Modeling Organizational PositionsChapter 2

p. 23

2.1 Social Structure as a Theoretical and Methodological Problem

Chapter 1 outlines how organizational theorists have engaged a set of issues

entailed in the problem of assessing the social structure of economic exchange. These

theorists identify two critical components of positional accounts of social structure—the

boundaries of the positional contexts in which organizations act, and a way of assessing

the similarity of organizations assigned to a given positional context. In the previous

chapter I argue that the blockmodeling approach proposed by White, Boorman, and

Breiger (1976) is a methodology that can be applied to the analysis of organizational

relations in a way that begins to incorporate both of these ideas. While this

blockmodeling approach addresses the issue of how contexts shape the behavior of actors

within their boundaries, it does not assess the relationship between an actor and its

context in a way that is meaningfully interpretable.

Consider as an example the analysis by White, Boorman, and Breiger (1976) of

the pattern of relations of positive affect in a monastery. White and his coauthors suggest

that the monks in this study might be meaningfully divided into a number of groups,

based on their expression of positive affect toward one another. To the extent that a

group is a social context that is important in determining emotional orientations, group

memberships should be able to provide some information about whether or not a monk

might like another monk, based on their respective group memberships. Following

Sampson’s (1969) original analysis, White and his coauthors identify a three-group

model, and labeled these groups as the Loyal Opposition, the Young Turks, and the

Outcasts. The results of their blockmodel analysis suggest, for example, that the Outcasts

p. 24

express positive affect towards the Young Turks, but that the Young Turks do not

reciprocate this affection.

One objective of this analysis would be to use a set of social contexts and their

associated boundaries to identify the ideal typical behavior of the actors embedded in a

given context. This objective can complicate the interpretation of the results of a

structural analysis. The three-group model proposed by White and his co-authors

essentially claims that there are three social contexts that actors can belong to, and it

makes specific claims about interaction within these contexts. For example, one claim of

this model is that monks acting in the context of the Outcast group will express positive

affect towards monks acting in the context of the Young Turks group. While this claim is

true in very broad terms, it is certainly not descriptive of the relationship between every

monk in the two groups. Models generated by these analyses are rarely perfect in this

sense, and as a result, empirical social scientists are left with the task of choosing

between alternative models. In assessing the appropriateness of blockmodels of the

social network of these monks, this choice is based on two questions. One question

concerns whether the three contexts identified are the three contexts that best describe the

social structure of these monks. Alternatively, a structural sociologist particularly

interested in defining the boundaries of the contexts of interaction might ask—as White,

Boorman, and Breiger did (1976: 751-2)—whether or not the three-group model

proposed by Sampson might be usefully decomposed into a more fine-grained partition.

A somewhat different objective of this analysis would be to develop a model

intended to predict individual behavior within the particular contexts identified as a part

of a given social structure. Such a model might be used to characterize the ways in which

p. 25

the behavior of actors in a particular social context is similar to the ideal-typical behavior

expected of actors within that context. One advantage of this approach is that it allows a

claim about social structure to be qualified in such a way that it might be empirically

assessed. Rather than simply claiming that the Outcasts express positive affect toward

the Young Turks, a model derived using this approach might express the claim that

monks from the Outcast group have a certain probability of expressing positive affect

toward the members of the Young Turks group.

These two objectives represent qualitatively different approaches to modeling

social structure. The first represents what might be labeled as a descriptive modeling

approach, in that its objective is to describe the structural regularities in individual

behavior. Descriptive models, in turn, can be evaluated with respect to their ability to

effectively describe the behavior they are based on. The second objective is consistent

with what might be termed a predictive modeling approach. Predictive models describe

structural regularities in individual behavior as well, but they also make explicit

predictions about the behavior of the actors that they model. While the notion of

assessing how well a model describes a given set of behavior is relatively flexible, the

assessment of how well a model predicts behavior is a somewhat more narrowly defined

metric. As I will argue, this allows predictive models to be assessed in a manner that is

less arbitrary and more powerful than methods that are frequently used to assess

descriptive models.

While these two modeling approaches differ in significant ways, they both

address two fundamental measures of the quality of the relationship between a structural

model and an observed set of behavior. The first of these is the accuracy of the claims

p. 26

made by a given model. A descriptive model is accurate to the extent that it provides a

good description of the behavior it is based on. In common sense terms, a descriptive

model that claimed that the Young Turks expressed positive affect toward the Outcasts

would be less accurate than one that claimed no such expression had been observed.

Similarly, a predictive model that estimated the probability of observing a bond of

positive affect from an Outcast directed at a Young Turk as 0.14 would be more accurate

than a model that estimated this probability as 0.86. In addition to being evaluated in

terms of their accuracy, structural models can be evaluated in terms of their complexity.

An in-depth discussion of assessing the complexity of a model in the general case will be

addressed in the following chapter, but in the case of Sampson’s monks, most measures

would assess a model with more groups as more complex than a model with fewer

groups.

In this chapter, I review the descriptive and predictive modeling approaches used

by organizational researchers interested in assessing social structure, and how each of

these approaches engages (or fails to engage) the ideas of accuracy and complexity in

their determination of the most appropriate structural model. In particular, I show how

the meaning of closeness in a predictive model is tied to a specific claim grounded in

probability theory, while the assessment of closeness in a descriptive model is based on

less powerful and essentially arbitrary measures.

2.2 Descriptive Modeling Approaches

One of the objectives of any structural analysis is to narrow the field of possible

structural models down to the smallest number of models—ideally just one. This process

p. 27

inevitably involves some explicit or implicit measure of “goodness-of-fit” that can be

used to determine which model or models should be selected. Goodness-of-fit measures

for structural models typically take the accuracy of a model with respect to the data into

account, and are sometimes affected by the complexity of a model. The choice of

accuracy and complexity measures, therefore, can fundamentally affect the way

variations in the data affect the way models are chosen. A review of empirical research

done using the descriptive modeling approach will demonstrate that the goodness-of-fit

measures used are generally not explicit with respect to this issue. As a result, it is

difficult to establish whether the model selection rationales implied by these studies are

consistent with the theoretical frameworks they are intended to empirically support.

A significant amount of methodological work has been done in the area of

blockmodel analyses of social networks. The distinction between descriptive and

predictive modeling approaches is made particularly clear in this body of work, because

of the relative clarity with which goodness-of-fit measures are made explicit. Wasserman

and Faust (1992) provide a comprehensive review of these goodness-of-fit measures,

some of which are considered here. In order to discuss these measures in detail, it is

necessary to briefly review some of the mechanics of blockmodel analysis, and to

introduce some terminology.

Social networks are frequently characterized by a matrix X, where the elements of

the matrix xij correspond to social ties. The assignment xij = 1 corresponds to a case in

which an actor i sends a tie to an actor j. A descriptive blockmodel θ is composed of a

mapping of actors to positions and an image matrix. The mapping φ(⋅) maps every actor i

to a position Br, typically on the basis of some measure of structural equivalence, such

p. 28

that if φ (i) = φ (j), then actors i and j are structurally equivalent. The image matrix B is

comprised of elements brs that represent hypotheses about the proposed social structure of

the network. In the strictest terms, the assignment brs = 0 represents the hypothesis that

there are no ties from actors in position Br to actors in position Bs. A descriptive

blockmodel is completely characterized by the mapping φ(⋅) and the image matrix B.

Measures that characterize a position or the relation between two positions figure

centrally into goodness-of-fit measures for blockmodels of social networks. Wasserman

and Faust (1992) denote the number of actors in a position Br as gr, and define grs as the

number of possible ties from actors in position Br to actors in position Bs as

€

grs =grgs if gr ≠ gsgr(gr −1) if gr = gs,

(2.1)

corresponding to the assumption that there are no ties between actors and themselves.

They go on to define the density of ties from a position Br to a position Bs as

€

Δ rs =xiji∈Br , j∈Bs

∑grs

. (2.2)

Assessing the goodness-of-fit of a blockmodel also requires some notion of the

pattern of ties predicted by the model. Wasserman and Faust (1994) refer to this as a

target blockmodel denoted X(t). In general, the network of ties predicted by a blockmodel

p. 29

can be referred to as the estimated network

€

ˆ X . The entries

€

ˆ x ij of this model can be

computed as

€

ˆ x ij = bφ ( i)φ ( j ). (2.3)

where

€

ˆ x ij  = 1 means that the blockmodel θ = (B, φ) predicts that there should be a tie

between actor i and actor j.

Wasserman and Faust use these definitions to outline several different goodness-

of-fit measures for descriptive blockmodels. Each of these measures has slightly

different implications for the way in which the accuracy of a model and the complexity of

a model contribute to determining which models are selected. Two of these measures are

relatively straightforward, effectively taking the form of a city-block distance measure.

The measure δx1 measures the distance between the observed tie densities and the

predicted tie densities, while δb1 measures the distance between the observed block

densities and the predicted block densities.

€

δx1 = ˆ x ij − xiji≠ j∑ (2.4)

€

δb1 = ˆ b rs − brsr≠s∑ . (2.5)

While these two expressions are very similar, the measures reflect a subtly different

balance of accuracy and complexity. The actor-level measure δx1 is effectively a direct

measure of model accuracy—every deviation of the predicted set of tie values is treated

p. 30

with the same weight. The complexity of the model, as a function of the number of

blocks, does not influence this measure either directly or indirectly, net of the ability of a

more complicated model to predict tie values with greater accuracy. The block-level

measure δb1 differs from the actor-level measure in that it weights deviations from

predicted ties by the size of the block. While this is a subtle distinction, it indirectly

causes the measure to factor in complexity effects, albeit in a less than straightforward

way.

With a bit of algebraic manipulation, Equation 2.5 can be rewritten as

€

δb1 =ˆ x ij − xiji∈Br , j∈Bs

∑grsr,s

∑ . (2.6)

In a given block, the value

€

ˆ x ij − xij will never change sign. In a block where brs = 0, this

value will either be 0 or –1, and in a block where brs = 1, this value will either be 0 or 1.

Either way, taking the absolute value of this expression inside the interior summation is

equivalent to taking it outside of the summation, as this interior sum is simply a measure

of the absolute value of the difference between the predicted and actual density in a

block. As a result, Equation 2.6 can be rewritten as

€

δb1 =ˆ x ij − xiji∈Br , j∈Bs

∑grsr,s

∑ =ˆ x ij − xij

gφ (i)φ ( j )i≠ j∑ . (2.7)

p. 31

This formulation of the block density measure makes the weighting of tie-level deviations

by the size of the block explicit. Moreover, it begins to show how the complexity of the

model can affect this measure of goodness-of-fit. If we assume that actors are evenly

distributed into positions, and denote the number of positions in a blockmodel as |B|, then

the average position Br has g/|B| actors in it, and the value gφ(i)φ(j) can be estimated as

g2/|B|2. This means that Equation 2.7 can be estimated as

€

δb1 ≈ˆ δ b1 =

B 2

g2ˆ x ij − xij

i≠ j∑ =

B 2

g2 δx1. (2.8)

A descriptive blockmodel with |B| positions is parameterized by |B|2 brs values.

Accordingly, this measure assesses the complexity of a model as a function of the

number of parameters |B|2, and as a function of the total number of possible ties g2.

Blockmodels of higher complexity in this sense are penalized in the complexity term,

balancing out the extent to which they are positively evaluated for being accurate. If

measure δx1 is taken to be a measure of model accuracy, then the measure δb1 does, in

some sense, incorporate both accuracy and complexity.

In addition to these two relatively straightforward measures, Wasserman and

Faust review other measures of descriptive blockmodel goodness-of-fit. The Carrington-

Heil-Berkowitz measure (Carrington, Heil and Berkowitz 1979; Carrington and Heil

1981), denoted here as δb2, is conceptually similar to a χ2 measure. It assesses the fit of a

blockmodel based on an α-fit criterion, such that a block is assigned to be a zero-block

p. 32

only if the density in that block is less than α. Carrington, Heil and Berkowitz define a

quantity trs as

€

trs =1 if Δ rs <α

1−αα otherwise.

(2.9)

This quantity, in turn, can be used to define the goodness-of-fit measure δb2 as

€

δb2 =grs

g(g −1)r,s∑ Δ rs −α

αtrs

2

. (2.10)

The second term in the summand in Equation 2.10 is, like the measure δx1, a measure of

the accuracy of a model. If the actual set of ties in a social network matches the

predictions of the blockmodel exactly, this term is equal to 1, and it is minimized to the

extent that the ties do not match the hypothesized pattern. Given the assumption that the

average position Br has g/|B| actors in it proposed above, the first term in the summand

effectively becomes |B|2, a count of the number of parameters in the model. However,

rather than balancing accuracy and complexity, increased complexity and increased

accuracy both raise the value of this measure. As such, it cannot be used to choose a

model that balances accuracy and complexity.

A final measure Wasserman and Faust review is an actor-level matrix correlation

measure δx3 (Panning 1982). If we define

€

x ij as the mean of all ties

€

xij , and

€

ˆ x ij as the

mean of all predicted ties

€

ˆ x ij , then the matrix correlation measure is defined as

p. 33

€

δx 3 =(xij − x ij )( ˆ x ij − ˆ x ij )i≠ j∑

(xij − x ij )2

i≠ j∑( )1/ 2

( ˆ x ij − ˆ x ij )2

i≠ j∑( )1/ 2 . (2.11)

The matrix correlation δx3 effectively measures the pair-wise correlation between the

actual tie values in a network and the ties values predicted by a blockmodel. As such,

this measure is, like the measure δx3, basically a measure of the accuracy of a blockmodel.

Net of the ability of a more complex blockmodel to more accurately predict ties, this

measure does not incorporate the complexity of a blockmodel.

The performance of these four measures can be demonstrated by considering the

network of expressed affect between the monks studied by Sampson (1969). Table 2.1

shows how each of these measures assesses the fit of four candidate blockmodels. The

subscript of each model corresponds to the number of blocks in the model. The models

θ3 and θ5 correspond to the three and five-position models proposed by White, Boorman,

and Breiger (1976), respectively. The model θ1 is a blockmodel with a single position,

and the model θ18 is a blockmodel with each actor assigned to his own position.

Model δx1 δx3 δb1 δb2

θ1 250 n/a 0.82 0.01θ3 78 0.49 2.41 0.39θ5 71 0.52 6.35 0.48θ18 0 1.00 0.00 1.00

Table 2.1: Descriptive Blockmodel Goodness of Fit Measures

p. 34

The results in the first two columns of Table 2.1 demonstrate how the two actor-

level goodness-of-fit measures δx1 and δx3 both behave as measures of model accuracy.

As the models become successively more fine-grained, they become more accurate, as

these measures indicate. As the effects of complexity and accuracy work in the same

direction as incorporated in the Carrington-Heil-Berkowitz measure δb2, it is impossible

to determine from these data the extent to which complexity and accuracy independently

affect the measure. It is clear, however, that the measure rewards models that are

complex and accurate relative to models that are simple but inaccurate. Of these four

measures, only the block-density measure δb1 balances complexity and accuracy. For

instance, it evaluates the three-position model θ5 more favorably than it does the five-

position model θ3, even though the five-position model is more accurate. Still, all four

measures evaluate the fully saturated eighteen-position model θ18 as the one that fits the

data best. That the results of these analyses do not reflect Sampson’s intuitive insights

about the structure of this group does not alone imply that these measures should be

dismissed as inadequate. A measure that cannot produce a result that suggests there is no

structure in a social system would clearly be problematic. Rather, all four of these

measures are inappropriate for determining structure in this way because they will never

evaluate a fully saturated model less favorably than a model of lesser complexity.

The examples of descriptive modeling approaches presented here all relate to

modeling group structure using social networks. While the critique presented here is

directed at descriptive blockmodeling approaches, it can be directed at any descriptive

modeling analysis in which the way that the accuracy and complexity implications of the

model selection criterion are not made explicit. For example, in their analysis of career

p. 35

systems, Stovel, Savage, and Bearman (1996) use a blockmodeling approach to cluster

career paths, and argue that a career structure based on five groups is appropriate given

their data. Their empirical support for this choice (1996: fn. 18) is based on the fact that

the mean within-block distances at this level of analysis are smaller than the mean

between-block distances, at a statistically significant level. This metric essentially

assesses the accuracy of the model of career systems without considering their

complexity. If the set of careers was partitioned into a successively larger number of

subgroups, the statistical significance of the difference between within-block and

between-block distances would grow monotonically until each career path were

partitioned into its own group. The fact that Stovel, Savage, and Bearman choose a five-

block model rather than a fully-saturated 80-block model is at least consistent with the

possibility that they sought to balance the complexity of their model against the accuracy

of their measure.

2.3 Predictive Modeling Approaches

Predictive modeling approaches are distinguished from descriptive modeling

approaches in that they provide explicit accounts for the mechanisms by which actors

diverge from these structural regularities. Many descriptive models of social structure

can be straightforwardly transformed into predictive models by making the assumptions

embodied in a descriptive model explicit. Predictive models of exchange in a social

network are generally referred to as stochastic network models, and predictive models

that take into account the assignment of actors to categories are termed stochastic

blockmodels. In this section I discuss these stochastic blockmodels and compare them to

p. 36

descriptive blockmodels with respect to their ability to assess the structure of exchange in

a network.

2.3.1 Stochastic Blockmodeling

The term stochastic blockmodeling can be used to refer to an entire class of

models that assign a probability p(x|θ) to the observation of a particular pattern of

network ties x given a set of model parameters θ. While there are a number of

researchers who have presented stochastic blockmodeling approaches (Wasserman and

Pattison 1996; van Duijn, Snijders, and Zijlstra 2004), I focus here on two models that are

particularly germane to the kinds of exchange in the networks that will be empirically

investigated in this dissertation. Both of these models are based on a set of ideas drawn

out of the basic p1 stochastic graph model (Feinberg and Wasserman 1981; Holland and

Leinhardt 1981), the details of which I present below.

The p1 stochastic graph model is an extension of a basic Bernoulli graph that

attempts to take into account the fact that some actors are relatively more likely to engage

in exchange than others, and that in some cases, actors may be likely to reciprocate the

exchange behavior that is directed at them from other actors. Holland and Leinhardt base

their model on the assumption that, net of a set of structural parameters, the exchange

behavior in a dyad Dij = (xij, xji) is independent of the behavior in all other dyads in a

network. They derive an expression for the probability of the observation of a given

pattern of dyadic exchange based on the likelihood that the dyad reflects a mutual,

asymmetric, or null pattern of exchange. This distribution is termed the MAN

distribution, such that for a given dyad Dij:

p. 37

mij = P(Dij = (1,1)) i < j, (2.12)

aij = P(Dij = (1,0)) i < j, (2.13)

aji = P(Dij = (0,1)) i < j, (2.14)

nij = P(Dij = (0,0)) i < j, (2.15)

and

mij + aij + aji + nij = 1, for all i < j. (2.16)

The authors use this formulation to show that the probability of the observation of a given

network of ties can be expressed as:

€

P(X = x) = mijxij x ji aij

xij (1−x ji )

i≠ j∏

i< j∏ nij

(1−xij )(1−x ji )

i< j∏ , (2.17)

which can be expressed in an exponential form as:

€

P(X = x) = exp{ ρij xij x ji + θij xij} niji< j∏

i≠ j∑

i< j∑ , (2.18)

where

ρij = log((mijnij)/(aijaji)) i < j (2.19)

and

θij = log(aij/nij) i ≠ j. (2.20)

Holland and Leinhardt explain that the parameter ρij governs what they term the

“force of reciprocation”, that is, the increase in the log-odds of the likelihood that a tie

will be sent from an actor i to an actor j (xij = 1) if there is a tie sent from the actor j to the

p. 38

actor i (xji = 1). Similarly, they explain that the parameter θij measures the increase in the

log-odds of the likelihood that a tie will be sent from an actor i to an actor j (xij = 1) in the

absence of a tie from the actor j to the actor i (xji = 0). The family of networks described

by a full set of these parameters is not estimable, so Holland and Leinhardt propose a

model based on a restricted set of parameters such that

ρij = ρ (2.21)

and

θij = θ + αi + βj. (2.22)

In other words, they restrict reciprocity to act in a constant way across all dyads, and

force the asymmetric choice parameter θij to be a function of the productivity of the

sending actor αi, the attractiveness of the receiving actor βj, and the mean choice

tendency θ. It is also worth noting that the expected value of the logit is determined by

this function as well, such that

E(log(pij/(1-pij)) = θ + αi + βj, (2.23)

where

pij = P(xij = 1). (2.24)

Collectively, these formulations can be summarized by noting that, for a binary-

valued network with no reciprocity, the p1 model predicts that the expected value of the

logit of tie values is an additive function of the overall tendency of ties to exist in the

p. 39

network θ , the productivity of the sender of the tie αi, and the attractiveness of the tie

recipient βj. In networks where the reciprocity ρ diverges from zero, the likelihood of a

tie to be sent from an actor i to and actor j will be increasingly determined by whether or

a not a tie is sent in the opposite direction, to the extreme case where ρ = ±∞, and a

network becomes completely symmetric (or asymmetric), wherein the tendency of a tie is

completely determined by this reciprocal behavior.

2.3.1 The p1 Stochastic Blockmodel

The p1 distribution is useful for characterizing the probabilistic structure of graphs

and network in a general sense, but it does not provide a mechanism for explicitly

modeling the influence of the group structure of actors on the likelihood of exchange

behavior. While a variety of stochastic blockmodeling approaches have been proposed to

achieve this aim (Holland, Laskey, and Leinhardt 1983, Anderson, Wasserman, and Faust

1992) the p1 stochastic blockmodel proposed by Wang and Wong (1987) most fully

achieves this objective in the context of the p1 random network distribution.

The principal contribution of the p1 stochastic blockmodel to the basic p1

distribution is that it allows the asymmetric choice parameter to be determined in part by

the group memberships of the sending and receiving actors involved in a dyadic

exchange. If the sending actor i is a member of a block labeled r, and the receiving actor

j is a member of the block labeled s, then Equation 2.22 above can be expanded as

θij = θ + θrs + αi + βj, (2.25)

p. 40

where θrs corresponds to the relative excess tendency for actors in block r to direct

choices toward actors in block s. This simple extension allows a p1 stochastic

blockmodel to capture the effect of different assignments of actors to groups on the

likelihood of observing a particular pattern of network exchange.

2.3.2 The p1R Stochastic Blockmodel

The exchange behavior that many stochastic network analytic approaches attempt

to model is essentially dichotomous—the outcome of interest is simply whether a focal

actor chooses a particular actor or not. While many kinds of social exchange behavior

can be reasonably modeled as dichotomous outcomes, there are clearly some kinds of

behavior for which reduction to a dichotomy would represent a fairly severe limiting of

the expressive range of the phenomena.

There are surprisingly few stochastic network models that can be used to measure

non-dichotomous exchange behavior. The principal analytic strategy taken by these

models has been to move from only considering the likelihood of an exchange taking on

a single (dichotomous) response level to considering the likelihood of an exchange taking

on one of a number of response levels. Wasserman and Iacobucci (1986) introduce an

early model along these lines that expands the p1 model to the analysis of networks where

relations take on one of C discrete values. Anderson and Wasserman (1995) generalize

this model by considering the interactions between response levels in addition to their

first-order effects.

There are a number of empirical phenomena that might effectively be analyzed

using a model based on ordinal or categorical relations. As an example, Wasserman and

p. 41

Iacobucci (1986) analyze networks of behavior expression in which the frequency

between two actors is characterized as “rarely”, “sometimes” or “frequently”. While the

models proposed by Wasserman and co-authors can straightforwardly be applied to these

phenomena, there are other relational behaviors that are not so easily reduced to ordinal

or categorical responses. In particular, these categorical models do not correspond well

to networks that represent resource flows. Networks that represent the flow of individual

migrants between cities or nations, investments between firms or nations, or goods and

money between industries (Burt 1983) exemplify these resource exchange networks. In

many of these cases it would be difficult to generate the theoretical logic that would

support modeling a level of exchange that is fundamentally continuous as a categorical

variable.

The p1R stochastic network model presented here departs from these categorical

models in that it explicitly models network exchange as a continuous variable. One of the

most significant differences between binary and real-valued networks, of course, is that

ties in binary networks can only take on two values, while exchange levels in real-valued

networks can take on any of a continuous range of values. As a result, the distribution of

tie values in a random real-valued graph that underlies such a network is a bit more

complex than the relatively simple one-parameter Bernoulli graph that underlies a binary

network. For positive real-valued exchange networks1, a relatively simple approach

would be to assume that the tie values are log-normally distributed with mean θij and

1 This approach can also be used for non-negative real-valued exchange networks if allzero-valued exchanges are assigned some value smaller than the lowest observed non-zero tie value. This approach may be particularly valid in those cases where zeroes in thedata represent exchange levels that were too low to report, rather than exchanges thatwere actually zero.

p. 42

variance σij2. The model can be further simplified by assuming that the variance of this

distribution is constant across the network, such that σij2 = σ2.

Given these assumptions, it is possible to expand the governing equations of the

p1 model to a model for real-valued exchange networks. Equation 2.23 above represents

the expected value of exchange between two actors in a dichotomous network. This

equation can be expanded to model real-valued exchange as follows:

€

E(log(xij )) = θ +(α i + β j ) + ρ(α j + β i)

1+ ρ, (2.26)

where the reciprocity parameter ρ ranges from 0 to 1. This formulation allows Equation

2.26 to reduce to Equation 2.23 when there is no reciprocity in an exchange network, and

it forces the expected value of an exchange E(xij) to be equal to the expected value of

exchange E(xji) when reciprocity is at its maximum of 1. Equation 2.26 can be expanded

to a governing equation for a p1R stochastic blockmodel by introducing a block parameter

θrs as follows:

€

E(log(xij )) = θ + θrs +(α i + β j ) + ρ(α j + β i)

1+ ρ. (2.27)

This formulation allows the p1R stochastic blockmodel to capture the expressive

range of models that the basic p1 stochastic blockmodel does in the context of real-valued

network exchange. To the extent that p1 stochastic blockmodels are appropriate for

empirical investigations of the organizational position metaphor in the context of

p. 43

dichotomous exchange, p1R stochastic blockmodels should therefore be appropriate for

studying organizational positions as defined by real-valued exchange.

2.3.3 Stochastic Blockmodel Goodness of Fit Measures

Comparatively speaking, there are many fewer goodness-of-fit measures for

stochastic blockmodels than there are for descriptive blockmodels. The p1 and p1R

stochastic blockmodels presented here are like many stochastic blockmodels in that they

can be used to produce a set of predicted tie values

€

ˆ x ij in addition to assigning a

probability p(x|θ) to any observed pattern of ties. Wasserman and Faust (1992) argue that

the likelihood-ratio statistic G2 is an appropriate goodness-of-fit measure for stochastic

blockmodels that can be characterized in this way. They determine the value of this

statistic in the context of a stochastic blockmodel θ as

€

Gθ2 = 2 xij log(xij / ˆ x ij ).

i, j∑ (2.28)

Wasserman and Faust argue that this is a reasonable goodness-of-fit metric for the

assessment of stochastic blockmodels that assume the dyadic independence of tie values

net of the structural parameters of the model. Under this assumption, they argue that this

€

Gθ2 metric is distributed as χ2, and as such can be used to compare the goodness of fit of

models of different complexity, as long as these models are nested by evaluating the p-

value of the

€

Gθ2 measure given the degrees of freedom in each model. The implication of

this approach is that the model that should be chosen for a given network is the most

complex one for which the p-value is still insignificant. Wasserman and Faust argue that

p. 44

an alternative to this approach is to use a normalized

€

Gθ2 metric, where the measure is

simply divided by the degrees of freedom. Following this logic, the model that should be

selected is the one with the lowest normalized

€

Gθ2.

Both of these proscriptions position the

€

Gθ2 metric as a goodness-of-fit measure

that balances the accuracy of the data with respect to a model against the complexity of

the evaluated model. The statistical rationale supporting the use of this measure for the

purpose of model selection highlights the distinction between descriptive and predictive

modeling approaches. While the predictive modeling approach can bring the power of

statistical analysis to bear upon the problem of model selection, statistical measures such

as the

€

Gθ2 metric cannot comprehensively address all of the issues this problem presents.

One inherent problem with this approach is that the probability theory underlying these

measures is based on the assumption that only one model is being evaluated. As

Wasserman and Faust (1992: 703) note, “This theory should be applied only to a priori

stochastic blockmodels, because the ‘data mucking’ that must be done to fit their a

posteriori counterparts invalidates the use of a statistical theory.” If the objective of the

model selection task is to compare a wide range of models to determine the one that is the

best representation of the data, then these approaches cannot be used.

2.4 Conclusions

The blockmodeling approaches outlined in this chapter provide formal methods

for addressing some of the issues raised by the organizational position metaphor.

Descriptive models and their associated goodness-of-fit measures are useful for

beginning to think about how to assign actors to organizational positions on the basis of

p. 45

their degree of structural equivalence. Given a particular level of aggregation, these

methods can be useful in identifying good ways to partition actors into their respective

contexts, and as such identify the boundaries of these positions. Descriptive blockmodel

goodness-of-fit measures can be helpful in evaluating the relative fit of one partitioning

relative to another, but the theoretical meaning of these measures is not precisely clear.

While descriptive blockmodeling approaches provide some purchase on the

problem of identifying the boundaries of organizational positions, predictive approaches

provide a way of thinking about the idea of closeness as implicated by the organizational

position metaphor. Specifically, in assessing the likelihood of observing a particular

exchange between organizations, a predictive model allows a researcher to directly assess

the extent to which the behavior of a given organization is close to the aggregate behavior

of other organizations located in the same position. Under this modeling approach,

organizations embody the positional idea of closeness explicitly to the extent that they are

likely to engage in a particular pattern of exchange behavior.

The conclusions that can be drawn from a predictive blockmodel may, in fact, be

richer and more informative about structural processes than those reached through a

descriptive modeling approach. The logic that Wang and Wong (1987) apply to

analyzing the impact of gender on the production of friendship relationships could

straightforwardly be extended to the analysis of the impact of industry structure on the

exchange of goods and resources between firms. The p1R stochastic blockmodel

introduced in this chapter aims to extend the analysis in exactly this way. A descriptive

blockmodel analysis of this exchange network might produce a set of ways to partition

organizations into industrial positions based on the similarity of their patterns of

p. 46

exchange behavior. A stochastic blockmodel analysis, on the other hand, would use one

of these industrial classification schemes to indicate the extent to which the economic

exchange behavior of individual firms is related to the aggregate exchange behavior of

other firms within their respective industry position.

While a predictive blockmodel analysis might be useful in assessing how well a

given set of exchanges between organizations corresponds to a particular industry

structure, it would not by itself be able to unequivocally identify the boundaries of a set

of organizational positions. The fundamental problem with both the descriptive and the

predictive modeling approaches is that neither provides a transparent facility for directly

assessing the complexity of a model of social structure. Predictive models of social

structure like the p1R stochastic blockmodel are more useful than descriptive models

because they define the accuracy of a model in terms of the probability that it will predict

the observed pattern of behavior, rather than in terms of an arbitrary metric. While

predictive models provide this statistically grounded rationale for evaluating the accuracy

of models, they do not provide such a rationale for the direct assessment of model

complexity.

In the following chapter, I introduce a method drawn from the field of information

science that specifically can be used to answer this question. This method allows the

problem of assessing model complexity to be laid out in terms of a formal and transparent

probabilistic theory. When taken in combination with the predictive models of social

structure outlined in this chapter, this method can be used to empirically identify

organizational positions, and move the organizational position metaphor in the direction

of an organizational position construct.