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Relations, Species, and Network Structure*
Department of Sociology, University of South Carolina
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 do 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.
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 4 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 1. 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 do not depend the labels attached to the nodes.
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():
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.
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:
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.
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 1. 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 1 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 1 if the original relation was valued). More details about each of the networks can be found in the Appendix.
Table 1. Description of Networks
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:
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 animal since only humans provided reports of their ties to others. Second, we categorize the relation as either positive or negative. Grooming, advice seeking, liking, etc. are considered positive, whereas dominance, agonistic encounters, and disliking are negative. This leads to four types: observed positive, observed negative, reported positive, or reported negative.
Figures 2-6 display results of the correspondence analysis. Figure 2 shows the location of each data set in the first two dimensions. The closer together two networks are the more similar are their predictions for the other networks in the collection. Thus, for example, "baboonnm2" is relatively far from "kids1" and so the two networks make very different predictions for other networks. Figures 3 through 6 analyze the location of the networks based on type of animal and type of relation. We present 68.2% confidence ellipses around the networks, centered on a category’s means along the first two dimensions with orientation determined by the covariance of the scores of the category’s networks on the two dimensions. Larger ellipses mean more variability in the location of networks of a certain type in the two dimensional space. For mutually exclusive categories like, say, humans and non-human primates, the smaller the overlap of the respective ellipses, the more distinctive is the region in two dimensional space occupied by networks in one category as opposed to the other.
Figures 2-6 display results of the correspondence analysis. Figure 2a shows the location of each network in the first two dimensions. The color and shape of the points code whether the valence of the relation is "positive" or "negative" (as defined below), whether it was recorded by observers or reported by network participants, and the kind of animal involved (human or non-human primate). Figure 2b is another version of the same figure, but with each point labeled by the network it represents. These labels and descriptions of the networks are in Table 1. In both figures 2a and 2b the closer together two networks are the more similar are their predictions for the other networks in the collection. Thus, for example, "baboonnm2" is relatively far from "kids1" and so the two networks make very different predictions for other networks. Figures 3 through 6 analyze the location of the networks based on type of animal and type of relation. We present 68.2% confidence ellipses around the networks, centered on a category's means along the first two dimensions with orientation determined by the covariance of the scores of the category's networks on the two dimensions. Larger ellipses mean more variability in the location of networks of a certain type in the two dimensional space. For mutually exclusive categories like, say, humans and non-human primates, the smaller the overlap of the respective ellipses, the more distinctive is the region in two dimensional space occupied by networks in one category as opposed to the other.
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