters expand: We take expectations on both sides and define y t ϭ E[Y t ], x t ϭ E[X t ] and z t ϭ E͉⌸ t ͉. We also know from the proof of Theorem 1 that (34) lim t3ϱ E͓⌺ ʦ⌸t ͓D͑,t ϩ 1͒ Ϫ D͑,t͔͔͒ z t ϭ 2 1 a͑C͒ .
We next take first differences of equation (37): We assume that u(t) ϭ 1 at first. We then have a simple difference equation of the following form: When we solve the characteristic equation, we get a solution of the form: (41) It can be shown that u(t) is sufficiently close to 1 for small enough ␦ that it does not affect this solution to the difference equation. From this solution equation (30) follows. Next, we have to relax the simplifying assumptions we made for the coupled process.
• Agents can start a new cluster through their weak link only once. Agents start new clusters at rate ␦ and become unavailable for starting a second cluster. However, we have just shown that the population of infected agents expands at a rate proportional to ͌ ␦. Hence, we again get a solution as in equation (41). • Not every agent inside the convex hull is infected. Whenever an agent at the boundary communicates with a distant agent, a "gap" is created which fills up with probability F(U គ ϩ C) in each time period. For each the ratio ( y tϪ )/y t 3 1 as ␦ 3 0: an arbitrarily large share of infected agents live in clusters of vintage for small ␦. By choosing large enough, we can ensure that the share of infected agents inside the convex hull of these clusters converges to 1. Hence we again get solution (41).
• An agent's weak link can become infected before the agent can infect that link herself. At this point we use the fact that we only model the evolution of the system until a share ␤ of agents has become infected. That implies that an agent can infect another agent through her weak link at least with probability (1 Ϫ ␤)␦/A(C ) which gives us formula (30). • Clusters can overlap. To deal with this contingency, we again use the fact that the share of infected agents is at most ␤. Assume that the y t infected would be randomly distributed along the circle-in this case the average distance between them would be at least 1/␤. But since clusters grow around the boundaries, the average distance between the boundaries of the z t clusters is also at least 1/␤. Two neighboring clusters can grow together if their boundaries are less than eight agents apart. The probability for this event is O(␤). Hence a share O(␤) of nonoverlapping clusters disappear in each time period which again gives us formula (30).

Step II
For the second step we create a coupled process that governs the evolution of the system after a share ␤ of agents has become infected. We bias the evolution of this process against the spread of the news-therefore the process provides a lower bound on the true process. Note that after a share ␤ of agents has become infected a share ϭ ␤(1 Ϫ O(␦)) of agents can spread news through their weak links (see step I). We call this set of agents I and the set of agents they are linked to N I . The following rules govern the coupled process.
1. Weak links generating from agents outside of the set I cannot spread news. 2. Each agent only sends news to her direct neighbor to the right if she collaborates with her and this agent does not belong to the set N I . The agents in the set N I are on average a distance 1/ apart and subdivide the circle into fragments on which the coupled process develops independently. The expected waiting time to cover each fragment is bounded above by 1/␦ ϩ 1/(1/F(U គ ϩ C)). By the law of large numbers the average time WЈ it takes for news to reach all agents on the circle in the coupled process is the same, and hence finite. But since the coupled process systematically dis-criminates against the spreading of news the average time for news to infect all agents is also finite. QED WESLEYAN UNIVERSITY HARVARD UNIVERSITY AND NATIONAL BUREAU OF ECONOMIC RESEARCH