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OCR for page 174
Local Rules and Global Properties:
Modeling the F,mer~f~n~f' of NPtwnrk
^^ Structure
Martina Moorish
University of Washington
Departments of Sociology and Statistics
Abstract
This paper reviews the interaction between theory and methods in the field of network
analysis. Network theory has traditionally sought to use social relations to hncl~e whet
~ ~ A: ~ 1 _ A: ~ ~ . , 1
~ ~e, _ ,, ^~ ~ ~
~io`;~m s`;~;nr~s rerer to as one m~cro-macro gap: understanding how social structures are
formed by the accumulation of simple rules operating on local relations. While early
network methods reflected this goal, the bulk of the methods developed later and
popularized in computer packages were more descriptive and static. That is beginning to
change again with recent developments in statistical methods for network analysis. One
particularly promising approach is based on exponential random graph models (ERGM).
ERGM were first applied in the context of spatial statistics, and they provide a general
framework for modeling dependent data where the dependence can be thought of as a
neighborhood effect The morle.l~ r.nn ha. 7~1~f~f1 1~ A~mmn^c`= Ill ^~1~~r~-lr ~f_~r~~]
properties into the effects of localized interaction rules; the traditional concern of the
field. An example is given using an HIV transmission network.
~ ~^ A_ _v_~ —~ at— V V ~i ct11 11~L W JO— i:>l~1 U~I.Ul al
Where do networks come from?
There are really two questions implied in this simple query. The first is about the
underlying process that gives rise to the patterns of links among nodes in the population -
this requires us to think about a dynamic mode] that links local processes to a global
outcome. The second is the inverse question: how to infer the underlying process from
the patterns we observe - this requires statistical methods for sampling. estimation and
inference. Network theory was explicitly focused on the first question. The current
developments in the field are making progress on the second. And the most welt
developed and commonly used methods in the field, while not particularly well suited to
answering either question. have provided some important intermediate tools.
Modern (post 1940) social network theory explicitly focused on the link between
local dynamics and ~,lobal structures. One root can be traced back to social anthropology
and exchange theory. In this research the focus is on how rules governing permissible
' This work was supported in part by NIH grants RO 1 HD4 1 877-01 and RO ~ DA I 283 1-02.
174
DYNAMIC SOCIAL NETWO~MODEL~G ED ISIS
OCR for page 175
exchange partners (for, e.g., gift giving, or marriage) cumulate up to determine the
overall structure and stability of an exchange system (White 1963; Levi-Strauss 1969;
Yamagishi and Cook ~ 993; Bearman ~ 9961. Another root can be traced back to the
social psychology and balance theory. In this research, the focus is on the how the
requirement of balance for positive and negative affect among three actors - e.g., my
enemas friend is my enemy- cumulates up (Heider 1946; Granovetter 1979; Chase
~ 982~. In both cases, the key findings have been that some simple local rules may lead to
overall network structures that display striking regulanties, like linear hierarchies, stable
cliques, or stable cycles of exchange. Other rules do not have these systemically stable
impacts, and this distinction leads to an evolutionary hypothesis about the differential
survival of rules over time, i.e., to links with game theory.
Given this early focus, one might have expected the field to develop a
methodology for systematically exploring such dynamics, but this did not happen. Nor
did the inverse statistical problem, of inferring the rules from the patterns, receive much
attention. It is tempting to attribute this to inadequate computing and statistical tools
available at the time (from the ~ 940s to the ~ 980s). And to the impact of the progress
macle in the (less demanding) linear model framework. Whatever the reason, however,
the methodology that did develop in the field of network analysis focused on static
descnptive measures, rather than the dynamic model.
The descriptive approach drew inspiration from mathematical graph theory, using
tools from linear algebra to manipulate the adjacency matnx, and focusing on issues of
clustering and connectivity. These tools have become the heart of the field, providing a
rich framework for thinking about networks and a wide range of summary measures to
represent both the network position occupied by specific nodes, and the overall network
structure. Almost all of the classic network measures - paths, cycles, density, centrality,
structural equivalence, cliques, blockmodels, role algebras and bicomponents -- owe their
development to researchers working with these descnptive tools. Textbooks and
computer packages for network analysis typically have these measures at their core.
They have become the common language for network analysis, defining the basic
features of networks and helping to develop our intuitions about the complex relational
structures we seek to understand.
The statistical issues - developing pnncipled methods for estimation from
samples, and quantifying the uncertainty in the estimates - have been addressed in a
limited way in this descriptive context bv eschewing' trarlitinnn1 mn~l~l hack ctotictira1
methods. Model-based methods rely on the assumption of independent observations to
obtain tractable likelihood-based estimation and inference. In network analysis,
independence is not something one would want to assume, since understanding the
dependence among observations is actually the primary task. The growth in computing
power has given rise to statistical methods that rely on resampling rather than likelihood
inference, e.g., the bootstrap, jackknife, and permutation tests. These have been readily
adopted by network analysts, finding their way into tests like quadratic assignment
procedures (QUAP) for sociomatrix regression. In recent years, Markov Chain Monte
CarIo (MCMC) algorithms have been developed for complex estimation problems, and
these are now providing the tools needed to return to a model-based framework.
DYNAMIC SOCIAI~ NETWORK A1Ol9E~ING~D TRYSTS
175
OCR for page 176
Statistical Models for Networks
Progress in model-based statistical methods for network analysis has picked up
momentum in recent years, due to a combination of theoretical developments, advances
in statistical computing, and innovations in data collection. Model-based statistical
estimation and inference would seem to offer exactly the right tools to answer the original
inverse question raised above, because it is based on an underlying stochastic model of
the population process. To be useful, however, two things are needed: (~) an appropn ate
model for the process and the dependencies in it, and (2) a method of estimation that
works with dependent observations.2 This would seem to be fairly straightforward, but
the history of the field makes it clear it is not.
There is general agreement that the exponential family of distributions provides a
good framework for modeling the probability that a random graph (or sociomatrix) X
takes the observed value x. Holland and Leinhardt (1981) were the first to propose using
this model for networks, noting that it was a natural forth because the sufficient statistics
were explicitly tied to parameters of interest, like indegree, outUegree, mutuality and etc.
The general form of the model is given by:
P(X = x) = exp {Z 0,, f (Xjj )} / C(X)
(1)
where the sufficient statistics, f (I ), are typically counts of links and of products of
links, the All are the parameters of interest, and coax) is a normalizing constant. This
mode} had been proposed 20 years earlier by Bahadur (1961) as a general representation
for multivariate binomial distributions. It is a completely general model, and can fit any
graph perfectly by using all (n)_ 1 possible parameters (where n is the number of nodes).
One can think of the sufficient statistics as defining, the neighborhood of dependence.
Conditional on the rest of the graph, the dependence between observations is restncted to
the other points in this neighborhood. The heart of the modeling effort is to specify (and
test) parsimonious representations of the neighborhood. To focus on this, we will strip
equation (~) down to its modeling core:
IJ
~~t,f(Xjj)
i=l
and work through the models that have been proposed in various contexts.
(2)
Much of the statistical theory for ERGM was developed and applied in the
context of spatial statistics by Besag (19741. The simplest spatial models represent
observations as points on a lattice, and assume that only the nearest neighbors have an
influence on the status of a site. For example, imagine an agricultural plot, divided into a
grid along two orthogonal axes (see Figure I.
~ It is important to keep in mind that the unit of analysis here is the link, not the node, and the dependence
is between links, not between nodes.
176
DYNAMIC SOCIAL fJETWORKMODELIl`JG AND ANALYSIS
OCR for page 177
Figure I. A lattice model for a "nearest neighbor', spatial process. The dark site in the
center is the site whose status is of interest; the grey sites indicate the neighborhood of
dependence.
Each square in the and has 4 adjacent sites in its neighborhood. Nearest neighbor
ERGM models mimic the classic Ising mode] from statistical mechanics, representing the
status of a site (say, infected with a fungus) as a function of:
Mali j~p~xi~jxi~kaj+k k={ i,])
. . . .
... ...
The spatial dependence among sites is captured by p, which is considered a nuisance
parameter, and the parameter of interest is a, which represents the spatially adjusted
propensity for infection.
When applying these spatial models to networks, both similarities and differences
in the nature of the data should be kept in mind. In both cases, the data are often
represented in matrix form - the lattice and the adjacency (or socio-) matnx. But the
meaning of the indices that order the matrix are quite different. The "site" in the spatial
context is a dyed (not a person) in the network context. The Cartesian coordinates that
define the and in the spatial context become persons in the network context. This latter
difference has important implications. In the spatial context, the spatial metric is
exogeneous, and the lattice is fixed. In the network context, the rows and columns of the
adjacency matrix have no intrinisic ordering, and may be permuted at will without
changing the pattern of ties among persons, or the neighborhoods that define dependence.
While physical space may influence the pattern of ties, it is not the only factor. As a
result, the notion of "space" in network models is largely endogeneous and
neighborhoods are typically not defined in terms of proximity in the matrix. Because the
nature of the "spatial" dependence is the primary focus of interest in network analysis,
the parameters that define this dependence are not nuisance parameters, but the
parameters of most interest. This also changes the kinds of models that are appropriate
DYNAMIC SOCIAL NETWORK MODELING ED ISIS
177
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Representative terms from entire chapter:
dynamic social
for networks. The models can be based on exogeneous non-spatial attributes of the nodes
(e.g., age, race, sex, etc.), or on the propensity for certain kinds of configurations in the
network (e.g., tnad effects, cycles and paths). Alternatively, the models may seek to
represent neighborhoods in terms of a completely endogeneously defined latent space
(with methods similar in spins to earlier blockmodeling (White, Boorman et al. 197641.
Each of these approaches can be found in the literature.
The first application of ERGM like models to networks was Holland and
Leinhardt's p] mode! for directed graphs. The core of this mode} takes the form:
n '`
~_, . ~ n
,/ ~ Dip+ ~ ,? ~ pjX~j + pI XijXj,
i=1 j=1 i
parameters will also grow. This is somewhat like a f~xed-effect model in economics,
where a term is fit to every respondent in a setting with repeated measures. Here, as
there, these are essentially nuisance parameters.
Homogeneity constraints are often imposed for parsimony, but they can also be
used to represent exogenous covanates. For example, one can specify ;,roup-specif~c,
rather than individual-specific parameters for indegree and for outUegree (Fienberg,
Meyer et al. 1985; Wang and Wong 19871:
~ Ok ~ Hi+ + ~ p~ ~ X+ j + ~ (k! ~ X;j + pit Xij-~ji k = I, G. ~ = l ? G (5)
ink jel iek,,je!
There are now G indegree and outUegree parameters, rather than N. This can lead to a
substantially more parsimonious mode! if the number of groups is much smaller than the
number of persons. The ~ parameters are used to specify the level of mixing within and
between groups. As in other log-linear modeling settings, the mixing can also be
parsimoniously modeled to represent patterns of interest (Morns 19911.
The next extension of ERGM in networks took on the task of modeling dyadic
dependence. The ability to mode! dyadic dependence is the single most important feature
of the ERGM framework, both because it is theoretically appealing, and because it is
statistically innovative. The range of possibilities it opens up, however, is daunting.
Without the physical space embedding, the notion of "neighborhood" is pretty much
unconstrained. Frank and Strauss (Frank and Strauss 1986) were the first to exploit the
more general form of the ERGM to represent dyadic dependence in the context of
networks. The approach they took was a natural, if somewhat mechanical, first step:
links are dependent if they share a node. The neighborhood is Markovian in the sense
that links must be directly adjacent to be dependent, so Frank and Strauss referred to the
model as the Markov Graph. Using the Hammersley-Clifford theorem, the sufficient
statistics for this mode! can be shown to be products of links that represent the stars of
various sizes in the graph, and triangles. A "star" is a cluster of linics with a single central
node. For an undirected graph the general Markov mode! is:
,2 - 2
IS. + ~ [ij~XijXji,X~;
k=]
where Sk represents a star of size k, so a product of k links. Frank and Strauss then
investigate a simpler version of the Markov anodes, restricting the configurations to
so arid so -- edges and 2-stars, and imposing an homogeneity constraint on the tnangle
parameter a. Wasserman and Pattison (Wasserman and Pattison 1996; Pattison and
Wasserman 1999) returned to the more general form. which they christened pi, in honor
of Holland and Leinhardt's pioneering work.
The models above are based on explicit measured covariates, though in some
cases the covariates are themselves part of the process being modeled. Another
approach, with roots in classic multivariate analysis, seeks instead to represent the latent,
unmeasured space implied by the pattern of network ties. In this purely descriptive
DYNAMIC SOCIAL NETWO=MOD~WG ED ^^YSIS
(6)
179
180
model, the aim is to find a latent space in which the probability of a tie vanes with the
distance between the two nodes (Hoff, Raftery et al. 20021. Conditional on this metric,
the dyads are independent.
a-~Zi - (Jo
(7)
The z parameters represent the latent spatial locations, and need to be estimated. Without
further dimensional reduction, the number of parameters required to map this latent space
becomes quite large. More parsimonious representations can be tested. Explicit
covanates can also be added.
Despite the enormous flexibility the ERG framework has brought to the field,
virtually all of the published applications of these models have either used a variant of the
original p] model, or the simplified Markov graph. This is somewhat curious, as both are
pretty mechanical rendenngs of what one might think of as the "neighborhood" of
influence, and neither is particularly well suited to representing the kinds of processes
traditionally of interest to network modelers.
The lack of mode! development is probably due to the extremely challenging
technical problems that accompany the estimation of these models.3 Another problem is
the lack of data, or at least of the type of data currently required. None of the models has
been specified in a way that missing data can be handled, so it is still necessary to have
the equivalent of census data on a network - data on every node and every link. Such
data do exist, but they are not common, and this has been a major obstacle to all forms of
network analysis. Finally' it may also be that our ability to think, empincally, in terms of
positions and network structure has atrophied as we have waited for the right tools to
become available. Whatever the reason, there has been remarkably little application of
these modeling tools to data.
Models are the bridge between theory and data. And while there is a certain
attraction to simple abstract forms, like Markov graphs, small world graphs, or scale free
networks, the simplification we seek will be embedded in each substantive context. In
the rest of this paper ~ will show how these models can be used to formalize the
investigation of how a global network structure might cumulate up from simple local
rules in a specific case.
' Other papers in this volume will be addressing these in detail, so they do not need to be covered here.
DYNAMIC SOCIAL NETWORK MODELING^D CYSTS
From local rules to global structures: Partnership networks and the spread of HIV
Over the past two decades, the epidemic of HIV has challenged the
epidemiological community to rethink its paradigms for understanding the risk of
infectious disease transmission, both at the individual level, and at the level of population
transmission dynamics. Opinion has rapidly converged on the central importance of
partnership networks. Because people acquire infections from their partners, it is not
only a person's own behavior that puts them at risk, but the behavior of their partners, and
more generally, the persons to whom they are indirectly connected by virtue being
connected to their partner. At the individual level the risk of infection is determined by
position in the overall transmission network. While individual behavior plays a primary
role in establishing this position, it is not the exclusive determinant. To some extent,
even the notion of "individual behavior" itself is open to question here, since all
behaviors can only be adopted with a willing partner, which has important implications
for both modeling and behavioral intervention strategies. At the aggregate level, too,
transmission dynamics are strongly affected by networks. The network of partnerships
channels the spread of an infection through a population, amplifying or impeding the
spread relative to a hypothetical randomly mixing population. At both the individual and
the population level then, our ability to quantify the risk of transmission for HIV or other
sexually transmitted or blood-borne infections, depends on our ability to measure and
summarize the transmission network.
One of the greatest obstacles to using network analysis in this context is the
complete data requirement. In the context of the sexual and needIe-sharing networks that
matter for HIV, such data collection is impossible. The Question. then. its whether we ran
~~,_1~ _ ^~ 11 ~ 1~ ',t ~~
GApl~] tIlO overall network secure won a small number of partnership formation mIes
that operate at the local individual level. If this is true, it will radically simplify data
collection needs, and give network analysis a central place in the tools of sexually
transmitted infection (STI) research and intervention.
There is some reason to think that such local rules may govern the network
structures of interest. At the population level, the findings from recent mathematical
modeling suggest that two general types of structures are important: selective mixing
patterns among different groups (Hethcote and Van Ark 1987; Sattenspie! 1987; Garnett
and Anderson ~ 993; Moms 1996), and the timing and sequence of partnership formation
(Watts and May ~ 992; Monks and Kretzschmar ~ 9971. These general insights come from
different types of mathematical simulations, including both simple deterministic
differential equations (sometimes called compartmental models) and stochastic
microsimulations of a population of interacting individuals (sometimes called "agent
based models"~.
Mixing refers to assortative and disassortative biases in the joint distribution of
partners' attributes. Examples include the degree of matching on race' age, sex, social
status, sexual orientation or level of sexual activity. Assortative biases can create patchy,
clustered networks, which tends to increase the speed of spread within groups, and slow
the spread between them. If the groups also vary in activity level, the resulting
DYNAMIC SOCKS NETWO=MODE~G ED CYSTS
181
distribution of infection can be very uneven. In the extreme, prevalence may rise to high
endemic levels in some groups, while other groups remain infection free. Disassortative
biases typically have the opposite effect, ensuring rapid spread to all groups. For
example, preferential linking between highly active persons and less active persons
ensures that the latter are reached more quickly in the epidemic. An example of this is
the traditional double standard, where men can have multiple partners, but women only
one.
The timing and sequence of partnership formation refers to the pattern of start and
end dates for partnerships over an individual's lifetime. If the partnership intervals
defined by these dates are strictly sequential, the pattern is called serial monogamy. If
the intervads overlap, the pattern is called concurrency. Serial monogamy retards the
spread of infection in two primary ways: it locks the pathogen in a partnership for some
time after transmission, and it ensures that only later partners are at indirect risk from
earlier ones. With concurrency, by contrast, one's partner can have other partners who
in turn have other partners, and so on. Instead of a monogamous population of dyads and
isolates, concurrency creates a potentially large connected component for rapid pervasive
spread. In addition, because earlier partners need not be dropped as later partners are
added, the infection can be spread both in both directions: from earlier to later partners,
and from later to earlier ones.
Almost all of our understanding of the effects of these network patterns on HIV
spread have been based on simulation. And with few exceptions, the simulations have
created network effects indirectly, by varying parameters of some convenient
mathematical function to produce a change in the simulated networks. Network patterns
are thus outcomes of the model, rather than inputs. This strategy has been enormously
valuable for orienting research, and it laid the groundwork for future progress. But it has
also limited our ability to place this work on a Burn empirical footing. and to anantifv the
risk in any specific network.
--rat =, ~ =~ .~ .~_
Linking network data to network simulation requires a statistical bndge: a
modeling framework that enables the key structural parameters to be estimated from
network data, so that these can be used to directly drive a simulation. ERGMs have the
potential to do this.
The simplest network statistic is the total number of links, represented as
Xij, which provides information on the density of the network - in this context,
i= ~ I n I. j
OCR for page 183
systematic method for summarizing the key structural features of an empincal network,
with a framework for comparison and testing.
While estimation techniques are often of little interest to non-statisticians, this
case is an exception. The constant in equation (~) requires a calculation that makes
simple maximum likelihood estimation impossible for graphs larder than about 20 nodes.
To avoid this, early applications of these models used an approximation called maximum
pseudolikelihood estimation (MPLE). The problem was that the estimates produced by
MPLE were of unknown quality. In the last few years, researchers have turned instead to
computationally intensive Monte-CarIo Markov Chain (MCMC) estimation methods
which allow the true likelihood function to be maximized. In addition to providing, more
accurate estimates, this approach is particularly interesting for our purposes because the
MCMC method effectively simulates the network in order to maximize the likelihood.
We can, however, just as easily use the MCMC algorithm to simulate the network given
the parameter estimates, and this provides the ideal solution to the problem of lining
network data to the network simulation. One can estimate the network parameters from
data, and then use the same model, with the empirically based parameter estimates, to
drive a simulation of the network with an infection spreading through it. The MCMC
aIgonthm provides the engine for both tasks. This makes it possible for the first time to
directly control the network structures in a simulation so that they "look like" the
networks we observe in different data sets.
A Model-Based Hypothesis
We are now in a position to test a different kind of hypothesis about the role of
networks on disease spread. The research of the previous decade has suggested that
attribute-based mixing and levels of concurrency have a large impact on network
structure and transmission dynamics. Now we can ask whether these are the only
features that matter.
One the one hand, there is a good theoretical basis for this hypothesis. For the
type of partnerships that spread STI it seems reasonable to presume that people make
decisions about which partners to choose based on preferences and norms that operate at
the local level. That is, we choose partners because they are the night sex, age, race and
status, and we often care if they have other partners. It is unlikely that people form
partnerships thinking "If ~ choose this partner ~ can shorten the number of steps between
me and a randomly chosen person on the west coast" or '`I'd like to complete as many 5-
cycles as possible with a single partnership". This may seem obvious at one level, but the
implications may not be as obvious, and they are quite striking. If simple local rules
govern partner selection, then these also determine the aggregate structure in the network:
what looks like an unfathomably complicated system is, in fact, produced by a few key
focal organizing pnnciples. By extension, these simple local rules are also. therefore, the
key behavioral determinants of disease transmission dynamics on the network.
There are also important practical implications of this hypothesis. Both mixing
and concurrency are network properties that can be measured with local network
sampling strategies. If it turns out that these two local rules explain most of the variation
DYNAMIC SOCIAL NETWO=MODEL~G ED CYSTS
183
in network structure that is relevant to disease spread, then we have a simple inexpensive
way to measure network vulnerability routinely in public health surveillance. And they
describe simple behavioral rules that people can be taught to recognize and change.
So the last piece of the puzzle is to determine how to test the hypothesis that
mixing anc concurrency contain all of the intorrn~tion we nosh tm l~n^~xr tin -~l,~q`= the
spread potential in a network.
to what we want to fit.
~ ^ ,' ~ _~ I ·~4V YV ~w ~ Y At L~1~
We seek a "goodness of fit'' tests but one that is tailored
Transmission potential in a network is determined by the network connectivity.
Connectivity can be measured in a number of ways, but two simple measures are the
properties of reachability (is person i connected to person j by a path of some length ?)
and distance (what is the length of that paths. Reachability and distance represent
epidemiologically relevant "higher r)rcl~r" network nrr~n~rti-c that cimr~lm ~^A^lc, ~h~.ll'1
~ ~1 ~ 1 ~~ r
___, . ^.~ —~ t~ ~~_~ ~~ ~~ ~4111~1~ lilV~l~l~ OllVUlU
oe ante to reproduce. we can develop fit statistics based on these higher order network
statistics to test the models that include parameters only for mixing and concurrency. This
makes it possible to test whether the local organizing features represented in the model
reproduce the larger structural features of the network. We can therefore develop formal
tests for the hypothesis that mixing and concurrency capture the epidemiologically
relevant vacation in network structure. We can also use the MCMC simulation engine to
verify whether these features are sufficient for establishing the epidemic potential in a
network.
Conclusion
The new tools for network modeling provide us with the ability to empirically test
a question that has both important practical implications, and deep theoretical roots. The
generalizable findings from this work will not necessarily be monolithic. There is no
reason ~c, inane anal attnoute mixing and concurrency are the local rules that drive all
network structure. What is generalizable, though, is that the models for networks should
_
. ^. . .. , , , . . ~
be rooted in the scientific context that they seek to explain. For this, one does not need a
one size fits all approach, one needs a flexible class of models that can be tested against
data in principled ways. ERGMs provide the basis for this kind of emp~ncally-based
network analysis, and should become widely used in the years to come.
184
DYNAMIC SOCIAL METWO=MODELING AND ANALYSTS
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