Meeting Strangers and Friends of Friends

To many people, the years spent tangled in the twisted realm of high school romance may have seemed like hard time served in prison. New work by Brian Rogers (Professor of Managerial Economics and Decision Sciences at the Kellogg School of Management) and Matthew Jackson (Professor of Economics at Stanford University) shows that patterns of social relationships found in these and other seemingly distinctive worlds are in fact surprisingly similar. Their recent work published in the American Economic Review shows that social networks share some core features that if manipulated slightly can foster growth in vastly different networks capable of widely varied functions.

“It was becoming clear since about 2000 that large, socially generated networks had macroscopic structural features that were similar to one another, even though they were generated in very different contexts,” Rogers said, explaining the motivation for this work. “There probably are some more or less simple explanations about how the networks were formed.”

Émile Durkheim, often called the “father of sociology,” proposed that social phenomena arise when individuals’ interactions lead to effects that can’t be explained solely by those individuals’ characteristics. Social network analysis thus focuses on the various types of relationships that link individuals and organizations. Encompassing everything from sibling rivalries to international politics, from commerce to the transmission of data and disease, the structures of social networks are critical in determining how groups function. For example, a 1967 study by the social psychologist Stanley Milgram showed that a typical person could pass a message to another random person through an average of five intermediate acquaintances. This phenomenon, which came to be known as “six degrees of separation,” is one of the most widely recognized descriptions of social networks.

Prison friendships and high school romances, the two most purely social networks, were accurately modeled by an almost entirely random process.“For economists [the determination of random social networks] is kind of a weird project, because the model is not particularly economic. We’re not modeling decisions based on a utility maximization problem,” said Rogers. “The background for this work comes from a lot of different areas, like computer science, statistical physics, and sociology. Our model can be seen as a combination of two different models. One is preferential attachment. The second is from pure math, the study of random graphs.”

Preferential attachment means that the more connected a person is, the more likely she is to receive new links in the future. For example, when a newcomer enters a community, he is more likely to become acquainted with well-known, highly visible people rather than with relatively obscure hermits. A random graph, as the name implies, is a graph that is generated by some unsystematic process, like letting a series of coin flips determine the strokes of a pen on a page.

In designing their model of social network formation, Rogers and Jackson had to ensure that it emulated several structural features typical of real social networks. “Our contribution is to more or less put the pieces together in the right ways—ways that seem to be empirically realistic,” said Rogers. For example, people in social networks can be connected by a relatively short path via links with co-acquaintances, or friends of friends. Individuals who have been in social networks longer typically have more links, a result of growth over time. Clustering, or cliquishness, is relatively high in social networks, as people meet one another through common neighbors. Social networks also have fairly large numbers of people at both extreme ends of the popularity scale—“like people on MySpace who have 300 friends, and then a significant percentage of people who have very few connections,” as Rogers put it.

The Rogers-Jackson model consists of a virtual space initially populated with individuals, or “nodes.” At each successive point in time, a new node is added to the population and picks at random some others to link to. Having established some initial, random relationships with “parent” nodes, an individual node can then link to some of the others with which it shares some common parent nodes. Over time, these relationship webs grow, a mixture of random acquaintances and friends of friends.

Rogers and Jackson were intensely interested in testing the ability of their model to mimic a variety of well-studied, real-world social networks. “We wanted to get networks from widely varied applications, collected in lots of different ways, because that’s one of our main points,” said Rogers. They found data that described who talked to whom over the HAM radio airwaves and which pages linked to one another on the University of Notre Dame website. They tracked “family trees” of research papers that have referred to Milgram’s “small world” work from the 1960s and looked at a network of economists to see who authored what study with whom. Studies of prison inmates and high school Romeos and Juliets revealed how hard-luck friendships and hormone-charged romances blossomed, respectively.

The researchers systematically varied a number of the model’s parameters to see which ones led to the development of networks that shared features with those real-world networks. For example, Rogers and Jackson varied the ratio of links that were formed randomly versus through a network, akin to assessing an individual’s level of comfort or shyness in dating a stranger from a pub versus being set up on a blind date through a mutual friend. They also varied the overall number of links that a new node is likely to form by tweaking the gregariousness or introversion of a newcomer and the warmth or iciness of the community in welcoming new members.

Adjusting the model to imitate real networks revealed substantial dissimilarities in how networks are formed in different settings. For example, connections among co-authors are more than eight times as likely to be formed at random compared to connections on the Notre Dame website. The ratio of random to network-based meetings is only 0.57 for the Notre Dame Internet data, less than an eighth of the 4.7 found in the network of co-authors.

Figure 1: An example of a small co-authorship network depicting collaborations among scientists in the Santa Fe Institute. The nodes represent scientists and lines between them indicate that they co-authored a paper during 1999 and 2000. The network is split along communities that correspond to disciplines.

Source: M. Girvan and M.E.J.Newman, “Community Structure in Social and Biological Networks,” Proceedings of the National Academy of Sciences, June 11, 2002, 99(12): 8271-8276.

Surprisingly, prison friendships and high school romances, the two most purely social networks, were accurately modeled by an almost entirely random process. In contrast, the Internet links and the web of references to Milgram’s research are heavily dominated by network-based associations.

Having devised a model that explains how social networks might evolve differently and then fine-tuning some of that model’s parameters, Rogers and Jackson sought to learn which structures made “better” networks. They outlined some basic assumptions about benefits to be gained from networks and used the model to determine the varying impacts of different structures on a network’s efficiency in offering those benefits to its members.

“Each node gets some benefit from his position in this network—for example, by acquiring information [and] professional contacts,” said Rogers. “If I then give you a new network with more connections, you increase everyone’s links, that benefit increases, everyone is happier.” This relates directly to a key parameter of the model, reflecting the impact of increasing the expected number of links that a new, more gregarious node may form in a more welcoming community. In this respect, the members of the Milgram-citation network, who averaged 5.0 links, enjoyed the most benefit, while the average high school Casanova was relatively out of luck, averaging only 0.83 links.

Rogers and Jackson also identified the benefits associated with the ratio, R, of random versus network-based meetings. “[In] diminishing marginal returns, more links are better, but once you have 100 links, one more doesn’t do you much good,” said Rogers. “If utility has that property, and your network is described by a higher number R, then utility will be higher.” In this sense, the networks of prison friendship and high school romance were far and away the most useful, as their network structures were almost exclusively reliant on random connections.

In summary, Rogers said, “there are other models that combine these ideas, but in slightly different ways, so they either can’t prove some aspects, or it can be proved that they miss pretty badly.”

Looking to the future, he envisioned some possible avenues to explore and apply this work further. “This is one small piece that would be needed to make it useful on a broad scale. But to do that, you need a few things,” he said. “You need an understanding of how the formation process at the individual level translates into network properties. Our work does that. You need to know what motivates people to make connections in certain ways. How do you connect these pieces?”

“For example,” Rogers concluded, “we might want a job market that allows individuals to exchange information in certain ways, so how might that improve the unemployment rate? We might want to create incentives such that when people make decisions in their own best interest, it will feed into a structure that we can show to be most efficient and useful.”

Further reading:
Milgram, Stanley (1967). “The Small World Problem.” Psychology Today, 2(1): 60–67.