Relations, Species, and Network Structure

Relations, Species, and Network Structure* John Skvoretz Department of Sociology, University of South Carolina Katherine Faust Department of Sociology, University of California, Irvine * For their encouragement and suggestions on the research, we thank H Russell Bernard, Linton Freeman, and A Kimball Romney Discussion with Tom Snijders on the p* models was most helpful We also thank Mike Burton for suggesting the matrix permutation approach On a more personal note, we would like to acknowledge and celebrate the influence of Linton Freeman on our careers On a visit to Lehigh University in the Fall of 1968 to give a talk, Lin advised John, a double major in Sociology and Mathematics, to his graduate work at Pittsburgh with a young professor named Tom Fararo, thereby setting in motion a life-long interest in networks and structure And, it was after Lin joined the School of Social Sciences at the University of California, Irvine in 1979 as dean and catalyst for the Social Networks Program that Katie's research interests turned to social networks It was also Lin who encouraged Katie to go to the University of South Carolina, thereby making possible the collaboration that led to this research ABSTRACT: The research we report here tests the "Freeman-Linton Hypothesis" which we take as arguing that the structure of a set of relational ties over a population is more strongly determined by type of relation than it is by the type of species from which the population is drawn Testing this hypothesis requires characterizing networks in terms of the structural properties they exhibit and comparing networks based on these properties We introduce the idea of a structural signature to refer to the profile of effects of a set of structural properties used to characterize a network We use methodology described in Faust and Skvoretz (forthcoming) for comparing networks from diverse settings, including different animal species, relational contents, and sizes of the communities involved Our empirical base consists of 80 networks from three kinds of species (humans, non-human primates, non-primate mammals) and covering distinct types of relations such as influence, grooming, and agonistic encounters The methods we use allow us to scale networks according to the degree of similarity in their structuring and then to identify sources of their similarities Our work counts as a replication of a previous study that outlined the general methodology However, as compared to the previous study, the current one finds less support for the Freeman-Linton Hypothesis "My overall goal is to learn something basic about the foundations and consequences of the sociability of social animals." Linton Freeman, 1999, Research in Social Networks (http://eclectic.ss.uci.edu/~lin/work.html) " just as the physical differences between men and apes diminish in importance and cease to be a bar to a relationship when they are studied against the background of mammalian variation, the differences in behavior diminish in importance when they are seen in their proper perspective." " human and animal behavior can be shown to have so much in common that the gap ceases to be of great importance." Ralph Linton, 1936, The Study of Man (New York: The Free Press) Introduction The passages of Ralph Linton quoted above suggest that the behavioral commonalities between humans and animals are substantial The claim would extend to social behavior, in particular, behavior in regard to others of the same species, "the sociability of the social animals." This view is echoed in Lin Freeman's work That is, both authors would contend that the networks of baboons and school children, of cattle and bank clerks, and of fraternity brothers and ponies would be similarly structured whenever the nature of the behavior defining the connections was common to both networks In this paper we explore what we will call the Freeman-Linton Hypothesis, named after the scholars quoted above In particular, we examine 80 different networks from three types of species (humans, non-human primates, and non-primate mammals), varying in size from to 73 units Many distinct types of relations are included: from liking, influence and grooming to disliking and victory in agonistic encounters Our specific research question is whether patterning in a network can be better predicted by type of animal or type of relation The Freeman-Linton hypothesis leads us to expect that type of relation will matter much more than type of social animal To investigate this hypothesis requires a methodology that allows the comparison of many networks even though they may vary dramatically in size, in type of social animal, and in relational contents The methodology should provide an abstract way of characterizing the structure of a network apart from the particular individuals involved It should also provide a set of guiding principles for what it means to say that two networks are similarly structured The method we build on has been described in detail elsewhere (Faust and Skvoretz forthcoming) In the next section we outline the steps in that method We then apply it to our networks, replicating the original analysis, which was restricted to a smaller set of networks (42 in number) We also extend the original analysis to consider systematically sources of variation in network structuring among networks of different species and different types of relations We conclude the paper with a discussion of directions for future work with particular attention to the theoretical questions our project may address Representation of the Structural Signature of a Network Faust and Skvoretz (forthcoming) propose a method that allows researchers to measure the similarity between pair of networks and to look at the overall patterning of similarities among a large collection of networks from diverse settings Their basic argument is that two networks are similarly structured, that is, have the same structural "signature," to the extent that the networks exhibit the same structural properties and to the same degree One way to quantify the magnitudes and directions of network's structural properties is to use a statistical model In that case, two networks are similarly structured if the probability of a tie between i and j is affected by the same set of structural factors to the same degree in both networks To explicate this idea, consider a single structural factor, say, mutuality and two networks: A is a network of advice ties between sales personnel and B is a network of helping relations between blue-collar workers Mutuality, the tendency for actor i to return a tie to actor j if j sends a tie to i, might be one structural factor that affects the probability of a tie between two actors in either network A or network B Tendencies toward mutuality have long been a concern of social network analysts (Katz and Powell 1955; Katz and Wilson 1956) and the measurement of mutuality remains a focus of contemporary research (Mandel 2000) It is a "structural" factor because it refers to a property of the arrangement of ties in any pair in the graph rather than to properties of the individuals composing the pair With just this one factor, Faust and Skvoretz would propose that networks A and B are similarly structured if a tendency toward mutuality is present or absent in both networks and to the same degree Specifically, their method calibrates the strength of such structural tendencies in terms of measures of impact that are invariant across networks that differ in size and overall density Therefore, strictly speaking, networks A and B are similarly structured if the standardized tendency toward mutuality is identical in both networks Of course, with just one structural factor, fine discriminations among the structural patterns in different networks are just not possible Networks that may be structurally distinct for other reasons (such as different tendencies towards transitivity) would be classed as similar because only one structural factor, mutuality, has been taken into account As additional factors are considered, finer and finer discriminations among entire sets of networks become possible But these finer distinctions require measuring multiple structural properties of the networks One could amass a collection of graph-based indices calculated on each network (mutuality, transitivity, ) and then compare these collections, but a more coherent approach is to estimate a set of effects simultaneously in the context of a statistical model for the network Thus the first step in the comparison methodology proposed by Faust and Skvoretz (forthcoming) is to estimate statistical models for the probability of a graph in which the set of predictor variables is expanded beyond simple mutuality Until recently, no statistical models were able to incorporate any structural effects beyond mutuality However, with the development of family of models known as p* such investigations became possible (Anderson et al 1999; Crouch et al 1998; Pattison and Wasserman 1999; Wasserman and Pattison 1996; Robins, Pattison, and Wasserman 1999) Faust and Skvoretz use a p* model that includes six structural properties: mutuality, transitivity, cyclical triples, and star configurations (in-stars, out-stars, and mixed stars) as illustrated in Figure The model is based on what Frank and Strauss (1986) call a "Markov" graph assumption This assumption stipulates that the state of a tie between i and j can only be influenced by the state of a tie between two other actors if at least one of these other actors is i or j Put another way, there is no impact "at a distance," meaning that the state of the tie between x and y cannot impact the state of the tie between w and z if x and y are complete different persons than w and z Furthermore, the model assumes that the Markov graph effects are homogeneous, that is, unrelated to specific labeled identities of actors Thus these effects are "purely structural" in that they not depend the labels attached to the nodes Figure Network Properties Included in the p* Models a Mutual b Out 2-star c In 2-star d Mixed 2-star e Transitive triple f Cyclic triple A p* model expresses the probability of a digraph G as a log-linear function of a vector of parameters , an associated vector of digraph statistics x(G), and a normalizing constant Z( ): (1) The normalizing constant insures that the probabilities sum to unity over all digraphs The parameters express how various "explanatory" properties of the digraph affect the probability of its occurrence The explanatory properties of the graph include the structural factors, like mutuality and transitivity mentioned above The model we use stipulates that the probability of a graph is a log-linear function of the number of mutual dyads, the number of out 2-stars, the number of in 2-stars, the number of mixed 2-stars, the number of transitive triples, and the number of cyclical triples If the resulting parameter estimate for a specific property is large and positive, then graphs with that property have large probabilities For example, if mutuality has a positive coefficient, then a graph with many mutual dyads has a higher probability than a graph with few mutual dyads Or, if the cyclical triple property has a negative coefficient, then a graph with many cyclical triples has a lower probability than a graph with few cyclical triples Thus, the resulting parameter estimates associated with the structural properties capture the importance of these properties for characterizing the network under study The set of parameters forms the structural signature of the network.[1] The equation (1) form of the model cannot be directly estimated Rather the literature proposes an indirect estimation procedure in which focuses on the conditional logit, the log of the probability that a tie exists between i and j divided by the probability it does not, given the rest of the graph (Strauss and Ikeda 1990; Wasserman and Pattison 1996) Derivation of this conditional logit shows it to be an indirect function of the explanatory properties of the graph Specifically, it is a function of the difference in the values of these variables when the tie between i and j is present versus when it is absent, as specified in the following equation: (2) where G-ij is the digraph including all adjacencies except the i,jth one, G+ is G-ij with xij=1 while G- is G-ij with xij=0 In the logit form of the model, the parameter estimates have slightly different interpretations For instance, if the cyclical triple property has a negative coefficient, then in the equation (1) version, we may say that a graph with many cyclical triples has a lower probability than a graph with few cyclical triples In the equation (2) version, the interpretation is that the log odds on the presence of a tie between i and j declines with an increase in the number of cyclical triples that would be created by its presence (Technically, however, interpretation is best phrased in terms of the probability of the graph.) The importance of the logit version of the model lies in the fact that, as Strauss and Ikeda (1990) show, the logit version can be estimated, albeit approximately, using logistic regression routines in standard statistical packages.[2] The significance for our problem of identifying the structural signature of a network is that it is possible to build and estimate models that capture multiple structural effects We are no longer limited to a structural signature built on only one or two factors In the research we report in the next section each network has a six-dimensional signature defined by the parameter estimates for the effects of the six structural factors diagrammed in Figure We also present several ways to compare the signatures of different networks, looking for similarities and differences One of these ways extends the work of Faust and Skvoretz (forthcoming) who use parameter estimates from different networks to generate sets of predicted tie probabilities for focal networks and then compare the sets of predicted probabilities using an Euclidean distance function Another way, new to the present research, explores the structural signatures based directly on the parameter estimates In all comparisons, we seek to assess the tenability of the Freeman-Linton hypothesis Specifically, we want to compare the structural signatures of human networks to the structural signatures of the networks of other species If we find, in fact, that the signatures differ, we want to see how much of the difference can be accounted for by "controlling for" relational type That is, the Freeman-Linton hypothesis would predict that any difference in the aggregate between human networks and the networks of other species would disappear once we take into account relational type In other words, the hypothesis holds that the nature of the behavior defining the connections, not species of social animal, is the fundamental factor determining a network's properties and thus its structural signature These are the implications of the hypothesis we seek to evaluate Comparisons of Structural Signatures Table lists the 80 networks we use to evaluate the Freeman-Linton hypothesis and to illustrate our methodology of comparison The networks range in size from four colobus monkeys to 73 high school boys The ties composing the networks also vary from advice relations and friendship ties to victories in agonistic encounters Each of the networks that we compare is represented by a 0,1 adjacency matrix (created by dichotomizing all non-zero entries equal if the original relation was valued) More details about each of the networks can be found in the Appendix Table Description of Networks Label Description N Type of Positive or Observed or Animal Negative Reported Relation Relation baboonf baboonm1 baboonm2 baboonm3 banka bankc bankf banks bkfrac bkhamc bkoffc bktecc camp92 cattle cole1 cole2 colobus1 colobus2 dominance between baboons (Hall and DeVore) 10 primate negative observed dominance between male baboons (Hall and DeVore) primate negative observed dominance between male baboons (Hall and DeVore) primate negative observed 21 primate negative observed 11 human positive reported 11 human positive reported close friends in a bank office (Pattison et al.) 11 human positive reported satisfying interaction in a bank office (Pattison et al.) 11 human positive reported rating of interaction frequency in a fraternity (Bernard et al.) 58 human positive reported rating of interaction frequency between ham radio operators (Bernard et al.) 44 human positive reported top rank order of interaction frequency in an office (Bernard et al.) 40 human positive reported top rank order of interaction frequency in a technical group (Bernard et al.) 34 human positive reported top rank order of interaction frequency in "Camp" (Borgatti et al.) 18 human positive reported contests between dairy cattle (Schein and Fohrman) 28 mammal negative observed friendship at time between adolescents (Coleman) 73 human positive reported friendship at time between adolescents (Coleman) 73 human positive reported non-agonistic social acts between colobus monkeys (Dunbar and Dunbar) human positive observed non-agonistic social acts between colobus monkeys (Dunbar and Dunbar) human positive observed outcomes of agonistic bouts between male baboons (Hausfater) advice in a bank office (Pattison et al.) confiding in a bank office (Pattison et al.) colobus3 eiesk1 eiesk2 eiesm fifth fourth ka kapfti1 kapfti2 kf kids1 kids2 medical newc0 newc0n newc1 newc1n newc10 newc10n newc11 newc11n non-agonistic social acts between colobus monkeys (Dunbar and Dunbar) human positive observed EIES data, rating of acquaintanceship (Freeman and Freeman) 32 human positive reported EIES data, rating of acquaintanceship (Freeman and Freeman) 32 human positive reported 32 human positive observed friendships between fifth graders (Anderson et al.) 22 human positive reported friendships between fourth graders (Anderson et al.) 24 human positive reported 21 human positive reported instrumental work relations in a tailor shop, time (Kapferer) 39 human positive reported instrumental work relations in a tailor shop, time (Kapferer) 39 human positive reported 21 human positive reported initiated agonism between children (Strayer and Strayer) 17 human negative observed dominance among nursery school boys (McGrew) 19 human negative observed 32 human positive reported top rankings of friendship in a fraternity, week (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week 10 (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week 10 (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week 11 (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week 11 (Newcomb) 17 human negative reported EIES data (Freeman and Freeman) advice between managers (Krackhardt) Krackhardt, friendship between managers physicians (Coleman, Katz and Menzel) newc12 newc12n newc13 newc13n newc14 newc14n newc15 newc15n newc2 newc2n newc3 newc3n newc4 newc4n newc5 newc5n newc6 newc6n newc7 newc7n newc8 top ranking of friendship in a fraternity, week 12 (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week 12 (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week 13 (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week 13 (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week 14 (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week 14 (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week 15 (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week 15 (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week (Newcomb) 17 human positive reported bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported top ranking of friendship in a fraternity, week (Newcomb) 17 human positive reported newc8n nfponies prison rhesus1 rhesus2 rhesus4 rhesus5 rhesus6 sampdes sampdlk sampes sampin samplk sampnin sampnpr samppr third vcbf vcg vcw bottom rankings of friendship in a fraternity, week (Newcomb) 17 human negative reported 13 mammal negative observed 67 human positive reported fights between adult female rhesus monkeys (Sade) primate negative observed fights between yearling rhesus monkeys (Sade) primate negative observed fights between adult rhesus monkeys (Sade) 10 primate negative observed fights between adult rhesus monkeys (Sade) 12 primate negative observed fights between adult rhesus monkeys (Sade) 10 primate negative observed 18 human negative reported 18 human negative reported 18 human positive reported 18 human positive reported 18 human positive reported negative influence between monks (Sampson) 18 human negative reported negative praise (blame) between monks (Sampson) 18 human negative reported 18 human positive reported friendship between third graders (Anderson et al.) 22 human positive reported best friends between seventh graders (Robins et al.) 29 human positive reported get on with between seventh graders (Wasserman and Pattison 1996) 29 human positive reported work with between seventh graders (Wasserman and Pattison 1996) 29 human positive reported threats between ponies (Tyler) friendship in a prison (MacRae) disesteem between monks (Sampson) dislike between monks (Sampson) esteem between monks (Sampson) influence between monks (Sampson) liking between monks (Sampson) praise between monks (Sampson) First, for each data set, we estimate the standardized coefficients for a p* model that expresses the conditional probability of a tie as a function of six structural factors: mutuality, out 2-stars, in 2-stars, mixed 2-stars, transitive triples, and cyclical triples Second, we use these standardized parameter estimates and the standardized change scores in these structural factors to calculate the predicted probability of a tie in each i,j pair in each data set using as coefficients the parameter estimates from its own model and from each of the remaining 79 models Thus for each data set, we have 80 sets of predicted probabilities, one from each set of parameter estimates including the set of estimates from the focal data set itself The third step uses the Euclidean distance function: (3) where d(t,y) is the distance between a target network t and a predictor network y, pt(i,j) is the probability of the tie between i and j in network t calculated from its own p* estimates, py(i,j) is the probability of the tie between i and j in network t predicted by the p* parameter estimates from network y, and gt is the size of network t The distance is a (dis)similarity score between the predicted probabilities from the estimates derived from t, the target network itself, and the predicted probabilities from the estimates derived from y, one of the other 79 networks The 80 by 80 matrix of dissimilarity scores is the input data for two of our three comparisons of network structural signatures The first operation follows the methodology of Faust and Skvoretz (forthcoming) and uses correspondence analysis to represent the proximities among all of the networks The resulting configuration is interpreted in light of the type of social animal and the type of relation The second operation uses matrix permutation tests to model the dissimilarity scores as linear functions of predictor variables including type of social animal and type of relation The third comparison of the structural signatures of the 80 networks directly inspects the standardized parameter estimates themselves, comparing their mean values across categories of animal type and relation type Correspondence analysis results Correspondence analysis involves a singular value decomposition of an appropriately scaled matrix Entries in the input matrix are divided by the square root of the product of the row and column marginal totals, prior to singular value decomposition Correspondence analysis is used because it does not require symmetric input data Since correspondence analysis requires that data refer to similarities rather than dissimilarities, we rescale the Euclidean distances by subtracting each from a large positive constant prior to doing the correspondence analysis (Carroll, Kumbasar, and Romney 1997) The matrix of similarities we analyze is not symmetric, that is, the distance between network x's prediction for network y and network y's prediction for its own data does not, in general, equal the distance between y's prediction for network x and network x's prediction for its own data In the following graphs we present the column scores from correspondence analysis of the matrix of similarities among the networks Column scores show similarities among networks in terms of the predictions they make for other networks Thus in the figures two networks are close together if they similarly predict other networks in the collection The following graphs show the results of the correspondence analysis in the aggregate and then disaggregated by species and type of relation Species is a categorical variable taking on three values, humans, non-human primates and non-primate mammals We highlight the contrast between humans and non-human primates because we have relatively few (only 2) networks among mammals in our set of 80 cases Relations are first categorized by how they were collected: observation or reported by respondent Obviously this is confounded with the type of Figure 12 Box Plot of In 2-Star Parameters from p* Model by Type of Animal and Whether Relation is Positive or Negative Figure 13 Box Plot of Mixed 2-Star Parameters from p* Model by Type of Animal and Whether Relation is Positive or Negative Figure 14 Box Plot of Transitive Triple Parameters from p* Model by Type of Animal and Whether Relation is Positive or Negative Figure 15 Box Plot of Cyclic Triple Parameters from p* Model by Type of Animal and Whether Relation is Positive or Negative Figure clearly shows that positive relations are similarly structured among humans and among primates Perhaps the sole exception is with respect to the transitive triples effect, which is modestly positive in the human networks and absent completely in the positive primate networks However, since there are relatively few positive primate networks, small differences in average parameter value should not be over interpreted Figure 9, on the other hand, clearly shows that negative relations among humans are structured quite differently than negative relations among primates With the exception of the out 2-stars effect, the effects of the six structural factors differ considerably between the kinds of species First, negative relations tend to be modestly mutual among humans but to exhibit anti-mutuality among the primates Second, among primates many transitive triples of negative ties tend to enhance graph probability, while among humans, many would tend to depress graph probability Third, among human negative relations there is no effect of cyclical triples, while among primate negative relations, the effect is clearly negative, meaning that graphs with fewer cyclical triples have higher probability Human negative relations tend not to be transitive (nor particularly cyclical), while primate negative relations tend to be both transitive and anti-cyclical Fourth, the effect of in 2-stars on graph probability is strongly positive among humans but modestly negative among primates, meaning that indegree variance in negative relations contributes to graph probability among humans but depresses graph probability among primates Finally, there is no effect of mixed 2stars among human negative relations, but there is a negative effect among primates That is, among primates nodes tend to be either sources or targets of negative ties, but not both Overall then we must conclude that the Freeman-Linton hypothesis is confirmed with respect to positive relations but not with respect to negative relations That is, species makes little difference in the "structural signatures" of positive relations However, the "structural signatures" of negative relations differ substantively in many ways between humans and primates Figures 16 and 17 graphically demonstrate the difference between positive and negative relations Figure 16 is based on a correspondence analysis of the distances between networks of positive relations among humans and primates, and Figure 17, on the distances between the negative relations Figure 17 clearly shows the separation between the kinds of species in two dimensions The 68.2% confidence ellipses not overlap at all Figure 16 shows that the positive human networks are embedded within the positive primate networks, the confidence ellipse for the human networks is completely within that for the primate networks Figure 16 also shows more research needs to be done There are only three positive primate networks and they are quite variable and so produce a very large confidence ellipse Furthermore, they appear to lie on the "outskirts" of the human positive networks although in different directions from human networks' centroid The pattern suggests that further research may find that the Freeman-Linton hypothesis will not be sustained even for positive relations Figure 16 Confidence Ellipses for Type of Animal Overlaid on Correspondence Analysis of Similarities between Positive Relations from p* Model Parameters, Column Scores Figure 17 Confidence Ellipses for Type of Animal Overlaid on Correspondence Analysis of Similarities between Negative Relations from p* Model Parameters, Column Scores Discussion The research we reported here had two aims: first, to replicate and extend methodology for the comparison of networks first introduced in Faust and Skvoretz (forthcoming) and, second, to evaluate evidence bearing on what we called the "Freeman-Linton Hypothesis." This hypothesis, suggested by observations of Linton Freeman and Ralph Linton, argues for no fundamental difference between the networks of relations among humans and the networks of relations among other primates or even among other nonprimate species More precisely stated, once relation type is taken into account, the networks are expected to be similarly structured, or in our terms, have similar "structural signatures." To conclude, we comment on two points: the tenability of the Freeman-Linton hypothesis and its value to research, even if it appears to be difficult to find empirical confirmation, and the general theoretical importance of a research program that systematically compares networks The Freeman-Linton hypothesis fails to be confirmed with respect to negative relations in our collection of networks That is, negative relations among primates have significantly different structural signatures than negative relations among humans How seriously should we take this disconfirmation? If we knew that the nature of the behavior defining the connections was common to both networks, we would be forced to abandon the hypothesis But, of course, we not know this for a fact indeed, can never know it for a fact Rather, the disconfirmation forces us to probe more deeply into the difference between the behaviors defining the negative relational networks in humans and the behaviors defining the negative relations among primates There is a difference Among primates, the negative ties are generated by success in agonistic encounters, conflictual interactions with winners and losers Among humans, most of the negative ties in our collection are generated by questions about the respondents' feelings towards others: they dislike or hold in low esteem other group members This indicator is neither behavioral nor "zero-sum" in the way an agonistic encounter is Given the ties we have analyzed, it is thus not surprising that negative relations among primates differ from those among humans once we realize just how different the ties are in behavioral contents Therefore, if we had networks among humans based on success in agonistic encounters, we expect that they would have very similar structural signatures to our negative networks among primates.[5] On the other hand, if we had networks based on avowed disliking or disesteem among primates, we expect that they would have structural signatures similar to our negative networks among humans We argue that the real value of the Freeman-Linton hypothesis lies not in whether it is empirically supported, but in its use as a tool to refine the conceptual categories by which we classify modes of social relatedness Empirical disconfirmation of the hypothesis, which we face in the current work, is a spur to more carefully delineate the fundamental dimensions along which social ties differ In the current research we have used rather crude classifications based on the valence of the tie, positive or negative, and on how data defining the tie were collected, by observation or by respondent report Even though this classification scheme is rough-cut, it still helps us to order the data on the 80 networks in a consistent fashion But it does not provide a complete account and thus supplies an impetus for further research based on more refined classification schemes Methodological concerns drive the current project's comparison of networks Our primary interest is the mechanics of how such a comparison might be accomplished, particularly, when the networks vary substantially in size, in relational type, and in species over which the relation is defined Yet the results create a theoretical agenda by raising questions about the theoretical reasons for similarities and differences in the structural signatures of networks We can illustrate this point with some examples Consider, for instance, the studies of work groups that often ask about the advice network among co-workers, that is, who goes to whom for advice Selection of advice seeking as an important social tie may have its roots in Peter Blau's early study of advice-seeking in a bureaucracy as a social exchange phenomenon (Blau 1955) Theoretically, one might argue that the nature of the advice seeking relation is revealed in the structural properties that affect the probability of particular graphs of the advice relation Thus we might expect advice networks in different organizational settings to have similar structural signatures Empirical research may reveal some commonalities but also some variants, thus establishing a problem for theory: what accounts for the commonalities, that is, an articulation of what is the "nature" of the advice seeking relation, and what accounts for its variants, for instance, the advice relation may differ in characteristic ways across cultures or across organizational forms or industries Further, the comparison of the structural signature of advice relations with other types of relations, such as friendship, may help illuminate the social and social psychological principles that differentiate such ties Clearly, the populations we study recognize such ties as different As researchers, we "know" such ties are different if for no other reason than that they, typically, link up different pairs of persons in our study population But to account for these differences and to even consider how to measure such differences are tasks that have not been as high on the social network research agenda as they perhaps should be Attention to these issues should give us a deeper understanding of the range and analytical types of relatedness among individuals Other territory opened for exploration would be any theoretical studies for which "ideal" types of networks are important conceptualizations in the theoretical enterprise An example here is the work of Markovsky (1998) on solidarity Markovsky hypothesizes that the solidarity of a group in terms of the network of communication ties among group members depends on the nature of the group Different types of groups have different "referent" networks against which their solidarity should be calibrated In Markovsky's thinking, a referent network is a pattern of actors and ties that identifies a class of networks He gives as an example a "cult" in which the referent network is "ten or more followers, each with ties of adoration and social attachment directed toward a leader, and each with ties to at least one other follower" (Markovsky 1998: 357) The solidarity of a group depends on the extent to which its interaction pattern approximates the referent pattern for its type This conceptualization is, clearly, comparative: to assess the solidarity of cult X we need to compare its interaction pattern with the referent pattern Our idea of "structural signatures" clarifies the nature of the comparison and says that the solidarity of cult X is measured by the similarity in the structural signatures of cult X and the cult referentnetwork Furthermore, the idea of structural signatures permits measuring degrees of solidarity that is, the extent to which cult X approximates the ideal pattern These theoretical dimensions to the comparison problem are independent of the specific methodology by which networks are compared These questions could have been asked several decades ago, but the answers, that is, the profile of structural effects that could be incorporated into the signature, were very limited Advances in the statistical modeling of networks have made it possible to expand vastly the elements composing a network's structural signature Yet we must recognize that this expansion is not the end of the story Future advances in statistical modeling will make the identification of a network's structural signature ever more precise There is no doubt that more precision will improve upon and possibly change the findings we have reported But more precision will not nullify the importance of the theoretical issues that network 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Psychometrika 61: 401-425 Appendix: List of Data Sources This appendix lists the 80 networks, describes the relations, gives a reference for the source of the data, and reports the label used in tables and figures Where data are published, the table number and page of the source are given • • • • • • • • • • • • • baboonf: dominance interactions between female and one adult male baboons (Figure 38, page 69, Hall and DeVore 1965) baboonm1 and baboonm2: dominance between male baboons (Table 3-2, page 60, Hall and DeVore 1965) baboonm3: outcomes of agonistic bouts between male baboons (Table XI, page 39, Hausfater 1975) banka: advice in a bank office (Table 5, page 558, Pattison et al 2000) bankc: confiding in a bank office (Table 5, page 558, Pattison et al.2000) bankf: close friends in a bank office (Table 5, page 558, Pattison et al.2000) banks: satisfying interaction in a bank office (Table 5, page 558, Pattison et al.2000) bkfrac: rating of interaction frequency in a fraternity Originally coded - 5; recoded 4,5 =1, 11=0 (Appendix A, Bernard, Killworth, and Sailer 1980, also available in UCINET, Borgatti, Everett, and Freeman 1999) camp92: top rank order of interaction frequency in "Camp" Originally a complete rank order from to 17, recoded 1-6=1, >6=0 (available in UCINET, Borgatti, Everett, and Freeman 1999) cattle: contests between dairy cattle (Figure 1, page 49, Schein and Fohrman 1955) • • • • • • • • • • • • • • • • • • • • • • • • • • • • cole1: friendship at time between high school boys (Table 14.5 (a), page 450, Coleman 1964) cole2: friendship at time between high school boys (Table 14.5 (b), page 451, Coleman 1964) colobus1: non-agonistic social acts between colobus monkeys in a small group (Table I, page 86, Dunbar and Dunbar 1976) colobus2: non-agonistic social acts between colobus monkeys in a small group (Table I, page 86, Dunbar and Dunbar 1976) colobus4: non-agonistic social acts between colobus monkeys in a large group (Table II, page 87, Dunbar and Dunbar 1976) eiesk1: EIES data, rating of acquaintanceship Recoded 3,4=1,