Duncan J. Watts. Six Degrees: The Science of a Connected Age. New York: W.W. Norton & Company, 2003, 368 pp., $27.95 paperback.

 

Review written by Phillip Bonacich, bonacich@soc.ucla.edu, University of California, Los Angeles

 

Six Degrees: The Science of a Connected Age is a useful introduction to the impressive work accomplished by Duncan Watts and his colleagues since Watts’ last book, Small Worlds: The Dynamics of Networks between Order and Randomness. In Small Worlds, Watts introduced sociologists to an exciting set of new mathematical tools for the analysis of the small world phenomenon. The most impressive part of the earlier book was its careful and productive definition of a small world--one in which the local density of choices was high (indicating cliquishness) but paths were short (as in a random graph). This was coupled with the establishment of methods for continuously disturbing locally dense graphs until they became random and the finding that density declined only after path lengths declined, so that small worlds could exist even without “intelligent design.” After exploring this phenomenon, Watts suggested some of its implications for the spread of disease, evolution of cooperation, and other network phenomena.

 

In his new book, Watts describes in a non-technical language the work that he and his associates have done in extending these models. Although the original papers in physics journals are worth reading, they are addressed to an audience more interested in percolation theory and Bose-Einstein condensation than in the diffusion of innovations. Watts is a bridge. The book is full of interesting and thoughtful suggestions for how these models can shed light on complex and mystifying human network behavior.

 

Chapters 1 through 3 review the models covered by Small Worlds: features of the two apparent extremes--local solidarity and macro-level fragmentation on the one hand and random nets with their short paths but complete lack of local clustering on the other--can be synthesized in the form of small world graphs with the introduction of a few random connections. Chapter 4 combines a shrewd criticism of Barabási’s scale-free approach (2002) with an exposition of what Watts and his colleagues have recently accomplished. An affiliation network is a bipartite graph of people and the groups to which they belong. In a random affiliation network all connections between individuals and groups are assigned randomly. Watts shows that the implied networks between individuals based on their co-memberships are small world networks--both clustered and with short paths. Watts also shows that this model fits some data on interlocking directorates quite well.

 

Chapter 5 presents some models that Watts and his colleagues have developed to study how individuals can navigate their small-world networks. The short paths available in these networks are more useful to individuals searching for jobs if the networks can be used to connect to particular others. This chapter presents two models of conditions under which these short paths can easily be found. Both models show that the searchability of networks is affected by the number of ways individuals categorize one another, and that the use of too many or too few categories impedes search. This suggests that societies with too few or too many sources of identity may both be difficult to mobilize on the basis of new identities.

 

Chapter 6 describes the ways in which epidemics can spread in different types of networks. Standard models of epidemiology have been developed using the well-known mathematics of random (“well-mixed”) populations. In random networks, diseases become epidemics only if infectiousness reaches a critical point; otherwise they disappear. In scale-free networks, on the other hand, there is no critical point and diseases that are not epidemics may not disappear. Hubs are easily infected and in turn infect many others. The last part of the chapter summarizes work on the vulnerability of networks of different design to failure and to attack.

 

Chapters 8 describes some exciting ideas about the dynamics of information cascades. Particularly interesting is Watts’ outline of the two radically different ways in which information cascades (and social movements) can fail: either through the sparseness of the network, in which case a movement will fail to diffuse because of infrequent contact, or because the network is too dense, in which case the movement fails because members are too well anchored in other relationships to be susceptible. Chapter 9 suggests a solution to the vulnerability of centralized hierarchical organizations to the loss of core members: “multiscale” networks that are “ultrarobust” with respect to failure.

 

Sociologists will have to adjust to Watts’ orientation to models and data, an orientation perhaps one more characteristic of physicists. His primary goal is to devise a simple but plausible model from which interesting and suggestive consequences can be drawn mathematically. There is a surprisingly casual attitude about the realism of the model--the tradeoff between fidelity and simplicity is much more in favor of the latter than we are used to. These are the features--barely plausible assumptions accepted because of their mathematical power--that many sociologists find irritating in economists: Yet, they are not irritating in this book. The reason is, I think, that Watts’ models are not dogma but are clearly tentative and under construction.

 

Another function of Six Degrees is to make clearer the distinctive contributions of the Watts camp and that of Albert-László Barabási, described in his recent book Linked: The New Science of Networks. Barabási and his colleagues have developed the mathematics of scale-free networks, while Watts has elaborated small-world networks. The unbiased reader will conclude that both scale-free and small-world models are useful under different circumstances and that both are welcome additions to our toolkit. Small world models are appropriate when the underlying reality is one of tightly clustered groups connected by a small number of weak ties. Ordinary social life, in which individuals belong to tightly clustered groups of friends and relatives supplemented by acquaintances, may be better fitted by this model. In scale-free networks, on the other hand, geography or location within a social space is not the primary factor determining connections. Rather, actors wish to connect to well-connected others. Thus, the WWW, where popular sites become even more attractive, is well suited to analysis by scale-free networks.

 

Some social network researchers have faulted Watts and Barabási for ignoring previous work in the area of social networks (for example, see Pool and Kochen 1978 for an intimation of the small world model). Part of this impression may well be due to the arrogance of physicists, but I think that there is a more important reason. Their new work concerns the emergent properties of extremely large networks. Examples include the internet, collaboration networks among researchers, networks of disease spread, and large ecological networks. Until recently, data on networks of this size were not readily available. Watts’ (and Barabási’s) models will be useful to only some network researchers. They will not be useful, for example, to those who study classroom sociograms, exchange networks, organizational and individual networks in small communities, or perhaps to any researcher whose network can be printed in the pages of a journal.

 

We are lucky that physical scientists and mathematicians have become interested in social networks. Of course we feel slighted; not all of our contributions will be noted. But this is the cost of moving onto a very much larger intellectual stage. The mathematical tools (generating functions, percolation) are powerful. The usefulness of these models ultimately depends on social scientists with substantive interests. There are suggestive applications to a variety of phenomena, from social movements and the diffusion of information to the structure of interlocking directorates. What is needed now is the complementary expertise of other sociologists to make use of these models. The invasion of the physicists will not stop. If we are lucky they will continue to offer new mathematical and conceptual to enrich the study of social networks.

 

Bibliography

Barabási, Albert-László. 2002. Linked: The New Science of Networks. Cambridge, MA:

Perseus Publishing.

Pool, I. de Sola, and M. Kochen. 1978. Contacts and Influence. Social Networks. 1:1-51.

Watts, Duncan J. 2003. Six Degrees: The Science of a Connected Age. Cambridge, MA:

W.W. Norton.

Watts, Duncan J. 1999. Small Worlds: The Dynamics of Networks between Order and

Randomness. Princeton, NJ: Princeton University Press.