We have seen three pure forms of organization—three ways in which structure elegantly emerges from relationships. We found a tendency for certain types of relationships to move toward one of these forms, although there were always possibilities for alternate structures due to inherent ambiguities or polyvalences in the content of relationships. In chapter 2, considering the ever-popular relation of “choosing as a friend,” we found that choices could be organized according to a “popularity tournament,” in which those already disproportionately chosen are disproportionately likely to be chosen by others. Here the organization is subjectively understood in terms of the qualities of individuals—the social organization of relationships seems to disappear into the bodies of persons. Following Harrison White (1992) I will call this mode of organization “selection” (see figure 5.1).1 Not all asymmetric relationships will be organized in such a fashion, but the range of environmental conditions under which the popularity tournament can emerge is rather broad, and hence it is both common and robust.
In contrast, wholly mutual relationships are more compatible with the development of isolated dyads. Again, following White, I will call this “commitment.” Perhaps the best example of this is marriage, although we have seen that mating systems can vary along related dimensions as well.2 Moving from the corner of the lower left point of figure 5.1 (“Commitment”) upward we might And the rating and dating system (one that organizes by individual characteristics as if it were a choice process) about halfway between “Commitment” and “Selection.” Equivalently, one could imagine moving from normal nominations of friendships toward this same structure by starting from pure “Selection” (the very top point of the figure.) and moving downward and to the left by increasing the degree of commitment (e.g., “going out”).3
Figure 5.1. Three pure forms
Finally, inherently antisymmetric relationships are more compatible with the single “dominance” order (the lower right point of figure 5.1). This structure is in many ways isomorphic to the pure popularity tournament, but participants are likely to have a subjective conception of the transitivity requirement. It may be somewhat perplexing if someone who admires someone who admires me does not, himself, admire me. But it can be seriously troublesome if someone I dominate dominates someone who dominates me. We saw such a necessary focus on triadic relationships emerge with the antisymmetry of relationships when we began with the case of mating structures (down and to the left). This is the difference between restricted and generalized exchange—moving further right (toward “Dominance” from “Commitment”) means renouncing some of the emphasis on dyadic exclusivity (and consequent symmetry) and moving toward antisymmetric relationships that spread out across the group.
Thus there are pure forms of organization, but human beings revel in impurity. We have seen some of the reasons for this in the previous chapter; let us review why, for one, dominance orders are so rare. We found that provisioned primates tended to form dominance orders because everyone wanted to be near the bananas. Absent the bananas and some would just walk away, leading to an inability to determine which animal “outranked” the other. It is not that human beings are never “caged” so that they have to encounter one another nor that they are never left to fight things out without a possible change of venue. But it is the case that these two rarely coincide. That is, the people who have the bananas also have procedures for conflict resolution other than potentially unbounded physical violence. Where there is no appeal other than through the fist, there usually aren’t enough bananas to keep us in one place. As a result, when there are antisymmetric relationships of agonism, they are not complete—A and B need not interact and establish a relationship of dominance simply because there is some C to whom both are connected via a dominance relationship.
And this is why the resulting relationships are messy—they move away from the position on the lower right (“Dominance”) of the above figure, introducing the voluntary selection characteristic of the popularity tournament. And this voluntary nature leads to the possibility of horizontal as well as vertical differentiation.
In the previous chapter, we found little evidence that dominance orders were a prominent form of social organization among human beings. This was not because we found human relations to be uniformly egalitarian or mutual. On the contrary, we examined cases of hierarchical relations, but found them falling short of a pure order for two reasons. The first was that a dominance order requires displays of submission that are unlikely to be emitted if people have a good chance of having their conflicts resolved without ever-escalating physical violence. The second, and related, reason was that people are insufficiently caged—with an exit option, it is possible for two persons never to enter into the bout that would lead one to be victorious over the other.
Let us, then, relax our understanding of the antisymmetric relationship in question, and see if we find similarly relaxed structures appearing. That is, when A can dominate B and B cannot remove himself or somehow avoid A, A will dominate B. If B has the opportunity of exit in some sense (which may or may not involve physical removal), then even if A can dominate B, A may not end up having this relationship with B. There is now some element of B’s choice. In this case it seems that we are moving more toward something like “influence.” When we say that B is influenced by A, we often mean an antisymmetric relationship that is based on B’s recognition of A’s superiority, but a recognition that must be freely given.
To anticipate, we can imagine bringing horizontality into our structure simply by beginning with a dominance order and then allowing some persons to politely decline to be dominated. The overall ordering of persons from low to high does not change, but there is a dramatic change in the social structure—that is, both in the form and in the content of the set of relationships. In terms of form, we move from a simple line to a more complex structure with both horizontal and vertical differentiation. In terms of content, we move from despotism to consent, from dominance to influence.
We will, as before, reach this conclusion inductively, by starting from relationships of influence and seeing how they tend to be structured. Here we shall use the term to denote only one person’s changing her cognitions after encounter with those of another person, and not how one person gets another to do what the first wants. While influence will serve as the focal point for these analyses, it is worth emphasizing that the class of relationships that will take on similar structural forms is much larger and ill-defined—there is a degree of influence in many relationships that would be called friendship or acquaintance—and indeed these structures may be considered the default for many informal relationships. They are characterized by two environmental conditions—vertical differentiation and choice—that are much more widespread than those required for cliques, exchange structures, or pecking orders. And the resulting structures are, as we shall see, particularly weak in contrast to seemingly similar structures.
Interpersonal influence has been a fundamental topic in sociology for so long that it is hard to imagine that there is anything new to say about it. When it seems that novelty is being introduced, it is often a result of the term influence being used differently, for the word can refer to a number of relationships that are substantively different, even if they shade into one another. On the one hand, we sometimes use the word influence to indicate implicit resistance. Thus if campaign donations allow corporation X to influence some politician, we assume that in the absence of this conduit of influence, the politician would not wish to act in a manner advantageous to X. At the opposite extreme we may imagine recipients of influence to require no incentive to overcome such implicit resistance. For example, we might consider much of peer influence to occur in circumstances in which it is of the greatest interest to some person to ascertain what others are doing or thinking and to act or to think accordingly.4
Second (and partially independently), in some cases we do not care from whom we get some information or opinion that can influence us. In other cases, however, we will only accept influence from someone who has a certain position, or who has a certain relation to us. This may be because others are not considered qualified, or it may be because certain positions bring with them some compulsive or persuasive power (e.g., priesthood), or because we simply do not trust some people.
If we dichotomize each of our dimensions (degree of resistance and degree of selectivity of source), we can imagine four types of influence (also see McGuire 1999: 79). Most ideal-typical categorization schemes are less than perfect; the types offered below are perhaps even less so than usual, and one can probably think of cases that are not illuminated by this scheme. Nevertheless, it can serve as a useful starting point for exploring the creation of differentiated structures. We thus divide relations of influence according to whether the recipient of influence is assumed to be implicitly resistant to adapting his or her behavior or beliefs (“reluctant”) as opposed to being pre-disposed to adapt (“eager”); and according to whether the recipient is more likely to be influenced by some persons than others (“selective”) or equally liable to be influenced by all classes of others (“open”) as in table 5.1. Each resulting combination has been given an identifying name, though these distinctions do not correspond exactly to current social-scientific usage, in which these terms are used somewhat differently, and in many cases are considered interchangeable.
In the history of structural analysis the first relationships formalized were those pertaining to contagion, as these are the simplest. But this simplicity implied a lack of structure and so shed little light on the issue of what forms sets of influence relationships might take. Still, it was exciting to imagine taking principles first worked out in epidemiology and applying them to other cases in which persons are only distinguishable by their possessing or not possessing the transmissible unit in question (e.g., knowledge or fashion). Because of this background, I will refer to a person possessing whatever is to be transmitted the “carrier” or the “infected.” (Here one may follow the treatment of Coleman [1964: 42ff], though he considers this a diffusion process and uses the word contagion differently. My notation is also somewhat different at first.)
Given any two people, A and B, the probability that A will transmit whatever is in question to B may be considered to be the product of (1) the probability that A and B will encounter one another; (2) the probability that A is a carrier; (3) the probability that B is not; (4) the probability of transmission from a carrier to an uninfected person in any contact. If we call the first of these p and the last q, we find that the most likely number of transmissions in any period is pqx(N − x) where x is the number of carriers and N the total number of persons.5 This leads to the familiar S-shaped logistic curve such as the one graphed in figure 5.2. Early sociological studies finding such a temporal pattern led to excitement that the contagion model would be able to explain many cases of interpersonal influence.
But are all forms of interpersonal transmission of information contagious? One famous example of the use of this model—deliberately adopted from epidemiology—was Coleman, Katz, and Menzel’s (1957; also see 1966: 71) study of the adoption of the use of a new drug (an antibiotic) by physicians. This model, as we recall, assumes no social structure—contacts are random. Although Coleman et al. (1957; 1966: 136) believed that there was some structure to the set of doctor-doctor relationships, they proposed that there were two classes: there were some who were connected, and others who were isolated. The contagion model seemed to fit the connected doctors (the isolated could be treated as having a constant propensity to adopt the innovation).
Coleman et al. (1957; 1966: 57) actually had a second, and partially contradictory model, namely, that physicians varied in some continuous degree of “centrality” and that those “in the center of things” tended to adopt the new drug sooner than the less popular. The more peripheral would be influenced by the central, leading to the hub and spoke model we saw in chapter 2 and to which we will return shortly.6 But they liked the contagion model because they seemed to see evidence in the pattern of growth of adoptions over time.
Figure 5.2. Simple contagion
Unfortunately, it turns out that many conceptually distinct processes are compatible with the basic logistic curve.7 Indeed, the results are even compatible with an absence of any interaction at all.8 Anything that goes from around zero to some upper bound without decreasing is pretty much guaranteed to follow a similar curve.9 The actual process leading to change—and the presence or absence of structure—is not visible in the aggregate data.10
Thus we cannot conclude that we have a structureless (“open”) system merely because we see adoptions over time following an S-shaped curve. Further, even if we assume that the system is open, such data cannot help us distinguish between the cases in which recipients are eager (contagion) as opposed to those where they are reluctant (pressure). In this latter formulation, we propose that recipients have some resistance that may be broken down when they realize that everyone else is indeed “doing it” too. In the simplest case, we can imagine that there is some form of psychic pressure toward conformity that leads reluctant recipients to switch when they find themselves to be increasingly isolated and different. If so, we might expect the probability of successful transmission to be an increasing function of the number of persons who have already become infected. It seems that such a dynamic is frequently expected, at least in the United States, where parents assume that their children need to be explicitly instructed not to jump off the Empire State Building, even if everyone else does it.
It is easy to generalize the equations for the contagion model with the addition of one or two parameters that characterize the susceptibility to such pressure. For example, we can make the probability of one person adopting (given exposure to someone who has already adopted) a function of the number of other people who have already adopted.11 We then get a class of curves, all of which look very much like the original S-curve, but with a change in the “inflection” point, that is, where maximum increase occurs. For the pure diffusion case it is when the proportion of people who are “carriers” is half the population, since this maximizes the number of uninfected people who meet infected people. Given a situation of “pressure,” the inflection point is somewhat shifted to the right.12 Contagion will take somewhat longer to get started, and then catch up quickly once a certain point is reached, but given the fuzziness of actual data—and this has perhaps not been appreciated—there is probably no way to distinguish a pure contagion process from a “pressure” process, even though the two make very different arguments about the fundamental nature of influence.13
The contagion model thus always appears to fit changes that go from low to high and stay there. Further, contagion models generally imply an end-state of total saturation, at least for the transmission of cultural elements such as beliefs, tastes, or practices.14 Yet this seems implausible, as there may well be something about many of these cultural elements that makes it unlikely that they will ever completely saturate a population. There may indeed be some people who prove completely resistant to some disease, but this is different from the inherent divisiveness of certain cultural elements; as Bourdieu (1984 [1979]) might point out, a cultural element that does not distinguish between persons is hardly a cultural element at all (also see Bearman and Brückner 2001).15
Further, in cases in which we might actually hold that a contagion process comes rather close to the mark in terms of explaining the process of adoption, the curve of prevalence over time may not look at all like a classic contagion curve, since there are forms of contagion that shift into reverse. Lieberson (2000) has given important examples of these—certain elements diffuse and become more and more popular but then “undiffuse” and become less and less popular over time.16 That is, even where a simple contagion process takes place, people will find some way to throw it into reverse to make sure that there are always some who do, and some who don’t. In these cases, persons are differentiated and maintain their differentiation. The weakness of these models, then, is that they ignore the differentiation of persons.
Both the contagion and pressure models assume that all people are fundamentally indistinguishable—just as you can catch a cold from anyone who sneezes in your face, so the tension mounts for each additional adherent when you are not conforming to the group. The basic approaches can be generalized to take into account various forms of population heterogeneity (e.g., Morris 1993; Wiley and Herschkorn 1989), but most of these generalizations are open-ended methodological ways to encompass some sort of differentiation, should it exist. But what structural form is differentiation likely to take for influence relationships?
There are, we recall, two factors that determine the probability that A will transmit to B, the probability that they will interact (p) and the probability that there will be transmission given interaction (q). Thus we can divide selectivity according to whether the process is “p-differential” in that some dyads are more likely to interact than others, as opposed to “q-differential” in that transmission is more likely between some pairs.
As a first approximation, we may propose that when people are eager for new information, p-differentiation is most important, and when they are reluctant to have their opinions or ideas changed, q-differentiation is more important. The logic here is that if you would be glad to get some piece of information, say, you will adopt it from the first carrier to come along, while if you are reluctant to change your ideas or adopt new ones, you will only do so when these are transmitted by certain people. You may encounter others, but you will not be persuaded.
I propose this only as a first approximation because it is not difficult to envision scenarios that do not fit this distinction neatly; indeed, since even reluctant people cannot have their minds changed by people they do not meet, p-differentiation will affect the actual spread of ideas or information where respondents are reluctant. But this distinction is sufficient to structure our investigation of the results of introducing differentiation to contagion models.
Diffusion
Let us thus consider the case of p-differentiation for eager listeners, which I have called cases of diffusion (in contrast to contagion). There are a number of coherent approaches for the general case of diffusion through social networks (including those that allow persons to have different degrees of stubbornness in terms of the number of their friends who must adopt some innovation before they go along [see especially Valente 1995: 73]). Such approaches treat the structure of relationships as given, and thus make reference to some other structured relationship (for example, friendship, organizational roles, family, etc.). But we may be able to say something about the forms that relationships established solely for the purpose of diffusing cultural elements might take.
First, because all recipients accept influence from those with whom they come into contact, there is no reason to expect that the relationship of transmission will be anything other than fully symmetric. The only formal issue is our ability to compile different pairs of persons’ likelihood of interacting in such a way that we can see some structural principles emerging. As a result, pursuing this issue will lead us to replicate the logic of chapter 2, which examined mutual associations and ended up focusing on two sorts of web-structures, those with relational and those with spatial logics.
Spatial models are perhaps more appealing for the case at hand, for it is easier to adapt them to cases in which relationships are continuous as opposed to either present or absent. Thus we might expect people not only to be more likely to have had some interaction the “closer” they are, but to interact more frequently the closer they are. Most simply, geographical space directly affects diffusion; while it is not the only factor, a geographical model seems to account for the diffusion of certain cultural preferences (see Harton and Latané 1997 for some examples). Taking “social distance” into account may further increase the plausibility of a spatial representation for a diffusion process.17
Such spatial models may be used to account for diffusion, by maintaining that the contact that allows cultural elements to be transmitted is more likely to occur across short distances of social space. This is in itself not very interesting; what is more interesting is that this can lead to aggregate distributions that have certain structural properties; by throwing a wrench into the simple contagion model, these distributions may stop short of total consensus and display internal organization (see Abelson 1979; Harton and Latané 1997).18 When we leave geographical space, and simply postulate a latent “social space” to parsimoniously explain patterns of adherence or transmission, it turns out to be quite easy to use a spatial model to account for diffusion when in fact no diffusion need be invoked to explain the spread at all. That is, we may assume that if A and B hold some competing versions of an idea or taste, say, and person C is “closer” in social space to A than to B, C will tend to receive A’s version and not B’s. But since the spatial logic is itself predicated on the likeness of A and C, A’s influence may be redundant, as C may prefer the cultural innovation for the same reason that A does. Whatever it is that they have in common, and that leads them to be in the same area of this space, will lead to C’s adoption without A’s “influence.” (This point underlies spatial analyses by market researchers; the key theoretical underpinnings of this are given by Spiegel 1961.)
Even apparent diffusion across geographical space may also simply be due to the slow movement of a source that independently spreads innovation like rats spreading the plague, or to likeness that happens to be geographically distributed, and not, in either case, interpersonal influence. One nice example is people all putting up their umbrellas at the same time, but there are less obvious cases. Consider the diffusion of home snowblowers.19 The snowblower was first invented in 1925 by Arthur Sicard, (significantly) a Canadian farmer struggling to get his product to market; for many years it was a full-scale truck, not suitable for home use. (The first “walk behind” snowblower was then invented in 1951 by the Toro company.)
Unfortunately, we lack data on the adoption of this device, but it seems quite reasonable that if we were to examine aggregate sales by province and state, it might appear that spatial diffusion is responsible, since more would be sold in the northern latitudes at first, with sales in ever more southern latitudes increasing regularly. Thus one might assume from the data that Minnesotans learned of the innovation from Canadians, Wisconsinites and Iowans from Minnesotans, Missourians from Iowans, etc. But this could also result from the price being relatively high at first and the product only attractive to those with regular and heavy snowfall. As the price declined, we might expect the “threshold of snowfall” at which a snowblower became attractive to decline as well.
In such a case, the data have not been produced by a diffusion process. But a model that assumes that diffusion did occur, and was more likely the closer people were to one another in physical space, would fit the data quite well. Similarly, models that position people in a social space also often have a deceptive ability to “account” for diffusion.
Thus spatial-diffusion models have the problem that they seem to fit more cases than they should. A further drawback of a spatial model is that influence must be seen as inherently symmetric (see Erickson 1988:109–15). While the transmissible thing in question cannot go from an uninfected person to an infected person, but instead must go the other way, the contact itself is symmetric and the transmission only depends on who is infected and who not. Generalizing to the case in which more than one thing may diffuse, if one person is highly influential because she has many ties to others, she is also highly susceptible to influence. There is nothing in the distance between two persons that can lead us to determine that the relationship of influence should go one way and not another. The spatial position that is closest to overall influentiality is then “centrality”—being close to many other people.
Now it is possible to modify spatial analysis to introduce some asymmetry, as done by Friedkin (1998) in his study of scientific communities.20 Friedkin allows some “central” persons to be more influential than others by giving people a differential “self-weight,” that is, an innate resistance to influence (Friedkin 1998: 169, 195). Thus the various central players do not necessarily influence one another, despite their proximity.21 But since one always is with oneself, any differential self-weight must be seen as a q-differential phenomenon and does not get us further in understanding the nature of p-differential structures.
Finally, the spatial model may more fundamentally fail when the thing being diffused is information that others are eager to possess, for it assumes that the underlying probability of contact is a cause of information transmission, and not an effect. But if B is indeed eager to hear some type of information, possession of this information may allow A, normally “far” in space from B, to make contact. Suggestive evidence comes from the fascinating experiment conducted by Festinger, Schachter, and Back (1963 [1950]: 130), who planted rumors so as to trace the path of their diffusion. They found that transmission tended to follow lines of friendship, to stay close in social space. But the rumor also jumped across space, as people made contact with members of the governing council to whom they did not normally talk. As Ryan (2006: 230) emphasizes, “diffusion” generally misstates the process, which is one of deliberate notification.
Thus information transmission may move the opposite way from influence generally understood. Knowledge is power, at least to some degree. A person with knowledge certainly may, as Festinger et al. thought, cough it up the chain because it is “relevant” to the leader or simply because people are supposed to pass on all information upward. But even in the absence of such relevance, we might imagine that a person who possesses a piece of rare information experiences a temporary boost in status, such that she may initiate communication with someone who is of higher status in some setting, or is of equivalent status but not sufficiently well known to justify contact under most conditions (see, e.g., Blau 1955: 130; Erickson et al. 1978: 79). Even in a situation in which rank is formalized, the possession of information can itself alter one’s position, and underlings with special information can command nominal superiors (Feld 1977: 79). Eager recipients may allow those possessing information to “tunnel” across what otherwise would be relatively great distances of social space, thereby defeating the utility of a spatial approach. Because spatial structures do not qualitatively distinguish vertical from horizontal separation, the changed distance that may arise when a person of low status possesses vital information cannot be adequately accounted for.
In sum, any vertical differentiation is likely to lead to a problem for a spatial model of diffusion of information, for two reasons. First, distance in a space is symmetric, which implies a fundamental horizontality between points. That is, it is as easy to get a piece of information located at point A to point B as it is to get a piece of information from point B to point A. In real life, however, we might reasonably expect that this is not always so. Second, the possession of information may itself change the social distance separating lower downs from higher ups.
Figure 5.3. Hub-and-spoke structure
Center-Periphery Models
We may be able to capture some of what seems reasonable about the spatial metaphors, namely the idea that Coleman et al. had that some persons were more in the “center of things” than others, without adopting a strictly spatial logic. In the simplest form of this, we can use the reasoning of chapter 2 that saw one form of the web as leading to a differentiation of persons into hubs and spokes. Each spoke is attached to only one hub; hubs are attached to both spokes and to other hubs, but no hub is attached only to other hubs. (That is as much as to say that no hub is a “second-order hub,” a “hub’s hub.”)22
In figure 5.3 the hubs are the larger blank circles, and the spokes the smaller, filled-in circles. This is basically the two-class model that has come to be assimilated to Merton’s (1949) distinction between “cosmopolitans” and “locals” (cf. Fischer 1978). The cosmopolitans are mediators, those who are the first to learn of and synthesize new ideas, information, or opinions, and who then transmit these to the locals (we have seen this in Coleman et al.’s work; the best example is Katz and Lazarsfeld 1955; on evidence for such structures, see Lai and Wong 2002: 51).
The overall structure is clearly a sensible one, familiar to many of us from contemporary domestic airline travel, which is oriented toward a multihub system, though we see deviations from the pure case, as there are some spokes that connect to more than one hub. Another example—and one that accentuates a particular combination of egalitarian communication between hubs, and authoritative direction within wheels—may be found in modern experimental science. Each laboratory is generally an extremely hierarchical form, though the number of levels and other particularities generally depend on the research setting (for example, firm versus university, large versus small, United States versus Europe). Communication, however, often occurs directly between hubs in a relatively informal way. This is true even though there are many institutional forms (for example, journals, meetings, societies) set up directly for the purposes of facilitating communication across work groups.23
Such structures arise in other realms (though often without the authoritarian relation between hub and spokes inside the wheel), because there is one thing they are very good at, which is robust communication.24 Whether it is sending persons or signals from one place to another, multihub structures are excellent at getting things from any one particular place chosen at random to any other. This is in contrast to some other structures we will examine, which are good at getting an order from one particular place known in advance to every other place (command trees), or at diffusing news in general (webs or power grids), but not in terms of deliberate connection from any two places chosen at random. Given a trade-off between efficiency of communication and robustness in the face of damage, multihub systems are nearly ideal. They do involve redundancy (some people may get a message more than once if one is attempting to diffuse information throughout the network) and require somewhat more communicative work on the part of the hubs than is truly necessary, but these are generally small matters.25
To demonstrate this, imagine that we remove both diagonal lines connecting the hubs on the lower left and upper right in the picture above. This structure then has the same number of relationships as does the tree structure of figure 5.4. But in the ordered tree structure, it takes 4 “legs” (as flights from one node to another are sometimes called) to communicate from any one spoke to another spoke not attached to the same hub. If the apex is removed, the structure falls completely apart. But the nonordered multihub system can make most interwheel spoke-spoke communications in only three links, and the structure is relatively robust in the face of damage.26
Figure 5.4. Tree structure
Such structures are not only ideal to handle issues of dissemination of information when the source and recipient of that information are not easily specified in advance, but there is abundant anecdotal evidence that this is how people frequently create communication networks. For an example, the early use of the telephone in rural areas led to a hub-spoke system of between-town communication: a drug store in one town might be a communicative hub that linked to a drug store in another town (see Fischer 1992). Personal telegraph messages used a hub-spoke system, although the interaction connecting hub to hub (electric transmission through a wire) was different from that connecting hub to spoke (personal contact).27
Unlike airline routes, interpersonal communication structures are generally not planned, and it might seem unlikely that such a simple and efficient structural form would arise spontaneously. But it actually is extremely reasonable. First, assume that there are certain advantages to being connected. One has the ability to transmit and receive information. Further, assume that this advantage decreases with the characteristic path length of the network—tire number of connections that will usually be necessary to reach someone. Given all this, it is quite reasonable that people will tend to operate according to the heuristic “connect to the ‘hubiest’ person you can.” A “good hub” might be understood as one that gives one access to all other persons with fewer than some relatively low number of connections. We might also expect that people may also find the maintenance of too many relations tiring, and hence some hubs will reject overtures.28
As a result, it is quite reasonable to expect that independent social processes will result in a rough equilibrium where there are just enough hubs to answer pressing communicative needs. Interestingly, then, we may see the multihub structure, which partitions people into two distinct classes of hubs and spokes, arising even when there is no preexisting categorical differentiation. While there may be some differentiation in people’s subjective disinclination to be a party to many relationships, the multihub structure can arise even if all people are identically inclined.29
For example, imagine a structure formed by people coming together and wanting to make sure that they are in contact with others. While they do not particularly want to serve as hubs, they are unwilling to be too far removed from any other member (as would be the case if there were a large number of links required to send or receive a message from another). While they have an absolute limit to the number of people they are willing to be connected to, they would rather be connected to another hub (more likely to give them valuable information sooner, say) than to a spoke. Figure 5.5 below sketches the evolution of a multihub communication structure arising by the successive addition of persons who operate according to the following heuristics: (1) make sure you can reach every person with no more than two intermediaries; (2) consider connections to hubs (those with more than one connection) half as onerous as connections to spokes; (3) do not accept more than three spoke connections (or the equivalent, such as two spoke connections and two hub connections); (4) connect to the oldest member you can. Note that the dashed lines indicate a hub-hub connection—when person 3 becomes a hub at time 7, person 2 forms a new connection to her (since previously this would have brought 2’s total connections to 3.5, over the maximum, but no longer does so, now that person 3 is a hub.)
Thus if people are to spontaneously form a structure for the purpose of diffusing information or other cultural patterns, it is quite plausible that multihub structures will emerge even when there is no variation in people’s desires for communication frequency or timeliness.30 However, in the cases in which unanticipated information is diffused through general friendship connections, there is no reason to expect a multihub structure. Instead, it seems that communication follows a somewhat spatial logic, whereby the hubs are least likely to transmit information to those “far away” precisely because they are central—they are surrounded by similar people. While they have an advantage in communicating authoritatively to those around them, it is the more isolated, marginal persons who may utilize their weak ties and take information to other clusters (see Weimann 1982).
Figure 5.5. Evolution of multihub structure
In sum, center-periphery or multihub structures may reasonably arise when people are deliberately oriented to the spread of something such as information, but are in general averse to having many ties. The structure serves to channel interaction in a more or less efficient way given that the respondents are basically eager to receive information. The resulting structures are very similar to the webs investigated in chapter 2. They have some of the characteristics of small-world graphs, though they tend toward more organization (in particular, a distinction between hubs and nodes and, most likely, a flatter degree distribution).31 They are smaller than the set of acquaintance relationships, but unlike friendships, they may be actively constructed (that is, we seek out someone from whom to get information). Although hubs may transmit more to spokes than they receive, the relationship is, in principle, mutual, for no one would refuse a bit of information because they find the source unimpressive. For this reason, hubs have no special position over spokes—the structure cannot be turned to other uses, the way a set of influence relations based on power or authority might.
But what about when recipients are reluctant, such as when what is being transmitted is a belief that contradicts previously held beliefs? In such cases, mere exposure is not necessarily sufficient to ensure adoption. If differentiation is to enter, it is likely to enter in the form of some being more credible, authoritative, or persuasive than others. That is, in contrast to our hubs who differ from other people only in terms of their interactional pattern, we will expect to see people who differ in terms of how authoritative they are. We go on to examine such cases.
Influence Structures
Reluctant Recipients
In some cases people will tend to be reluctant to adopt new beliefs, if only because, as cognitive consistency theorists have emphasized (see Festinger 1957; Feldman 1966; and the essays collected in Abelson et al. 1968), changing one belief may require changing other beliefs, undertaking new actions, or revising one’s sense of self. Whether they justify it on pragmatic (conditional) grounds—being influenced by C is going to be good for me—or on normative (unconditional) grounds—I should always be influenced by C on these matters—reluctant adopters, we may hazard, will tend to make a distinction between those from whom they should accept influence and those whom they can safely ignore.
The nature of this division between the influential and all others will likely depend on what types of beliefs are being transmitted. In some cases, C is considered a reliable guide to A’s cognitions when it comes to interpretations and to adjudications between conflicting interpretations; it is only this “interpretation reliability” that I shall call “authoritativeness.” In other cases, B is merely considered reliable when it comes to matters of observations; this “source reliability” I shall call “crediblity.”
Now it may be impossible to make a theoretically perfect distinction between authoritativeness and credibility, as it is generally impossible to say where fact ends and opinion begins. Yet as actors we may be able to agree in practice as to where the division is, and it is this practical ability that will be related to structures of interaction. Even if there are intermediate cases along a continuum, we may distinguish cases in which person A considers person B credible but not authoritative from cases in which person A considers person B authoritative. The structures compatible with the first form of q-differentiation are not necessarily compatible with the second. We begin with credibility.
Credibility versus Authority
For person A to consider person B (but not necessarily person C) credible in some context or for some domain is for her to regard relevant statements from B as believable and, barring other credible information to the contrary, acceptable. For example, the American legal system is predicated on the assumption (correct or incorrect) that any adult without cognitive impairment is a credible witness. But such witnesses are (with exceptions generally pertaining to issues of character) not considered to have any particular authority. If a person claims that he saw a car go through a red light, this is considered credible unless contradicted by an equally credible witness, but despite his ability to make a potentially decisive contribution to an evaluation of guilt, he is not asked whether or not he believes the defendant to be guilty.
In contrast, expert witnesses are considered to have a different type of authority. They do not simply report that at 3:08 p.m. the Tuesday before that they saw a succession of bands on a computer printout—they state with authority that the two samples of DNA are from the same person or at least close family members. Similarly, a priest does not simply report that according to the Catholic Church (catechism #1857), a mortal sin must be done with full knowledge: like other experts, a priest hearing confession tells a parishoner when he is in mortal sin and what must be done to get out.
Structurally, the difference between authority and credibility is important, for it is reasonable to expect that it is easier for attributions of credibility to be mutual than for relations of authority to be mutual. Two citizens or two scientists can both expect the others’ reports to be credible, but if a priest allows himself to be guided by a bishop’s interpretation of some issue pertaining to church dogma, it seems unlikely that the bishop would also allow himself to be so guided by the priest.
Accordingly, there are cases in which q-differential credibility is organized in mutual fashion, most simply, in cliques. The earliest years of the Royal Society, as discussed by Shapin (1994), serve as an excellent example. No scientific discussion could take place unless a group could be assembled whose basic credibility was beyond reproach. One member could dispute an interpretation of an observation made by another member, but not (says Shapin) the observation itself. While it is possible to overstate this case, it seems plausible that the members drew on early notions of the qualitatively privileged position of honorable gentlemen to establish this clique as an equivalence relation (in the way in which the British used the equivalence relation “peer” in a distinctly exclusionary sense).
Although one would assume the credibility of other gentlemen, this was not a case of an open structure—there were others to whom such credibility was not extended. Thus common seamen might bring back many interesting observations but these would be treated differently from those returned by Royal Society members. It is certainly not the case, of course, that science as an institutional system continued to be reducible to cliques of saturated mutual relations of credibility.
This is because the very success of such a structure of equivalence, once set up, is its undoing. It is all very well to attribute blanket credibility to all clique members when one is dealing with matters of observation. But what happens once a set of equally believable persons are oriented toward the same observations? Sooner or later, we might imagine, there is a difference of interpretation regarding observations that all agree on. Since the issue cannot be settled simply by declaration on the part of the more authoritative, the community formed by this equivalence becomes oriented to the resolution of the difference. This resolution may be cooperative or it may be competitive, or both, but in any case, there are two results. The first is that statements of observation necessarily comprise a smaller and smaller portion of scientific interest. Discussion quickly zeros in on the work between observation and interpretation, and hence the base credibility of all participants becomes less and less important (also see Collins 1998: 533).
It follows (and this is the second result) that the process of community resolution, over time, leads some to be vindicated more often than others. The authoritativeness of the more-often-vindicated rises, in that would-be detractors without a record of vindication find themselves starting in any dispute with a handicap (if they did not start with one already). Even if there were no external forms of differentiation (e.g., political backing, personal magnetism, resources, and so on), we would probably find that a clique based on equivalent credibility would give rise to differential authority (see Latour and Woolgar 1979; Latour 1987). In sum, once again, we start from a clique and end up with a popularity tournament. We are hence drawn to examine q-differential structures that include some measure of vertical stratification of members into the more and less authoritative.
Hierarchical Models of Influence
Let us accept, then, that persons can be more or less authoritative. Because authoritativeness is (as Durkheim argued) an attributed quality, there may be disagreement about the authoritativeness of some person without this implying contradiction with our definition. Thus C may see B as more authoritative than A, while D believes A to be more authoritative than B. While we must briefly put this complication on hold, it is worth emphasizing that it does not necessarily undermine the project of looking for authority structures, for there can be a great deal of slippage between (on the one hand) the attributed authoritativeness of persons, and (on the other) an actual structure of relationships of authority—that is, not simply attributions of a characteristic, but a pattern of actual interaction.
Let us define such a relationship of authority to exist when A decides to treat B’s pronouncements as valid guides for her own belief. We can begin by assuming A will only establish such a relationship if A thinks that B is more authoritative than she. We are now interested in whether sets of such relationships of authority can take on a structural form, and if so, what this form may be. We will use this to derive more general principles regarding the relation between subjective conceptions and structure that arise with the introduction of horizontal differentiation via incomparability.
By a “structure,” we will mean a set of persons each of whom is tied to at least one other by a relation of authority such that the resulting graph is connected. If this structure is to be coherent, it seems reasonable that there is some sort of limit to the disagreement that persons may have about the distribution of authoritativeness. Let us call a set of (possibly disagreeing) views about the social distribution of persons that are compatible with the same structure of interaction “architectonic,” since they do not contravene the establishment of a structural model.
Our question as to the architectonic principles of influence structures may be seen as a specific aspect of a more general question as to how horizontality can arise in a relationship that is inherently asymmetrical, such as “A is more authoritative than I am.” We can begin to examine this more general question by considering simple conditions or heuristics that generate vertical structures, and then relaxing some assumptions.
The Introduction of Horizontality
We may take the simple case in which all persons are vertically ordered in terms of authoritativeness; all persons agree on the relative authoritativeness of one another, and a relationship of authority between A and B is established if and only if B is more authoritative than A. Since this is a general argument, we can consider “authoritativeness” one particular example of a personal attribute which we can denote ζ. Thus we may say (B → A) if and only if ζB > ζA. Such a case clearly implies the same transitive order that we examined in the previous chapter. Now let us consider how horizontal differentiation might be introduced into the order; as we allow for the relaxations that transmute dominance to influence, we will be able to derive a corresponding change in the structure of relationships.
We can begin with the simplest case, one in which all persons actually are vertically ordered but cannot correctly or reliably ascertain the difference in vertical position when they are rather close to one another. For example, we may imagine a group arrayed on a vertical dimension (figure 5.6, left), in which an antisymmetric relation is established whenever the distance between two persons exceeds some “just noticeable difference” (or JND, which we can imagine is some fixed quantity Δ). In other words, we now say that (B → A) if and only if ζB > [ζA + Δ]; the corresponding heuristic might be “accept the dominance of B if B is significantly above you.” The structure of this antisymmetric relation will not be a pecking order—instead, it will be what is known as a partial order, a set of elements and a relationship that is antisymmetric, reflexive, and transitive. The resulting partial order is displayed to the right of the order, toward the middle of the figure. This diagram is known as a Hasse diagram, as it does not include transitively implied relations: thus in the middle panel of the illustration below, from this we know that (1) → (5) since (1) → (3) and (3) → (5) and so this line is suppressed for the sake of clarity. As we can see, there has been an element of horizontality introduced by the de facto incomparability of adjacent persons. (This de facto incomparability is different from the structural incomparability we will examine shortly.)
If, however, the size of the JND is increased so that it is even harder to determine which of two persons is of greater authoritativeness (right panel), we decrease the number of “levels” in the structure while increasing the horizontal differentiation. As a first approximation, then, we might say that horizontality can be increased by fuzzier distinctions between persons. Indeed, there are probably many good examples that could be found, and not all require that the JND be due to perceptual limitations. For one, if the antisymmetric relationship in question is one of ritualized submission behavior, it might be that the original ranking pertains to fighting ability, and person B is aware that his or her ability to give (a stronger) person A a rough time for his or her money is sufficient to exempt B from ritualized submission. For another, higher- status persons, when giving direction to lower-status ones, often initiate touch and physically displace others; this is most commonly seen when status arises on the basis of age difference. Given a ranking in terms of age, it might turn out that while a thirty-two-year-old feels that it is appropriate for her to give directives and perhaps physically guide to anyone younger than sixteen years old, and a twenty-five-year-old feels that it is appropriate to give directives to anyone under twelve years old, the thirty-two-year-old would not feel quite comfortable giving a directive to a twenty-five-year-old or initiating touch.
Figure 5.6. Horizontality and JND
For these examples the resulting structure of interaction would form a partial order of the form seen above. Now in such a case, one can generally re-create the underlying order, simply by counting up the number of arrows leaving and entering any node (the out-degree and in-degree respectively).32 Indeed, the participants themselves may be aware of the underlying order. But such awareness will not necessarily lead the structure of interaction to increasingly approximate an order—the hypothetical persons discussed above may very well know that age orders them without this reducing the size of the JND.
Figure 5.7. Architectonic interpretations
Slightly further away from verticality are those partial orders that do not allow us (or participants) to re-create an ordering even if such an underlying ordering exists and was in part responsible for the creation of the observed relations. Such a structure arises when the formation of an antisymmetric relationship between two people (which I shall refer to as a “choice,” though forms of external assignment can also have this result) involves something other than the vertical positions of the two. For example, consider the partial order graphed in figure 5.7. If the antisymmetric relationship in question only forms from a higher to a lower person, there are a number of orderings that are compatible with this structure. It is possible that the ordering goes 1, 3, 6, 2, 4, 5 or that it goes 1, 2, 3, 4, 5, 6, etc. We know that 1 will be at the top and 4, 5, or 6 at the bottom, and we know that 2 will come before 4 and 5, and 3 will come before 6, but that is all. Even more importantly, we see that agreement among the members as to the underlying differentiation can be far from perfect without undermining the structural principles. Thus it might be that both persons 3 and 6 will hold that the true ordering is 1, 3, 6, 2, 4, 5, while person 4 holds that it is 1, 2, 4, 5, 3, 6. Thus person 6 justifies why she does not treat 2 as authoritative, and person 4 justifies why he does not treat person 3 as authoritative. These visions of the social world are contradictory in some ultimate sense, but as both are equally compatible with the structural ordering, they can be considered architectonic.
Structures such as the above, it will immediately be seen, will arise if persons are ordered (or orderable) and establish relations on the basis of this order, but at least one of the parties is free to decline the establishment of the relationship. More precisely, we may say (B → A) only if ζB > ζA (but we do not say “if” or “if and only if” as we did in the pecking orders of chapter 4). This element of choice may come from the introduction of a new kind of “distance” (in addition to vertical distance)—the sort of distance that becomes possible when people are uncaged. For example, person (2) may simply have no interest in establishing a relationship of any sort with (6), and if this is permitted, (2) and (6) never compare their vertical positions. Such structures, or ones closely related, I will go on to argue, may reasonably be expected to arise in a number of situations in which the relationship in question is one of influence, especially influenced considered as a result of authority.
Uncaged Influence
Influence Trisets
The single best case of an influence structure that emerged in an informal group outside of any institutional constraints is still found in William Foote Whyte’s (1981 [1943]) Streetcorner Society.33 This case, in addition to its own merits, is of special interest because it inspired both theoretical (Homans 1950) and methodological work (Friedell 1967) on influence structures; unfortunately, as we shall soon see, the theoretical work inspired was in contradiction to the methodological work. An example of one of the diagrams that Whyte drew to communicate the structure of influence relations among men who formed a loose gang is reproduced in figure 5.8.
A number of things are immediately clear; the first is that there is a vertical organization—Whyte clearly assumes we will read this as flowing from top (Tony Cataldo) down to the bottom. Second, there is horizontal organization as well: there are two main cliques, so that few people in one clique influence anyone in another. But the horizontal organization goes further than that—it seems that Carlo is further “toward” the lunch room clique than Mike, since he influences Charlie who influences a lunch room boy. Gus seems to fall “in between” the two cliques, as Whyte deliberately drew him between the cliques despite the fact that he only influences Chris.
Even more, the overall organization seems somewhat reminiscent of an organizational hierarchy. Friedell (1967) was struck by this similarity, and argued that such influence structures were a special type of algebraic structure that we shall examine shortly, a “tree.” A tree is a particular form of semilattice, which is in turn a particular form of partial order.34 Basically, Friedell tried to assimilate these structures to partial orders in which every two or more elements have a least upper bound—that is, a common superordinate—and hence all upper bounds of any subset are comparable to one another. To do this, Friedell had to drop certain relations of unclear direction. (His rendition is given in figure 5.9.) More important, since the binary relation defining a partial order is, as we recall, transitive by definition, Friedell made the strong claim that all transitive relations were present.35
Figure 5.8. Informal influence in Cornerville. Used by the kind permission of the University of Chicago Press.
Unfortunately, Whyte gave these diagrams without any explanation other than a cryptic legend, but in some cases transitively implied relations are explicitly drawn in, while in others they are absent (for a more detailed discussion see Martin 1998; Martin 2002). This suggests that Whyte did not believe that transitive relations were necessarily present. Homans (1950: 182ff) argued that such a lack of transitivity was quite reasonable given two seemingly contradictory forces at work. One the one hand, the leader tends to influence all others, but on the other hand, people tend to interact more frequently (and hence be influenced by) those closest in status to themselves. (We have seen support for this contention in our analysis of the camp dominance data in the previous chapter.) This implied that the leader’s influence tends to be mediated by lieutenants.
Figure 5.9. Friedell’s interpretation of figure 5.8. Used by the kind permission of Morris Friedell and the American Sociological Association.
When the relation of “influence” that Whyte was describing is thus understood to mean face-to-face influence, and not “ultimate” influence, it is quite clear why transitivities are not implied—Carlo does not have a relationship of influence with Dodo because he would not take the time to talk to Dodo personally on a matter of controversy. But Joe would pass along Carlo’s thoughts to Dodo. Even though the network is not a partial order under the relationship of influence, the members are partially orderable—Whyte is able to put them in a structure of top to bottom, with equality and separate lines of authority, including both incomparability and intransitivity.
While scholars of influence structures have tended to assume transitivity for reasons of mathematical elegance,36 it is unlikely that influence structures really are transitive (here also see March 1957: 226). Yet they may still have mathematical properties. In particular, they may be arranged on paper so that all influence relations flow downward; written in matrix form, they can be permuted so that the lower triangular half of the matrix consists of only “zeroes” (no relationship), hence we can call them “trisets.”37 Returning to the arguments regarding caging made at the beginning of this chapter, we may say that this is a structure that arises when we simply allow some people to reject relationships. It is like a dominance order with holes. Any relation that does exist is compatible with an ordering of all persons in terms of status, but not all relations compatible with this order do exist.
Such trisets may contain transitively implied relations, but they may not. It seems reasonable that if the transmission of information is costless and clean, it is economical to eliminate such transitively implied relations, just as Carlo would not bother to talk to Dodo when he could count on Joe to pass along his thoughts. If Carlo suspects that Joe does not reliably pass on Carlo’s thoughts, however, he may have good reason to open up a direct channel of communication (cf. Evans 1975). But because of the lack of transitivity, the horizontality in such structures can be distinguished from that which we saw earlier, namely the horizontality that arises because of simple difficulties in discriminating who is higher in status than whom. In these cases, those on the top must directly influence more people than those below; this is not necessarily true in trisets.
For another example, Mary Sisock (2008) asked a community of persons who owned forest land whom they asked for advice about various matters pertaining to dealing with their land and trees and to whom they gave advice; she also asked the same questions of some of the forestry professionals who worked in the area to get a rather complete view of the network of influence. A portion of that network is graphed below in figure 5.10 (excluded are persons who were named by only one informant and were not in the data set, as well as some small unconnected components).38 First, we can see that all persons can be arrayed so that no lines go upward—despite so many persons, there is not a single “cycle” whereby A advises B who gives advice to C who in turn gives advice to A.39 Second, we can see that contrary to the JND model, those at the top do not necessarily influence more people than those lower down. Finally, we see that the structure is intransitive: just because A gives advices to B who in turn gives advice to C, it does not follow that A gives advice to C. In fact, in the entire network (bracketing the one mutual relationship), only three transitively implied relationships are present, two involving the same mediator (O5).
At the same time, it is not necessarily the case that we can determine someone’s authoritativeness by their out-degree. Although DNR Forester 1 does have a special position of giving advice to many, O5 gives advice to far more people than do those from whom he gets advice. The structure, then, is a perfect example of a triset, implying that people can be perfectly ranked in terms of authoritativeness. We do not, however, know precisely what this rank is.
Figure 5.10. Influence relationships among forest landowners
We have found that when people are ordered in terms of authoritativeness, influence trisets may emerge in which this order is only partially apparent. If we compare this structure to the pecking orders examined in the previous chapter, we see not only the introduction of horizontality but an implicit change in the correlative subjective heurisitics. A pecking order of influence relations would result if people followed the rule “always accept influence from those higher in status than you.” A triset results if instead they follow the rule “never accept influence from those lower in status than you.” Since pecking orders do not tend to arise for relationships of influence, we may conclude that while influence relations may be status-conscious, people insist on retaining some right of refusal (though we do not know whether it is higher-status or lower-status members who retain this right, or both). The additional rule, “do not directly influence someone who is influenced by someone you already influence” produces mediated trisets, but since actually existing trisets tend not to be wholly mediated (some transitively implied relations are present), it may be better to say that persons tend not to directly influence those who are too far “below them.”
These heuristics allow for the sort of structure seen above, in which two or more lines go “into” the same person. Thus O13 near the bottom of the graph gets advice from another person in the data set, and industry fortester, and from a DNR forester. It seems that such an arrangement is quite consistent with informal influence structures. But consider what happens when influence “hardens” to the point in which the subordinate party has to accept the influence—when influence has passed over to command. Then one of ego’s super-ordinates may contradict a command given by the other superordinate, putting ego in a rather difficult position. Not suprisingly, such multiple subordination is generally forbidden in what we will call “command structures,” sets of relationships of command and control such as those seen in formal organizational hierarchies. Such relationships assume an environment of preexisting differentiation in which A is not simply attributed more “authoritativeness” by some B. Instead, there is some preexisting material inequality—A has access to or control over resources or outcomes that affect B. It is in this case that the tree structure Friedell discussed is likely to emerge.
The weakness of the influence triset for coordinating action goes beyond merely allowing for a person to receive contradictory orders. The triset’s horizontal organization makes a poor basis for the organization of other relations. In contrast, the tree structure is such as to encourage the formation of heuristics for action that reinforce its own clarity. Indeed, as we shall see, it is a sufficiently useful structure for coordinating action that, when it does not exist, we are often forced to invent it.
To summarize, contrary to some earlier analysts, we have not found informal influence structures to take on the same form of the tree structure that is (as we shall see in chapters 7 and 8) a recurrent aspect of formal organizations that involve stratification and coordination of action. Yet these trees are exceedingly useful (at least to someone). They arrange antisymmetric relationships so as to maximize unambiguous direction and control while also facilitating other heuristics for action that are wholly horizontal. In fact, the unambiguousness of the verticality allows for the emergence of equivalence relations, returning us to the themes of equality first raised in chapter 2.
The relation between verticality and equivalence can best be seen in kinship relations—at least kinship as it is represented according to unilineal descent. If parenthood is antisymmetric (such that if I am your parent, you cannot be my parent), and each person has only one parent,40 the tree structure below naturally arises. Of course, humans have more than one true parent, as their reproduction is not asexual, but if actors treat it as such, then relationships between men (or between women) can be cast as in figure 5.11.
Figure 5.11. Tree structure
What is so important about this structure is its implications for the organization of heuristics for action. In a technical sense, the vertical relationships of unilineal descent between generations induce a set of equivalence relations—and hence the possibility of a different social structure consisting of mutual relationships. Such structures can be formed by making “cut points” at some level; alternatively, they can be understood as sets of walks “upstream” and back “downstream.” Thus if we let the relation R mean patrilineal parentage, so that iRj means that person i is the father of person j, then C = RT (the transpose of R) is the relation “is a son of.” The compound relation CR defines the equivalence class of brothers (actually, “self and brothers”—all my father’s sons); the relation CCRR defines the class of male cousins (actually, “self, brothers, and cousins”—my father’s father’s sons’ sons).
The tree structure of descent thus induces a different structure composed of nested sets of wholly horizontal relations or classes (also see Sahlins 1968: 15, 21). Such an induced structure is shown in figure 5.12. Each circle can be considered a horizon of relatedness. All those within the small circles consider themselves brothers, and all those within the larger one consider themselves cousins. The induced relationships of brotherhood are denoted with dashed lines and some—only the ones connecting the two groups of brothers on the left, to avoid drowning the diagram in lines—of the relations of cousinhood are denoted with light dotted lines. More generally, we can induce a new relation (Tk) whenever we go K steps back and then forward in time, with each relation Tk being CkRk we can consider all those who are tied by a relation at level Tk to be Tk–equivalent. (Note that if two people are Tk–equivalent, they are also Tk + n–equivalent, where n is any positive integer.) These induced classes are indeed equivalence classes, which means that they are transitive—my brother’s brother is my brother, and my cousin’s cousin is my cousin (because your father’s son is your brother, and your son’s father is you, my father’s father’s son’s son’s father’s father’s son’s son is necessarily my father’s brother’s son, or equivalently, my father’s father’s son’s son, as am I myself).
Figure 5.12. Equivalence relations induced by figure 5.11
It is really the existence of these induced equivalence classes, and the collective action they facilitate, that gives the unilineal family tree its importance. The tree itself rarely exists as much of a social structure, because a large number of the relevant nodes refer to dead people who do not interact (see Zerubavel 2003: 57–79). The tree is then less often an actual structure of interaction than it is a guide to the parsimonious construction of nested relations of equivalence, and hence the facilitation of certain interactions (also see Firth 1963 [1936]: 328).
Not surprisingly, where descent is unilineal (especially patrilineal), social organization often follows the implied structure of nested circles (though see Barth 1981: 149f for a qualification). Perhaps the best example here are a number of North African societies such as the Berbers.41 Early Roman sources suggest that such tribes retained around five levels of organization: any person was a member of a family group (domus), in turn incorporated into extended family groups (familae), in turn incorporated into clans, in turn incorporated into subtribes, in turn incorporated into tribes, tribes which then formed confederations such as the Numidae (see Mattingly 1992: 36; Sahlins 1968: 24). In other cases, these classes may not be in continuous existence as corporate groups—they may be called into being to accomplish certain collective goals requiring coordination at a certain level, such as when the tribes of Israel were summoned to redress some collective outrage. Such groups can quickly assemble according to the simple heuristic “given an insult or injury coming from someone with degree of relation k to me, unite with all Tk − 1–equivalents against him.” (Among North and East Africans—it is attributed to a wide number of Bedouin and Sudanese groups—there is a saying: “I against my brother; I and my brother against our cousin; I, my brother and our cousin against the neighbors; All of us against the foreigner [Chatwin 2003: 201; cf. Sahlins 1968: 50f; Barth 1954: 166; Mair 1977: 37]).
Figure 5.13. Nesting of equivalences
In other words, from the perspective of any person—say, the double outlined circle in figure 5.11—the social order looks as arranged in figure 5.13. Inside the inner circle are ego’s brothers, all under the same father (the somewhat larger circle above the double outlined one); inside the outer circle are ego’s cousins, all under the same grandfather (the very large circle at the top), and so on, with successive circles of decreasing closeness. As Landé (1973: 106) writes, “Like the ripples from a stone dropped into a pool, the strength of kinship ties gradually declines as the genealogical and affective distance of kinsmen from ego.” It is this pattern of ripples that is phenomenologically important to the actors. Even should the grandfather be removed, the relative distance of the different persons can be maintained and can be the basis for alliance.
What is of crucial import is the degree of consensus as to distance—because of the transitivity of equivalence, all “cousins” agree that they are all closer to each other than they are to anyone who is not at least a cousin. We might say that subjective representations of unilineal descent are architectonic in that they all speak to the same social structure of action (Fortes 1953: 29). But where descent is bilineal, while things may appear similarly to any one ego, the horizontal relations do not form equivalence classes, and so each person has a unique kin structure. For example, my cousin’s cousin is not necessarily my cousin under bilineality: my mother’s brother’s daughter’s mother’s brother’s daughter is probably not my cousin.42 Such kinship then isolates each nuclear family, but also embeds it in a complex web of latent ties which may be activated.43 As a consequence, there is no possibility of extended kin forming a bounded group (without the entry of much more complicated mechanisms involving regularities in marital exchange)—each person has two lineages, and those who share one do not necessarily share another. As Bloch (1961 [1940]: 138) notes, this leads the kin system to be poorly suited to form the backbone of the social structure as a whole—to channel all relations.
Consequently, we are not surprised to find a rough association between patrilineal organization and institutionalized feuding. This is not to say that there are not bilineal societies in which interfamily hostility and feuding are present; clearly there are many. But this is to say that bilineality tends to allow for dampening of the normal oscillations of the vendetta.44 Unilineal trees, on the other hand, are more often associated with crippling series of violent exchanges, precisely because the tree is such a clear structure for forming alliances.
Such corporate organization recurs where strong states are lacking and descent unilineal. But it is not simply that an existing tree coming from a unilineal descent structure can facilitate alliances. It is also possible for persons to cognitively simplify their alliance relations by interpreting them as kinship relations, especially where kinship is already seen as unilineal (see Fortes 1953: 27). For one example, the tribal structure of ancient Israel, rather than a result of patrilineal descent of the “twelve children of Israel” (the man Jacob), was quite possibly a federation of tribes who then invented lineages justifying their mutual obligations. Thus a member of the tribe of Judah had greater responsibilities to come to the aid of a member his own tribe than a member of the tribe of Benjamin, and a greater responsibility to aid a member of the tribe of Benjamin than one descended from Jacob with a different mother, and even less interest in supporting a member of one of the neighboring tribes said to descend from Esau (as opposed to Jacob). Still more distant would be those descended from Ishmael. The Nuer similarly re-create lineages as they adopt new members to best make sense of the current alliances.45 In a word, the tree structure is quite useful, and we have found that it cannot be derived from the structures that emerge from influence relationships.
In some sense, influence as a relationship is not hierarchical enough to establish equivalence. The choice element makes it impossible to use common influence to rationalize the horizontal social distance of allies, in what we can consider “social cladistics” (cladistics being the branch of evolutionary science that attempts to graph degrees of likeness through tree structures [cf. Zerubavel 2003]). But we cannot choose our fathers. In part because of this lack of choice, the tree structure can also be used to express certain ambiguities in vertical relations. In particular, the kinship tree can explain a relation of support from a higher-ranking person or family or clan to a lower-ranking one. Such relations are often called “patronage” relations, and it is significant that the word patron comes from the root “father.” The patronage relationship allows unequals to assimilate their relationship to a well-understood familial one, thus justifying restraint from naked exploitation and resentment. Patronage relations can lead to actual adoption or merely the invention of more distant pseudo-familial connections (thus in the mid-twentieth century the Bobo tribesman who relocated to Mali generally became the client of a local protector and assumed the ethnic identity of this patron [Lemarchand 1972: 70]).46
The tree structure, then, facilitates collective action where other corporate groups are lacking. In some formal structures, they are deliberately constructed, but they may also emerge spontaneously. In many cases, such structures arise because of patterns of unilineal kinship. But in other cases, this same structure arises for other vertical, antisymmetric relationships. But this is not when the antisymmetric relationships express some sort of endogenous inequality—an inequality of choice, as in the popularity tournament or its vertical cousin, the influence triset. Instead, it is when the relationships are being formed against an exogenous backdrop of serious material inequality. Such an environment is an extremely common one, and it is therefore not surprising if such structures are either common or important or both. Indeed, we will find that the resulting social structures are generally the crucial building blocks for the largest political structures that first impressed Spencer and Comte. We go on in the next chapter to examine their emergence.
1 A note on how I adopt this terminology without the underlying scheme: first, White calls the forms of organization “disciplines” and considers the dimensions of dependence, involution, and differentiation (briefly discussed in chapters 2, 3 and 4) to be something that, in some ways, appear “in between” the disciplines (thus if we were to put the disciplines as apexes in a triangular chart, these dimensions would appear as the sides). The exposition here loses this formal elegance and comes closer to lumping the two together (in effect, rotating the dimensions by nearly 120°). Thus the correspondence to White’s terms falls quite short of a proper allegory, but much of the logic of trade-offs between different forms holds. Second, White considers selection characteristic of the “arena,” the discipline in which (among other things) there is a tendency to transitive closure. I argue that all strong forms of organization are compatible with transitivity, but that the transitivity of selection is not a subjective heuristic but instead should be understood as a side effect of the stratification of persons.
2 White thinks more of producer markets than marriages here, which he subsumes under the discipline of the “interface,” but he also highlights the importance of inequality and mutuality to the relationships.
3 Thus one could perhaps redraw the figure below as three triangles (with an empty inverted triangle in the center), an upper one for asymmetric relationships like friendship choices (the base of which is a horizontal line midway between the apex and base of the larger triangle), a left and lower one for symmetric relationships such as marriage (the apex of which is midway on the left side of the larger triangle) and a right and lower one for antisymmetric relationships such as dominance (the apex of which is midway on the right side of the larger triangle).
4 Kant (1991 [1797]: 250) makes this distinction between a willful and free sharing in another’s feelings on the one hand and an unfree sharing due to susceptibility that involves communicability.
5 If the number of carriers is x out of a total number of N persons, then the probability that j is not a carrier is (N − x)/N, and the probability that i is a carrier given that j is not one is x/(N − 1). We subtract the one because since we know that j is not a carrier, the probability that i is one of the x carriers increases somewhat. We multiply this product by p and q to find that the probability of successful transmission to be pqx (N − x) / [N (N − 1)]. We then multiply this by the N(N − 1) possible dyads to figure out the probability that there is some transmission in a period. Calling pq k for short, this yields the familiar logistic expression of contagion in a fixed population: dx/dt = kx (N − x). Assuming that x = 1 at t = 0, this leads to x = Nekt/[N − 1 + ekt] (here see Coleman 1964).
6In support, Coleman et al. (1966: 114, 119, 123f) pointed to what seemed to be a greater tendency of those pairs who discussed matters with each other to adopt the drug at roughly the same time–at least in the first few months after release of the drug. This result, unfortunately, is probably spurious, due to the floor effect whereby no one could adopt the drag before it came out, and hence early adopters were more likely to adopt “close in time” to one another while late adopters were not. Coleman et al. made a second mistake: they equated being peripheral–receiving few choices from others as an influential person, a friend, a conversation partner–as equivalent to social isolation. But if influence flows as expected, it is not being named that means that Dr. X will convert to use of the new drag as a result of social influence, it is naming others–that is. being receptive to social influence.
7 Other processes leading to a vaguely logistic form include normative pressure, competitive concern, and increasing returns to market share (for example, it makes sense to buy the operating system that more people have) (see Van den Bulte and Lilien 2001), all of which suggest an “overdispersion” logic according to which people are more likely to zero in on the same choices than would be predicted under a random diffusion model. As Granovetter (1978) has shown, it is basically impossible to use aggregate data to distinguish a diffusion process from one based on varying thresholds. Valente (1995: 81) suggests that in some cases one may be able to see the difference between an “external” influence and a diffusion process by eyeballing the distribution, but there is little hope of demonstrating the reliability of such a procedure.
8 Coleman, Katz, and Menzel (1966: 163) recognized and discussed the possibility that this curve results from simple heterogeneity in propensity to accept innovations–if this propensity is normally distributed, the cumulative density function is basically the same as the logistic (see Valente [1995: 64] for a more thorough treatment).
9 Van den Bulte and Lilien (2001: 1410f, 1419, 1427) recently reanalyzed the setting of Coleman et al.’s study and found that the adoption of the innovation in question is better explained by advertising than by interpersonal influence, whether structured or unstructured. Coleman et al. (1966: 52, 53, 61) had reported that doctors themselves acknowledged the importance of commercial influences such as ads and pharmaceutical representatives (“the detail man”) and indeed ranked such influences much more important than influence from other doctors, but Coleman et al. emphasized personal influence in their theoretical account.
10 This despite the fact that Sorokin (1959 [1941]: 631f) had made precisely this point. Another example is Crane’s (1972: 24f) argument that scientific growth is a form of what I am here calling contagion, and hence that the number of persons in some particular research field who accept some new idea should be proportional to the number who have already adopted. While Crane admitted that a number of cases she examined did not fit the model, she concluded that there was rather good evidence of science following this pattern. Yet close inspection of the curves suggests something else. Even if we discount the fact that for all research areas there was a big boom in the 1950s and 1960s, such that those that had previously seemed to be at their point of maximum “saturation” again began growing—due to the expansion of academia in the postwar era, and not with the internal dynamics of any field—and focus on the curves that seem to fit the contagion model, we cannot distinguish between any number of reasonable processes of monotonic increase.
11 In the contagion model, we derived the probability of any person acquiring the innovation as pq[x(N − x)]. We might understand this pressure as generalizing the contagion model by allowing q = (1 + b)xaq0, where q0 is some constant base rate; if a = b = 0 we have the classic contagion model. (This is a more general generalization than that following Coleman’s [1964] contagious Poisson model. Thus it may be seen as the “overdispersed” equivalent of the normal diffusion process, just as the negative binomial is an overdispersed form of a Poisson process. This similarity is theoretically important and will be discussed below.) This leads to a new expression for the change as follows, namely dx/dt = p(1 + b)xaq0x(N − x). Calling pq0(1 + b)k* for short, this is then = k* xa +1 (N − x).
12 In the formulation given in the previous note, we find that the increase in infected is fastest when the proportion of the population that is already infected is (a + 1)/(a + 2) (and note that this is .5 if a = 0). Since dx/dt = k*xa + 1(N − x), d2x/dt2 = k* [(N − x)(a + 1) xa − xa + 1] = k* xa[N (a + 1) − (a + 2) x]. Setting this equal to zero leads to one non-trivial root where N(a + 1) = (a + 2) x or x = N(a + 1)/(a + 2). The b parameter affects the total likelihood of spread, not where the curve is inflected.
13 In some ways, this harks back to the failure of the first principled use of statistics in sociology, namely the application of Gaussian curves to demonstrate the existence of “social facts.” Adolfe Quetelet and his followers in French sociology argued that the normal distribution of many individual measurements demonstrated the existence of social law–something must constrain individuals if their aggregate distribution conformed to this mathematical construct. This basic idea formed the heart of Durkheim’s sociology and passed into Anglo-American sociology from there, but it is mathematically nonsensical. As William Lexis showed in 1879, the Gaussian is exactly what one would expect if there was no supraindividual constraint (Porter 1986: 249). Social constraint would be indicated by over- or underdispersion. Yet without an extremely strong theory of the precise nature of the process generating the data, it is well-nigh impossible to determine whether data show evidence of such constraint. So too with the contagion data.
14 When a disease is transmitted, the death of the infected and their replacement with younger uninfected can keep the equilibrium below saturation; when it comes to cultural transmission, transmission must be very spotty for normal turnover of population to prevent near saturation.
15 This is not true of all cultural elements; there are some practices (for example, those that relate to new technologies) that may not contain any intrinsic limitation to their diffusion.
16 For other studies of selective diffusion see Zachary (1977).
17 In chapter 2, we saw that the combination of these two types of space meant that weak ties of acquaintance were not likely to follow a spatial logic, since those close in either type of space are likely to be acquainted. But for stronger ties, the two distances may concatenate via an “AND” logic and hence lead to a spatial logic. For example, much (although not all) of the sexual interaction that can diffuse disease is likely to happen between people who are close both in geographical space and social space.
18 It is also interesting that such models imply only weak transitivity, as pointed out by Erickson (1988).
19 This example is hypothetical because longitudinal geographic data on snowblower use and/or sales is evidently so difficult to come by that the Nonroad Engine Emission Modeling Team of the EPA Office of Mobile Sources has created a computer program to estimate use based on climate data and housing stock. See their report NR-014, September 16, 1998. I am not making this up.
20 This example is more complex than others, as the relational space constructed also takes into account structural similarities in addition to the mere presence of ties.
21 While this might seem a bit arbitrary, one may reconceive space as a three-dimensional mountain range, where positions are not merely close to one another on the X and Y dimensions, but have different heights along the Z dimension, such that cultural elements will “roll” down from one person to another but not the other way around. Friedkin’s technique may be understood as a version of such a model.
22 Because contact is symmetric we here assume that influence is symmetric; it is, however, possible to envision a structure in which hub-spoke relationships are antisymmetric.
23 In a recent work, Karin Knorr-Cetina (1999: 180) suggests that in high-energy physics, a branch of science tied to extremely large experimental objects, the structure of work groups flattens considerably and leaders cease being apexes (those having an asymmetric relation to many others). Instead, they become mere hubs (those with symmetric relations to many others)–they are simply those “centrally located in the conversation conducted within the laboratory.” Her parallel investigation of molecular biology, on the other hand, reveals the more traditional triangles.
24 Some of the structures that we will investigate in the next chapter, such as the “big men” of certain Pacific Island societies, also tend to concatenate into multihub structures; it is therefore not surprising that Hage and Harary (1983: 38) report that the big men of Mount Hagen in New Guinea explicitly see themselves as “communication mediators.”
25 More intelligent systems that allow local structures to effectively contain knowledge of the structure as a whole can be more efficient than a multihub system, but these are difficult to establish without deliberate architecture.
26 It is worth noting that these structures must be understood as distinct from the more general small-world graphs we explored above; though multihub structures may in some cases have small-world properties, they need not (and probably do not when the degree of spoke-hub connections is high relative to the hub-hub connections). Though both are good at diffusing information, there is an essential difference in that small-world graphs come about in spite of tendencies toward social structure, while multihubs are themselves a form of structure.
27 Hedström et al. (2000) have recently discussed the importance of such structures for diffusion but distinguish the pattern of connections between hubs as a “mesolever” network distinct from the local networks revolving around each hub.
28 Although sociologists (e.g., Burt 1992), thinking of informal connections that cross-span formal organizations, have tended to stress the advantages that come from being able to broker information, there can be costs to being the person always expected to go to the source for some information, and then disseminate it to others. These costs may be expected to increase with the number of relationships actively maintained.
29 Yamaguchi (2002: especially 175, 178) has recently formalized such a model of hub structures arising as an equilibrium from individual choices.
30 For a different formalization leading to different results, see Jackson and Wolinksy (1996, especially 49f).
31 There has been a great deal of hoopla in social networks research in recent years over the class of large graphs with very skewed degree distributions–examples are the World Wide Web, the Internet Movie Data Base, and so on. These graphs tend to have many people with low degree, exponentially fewer with medium degree, still fewer of high degree, and a very small number with a very high degree indeed. Breakthroughs in the mathematical treatment of such large and scale-free networks have been extremely important in network analysis but have led some casual observers to imagine that social networks must have a similar form (since there are also a lot of people). While there have been some innovative attempts to estimate the distribution of the number of people that we know (e.g,. Bernard et al. 1989; Bernard et al. 1990; and recently Zheng et al. 2006), no one has any idea what the actual distribution of degree of friendship or acquaintance is, if only because we are not sure what it means to be “acquainted.” On social and cognitive grounds a scale-free distribution for either is implausible. (For acquaintance, a skewed normal distribution is most likely, and for friendship, an over-dispersed Poisson.)
32 In fact, this process leads to a special case of partial order where if we let the in-degree of person i be Di (remembering to count the transitively implied relations that are suppressed in the Hasse diagrams printed above) and denote the relationship in question as →, if i → j, then Di < Dj. If this condition holds or is approximated for some structure, this might be taken as suggestive evidence that there is an underlying ranking. That is, with proper discrimination we could say that if Di < Dj, then we know that there exists some ζ as defined above such that ζi > ζj. Also see Landau (1953) for further explorations of such a score structure.
33 There have been many network studies of influence over the past three decades, but most are influence relations within organizations, and thus do not fall under the class of the elementary structures considered here. However, even here, classic studies (e.g., Blau 1955: 129) find the same structural principles for influence relations. Other work on informal “ganglike” structures confirms the patterns described by Whyte (see, for example, Suttles 1968: 189f).
34 The word tree is used in different ways; in some cases in graph theory it refers to any graph that has N – 1 edges for N nodes. Here it will be used to denote the particular vertical structure in which any node has only one superordinate.
35 The ambiguity as to whether trees include all transitive relations is also found in the discussion of Hage and Harary (1983: 87).
36 An important exception is Harary (1959), who used the definition of tree that implied antitransitivity and demonstrated that this allowed for a measure of status that was equivalent to a recursively defined set of partially ordered cardinals.
37 Such structures turn out to have some interesting mathematical properties. In particular, if something diffuses through the influence relationships, the sets of possible states pertaining to which people hold some thing at any time forms an algebraic structure (a lattice closed under union) from which the triset can be reproduced (Martin 2002).
38 Persons in the data set are indicated by numbers; named professionals are given by their role. DNR stands for the Department of Natural Resources.
39 There is one, and only one, mutual relation, that between persons O8 and O22.
40 While it is possible to define a class of persons who all stand in the relation “mother” or “father” to some ego, it is also possible to make motherhood or fatherhood singular by definition.
41 A similar form of organization–families within clans, clans within bands, and bands within a troop–is also found in the extremely patrilineal organization of hamadryas baboons (see Kummer 1984; Kummer 1995: 144ff; Dunbar 1984: 16f).
42 Note that since we must assume two sexes in bilineal descent, my mother’s daughter is not necessarily myself (as I may be a man).
43 In some cases, such as the !Kung Khoisan, one family or set of siblings can be the center of a politico-territorial unit, with concentric circles formed by the sibling’s spouses, the siblings of these spouses, and spouses of these siblings, etc. (Lee 1972: 351; for the Pacific Northwest and a theoretical statement, see Ives 1998: especially 188ff; Sahlins 1968: 54f). The “ripples” that result from this initial structure then channel the possible corporate actors in ways that are heavily dependent on the precise nature of the founding.
44 For example, while the Greek Sarakatsani shepherds consider all those as far as their second cousins (under bilineal descent) kin, it is the nuclear family that is a strong corporate group. Thus although the Sarakatsani tend to assume that the interests of families are mutually opposed and should and do lead to violence, such violence is contained by the fact that only one’s immediate family have an obligation to take on revenge killings (Campbell 1964: 36, 42, 9, 41, 96, 55).
45 On the ancient Greeks creating a similar scheme of descent see Bury (1955 [1900]:79).
46 The tendency of European colonists and well-armed explorers to be called “father” by those they overawed–often taken by the Europeans as a sign of the childlikeness of the natives and the legitimacy of European rule–was generally due to an assimilation of the Native-European relation to existing client-patron ties which were given familial terms (Jennings 1984: 7f, 40f, 161; for an example pertaining to the Algonquian, see Anderson [2000: 15]). These terms were taken quite seriously in that “son,” “grandson,” “nephew,” or “brother” connoted different political relationships (and due to different child-rearing practices, to the Iroquois “father” did not connote the capacity to command), and disagreement as to which kin-term characterized the relationship could cause a serious breakdown in relations between Indian nations or between Indians and settlers. (On the attempt of Chinese rulers to identify feudal and familial terms during the Western Chou period, see Hsu [1965: 3, 7, 52]; on the flexibility of the Thai use of kinship terms to create, define, and express closer relations, see Kemp [1984: 60] and compare McLean [1996]).
