My Old Book Review of Six Degrees

I wrote a review of Duncan Watts book on social networks, for a class and thought I might as well share it here, even if it is a little out of date:

Networks are everywhere.  In the first chapter of Six Degrees Duncan Watts notes that gossip, power outages, epidemics, even properties of the human brain such as consciousness all emerge from the interaction of their constituent elements.  Having provided this motivation, Watts spends much of first half of the book discussing what he knows best, “small world” networks.  In the second half he presents a network perspective for a wide range of topics such as… epidemics, externalities, speculation, social decision making, and organizations.

Like many academics marketing books to non-academics, Watts skillfully weaves his personal story with the science.  His personal story is not only provided to keep laymen interested.  Watts is now a member of the sociology department at Columbia University, but one can’t help but wonder whether he identifies as a sociologist?  How would other members of the discipline respond to a youngster whose PhD is in theoretical and applied mechanics who may never have read Durkheim?  His early collaborators were mathematicians, physicists, and computer scientists lodged in appropriate departments.  Watts though, has become a strong proponent of interdisciplinary science, and he respectfully acknowledges research that has been done in anthropology, sociology, psychology and economics.

His first foray in the social sciences was inspired by the “small world” phenomenon.  When two people are surprised to learn they have mutual acquaintances, someone often offers the cliche, “It’s a small world.”  In 1967, social psychologist Stanley Milgram decided to investigate how small the world really is.  He tasked randomly selected residents of Boston and Omaha with getting a letter to a stockbroker who lived in Massachusetts.  The rule was, they could only send the letter to people they knew on a first name basis.  Amazingly, the letters that reached their destination usually did it in just 6 steps.  This finding was then misconstrued and became the urban legend that there are six degrees of separation between any two people.  Despite the widespread interest in the small world phenomena, little progress was made understanding it over the next thirty years.

Watts got interested in this problem when he was a graduate student in theoretical and applied mechanics.  He and his advisor, Steven Strogatz, had been trying to understand how crickets’ chirping becomes synchronized without a conductor cricket.  Watts surmised that the timing of a cricket’s chirp must be influenced by where it is located and the other crickets it is listening to.  The ability to synchronize may depend on the structure of this network of crickets.  The relationship between network structure and network phenomena such as synchronicity suddenly seemed broadly important, and he was surprised to learn how little mathematical attention it had garnered.  Recalling the idea of “six degrees of separation,” Watts and Strogatz turned to social networks and set about building simple models.  Where Milgram had asked, “How small is the world?” they were now asking, “What does it take to make a world small?”  This reframing of the problem was fundamental to the contribution they were to make.

Watts and Strogatz settled on modeling just two facets of social networks.  One was the “small world” aspect, quantified as average path length (the number of links required to connect two randomly chosen people).  The second was clustering, the extent to which my friends overlap with my friends’ friends.  What makes small world networks surprising is that short path lengths and high clustering are inherently antagonistic.  Paul Erdös and Alfred Rényi rigorously proved that path lengths are short in networks with no inclination towards increased clustering, a random graph in the parlance of mathematicians.  At the opposite extreme, if everyone was friends with all of their friends’ friends, short path lengths would be impossible (in fact social groups would be completely disconnected from each other).  After countless computer simulations, Watts had two important results.  The alpha model captured the small world balance of path length and clustering.  The beta model showed that if a network was systematically clustered, to the point of fragmentation, just adding five random links (edges) halves the average path length.  He then began acquiring and examining network data sets.  Remarkably, Hollywood actor collaborations, the neurology of C. Elegans, the power grid of the Western United States,   interlocking boards of directors and the world wide web are all small world networks.

Next Watts reviews the work by Lázló Barabási, a physicist at the University of Notre Dame.  His major contribution is research on scale free networks.  Sociologists have long been concerned with questions surrounding the number of connections (degree) people have.  Barabási realized the importance of the degree distribution in a network.  The degree distribution of many networks is approximately Poisson but Barabasi showed that the degree distribution of other important networks follows the highly skewed power-law.  The distribution of wealth and the size of cities both fit this model.  Furthermore he showed that this distribution will follow if the future growth rate is linearly related to the present size.  This has obvious implications for these two examples and calls to mind Merton’s Matthew Effect.

Barabási’s book, Linked, is similar to Six Degrees in that is geared to the general public and reviews many of the most important advances in network scholarship.  Do Watts and Barabási overstate their case?  Rather than get bogged down in the semantic debate that is likely to arise from the claim to a “new” science, we should appraise the value of this line of research.  It clearly has potential but Watts himself sometimes alludes to the difficulties in achieving that potential.  Watts’ work is mostly theoretical.  Six Degrees offers a thought provoking network perspective on many topics but little help harnessing the theory in empirical work.  Perhaps Watts has provided ideas that creative empiricists will find ways to exploit, but there are methodological challenges that may prove to be stubborn.

Despite some important exceptions such as Granovetter’s Strength of Weak Ties traditional network analysis tended to take one of two approaches.  One was to focus on the relationship between social structure and network structure.  The other was to view network ties as sources of information or influence.  This means exploring the association between position in a network, and a node’s identity or power.  Watts is right to call attention to the fact that these approaches usually ignored dynamics: changes in the network structure (changes in network connections), and what individuals do on the network (search for information, spread rumors, make decisions).  Network data that captures these dynamics may be harder to come by.

Furthermore, large detailed datasets may be limited by the computational power available.  Even simple computer simulations can be very computationally demanding.    Threshold models of decision making, discontinuous phase transitions and cascades – many of the fundamental concepts in the study of networks are nonlinear.  Proving the existence of causal relationships is always a challenge but these complex systems make a hash of everything.  The measured effect of an independent variable, on average or at the margin, tells us little about the importance of that variable.

Despite a reasonable display of humility and respect, Watts might be criticized for the sociology he leaves out.  Neither space limitations, nor a rush to publication can justify the gaps in his otherwise helpful recommendations for further reading.  For example, Blau, Burt, Coleman, Homans, Laumann, Marwell and Oliver are conspicuously absent from the list.  Perhaps this observation should not be overanalyzed but it does brings us back to how Watts will be received by sociologists and what impact he and scholars outside the discipline will have on sociology.  It is hard for this reviewer to understand how anyone who reads this book could come away uncertain of the value of mathematics for theory development as well as empirical analysis.  Model building can simplify and clarify, enhancing our intuition.  Watts would never argue that all sociologists should drop what they’re doing and begin running computer simulations, just that we should be open to such approaches.  As he points out, “For any complex system, there are many simple models we can invent to understand its behavior.  The trick is to pick the right one.  And that requires us to think carefully, to know something about the essence of the real thing.”  Sociologists know something about the real thing.  That’s why we can’t leave all the modeling to physicists and economists.


3 Responses to My Old Book Review of Six Degrees

  1. eli says:

    hey mike,

    My question for you is really simple but I think important. Let’s suppose I’m convinced by your basic point here, that mathematical modeling of social networks is a useful tool. My question is then the following: what kinds of questions about social networks *cannot* be answered with mathematical models? Surely a thorough analysis of this kind of book would have to explain what kind of research questions are *ruled out* by the modeling methodology? I guess another way of putting this is to ask you: what can a qualitative sociologist — an ethnomethodologist for example — tell us about social relations that is lost here? You gesture in this direction but I want to know more.

    My intuition here is that individual social relationships — just take ours for example — have a kind of qualitative richness that can’t be accurately reduced to quantitative measures; social relations accumulate history and experience over time in ways that can be nonlinear and at any rate are distorted with numerical proxies. For example: I have plenty of other social relationships with other grad students who are sociologically similar to you, relations that have lasted similar lengths of time, that involve roughly similar kinds of interaction rituals, etc; but these other social relationships have quite different qualitative characters. I realize this seems far from the kinds of network effects that these models address, but aren’t there interesting sociological questions about the intricacies of deeply individualized social relations that are excluded by modeling kinds of approaches?

    But that is my own answer to the question I started out by asking and I’d love to have yours instead…and ps: I must say that I don’t really know anything about social network research in sociology — if you have tips for places to start, a few key articles or something, I would love to have some citations too. Just to get a sense of what kinds of things it addresses.

  2. Michael Bishop says:

    Eli, great question! it may be simple to ask, but as you surely know it isn’t simple to answer. Your request for examples is probably the best approach. Network analysis comes in all sorts of different flavors. Some of them are what would be called formal, or mathematical models (see my latest post: but most NA only uses straight forward statistical models (i.e. regression) which are equally applicable in non-network situations. Historically, some important NA has even been qualitative. I sat in on John Padgett’s course, and thought he did an excellent job of touching on many different approaches. You can find his syllabus here:

    I also took Ron Burt’s course, which was excellent, and completely different. He taught us how to do his own flavor of NA including all the nitty-gritty software issues. If you email me I’ll send you a zip with all the pdfs for both courses. Also, is far from perfect, but its not too bad.

    Now for your deep question. “What kinds of questions about social networks *cannot* be answered with mathematical models?” I’ll expand that to include statistical models because I can’t imagine using the former (theoretical tools) without the latter (empirical tools) but even that clarification won’t help me give you a satisfying answer.

    I believe that it is possible and often fruitful to integrate quantitative and qualitative research. I believe that integrating the two, or at least eliminating inconsistent conclusions, should be one of our scientific goals.

    I am fairly steeped in models, but they have given me fairly limited insight into my relationships. They give me relatively more insight into relationships other people have, because I lack the personal experience of those people. Even so, the models would be much less valuable if I couldn’t combine them with the thousands of direct observations I’ve made, conversations I’ve had, and ideas floating around in our culture.

    A couple common observations of advantages/disadvantages… in a given project qualitative researchers have, in some sense, more control over what data you collect and can often change their approach as they go along. On the other hand, purely qualitative research never has data on an enormous random sample of individuals spread out over a large area, with all the advantages that entails.

    The relationship between theory building, and empirical work is always complex, whether a researcher is primarily quantitative or qualitative. We may attempt an idealized approach to research, but so much of what we think and do is unconscious, and not captured by such idealizations (aka models). Right now I’m reluctant to try to say more, especially about ethnomethodology, because I think I’ll wince when I look it over later.

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