Social Structures

by johnmccreery

The title of this post refers to a book, Social Structures, by John Levi Martin (Princeton University Press, 2009). In it Martin adopts a classic stance I associate with Durkheim and Simmel; he attempts to rethink the nature of social relations starting with the simplest possible forms. The novelty of his approach lies in combining mathematical abstraction with awareness of the content, conventional meanings and motives, associated with relations.

Consider, for example, friendship, the paradigm case of social equality. A mathematical model of clique formation assumes three properties: (1) symmetry — if aRb then bRa; (2)reflexivity — aRa; and (3) transitivity — if aRb and bRc, then aRc. It requires friends to form groups in which everyone is friends with everyone else. The geometrical representation of a group of five friends is, then, a star drawn inside a pentagon.

A takes but a little reflection to realize why perfect equality as conceptualized in these terms is never found in actual human societies. As groups increase in size, the number of possible relationships is n(n-1)/2, where n is the number of members in a group. This number increases geometrically and soon exceeds the ability of individuals to maintain so many relationships. It is not surprising, then, that real-world social networks tend to be “small worlds,” here a technical term that refers to sparse networks with local clustering and a few cross-cutting ties that connect local clusters. By definition, however, the individuals involved in the ties between local clusters are no longer the equals of the other members of the clusters to which they belong. They have entered into relationships that are not transitive for other members of their clusters. They have also, it is commonly found, acquired access to information or goods that other members of their clusters do not enjoy. The original all-for-one, one-for-all equality is now compromised by “except of course for my other friend over there.”

At this point, I have only read a bit of the first chapter of Social Structures. But, as indicated above, I am finding it thought-provoking. How would people here feel if I were to rattle on as I read it?


25 Comments to “Social Structures”

  1. A first-time comment this, but, this sounds like something I quite possibly ought to read myself, and given that time is scant at the moment it would probably be of great use to me in deciding whether this is true if you so rattled. So this person would feel informed. Does that help?

  2. Yeah, I would love the rattling. I hadn’t heard of the book and the mathematical approach would not normally be to my taste, so you’d be a valued cultural broker – a translator from an alien cluster.

  3. Please do, this sounds great.

  4. The following are the notes I took as I continued to read Martin. The term “Pajek-speak” refers to the terminology used by the authors of Pajek, the free social network analysis program I use. The program was created by a team at the Unversity of Llubjana in Serbia, and in Serbo-Croation, “Pajek”means spider.


    p. 30

    “There are…two options for facilitating structure formation via the aggregation of cliques. The first is to introduce a second or perhaps also third type of relationship…. The second is to abandon the rigid transitivity requirement. Thus we allow some person(s) to be in more than one clique at a time.”

    In Pajek-speak, the first approach identifies bridges, relationships between clusters. The individuals who form bridges may become cut vertices, i.e., their removal breaks larger into smaller clusters.

    Martin’s second approach suggests a somewhat different idea, individual actors who function as welds, by belonging to more than one group. They resemble cut vertices but, given two groups, are members of both groups instead of forming a new type of relationship with someone in another group.

    p. 30-31

    “Allowing for welds not only requires a sacrifice of strict transitivity, but of strict equality as well: we see the transformation of a wholly horizontal to a potentially vertical structure. The verticality comes in the fact that welds are qualitatively different, and where there is difference there may be inequality.”

    That “may be” is important. in an organization being a weld may result in faster promotion, higher salary, or better evaluations (Burt (1992:2004). It may also mean having more work to do. Consider a secretary shared by two departments. In either case, difference introduces the potential for inequality.

    p. 31

    “In [the] simple example of a single weld, this verticality can be expressed as a categorical distinction between welds and regular nodes. But as we attempt to build a larger structure, we are likely to find this clear distinction dissolving into a differentiation that is only a matter of degree, not of kind. At the same time, we find that the structural components…lose their clear identity. Instead of a clear structure, we have only some degree of tendency toward structure—some degree along a continuum stretching from total randomness at one end to separate cliques at the other.”

    Martin describes these less clear structures as webs.


    I am getting ahead of Martin’s argument, but I find myself thinking about Chie Nakane’s Japanese Society, in which she draws a distinction between “frames” (particular, typically localized groups) and “attributes” (categories that cut across group boundaries). The distinction and the proposition that societies differ in whether frames or attributes are more important was suggested by Nakane’s fieldwork in India. There she was startled to find daughters-in-law arguing and winning fights with their mothers-in-law, something unthinkable in Japan. The families in which a woman is born and into which she marries are not, of course, cliques. They are far too highly structured internally for that. However, if, for the sake of argument, we ignore the internal structures and imagine them as cliques, we see married women functioning as welds. Depending, however, on whether frame or attribute is dominant, the weld will be weak or strong. Strong frames mean that a Japanese daughter-in-law is cut off from her natal family and has no one to support her if she argues with her mother-in-law. Strong attributes mean that an Indian daughter-in-law retains strong connections with her natal family, who are allies in her battles. Which brings me back to Martin and the question of how to deal with strong versus weak welds; I wonder if he addresses that.

  5. A mathematical model of clique formation assumes three properties: (1) symmetry — if aRb then bRa; (2)reflexivity — aRa; and (3) transitivity — if aRb and bRc, then aRc. It requires friends to form groups in which everyone is friends with everyone else.

    This is a strong requirement; it is the very definition of of an equivalence relation. Given say a population P of individuals, and letting cliques C(i) be subsets of P, where for any pair a and b in C(i), (1),(2), and (3) are satisfied. It follows from this that if an individual p is in clique C(i) and in clique C(j), then C(i) = C(j). P must consist of m completely disjoint cliques. A clique is an equivalence classes of individuals– (1),(2), and (3) guarantee that.

    I don’t think that this is justified even for troops of chimpanzees! But it is very much a reasonable place to start. I’ll take a more careful look at your notes later. Interesting stuff as always.

  6. Jacob’s elegant summary of Martin’s math points to the central thrust of Martin’s first argument. To be able to combine cliques to form a larger network requires relaxing the mathematical conditions that define perfect equivalence. When cliques are joined, those who become the welds or form the bridges between them are no longer equivalent to the other members of the original cliques. They are different, and with difference inequality appears. Inequality is, thus, inevitable in any social network too large to be a self-contained clique. How great the inequality is and the details of its content may vary; but the dream of perfect equality is demonstrably an illusion.

    In the next few pages of chapter 1, Martin introduces an important distinction between spatial and relational networks and how their logics differ. In spatially defined networks the likelihood of any two individual actors being related is proportional to the distance between them. In the limiting case, society is made up of groups whose members can only form relationships with members of immediately adjacent groups. In relationally defined networks, the likelihood of two individuals being related is proportional to the number of others with whom they share relationships. In this case, individuals are free to move about, and it takes only a few travelers to create a small world in which a small number of connections link everyone to everyone else.

    Extrapolating this argument to the East Asian societies with which I am familiar, most peasants may never leave their villages, but a relatively small number of travelers (merchants, entertainers, officials) create paths by which, other things being equal, the peasant’s complaints can reach an emperor, the products of his fields or craft can travel from one side of the empire to the other, and his life may be transformed by rumor or disease.

  7. I have to admit that the mathematics here is partly beyond me, but the extrapolations from it still interest me, so thankyou for the rattle. You might find the links off this post of mine interesting, as well as finding it useful material with which to assess my ignorance; it has some very similar conclusions to your last paragraph there.

  8. Just to quibble slightly because the language of math seems to demand precision, there are important distinctions among identity, equality, and equivalence, especially in a social context. So, no one is identical to me; I am equal to all other democratic citizens; and although we are not workplace equals, the janitors in this building may fairly assert that we are equivalent to the overall functioning of the institution.

    It also may be important to distinguish formal and substantive equality (although both will usually turn out to hinge on what principle of substance is selected). The U.S. Supreme Court has just recently decided that corporate individuals like drug companies and unions are formally equal to all other individuals for the purpose of free speech. Substantively, of course, corporate individuals are in a position to purchase much more ‘free’ speech than the average person.

  9. I really like the frames/attributes analytic as a way to see how people in ‘the same’ situation can actually be quite differently situated.

  10. @Jonathon

    Much of the math involved in this stuff is beyond me, too. One way I’ve found to tackle it is to come up with images that capture what the formulas imply. So, for example, I am trying to get a handle on this distinction between spatial and relational networks. To describe spatial networks, Duncan Watts talks about caves. I find it more compelling to think about a pearl necklace. Imagine that each clique (cluster) is a pearl. The critical point is that each pearl is directly connected only to the pearls beside it on the string. And, if there is no other way to reach another pearl but to follow the string, you can’t get from pearl No.1 to pearl No. 20 without passing through pearls No. 2-19 along the way. In contrast a relational network is like a cat’s cradle (the kid’s game twisting string around your fingers). As the fingers move the string around new relationships appear. At the start the little finger on one hand is not directly connected to the index finger on the other hand. A few moves later it is. I don’t know if these particular images work for you. But I’m sure that if you get in the habit of constructing an image to capture the essence of a formula, the math gets a lot easier to grasp.

    P.S. At my first Sunbelt conference, one of the most fascinating papers I heard was from a bunch of German historians who were doing a network analysis of the correspondence surrounding the battle between Newton and Leibniz over who had invented the calculus. The individuals writing to each other were the nodes, the letters were the lines connecting them. What they were excited about was discovering people who were central figures in the network who had not yet received much attention from historians. More grist for the Ph.D. mills!


    there are important distinctions among identity, equality, and equivalence, especially in a social context

    Of course there are. That is why I would like to draw attention to the difference between two uses of mathematical models. One approach assumes that the model is True or False and counts uncovered questions or data as evidence against it. The goal is to accept or reject the model. The other, the one that Martin is using, assumes that the model is only a model, a simplification that is easier to think about than the whole messy reality it purports to represent. Then, when the model fails to answer all the questions or cover all the data, the next move is to tinker with the model, to relax or alter assumptions to see if a closer fit is possible.

    Imagine an ideal marriage. Note that few marriages are ideal. What then? The first approach says, “Not ideal, get divorced.” The second says, “Let’s work on this.”


    I, too, think that the frames/attributes analytic is a wonderful thing. Too bad it got buried in Japan studies.

  11. Great distinction, John. The marriage analogy is interesting. I don’t think any optional relationship can or should last long with work as a functional condition. A bit of wiggling to find the right fit, sure. But when relationships are optional the right thing to do when a pretty low work threshold is reached is to wish each other well and go find someone who’s less work. If we’re stuck with people it’s a different story, of course, and so too with models. These among many other sorts of conditions that might affect whether we think of particular models as something to accept/reject or work on.

  12. There are, of course, models and relationships where the fit is just plain lousy. I wouldn’t start to model the flight of a bumblebee with a hypothetical bowling ball, and I do find as I get older that I am less willing to invest time in relationships that aren’t working out. That said, and taking the risk of sounding stuffy, I can’t think of anything I truly value in my life that hasn’t been a lot of work and occasional pain to get good at.

  13. Returning to Martin’s distinction between spatial and relational logics, I find a lot to think about here. As Martin himself notes, space is a powerful metaphor in social theory. Thinking of relationships as close or distant, up or down, thick or thin seems so natural that, as philosopher turned anthropologist, I have to explore the assumptions behind the metaphor.

    The fundamental assumption of spatial logic is that distance matters. The space in question may be an analytical space, but the rules that apply are the same as those in physical space. The likelihood and/or strength of a relationship is inversely proportional to the distance between the points.

    In contrast, a relational logic starts with relationships that, like Superman, can leap long distances at a single bound. Another image that comes to mind is the warp drive used for faster-than-light travel in science fiction. A familiar explanation starts with a map, a two-dimensional representation on a piece of paper where, if you have to keep your pencil on the paper to get from one point to another, the distance may seem long. Then, some smart ass picks up the paper and folds it. The two points are now right beside each other. Now space becomes an origami (Japanese folded paper sculpture). The question is no longer the distance required if we follow the surface of the paper. The question now is how many folds you have to cross, leaping from one point to another. Then, coming back to Martin, the likelihood and/or strength of a relationship is proportional to the number of other points on the paths that connect the points in question.

    Does this make sense to anyone but me?

  14. “As groups increase in size, the number of possible relationships is n(n-1)/2, where n is the number of members in a group. This number increases geometrically and soon exceeds the ability of individuals to maintain so many relationships.”

    –First a (very) small quibble: n(n-1)/2 is the lower bound on the number of possible relationships for a group of size n, provided we are willing to differentiate two relations which happen to have the same extensions in the population.

    –There do seem to be many practical and cognitive constraints operating here. I would like to point to a paper by Dwight Read published a few years ago. The gist of his thesis is that the relational nature of kinship allowed hominids to overcome the very problem you describe. To quote, from the paper’s abstract:

    “…Genealogical relations transcend the limitation of biological kinship as a basis for group coherency, but the combinatorial complexity of all possible genealogical relations becomes problematic with increase in group size. The latter was resolved, it is argued, through the construction of a computational system—a kinship terminology—whose conceptual complexity is independent of the size of a group.”

    Read, Dwight. 2003. The Emergence of Order from Disorder as a Form of Self Organization. Comput. Math. Organ. Theory 9, no. 3: 195-225.

  15. “Now space becomes an origami”


  16. What Dwight is talking about is the cognitive processing problem. Humans classify so as not to have to deal with an overwhelming number of unique combinations of elements. This is an interesting problem in itself and was recognized at least as early as Edward Sapir’s Language, where the great linguist observes that all language is conceptual since human beings cannot keep track of all the particulars we encounter. William James had talked about how we see the world as discrete objects instead of a “blooming, buzzing confusion,” and James himself may have been drawing on Kant’s response to David Hume, in which Kant asserts the necessity of categories to organize sense perceptions. To the best of my knowledge, this conversation begins with Plato and the Forms, of which the everyday world is only an approximation. What is interesting about Read’s contribution is that it is framed in a computational manner; so instead of mysterious particulars with unfathomable depths to plumb, we have a combinatorial explosion of a limited set of elements to organize.

    What Martin appears to be talking about isn’t the cognitive processing problem per se. It is rather the difficulty of maintaining more than a relatively small set of social relationships, i.e., having the time and resources to engage in the necessary interactions to keep them going.

    Where you are absolutely right, of course, is to point to the thinness implicit in the assumption that individuals are linked by only one relationship. One weakness of network analysis to date has been the difficulty of addressing multi-stranded relationships with the existing, still relatively primitive, mathematical models. I think, for example, of my relationship to my friend Jim Ennis, whom I met when he was a senior at Middlebury College and I was a new faculty member. Jim wound up as a full professor of sociology at Tufts and a valuable mentor when I became seriously interested in network analysis. But to understand how important was to me that he was living three miles away in Cambridge, MA, when we were there last year, it would help if you also knew that we have kept in touch for years. He met my daughter when she was still an infant and attended her wedding. His sons Noah and Sam introduced me to EV Nova, a computer game on which I once spent altogether too much time.

    Also, one of the theories I picked up in graduate school was Max Gluckman’s proposition that ritual is more elaborated in societies with thick, i.e, multi-stranded relationships in which the same people interact with each other in several different roles. A shaman, for example, might just be your uncle when you are hoeing a field together, someone totally different when doing his shamanic thing; the ritual surrounding shamanism would serve to mark the difference.

  17. You are right. They are not the same. I do not see these issues as entirely separate though.

    Read notes that one can take a pair of kin terms and calculate a third kin term, its relative product. The three terms map onto the genealogical space in a regular way, but the kin-term product allows one to leap across genealogical distances using simple calculations instead of by a full traversal of that space.

    Yes, this is a way to address issues of cognitive load. But it may be more than that: a way of generating or establishing relationships on the cheap; Kinship terms have, in some complicated way, associated with them certain roles and expectations, kinds of relationship schemas, which of course may be contested or ignored, but do establish a kind of minimal set of expectations. Because you are the son of my great uncle, you must be my cousin, and I yours, and thus, we will treat each other as cousins do. I needn’t have ever met you to know what, in some minimal sense, our relationship is. I might, for example, be able to go to your village and claim the rights of a kinsman on the basis of such a calculation, without ever having before made the effort to develop a relationship with you or anyone else in your village. In this way, kinship can cut across groups. Of course, kinship isn’t the only way to do this.

  18. Y’all may also be interested in (or already know) the work of Roger Gould; his last book, published posthumously, was “Collision of Wills: How Ambiguity About Social Rank Breeds Conflict,” and his first book was about the Paris communes, wherein he argued, IIRC, that proximity was more important that economic status.

    He was a good friend of mine, so I heard a lot about the Collision of Wills research as he was conducting it. (He was only 39 when he died.)

  19. @Jacob

    I might, for example, be able to go to your village and claim the rights of a kinsman on the basis of such a calculation, without ever having before made the effort to develop a relationship with you or anyone else in your village. In this way, kinship can cut across groups. Of course, kinship isn’t the only way to do this.

    This is also, by the way, one of those places, whether Nakane’s frames and attributes are relevant. Kinship is an attribute, and where attributes are strong can work in just the way you describe. Where frames are more important, the effectiveness of attributes like kinship may be significantly weaker. This is one of the major differences between Chinese families and Japanese households and the Chinese and Japanese institutions that build on these models.


    Could you tell us a bit more about the respects in which proximity was more important that economic status?

  20. A bit more on spatial logic.

    Martin points out that spatial logics may involve multiple dimensions. One might, for example, map positions in a space defined by (a) geographical and (b) social distance. Assuming Cartesian coordinates, in which the two axes intersect at right angles, the combined distance between two points can be defined as the hypotenuse of the right triangle, the line that joins (g1, s1)=the location of one point and (g2, s2)=the location of the other point. Its length, as determined by the Pythagorean theorem, is the square root of (g1-g2)*2+(s1-s2)*2=the sum of the squared differences. The testable proposition here is that the tendency of the actors represented by the two points to form a relationship is inversely proportional to this distance. That is, closeness on both dimensions increases the likelihood of the relationship; distant on both relationships decreases the likelihood of the relationship. The more interesting possibilities, however, are those generated by distances that are high on one dimension and low on the other. Martin suggests that the longer of the two distances will be the determining factor. Thus, for example, service staff (wait staff, janitors, etc.) may spend considerable time in geographical proximity to people from whom they are widely separated in terms of social distance. The likelihood of their forming strong relationships with those they serve is low, albeit higher than those whose services are performed at a greater geographical as well as social distance (migrant farm laborers or sweatshop employees in a different country, for example).

    Martin also mentions, however, that “spaces are not necessarily isotropic.” “Isotropic,” the fancy word at the end of this phrase means uniform in all directions. Real spaces are frequently lumpy. Thus, for example, in a study of neighbors living in university married student housing, those who shared a stairway were more likely to become friends than other next door neighbors who did not share the same stairway. Shared paths through the non-isotropic space of the housing complex were a better predictor of relationship than physical distance between apartments per se. In a more exotic example, we can easily envision people (hill tribes in Southeast Asia or in central New Guinea, for example) who live in villages strung out along valleys separated by high mountains and are, thus, more likely to interact with people from the same valley than those from they are cut off by the mountains between them.

    Speculating along these lines leads me to the work of Kevin Lynch and others on mental maps of cities and how these vary depending on whether people spend most of their lives in the same neighborhood or travel outside of the neighborhood for school, work or recreation.

    Lots to think about here.

  21. I think in the Paris book it was geographical proximity–i.e., neighborhood and one’s associates were more important than social status. For the other book, he did a lot of research in Corsica, IIRC, and read a lot of letters. I have “Collision of Wills” but not the other one. The google can provide more info, too.

  22. Incidentally, I’ve thought about the mental maps of my city quite a bit, as where one “can” go has a whole lot of racial subtext to it. Arnold Hirsch’s “Making the Second Ghetto: Race and Housing in Chicago, 1940-1960” is quite good as background on that, and I have a wealth of anecdata, given the things I’ve done and places of lived in the city.

  23. @Narya

    Please share your anecdata.


    Moving on now from cliques to “Webs and Wheels.”

    Outside of small worlds, “people can also be reasonably close if some special persons have structural positions that facilitate the indirect connections of others. In the simplest such structure [a hub and spoke or star structure], there are as few ties as is compatible with a connected structure—all persons except the special person have ties only to this person….We often see hub structures when a group has gelled around a particular charismatic individual. The group only exists so long as the hub person retains his or her charisma or activity.”

    I note as I read these sentences that the hub-and-spoke model plays an important role in the mathematics of social network analysis. It is the limiting case in which, given a structure with n actors, one has the maximum number of connections with others, n-1, while all of the others have the minimum possible number of connections, 1. These conditions make the hub-and-spoke model the starting point for calculations of density and degree in more complex structures.

    Martin continues,

    “Of course, many charismatic leaders show a marked disinclination to have anything to do with one another, and hence their structures do not form the building blocks for larger structures. But in other cases, hubs establish relationships not only with their own spokes but also with other hubs, thereby bringing their spokes into indirect connections.”

    It is hard to imagine, I observe, the emergence of a large organization without these indirect connections, and the core of any organization is likely to be composed of people who have, at least in part, their own followings. This, in turn, will generate tensions between the capstone charismatic leader and subordinates who may break away from the movement.

    I am also conscious when I write “at least in part” that, even in groups formed around charismatic individuals, events in which group members participate collectively are likely to lead to formation of cross-cutting ties. Indeed, I have speculated elsewhere that cult formation largely depends on the mutual reinforcement of followers’ shared belief in the leader. Without this mutual reinforcement, everything comes down to one-on-one encounters between leader and followers.

  24. Sharing of the anecdata is on my to-do list; remind me if I fail.

    W/r/t the hub thing, I know that I have tended to be a hub (yeah, big surprise, I know), and, the more I add odd Next Things to my life, the more wheels on which i get to be the hub. In some ways, the hub-ness ends when my participation ends, but there is another way in which I have been a kind of meta-hub, mostly having to do with my effect on women in their 20s. More anecdata, which I also will share.

    But both at a different time-place, because I must sleep.

  25. Just to let everyone know. I’ve got a paper based on the qualitative component of my big research project due at the end of March. I’m suspending my reading of Martin until I get that done.

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