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 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.
"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 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.
Figure 1. 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 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
Label
Description
N Type of Positive or
Animal Negative
Relation
Observed
or
Reported
Relation
baboonf
dominance between baboons (Hall and
DeVore)
10 primate
negative
observed
baboonm1 dominance between male baboons (Hall
and DeVore)
6 primate
negative
observed
baboonm2 dominance between male baboons (Hall
and DeVore)
6 primate
negative
observed
baboonm3 outcomes of agonistic bouts between
male baboons (Hausfater)
21 primate
negative
observed
banka
advice in a bank office (Pattison et al.)
11 human
positive
reported
bankc
confiding in a bank office (Pattison et al.) 11 human
positive
reported
bankf
close friends in a bank office (Pattison et
al.)
11 human
positive
reported
banks
satisfying interaction in a bank office
(Pattison et al.)
11 human
positive
reported
bkfrac
rating of interaction frequency in a
fraternity (Bernard et al.)
58 human
positive
reported
bkhamc
rating of interaction frequency between
ham radio operators (Bernard et al.)
44 human
positive
reported
bkoffc
top rank order of interaction frequency in
an office (Bernard et al.)
40 human
positive
reported
bktecc
top rank order of interaction frequency in
a technical group (Bernard et al.)
34 human
positive
reported
camp92
top rank order of interaction frequency in
"Camp" (Borgatti et al.)
18 human
positive
reported
cattle
contests between dairy cattle (Schein and
Fohrman)
28 mammal
negative
observed
cole1
friendship at time 1 between adolescents
(Coleman)
73 human
positive
reported
cole2
friendship at time 2 between adolescents
(Coleman)
73 human
positive
reported
colobus1
non-agonistic social acts between
colobus monkeys (Dunbar and Dunbar)
4 human
positive
observed
colobus2
non-agonistic social acts between
colobus monkeys (Dunbar and Dunbar)
5 human
positive
observed
colobus3
non-agonistic social acts between
colobus monkeys (Dunbar and Dunbar)
9 human
positive
observed
eiesk1
EIES data, rating of acquaintanceship
32 human
positive
reported
(Freeman and Freeman)
eiesk2
EIES data, rating of acquaintanceship
(Freeman and Freeman)
32 human
positive
reported
eiesm
EIES data (Freeman and Freeman)
32 human
positive
observed
fifth
friendships between fifth graders
(Anderson et al.)
22 human
positive
reported
fourth
friendships between fourth graders
(Anderson et al.)
24 human
positive
reported
ka
advice between managers (Krackhardt)
21 human
positive
reported
kapfti1
instrumental work relations in a tailor
shop, time 1 (Kapferer)
39 human
positive
reported
kapfti2
instrumental work relations in a tailor
shop, time 2 (Kapferer)
39 human
positive
reported
kf
Krackhardt, friendship between
managers
21 human
positive
reported
kids1
initiated agonism between children
(Strayer and Strayer)
17 human
negative
observed
kids2
dominance among nursery school boys
(McGrew)
19 human
negative
observed
medical
physicians (Coleman, Katz and Menzel)
32 human
positive
reported
newc0
top rankings of friendship in a fraternity,
week 0 (Newcomb)
17 human
positive
reported
newc0n
bottom rankings of friendship in a
fraternity, week 0 (Newcomb)
17 human
negative
reported
newc1
top ranking of friendship in a fraternity,
week 1 (Newcomb)
17 human
positive
reported
newc1n
bottom rankings of friendship in a
fraternity, week 1 (Newcomb)
17 human
negative
reported
newc10
top ranking of friendship in a fraternity,
week 10 (Newcomb)
17 human
positive
reported
newc10n
bottom rankings of friendship in a
fraternity, week 10 (Newcomb)
17 human
negative
reported
newc11
top ranking of friendship in a fraternity,
week 11 (Newcomb)
17 human
positive
reported
newc11n
bottom rankings of friendship in a
fraternity, week 11 (Newcomb)
17 human
negative
reported
newc12
top ranking of friendship in a fraternity,
week 12 (Newcomb)
17 human
positive
reported
newc12n
bottom rankings of friendship in a
fraternity, week 12 (Newcomb)
17 human
negative
reported
newc13
top ranking of friendship in a fraternity,
week 13 (Newcomb)
17 human
positive
reported
newc13n
bottom rankings of friendship in a
fraternity, week 13 (Newcomb)
17 human
negative
reported
newc14
top ranking of friendship in a fraternity,
week 14 (Newcomb)
17 human
positive
reported
newc14n
bottom rankings of friendship in a
fraternity, week 14 (Newcomb)
17 human
negative
reported
newc15
top ranking of friendship in a fraternity,
week 15 (Newcomb)
17 human
positive
reported
newc15n
bottom rankings of friendship in a
fraternity, week 15 (Newcomb)
17 human
negative
reported
newc2
top ranking of friendship in a fraternity,
week 2 (Newcomb)
17 human
positive
reported
newc2n
bottom rankings of friendship in a
fraternity, week 2 (Newcomb)
17 human
negative
reported
newc3
top ranking of friendship in a fraternity,
week 3 (Newcomb)
17 human
positive
reported
newc3n
bottom rankings of friendship in a
fraternity, week 3 (Newcomb)
17 human
negative
reported
newc4
top ranking of friendship in a fraternity,
week 4 (Newcomb)
17 human
positive
reported
newc4n
bottom rankings of friendship in a
fraternity, week 4 (Newcomb)
17 human
negative
reported
newc5
top ranking of friendship in a fraternity,
week 5 (Newcomb)
17 human
positive
reported
newc5n
bottom rankings of friendship in a
fraternity, week 5 (Newcomb)
17 human
negative
reported
newc6
top ranking of friendship in a fraternity,
week 6 (Newcomb)
17 human
positive
reported
newc6n
bottom rankings of friendship in a
fraternity, week 6 (Newcomb)
17 human
negative
reported
newc7
top ranking of friendship in a fraternity,
week 7 (Newcomb)
17 human
positive
reported
newc7n
bottom rankings of friendship in a
fraternity, week 7 (Newcomb)
17 human
negative
reported
newc8
top ranking of friendship in a fraternity,
week 8 (Newcomb)
17 human
positive
reported
newc8n
bottom rankings of friendship in a
fraternity, week 8 (Newcomb)
17 human
negative
reported
nfponies
threats between ponies (Tyler)
13 mammal
negative
observed
prison
friendship in a prison (MacRae)
67 human
positive
reported
rhesus1
fights between adult female rhesus
monkeys (Sade)
7 primate
negative
observed
rhesus2
fights between yearling rhesus monkeys
(Sade)
5 primate
negative
observed
rhesus4
fights between adult rhesus monkeys
(Sade)
10 primate
negative
observed
rhesus5
fights between adult rhesus monkeys
(Sade)
12 primate
negative
observed
rhesus6
fights between adult rhesus monkeys
(Sade)
10 primate
negative
observed
sampdes
disesteem between monks (Sampson)
18 human
negative
reported
sampdlk
dislike between monks (Sampson)
18 human
negative
reported
sampes
esteem between monks (Sampson)
18 human
positive
reported
sampin
influence between monks (Sampson)
18 human
positive
reported
samplk
liking between monks (Sampson)
18 human
positive
reported
sampnin
negative influence between monks
(Sampson)
18 human
negative
reported
sampnpr
negative praise (blame) between monks
(Sampson)
18 human
negative
reported
samppr
praise between monks (Sampson)
18 human
positive
reported
third
friendship between third graders
(Anderson et al.)
22 human
positive
reported
vcbf
best friends between seventh graders
(Robins et al.)
29 human
positive
reported
vcg
get on with between seventh graders
(Wasserman and Pattison 1996)
29 human
positive
reported
vcw
work with between seventh graders
(Wasserman and Pattison 1996)
29 human
positive
reported
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
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 nonhuman 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.
Figure 2. Correspondence Analysis of Similarities between Networks, Column Scores
Figure 3. Confidence Ellipses for Type of Animal Overlaid on Correspondence Analysis
of Similarities between Networks, Column Scores
Figure 4. Confidence Ellipses for Positive or Negative Relation Overlaid on
Correspondence Analysis of Similarities between Networks, Column Scores
Figure 5. Confidence Ellipses for Observed or Reported Relation Overlaid on
Correspondence Analysis of Similarities between Networks, Column Scores
Figure 6. Confidence Ellipses for Type of Relation (Observed or Reported, Positive or
Negative) Overlaid on Correspondence Analysis of Similarities between Networks, Column
Scores
Each of the classifications contributes to understanding the clustering of the networks in the two
dimensional space. Despite some overlap, human and primate networks are found in different
regions, and so too are positive vs. negative and observed versus reported networks. Positive
reported networks are clearly distinguished from negative reported networks - their confidence
ellipses do not overlap at all. Positive observed and negative observed overlap and both are
more variable than either of the reported relations. Problematic for the Freeman-Linton
hypothesis is the clear distinction between the networks of different species particularly humans
vs. nonhuman primates. The data remain problematic for this hypothesis in an analysis of
variance comparing column scores along the first three dimensions between categories of the
classificatory variables, as reported in Table 2. Table 2 uses the proportion reduction in error
(PRE) in dimension scores due to the categorical grouping variables, as measured by the
correlation ratio squared, . Type of animal is an important distinction along the first and
second dimensions of the correspondence analysis. The kind of relation, especially the four
types differentiating observed vs. reported and positive vs. negative simultaneously, is an
important aspect of all three dimensions. But it is difficult to maintain the position that type of
relation is more important than type of animal in distinguishing among the networks. To
investigate this issue, we return to the distances between the networks and examine them directly
to answer the question: if we control for relation type, does the effect of animal type disappear?
Table 2. Proportion Reduction in Error Measures ( ) for Correspondence Analysis
Dimensions by Type of Animal, and Type of Relation
Type of Animal1
Observed or
Reported
Relation
Positive or
Negative
Relation
Type of Relation2
1
0.16**
0.20**
0.13**
0.67**
2
0.34**
0.22**
0.36**
0.46**
3
0.04
0.01
0.02
0.17*
Dimension
*
**
1
2
p < .05
p < .01
Type of animal: human, non-human primate, non-primate mammal
Type of Relation: observed positive, observed negative, reported positive, reported negative
Matrix permutation test results. A matrix permutation test allows us to test directly whether
distances between networks are significantly smaller when they are measured on the same kind
of animal than when they are measured on different kinds of animals, whether they are smaller
when both networks express the same type of relation, and then, controlling for the kind of
relation, does the greater similarity (i.e., smaller distances) between networks from the same
species disappear. We take the matrix of distances as the dependent variable and regress it on
matrices which express hypotheses about species and relations as the basis for similarity between
networks, using the methodology described in Krackhardt (1988). To execute the test, we build
0,1 matrices in which the ij cell is coded 1 if networks i and j are of the same type, that is, are
measured over the same type of animal, or the same type of relation. We then use these matrices
as predictors with the dependent variable being the matrix of Euclidean distances between
predicted probabilities. We first consider the bivariate relationship between the various
classificatory variables and distance and then pass to a multiple regression analysis of the
distances. A permutation test is used to assess the significance of the regression coefficients.
The results are presented in Table 3.
Table 3. Standardized Regression Coefficients Predicting Distances between Networks
using Matrix Permutation Regression, 80 Networks
Model Model Model Model Model Model Model Model Model
1
2
3
4
5
6
7
8
9
Positive or
Negative Relation
Observed or
Reported Relation
-
-0.037* 0.174**
-0.037
-0.035 0.189**
0.108**
0.420** 0.414** 0.278**
-0.037 -0.031
0.121
Relation,
Observed/Reported
and
Positive/Negative
-0.327**
0.307**
Animal
-
-
-
**
**
**
0.499
r2
0.012
0.177
0.178
*
0.199
0.249
0.493
0.250
0.468
0.249
- -0.477**
**
0.468
0.250
0.275
p< .05
p<.01
**
Models 1, 2, and 5 present the bivariate results. All the coefficients are negative, meaning that
networks in the same category have smaller distances between them than networks in different
categories. Type of animal clearly gives us the most explanatory power with respect to
accounting for the distances between networks (r2=.249), followed by observed vs. reported
relations (r2=.177). Although the coefficient for positive vs. negative relations is significant and
in the right direction, this classification accounts for relatively little of the variation in distance
between networks (r2=.012). Model 4 is the most comprehensive model for relation type alone.
It has a main effect for both observed vs. reported and for positive vs. negative, and an
interaction effect between the two classifications. Even this model accounts for less variation in
distance than the classification by type of animal. It is important to note that the coefficient for
positive or negative relation changes sign from model 3 to model 4, indicating an instability of
results when the interaction between positive vs. negative relation and observed vs. reported
relation is included in the model.
Models 5 through 9 are the multiple regressions and their results are consistent. Namely, animal
type is a significant factor no matter what aspect of relational type is controlled. Model 9 is the
most comprehensive model with an r2 of .275 and in this model, animal type has the largest
standardized coefficient. While aspects of relation type, net of animal type, are associated with
distance between networks, the direction is opposite that expected for the positive vs. negative
classification. However, it is difficult to interpret this effect since it occurs in the context of an
interaction effect between positive/negative and observed/reported.
The conclusion from the matrix permutation tests is very clear. The Freeman-Linton hypothesis
fails to receive confirmation in these 80 networks. The effects of type of animal on similarity of
network do not disappear when type of relation is taken into account. In these 80 networks,
contrary to the results of previous research (Faust and Skvoretz, forthcoming), animal type
matters much more than relation type. We continue to explore reasons for the impact of animal
type by inspecting variation in the p* parameter estimates directly, that is, by directly comparing
the structural signatures of the networks.[3]
Analysis of parameter estimates. Table 4 presents some descriptive statistics regarding
parameter estimates from the p* models. Figure 7 graphs the mean parameter values for humans
vs. primates. Since only 2 of the 80 networks refer to nonprimate mammals, they are dropped
from further consideration. Relations measured on humans comprise 66 of the networks and
relations measured on nonhuman primates make up the remaining 12 networks. Table 4 shows
that a particular structural effect cannot be estimated in some networks. For instance, an out 2stars effect cannot be estimated in 32 of the 66 networks. The 32 networks refer principally to
the Newcomb data in which each individual has the same outdegree in our recoding of the
rankings respondents gave to others. With no variance in outdegree, no out 2-star effect can be
estimated. Among the 12 primate networks, mutuality cannot be estimated in two of them. In
general, when we found it impossible to estimate a p* model with all six effects in it, we
estimated reduced models with various combinations of five (or four, if necessary) effects,
selecting the specific combination with the best fit.[4]
Table 4. Descriptive Statistics for p* Parameter Estimates by Animal Type
and with Type by Positive or Negative Relation
Human
Parameter
Human
Primate Positive
Primate
Negative
Positive
Negative
Mutuality
Mean
SD
N
0.40
0.21
66
-0.55
0.81
9
0.44
0.22
45
0.32
0.18
21
0.55
0.44
2
-0.86
0.56
7
Out 2-Stars
Mean
SD
N
0.08
0.56
34
0.20
1.39
12
0.07
0.61
28
0.15
0.22
6
0.53
0.68
3
0.09
1.58
9
In 2-Stars
Mean
SD
N
0.49
0.55
66
-0.39
1.15
10
0.22
0.30
45
1.06
0.55
21
0.38
1.34
2
-0.58
1.12
8
Mixed 2-Stars
Mean
SD
N
-0.21
0.29
66
-0.68
1.27
9
-0.31
0.24
45
-0.00
0.27
21
0.02
0.18
2
-0.88
1.39
7
Transitive
Mean
SD
N
0.10
0.64
66
0.69
1.62
10
0.42
0.45
45
-0.59
0.41
21
0.09
0.58
2
0.84
1.79
8
Cyclic Triples
Mean
SD
N
-0.01
0.25
65
-0.53
1.19
8
-0.06
0.24
44
0.10
0.26
21
0.21
1.05
2
-0.78
1.21
6
Figure 7. Mean P* Parameter Estimates by Type of Animal
It is clear from the table and the figure that the mean parameter values for human relations differ
considerably from those for primate relations for several structural effects except for the out 2stars effect. Two differences are particularly dramatic - the difference in mutuality and the
difference in in 2-stars. The average level of mutuality among human relations is a positive
standardized coefficient about .40 while the average level of mutuality among primate relations
is a negative standardized coefficient near .55. Among human relations, in 2-stars also has a
positive effect but among primate relations, the in 2-stars effect is just as strongly negative. This
in 2-star effect captures how important indegree variance is to the probability of a graph, with
positive values indicating graphs with relative more indegree variance are more likely. Thus
indegree variance contributes to graph probability among humans but detracts from it among
primates. Less dramatic are the differences between humans and primates in the transitive and
cyclical triple effects. Among humans the average impact of both effects is near .00, meaning
that neither an abundance of cyclical triples nor an abundance of transitive triples enhance graph
probability among humans. However, among primates, the transitivity effect is strongly positive
while the cyclical triple effect is modestly negative, indicating that graphs with many transitive
triples and few cyclical ones have higher probability. Finally, in both populations the effect of
mixed 2-stars is negative, meaning that in human and primate relations analyzed, the tendency
for nodes to be either sources of ties or targets of ties, but not both, increases graph probability.
The Freeman-Linton hypothesis suggests that we now ask whether these differences between the
kinds of species arise because different types of relations are measured in each. For instance, if
negative, agonistic relations have strong transitive and anti-cyclical tendencies (the dominance
hierarchy effect), then if the primate networks were disproportionately negative ties and the
human networks were disproportionately positive ties, we would expect the average effects of
transitive triples and cyclical triples to be very different in the primate networks than in the
human networks. Figures 8 and 9 graph the mean parameter values separately for positive and
negative relations among humans vs. primates. Figures 10-15 are box plots that provide a more
complete view of the overall distribution of parameter values within kind of species by relation
type.
Figure 8. Mean P* Parameter Estimates by Type of Animal for Positive Relations
Figure 9. Mean P* Parameter Estimates by Type of Animal for Negative Relations
Figure 10. Box Plot of Mutuality Parameters from p* Model by Type of Animal
and Whether Relation is Positive or Negative
Figure 11. Box Plot of Out 2-Star Parameters from p* Model by Type of Animal
and Whether Relation is Positive or Negative
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 8 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 do 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 do
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: do 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 comparisons address.
One final issue of theoretical importance as it bears on the evaluation of hypotheses is whether
our sample of networks represents a complete range of typical social relations. Without a
definition of the "population" of social relations our sample is to represent, it is clear that we
cannot answer this question. It appears that many kinds of relations are underrepresented in our
sample, including both affectionate and agonistic behavioral encounters among humans and
affectionate encounters among nonhuman primates and nonprimate mammals. The extent to
which a non-representative sample of relational contents might lead to an incorrect conclusion
about the Freeman-Linton hypothesis (or about any other hypothesis about the basis for
commonalities among network structures) is worthy of further investigation.
References
Anderson, Carolyn J., Stanley Wasserman, and Bradley Crouch. 1999. "A p* primer: Logit
models for social networks." Social Networks 21: 37-66.
Bernard, H. Russell, Peter Killworth, and Lee Sailer. 1980. "Informant Accuracy in Social
Network Data IV: A Comparison of Clique-Level Structure in Behavioral and Cognitive
Network Data." Social Networks 2, 191-218.
Blau, Peter M. 1955. The Dynamics of Bureaucracy. Chicago: University of Chicago Press.
Burt, Ronald. 1991. Structure 4.2. Center for the Social Sciences, Columbia University.
Carroll, J. Douglas, Ece Kumbasar, and A. Kimball Romney. 1997. "An Equivalence Relation
between Correspondence Analysis and Classical Multidimensional Scaling for the Recovery of
Euclidean Distances." British Journal of Mathematical and Statistical Psychology 50: 81-92.
Coleman, James. 1964. Introduction to Mathematical Sociology. New York: Free Press.
Crouch, Bradley, and Stanley Wasserman. 1998. "A Practical Guide to Fitting p* Social
Network Models." Connections 21: 87-101.
Dunbar, R. I. M., and E. P. Dunbar. 1976. "Contrasts in Social Structure among Black-andWhite Colobus Monkey Groups." Animal Behaviour, 24: 84-92.
Faust, Katherine, and John Skvoretz. Forthcoming. "Comparing Networks Across Space and
Time, Size and Species." Sociological Methodology, forthcoming.
Frank, O., and D. Strauss. 1986. "Markov Graphs." Journal of the American Statistical
Association 81: 832-842.
Freeman, Linton. 1999. "Research in Social Networks." Available:
http://eclectic.ss.uci.edu/~lin/work.html [June 18, 2002].
Freeman, Sue C., and Linton C. Freeman. 1979. The Networkers Network: A Study of the Impact
of a New Communications Medium on Sociometric Structure. Social Science Research Reports,
No. 46. Irvine, CA: University of California.
Friedkin, Noah E. 1998. A Structural Theory of Social Influence. Cambridge, UK: Cambridge
University Press.
Hall, K.R. L., and Irven DeVore. 1965. "Baboon Social Behavior." Pp. 53-110 in Primate
Social Behavior: Field Studies of Monkeys and Apes, edited by Irven DeVore. New York: Holt,
Rinehart, and Winston.
Hausfater, Glenn. 1975. Dominance and Reproduction in Baboons (Papio cynocephalus), a
Quantitative Analysis. New York: S. Karger.
Kapferer, Bruce. 1972. Strategy and Transaction in an African Factory: African Workers and
Indian Management in a Zambian Town. Manchester: Manchester University Press.
Katz, L., and J.H. Powell. 1955. "Measurement of the Tendency toward Reciprocation
Choice." Sociometry 19: 403-409.
Katz, L., and T.R. Wilson. 1956. "The Variance of the Number of Mutual Choices in
Sociometry." Psychometrika 21: 299-304.
Krackhardt, David. 1987. "Cognitive Social Structures." Social Networks 9: 104-134.
_____. 1988. "Predicting with Networks: Nonparametric Multiple Regression Analyses of
Dyadic Data." Social Networks 10: 359-382.
Linton, Ralph. 1936. The Study of Man. New York: The Free Press.
MacRae, Duncan. 1960. "Direct Factor Analysis of Sociometric Data." Sociometry 23: 360-371.
Mandel, Micha. 2000. "Measuring Tendency towards Mutuality in a Social Network." Social
Networks 22: 285-298.
Markovsky, Barry. 1998. "Social Network Conceptualizations of Solidarity." Pp. 343-372 in
"The Problem of Solidarity: Theories and Models" edited by Patrick Doreian and Thomas
Fararo. Amsterdam, The Netherlands: Gordon and Breach.
McGrew, W. C. 1972. An Ethological Study of Children's Behavior. New York: Academic
Press.
Newcomb, Theodore. 1961. The Acquaintance Process. New York: Holt, Reinhardt andWinston.
Parker, J. G. and S. Asher. 1993. "Friendship and Friendship Quality in Middle Childhood"
Links with Peer Group Acceptance and Feelings of Loneliness and Social Dissatisfaction."
Developmental Psychology 29: 611-621.
Pattison, Philippa, and Stanley Wasserman. 1999. "Logit models and logistic regressions for
social networks: II. Multivariate relations." British Journal of Mathematical and Statistical
Psychology 52: 169-193.
Pattison, Philippa, Stanley Wasserman, Garry Robins, and Alaina Michaelson Kanfer. 2000.
"Statistical Evaluation of Algebraic Constraints for Social Networks." Journal of Mathematical
Psychology 44: 536-568.
Robins, Garry, Philippa Pattison, and Stanley Wasserman. 1999. "Logit Models and Logistic
Regressions for Social Networks: III. Valued Relations." Psychometrika 64: 371-394.
Sade, Donald S. 1967. "Determinants of Dominance in a Group of Free-ranging Rhesus
Monkeys." Pp. 99-114 in Social Communication among Primates edited by Stuart A. Altmann.
Chicago: University of Chicago Press.
Sampson, Samuel F. 1968. A Novitiate in a Period of Change: An Experimental and Case Study
of Social Relationships. Unpublished doctoral dissertation. Cornell University.
Schein, Martin W., and Milton H. Fohrman. 1955. "Social Dominance Relationships in a Herd
of Dairy Cattle." The British Journal of Animal Behavior 3: 45-55.
Strauss, David, and Michael Ikeda. 1990. "Pseudolikelihood Estimation for Social Networks."
Journal of the American Statistical Association 85: 204-212.
Strayer, F. F., and Janet Strayer. 1976. "An Ethological Analysis of Social Agonism and
Dominance Relations among Preschool Children." Child Development 47: 980-989.
Tyler, Stephanie J. 1972. "The Behaviour and Social Organization of the New Forest Pony."
Animal Behavior Monographs, Volume 5, Part 2.
Vickers, M., and S. Chan. 1981. Representing Classroom Social Structure. Melbourne: Victoria
Institute of Secondary Education.
Wasserman, Stanley, and Philippa Pattison. 1996. "Logit Models and Logistic Regressions for
Social Networks: I. An Introduction to Markov Graphs and p*." 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 1 - 5; recoded
4,5 =1, <4=0. (Appendix A, Bernard, Killworth, and Sailer 1980, also available in
UCINET, Borgatti, Everett, and Freeman 1999)
bkhamc: rating of interaction frequency between ham radio operators. Originally coded 1
- 9; recoded 7,8,9 =1, <7=0 (Appendix A, Bernard, Killworth, and Sailer 1980, also
available in UCINET, Borgatti, Everett, and Freeman 1999)
bkoffc: top rank order of interaction frequency in an office. Originally a complete rank
order from 1 to 39; recoded 1-13=1, >13=0. (Appendix A, Bernard, Killworth, and Sailer
1980, also available in UCINET, Borgatti, Everett, and Freeman 1999)
bktecc: top rank order of interaction frequency in a technical group. Originally a
complete rank order from 1 to 36; recoded 1-11=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 1 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 1 between high school boys (Table 14.5 (a), page 450, Coleman
1964)




























cole2: friendship at time 2 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, <3=0. (Freeman and
Freeman 1979; Table B.8, page 745, Wasserman and Faust 1994)
eiesk2: EIES data, rating of acquaintanceship. Recoded 3,4=1, <3=0. (Freeman and
Freeman 1979; Table B.9, page 746, Wasserman and Faust 1994)
eiesm: EIES data frequency of message sending, Recoded "1" if any message was sent.
(Freeman and Freeman 1979; Table B.10, page 747,Wasserman and Faust 1994)
fifth: friendships between fifth graders (Table 3, page 44, Anderson et al. 1999, data
from Parker and Asher 1993)
fourth: friendships between fourth graders (Table 3, page 44, Anderson et al. 1999, data
from Parker and Asher 1993)
ka: Each manager was asked who they went to for help or advice at work. (Krackhardt
1987; Data available in Wasserman and Faust 1994 and UCINET, Borgatti, Everett, and
Freeman 1999).
kapfti1: instrumental work relations in a tailor shop, time 1 (Matrix 1, pages 176-177,
Kapferer 1972)
kapfti2: instrumental work relations in a tailor shop, time 2 (Matrix 2, pages 178-179,
Kapferer 1972)
kf: Each manager was asked who they were friends with at work. (Krackhardt 1987;
Data available in Wasserman and Faust 1994 and UCINET, Borgatti, Everett, and
Freeman 1999).
kids1: initiated agonism between children (Figure 2, page 986, Strayer and Strayer 1976)
kids2: dominance among boys in a nursery school (Figure 5.5, page 125, McGrew 1972)
medical: ties between physicians (Coleman, Katz and Menzel, data available in Structure,
Burt 1991) newc0 to newc15: top rankings of friendship in a fraternity, weeks 0 through
15. Original data were complete rank orders, recoded 1-5=1, >5 = 0. (Newcomb 1961,
data available in UCINET, Borgatti, Everett, and Freeman 1999).
newc0n to newc15n: bottom rankings of friendship in a fraternity, weeks 0 through 15.
Original data were complete rank orders; recoded 11-15=1, <11 = 0 (Newcomb 1961,
data available in UCINET, Borgatti, Everett, and Freeman 1999).
nfponies: threats between ponies (Table XIV, page 122, Tyler 1972)
prison: closest friendships in a prison (Table 1, page 363, MacRae 1960)
rhesus1: fights between adult female rhesus monkeys (Table 1, page 105, Sade 1967)
rhesus2: fights between yearling rhesus monkeys (Table 2, page 107, Sade 1967)
rhesus4: fights between adult rhesus monkeys (Table 4, page 108, Sade 1967)
rhesus5: fights between adult rhesus monkeys (Table 7, page 110, Sade 1967)
rhesus6: fights between adult rhesus monkeys (Table 8, page 111, Sade 1967)
sampdes: disesteem between monks, time 4 (Table D14, page 470, Sampson 1968)
sampdlk: dislike between monks in a monastery, time 4 (Table D13, page 469, Sampson
1968, available in UCINET, Borgatti, Everett, and Freeman 1999)
sampes: esteem between monks in a monastery, time 4 (Table D14, page 470, Sampson
1968, available in UCINET, Borgatti, Everett, and Freeman 1999)
sampin: influence between monks in a monastery, time 4 (Table D15, page 471,
Sampson 1968, available in UCINET, Borgatti, Everett, and Freeman 1999)








samplk: liking between monks in a monastery, time 4 (Table D13, page 469, Sampson
1968, available in UCINET, Borgatti, Everett, and Freeman 1999)
sampnin: negative influence between monks in a monastery, time 4 (Table D15, page
471, Sampson 1968, available in UCINET, Borgatti, Everett, and Freeman 1999)
sampnpr: negative praise (blame) between monks in a monastery, time 4 (Table D16,
page 471, Sampson 1968, available in UCINET, Borgatti, Everett, and Freeman 1999)
samppr: praise between monks in a monastery, time 4 (Table D16, page 471, Sampson
1968, available in UCINET, Borgatti, Everett, and Freeman 1999)
third: friendship between third graders (Table 1, page 42, Anderson et al.1999, data from
Parker and Asher 1993)
vcbf: best friends between seventh graders (Table 3, page 385, Robins, Pattison, and
Wasserman 1999, from Vickers and Chan 1981)
vcg: get on with between seventh graders (Table 11, page 422, Wasserman and Pattison
1996, from Vickers and Chan 1981)
vcw: work with between seventh graders (Table 12, page 423, Wasserman and Pattison
1996, from Vickers and Chan 1981)
Footnotes
[1] In our analysis, as in Faust and Skvoretz (forthcoming) we use the p* modeling framework to
estimate the collection of parameters comprising the structural signature of each network. This
is by no means the only possible approach, but it has several advantages as compared to other
approaches currently available. In general the problem is to express, for each network, its
structural tendencies. Obviously there are numerous graph theoretic properties that could be
used for this characterization. One problem is to express these tendencies in a "metric" that is
comparable across networks of different scale (different sizes and densities). Using standardized
regression coefficients from the statistical model accomplishes this by expressing the effect of a
structural property in standardized units, and does so for all properties and all graphs. This
facilitates comparison across networks. A second problem is possible interdependencies among
structural properties. For example, an observed level of transitivity in a graph might be
attributable to the graph's high density or tendency for mutuality. Using a statistical model
which estimates effects of structural parameters simultaneously deals with this by expressing the
effect of a structural property, net of other factors in the model. Certainly other statistical
models (such as Friedkin's (1998) local density model) or quantifications of structural properties
are possible avenues for constructing structural signatures.
[2] However, using such routines assumes, contrary to fact, that the logits are independent.
Therefore, the model is not a true logistic regression model and statistics from the estimation
must be used with caution. Goodness of fit statistics are pseudo-likelihood ratio statistics, it is
questionable whether the usual chi-square distributions apply, and standard errors only have
"nominal" significance (see Crouch and Wasserman 1998).
[3] Given the pseudo-likelihood estimation underlying the p* model, we need to be cautious
about literal interpretation of the parameter estimates, particularly when they attain extreme
values. While we look at means of subsets of estimates to compare structural tendencies
between different kinds of networks, the reader should be aware that extreme values are suspect.
However, such comparison are the best we can do until there are better models and estimation
techniques.
[4] Among humans, it was possible to estimate the full model in 33 of the 66 networks. In the
other 33, a model with five parameters could be estimated. Among primates, the full model
could be estimated in six of the 12 networks, a model with five parameters in two, a model with
four in two, and a model with only two parameters in the remaining two networks. The two
mammal networks allowed for the estimation of the full model in one and a model with four
parameters in the other.
[5] There is some support for this speculation. We have two networks of observed negative
encounters between humans - initiated agonism among nursery school boys (labeled kids1 and
kids2, respectively). In Figure 2, these networks are in the lower left corner of the plot. They
are situated on the lower edge of the networks of observed negative encounters among primates
and mammals (on the left of the figure) and are clearly separated from the negative affective ties
among humans (on the right of the figure).