© F. Enke Verlag Stuttgart Zeitschrift für Soziologie, Jg. 6, Heft 4, Oktober 1977, S. 3 8 6-402
Social Structure as a Network Process*
Paul W.jHolland
Educational Testing Service, Princeton, New Jersey Samuel Leinhardt
Carnegie-Mellon University, Pittsburgh, Pennsylvania Sozialstruktur als ein Netzwerk-Prozeß
I n h a l t : Ziel des Aufsatzes ist eine Diskussion des Konzepts der sozialen Struktur sowie die Beschreibung eines mathematischen Modells zur Analyse sozialer Strukturen. Es wird argumentiert, daß eine möglicherweise fruchtbare Darstellung sozialer Strukturen erreicht werden kann, wenn man sich bestimmter stochastischer Netzwerk-Prozeß- Modelle bedient. Ausgangspunkt sind die strukturellen Eigenschaften elementarer sozialer Beziehungen als erster Schritt zu einem besseren Verständnis der Entstehung und Entwicklung von Strukturen sozialer Beziehungen.
A b s t r a c t : Our purpose here is to discuss the concept of social structure and to describe a mathematical framework for its study. We will argue that for these purposes, a potentially fruitful representation of social structure can be obtained using certain stochastic network processes. We concentrate on the structural properties of elementary, social relation as a first step towards understanding how structure in social relations arises and evolves.
1. Introduction
Our purpose here is to discuss the concept of social structure and to describe a mathematical framework for its study. We will argue that for these purposes, a potentially fruitful representa
tion of social structure can be obtained using certain matrix-valued stochastic processes. But before we can investigate this approach we need to make precise what we mean by social struc
ture.
The definition of social structure which we offer is not meant to encompass or summarize other views. It has an obvious local focus. We concen
trate on the structural properties of elementary, social relations as a first step towards unter- standing how structure in social relations arises and evolves.
* This report is part o f a continuing research series and describes work that is collaborative in every respect.
The order of authorship is alphabetical. We are grateful to Stanley Wassermann and James A. Davis for as
sistance in conducting the reported research. Support was provided by NSF Grant SOC 73-05489 to Carne
gie-Mellon University. Prepared for presentation at the International Conference on „Mathematical Ap
proaches in Social Network Analysis“, Bad Homburg, West Germany, March 1 7 -1 9 , 1977.
2. The nature of social structure
In the jargon of sociologists no term is more common, more important and less consistently applied than: “social structure” . Indeed, in preparing this paper we explored the sociological literature and informally interviewed other sociologists in an attempt to arrive at a coherent, precise and functional definition. Unfortunately, we obtained as many definitions as we had sources. While some have perceived strength in this diversity of opinion (MERTON 1975) we were distressed by it. To attempt to model mathematically a thing which everyone agrees is important but whose nature no one agrees on is an impossible task.
We believe it is important to be consistent in scientific research and, in the absence of a more compelling commonly accepted definition, we propose the following. Social structure is regular arrangements o f social relations that result from laws operating over time in a persistent though not necessarily unchanging fashion. It follows from this definition that two fundamental tasks for sociology are the identification of these social laws and the modeling of their dynamics. In this section we elaborate upon this definition.
2.1 Relations
When we study social structure we study relations,
P. W. Holland, S. Leinhardt: Social Structure as a Network Process 387 social relations. By social relations we mean social
ly meaningful relationships existing between pairs of individuals. Examples include friendship, in
fluence, kinship, authority, joint board member
ship, etc. For our purposes, relations are taken as givens. That is, we are not particularly interested in investigating the substantive meaning of rela
tions. We are satisfied to assume that their meaning and significance are empirically ba
sed. Since social relations are our unit of observation, however, we need to provide an intuitive appreciation for the concept. Thus, the statement that a relation obtains between two entities, a and b, means, for us, that there exists a relation, R, defined on a set of nodes, S, such that the ordered pair (a, b) is contained in R (denoted as usual by aRb)1.
This conceptualization of a social relation follows the usual set-theoretic definition of a binary rela
tion, a relation that either obtains or does not obtain between a pair of elements in a set (see, for example, HARARY, NORMAN and CART
WRIGHT 1965). With this use of elementary set theory we begin the development of a mathemati
cal framework for studying social structure. As we proceed we shall take care to check on the reasonableness of our formalization, on the assumed isomorphism between our mathematical constructs and the real phenomenon. Consequent
ly, we must ask whether it makes sense to talk about social relations in the same way that we talk about mathematical relations.
We often speak of social relations as tieing individ
uals together, bonding them to one another.
The above conceptualization of a relation provid
es this in the intuitive sense of an aggregation of the elements a and b. Thus, we have no problem thinking this way about the relation parent. If the S is taken as all living people we can define a relation, P, “is the parent of,” that is the set containing parents and children as ordered pairs, (a, b), such that if a is the parent of b, aPb for all a and b contained in S. The relation, P, is obviously directed. P is both a biological and a social rela
tion. Of course, it is a relation which may have 1 When we use the term relation we will be referring
to be entire set, R, containing all the ordered pairs each of whose members is contained in S. When we speak of a specific pair of nodes and focus on the relation between them we will use the term tie.
different socially meaningful implications in dif
ferent societies but it is a relation that exists in all societies.
The parent relation and other similar kinship re
lations seem to be in accord with the mathema
tical notion of a relation. We face no particular conceptual difficulty in thinking about the social relation brother or mother’s brother’s son or great-aunt as if they were mathematical rela
tions defined on a society’s population. One of the reasons that this representation is so readily accepted is that kinship relations have a quality of certainty and permanence to them. Being someone’s parent is not a matter of choice or discretion. It is a fact of birth. While one may disavow parentage, legally alter it, or not even be aware of it, the relation of natural parent remains unchanging.
If we conceive of a continuum of discretion and changeability, with ascriptive kin relations at one extreme, then interpersonal affective relations must surely be at the other extreme2. Friendship or sentiment ties are acquired at no particular point in the life cycle, are apparently subject to individual choice, and are certainly modifiable.
And yet they are social relations in the sense that we aggregate friends together informally as well as analytically. Consequently, in a given population we may think of the social relation friendship in terms of a set of ordered pairs of elements belonging to the population in much the same way as we conceptualized the relation of parent. The difference is that we expect to observe modifications in the friendship relation even after the passage of a very short period of time whereas, in the parent relation, we would be surprised to see much variation over short periods of time.
It is important to notice that we are also at ease in thinking of a population with more than one relation defined on it. Kinship ties co-exist with friendship ties. Obviously, individuals may be tied to one another in numerous socially meaning
ful ways. Indeed, some social relations may be functionally dependent on others (see, for example, 2 Formal authority and board membership are exam
ples of social relations lying between kinship and friendship along this continuum. Kinship relations may themselves also be ordered in this fashion.
388 Zeitschrift für Soziologie, Jg. 6, Heft 4, Oktober 1977, S. 386 -4 0 2 LO RRAINE and WHITE 1971, WHITE, H. C., S.
A. BOORM AN and. R. L. BREIGER 1976, and BOO RM AN, S. A ., and H. C. WHITE 1976).
To summarize: When we study social structure the unit of analysis is a social relation. Such rela
tions tie pairs of individuals to one another in the sense that if a particular relation exists between two individuals then it is reasonable to aggregate the two and represent them as an ordered pair, a member of the set defining the relation. Relations may vary in their expected permanence and the manner in which they are acquired3. Multiple relations may be defined simultaneously on the same population.
2.2 Regularity
Set theory is perhaps the simplest structural theory in mathematics. It is structural in the sense that aggregations are made without regard to the properties of the elements; the properties of the relations provide organization. These relational properties are essentially rules for predictions.
They are qualities that a relation possesses which condition it by informing us of its regular conse
quences. In other words, when a relation has a specified property, membership in the relational set is restricted. Perhaps an example will help.
There are numerous mathematical properties;
consider two: asymmetry and symmetry. If a relationship, R, is asymmetric then there are no elements, a, b in S with aRb and bRa. That is, it is not possible for the ordered pairs (a, b) and (b, a) to be in R simultaneously. Such an event is against the rule. Symmetry simply means that for all a, b in S aRb if and only if bRa. That is, whenever (a, b) occurs in R (b, a) must also occur in R, and vice versa. Thus, when a relation posses
ses a property such as symmetry the existence of a tie between a pair of elements carries with it a further implication (which may refer to the same or another pair).
A fundamental objective in studying social struc
ture is to determine whether we can identify regularities. We mean by regularities much the same thing as relational properties: qualities consistently associated with a social relation that 3 We have not spoken about the strength of a relation.
We assume relations are binary.
affect the way it ties individuals to one another.
These regularities are equivalent to social laws which operate to effect a structuring of individuals in the same sense that properties of relations structure relational sets. We call the regularities laws because they specify relationships among variables, relationships among ties between pairs of individuals. If social structure is more than simply an architecture of social entities, i.e., a temporary and ad hoc pasting together of indi
viduals, then we should expect to see social rela
tions operating regularly, lawfully conditioning the arrangement of ties between individuals.
Kinship again provides an unambiguous example.
The parental relation, P, discussed earlier, is clearly asymmetric. If aPb then bPa cannot be true. Thus, in a society no pair of individuals exists such that each is the other’s parent.
Empirically, we might ascertain this by inter
viewing all members of a society or by studying the society’s kinship rules. From each we could derive the empirical fact that one of the regular qualities of the parent relation is asymmetry.
Assumedly, we would not jump up and down having deduced this law. But why not? After all, it is a social law to the extent that it effects the arrangement of individuals and their ties to one another. Knowing that the relation parent exists between a pair of individuals certainly alters our expectations regarding whether the relation might obtain in the opposite direction. Perhaps, our lack of enthusiasm would be due to the facts that this is a commonly acknowledged property and one which is deterministic, there is no aspect of chance associated with its implication. If, however, we were discussing some other relation, say friendship, whose properties are neither so obvious nor so well determined, we might indeed have found something significant and worth calling a social law. The point, of course, is that if social relations such as friendship have proper
ties then these properties have implications for the social structure of a society just as the properties of kinship relations impact upon it.
In the case of kinship one of the most intriguing aspects of a structural analysis derives from the association of other non-kinship relations with kinship relations. Thus, understanding the kinship structure of a society can provide important insight into other features of the social structure.
P. W. Holland, S. Leinhardt: Social Structure as a Network Process 389 In some societies where classificatory kinship is
practiced, this objective is evident (see BOYD 1969). In contemporary western societies the gain is probably less.
In discretionary relations, such as friendship, the connection with other relations may not be obvious but, nonetheless they may exist and be consequential. The existence of a friendship relation between a pair of individuals may increase the likelihood of their interaction, of informa
tion diffusing, of influence flowing, of opinions being similar, etc. Notice, in these examples, that the consequence of the existence of a discretionary relation alters a probability of another relation, i.e., it conditions it stochastically.
To summarize: We expect social structure to be lawful in the sense that social relations possess specifiable properties which condition the ar
rangement of individuals, and structure their ties to one another. We expect these properties to hold throughout a society and we expect them to have demonstrable behavioral consequences.
2.3 Persistence
The notion of structure connotes something which is lasting and not ephemeral, something, in fact, which may outlast the individuals who are at any moment its constituents. And yet we are also aware of social change. Not only do individuals transit through positions in a social structure but the organization of relations itself manifest chan
ges over time.
What is it that changes and what persists? When we refer to the temporal continuity of social structure we are describing the relative permanence of the properties of relations, the laws of social structure, the rules describing relationships among relational variables. These properties persist and act over time to condition ties among pairs of individuals. In an actual society we expect the re
lation parent to be asymmetric at the beginning of an individual’s life and at its end. We expect it to possess this property from one generation to the next. Who is related to whom as parent chan
ges in an orderly fashion as generations advance but the relational property of asymmetry not does.
Over time the association of a social relation with other types of social relations may change, and
some properties of the relations themselves may change. However, we expect relational properties to change far more slowly than the individuals who are effected by them or the empirical ar
rangements they may at any moment engender.
When change in social structure occurs as a con
sequence of relational properties we talk of lawful change. Since these properties act over time we can expect change to be ongoing, an ever present aspect of the social environment. Indeed, in our view social change is fundamental;social structure is a consequence of lawful change. When we ob
serve continuity in structure over time we tend to assume that structure is unchanging. If a popu
lation has been in existence for some period and a relation has operated to structure it, most of the change due to the structuring process may be history. What remains may be a consequence of minor movements of individuals into or out of the population and residual fluctuations of ties.
Thus, what we perceive to be unchanging form may in effect be dynamic equilibrium, a situation in which the implications of a relation are effectively in balance.
To summarize: The properties of social relations have a persistence in time and act to condition the organization of relations among individuals over time. Consequently, at any moment, the social structure is a result of the operation of lawful properties. This social structure will change so that its future state will be related to its current state through the modifications that naturally result from the continued operation of relational regularities.
2.4 Configurations
Structure is synonymous with arrangement. In physics, one describes the structure of a crystal in terms of its lattice, which represents schema
tically the way the constituents are normally organized. The description covers all similar crystals because the structure is a consequence of the bonding properties of the crystal’s con
stituents, i.e., function of their physical rela
tions (and the environment). The lattice is a skeleton, the constituents are reduced to di
mensionless points and the skeleton diagrams the pattern followed by the relational bonds linking the constituents together. Because the structure
390 Zeitschrift für Soziologie, Jg. 6, Heft 4, Oktober 1977, S. 3 8 6-402 is a consequence of elementary physical properties
which we cannot alter we readily talk in terms of the characteristic lattice as if it were a consequence of physical laws. We may even try to determine these laws by examining the structure of crystals.
Social structure is not a crystal lattice and we do not intend to pursue the analogy further. But it does serve to emphasize the following point:
when we talk about structure we speak in terms of lawful arrangements. In social structure the issue for us is lawful arrangements of social ties because from their study we hope to deduce the nature of social laws.
The properties of social relations impose restric
tions on the arrangements of ties. The property of asymmetry provides an example. Let us dia
gram a relation, R, with an arrow and say that aRb is the same as the graphical configuration a-*b. If R is asymmetric and a-*b then we should not expect to see the relational configuration a^b.
Relations may simultaneously possess many pro
perties not all of which may be of analytical in
terest. The combined restrictions of multiple properties may further condition the kinds of arrangements or configurations of ties that can exist. For example, when a relation, R, is transi
tive, then the configurations (see HOLLAND and LEINHARDT 1975a)
j
may coexist. However, if the relation is transiti
ve and is also asymmetric, then the existence of one of these configurations precludes the other.
Certain groups of properties that can be posses
sed by a relation occur so commonly in mathe
matics and have such important consequences that a name exists for relations having them. Thus, a relation that is simultaneously reflexive, symme
tric and transitive is called an equivalence relation.
One that is reflexive, transitive and antisymmetric is called an ordering relation. An equivalence re
lation structures a set by inducing a partition on it of classes within which the relation holds. An ordering relation establishes a linear ordering among sets of equivalent elements.
Social relations appear to behave similarly.
Formal authority, for example, is usually assumed to be asymmetric, transitive and antisymmetric, i.e., it establishes a hierarchy of individuals.
Friendship is often presumed to be reflexive and transitive. When it is also symmetric it becomes an equivalence relation engendering cliques.
While the properties of relations affect the struc
ture the arrangements or configurations of rela
tions that result are the concrete aspects of struc
ture, the arrangements or configurations of rela
tions that result are the concrete aspects of struc
ture ; the aspects that usually come to mind when sented when it is used as an independent variable whose influence on other variables is under study.
Since the properties of relations act regularly to bring about lawful arrangements, observed con
figurations of ties will be those that do not yield contradictions, i.e., that follow regularly from relational properties.
If the essence of structure is arrangement then those arrangements that are of enduring scientific importance are the ones deriving from lawful processes. Arrangements that arise on an ad hoc, idiosyncratic basis tell us nothing about the laws of social structure. This is not to say that they have no impact on behavior. Rather, since they are not consequences of the regularities we call social laws, they provide us with no insight into these laws. In data on social relations, we observe particular arrangements of relations tieing individ
uals to one another. Not all possible arrange
ments of social relations are common. Some tend to be rare. This is to be expected if social relations possessed properties that conditioned their own arrangement making certain arrangements more likely to occur than others. Since these properties are probabilistic and act over time we do not expect to observe only lawful arrangements.
P. W. Holland, S. Leinhardt: Social Structure as a Network Process 391 Instead relational regularities operating over time
would tend to make certain arrangements more common. Change in arrangements would occur as discretionary choices were made, as individuals transited through the system and as other exo
genous events took place. Over the longer term, even the properties of the relations might change making some previously unlikely arrangements more likely in the future. In all cases, however, the relative prevalence and scarcity of particular relational configurations would be a consequence of lawful, regular processes. The research task, as we see it, is to find the laws that predict the con
figurations.
To summarize: Social structure is manifest in the orderly arrangement of social ties. The structural properties of relations condition the probability of occurence of each type of configuration.
Specific configurations are of interest to us be
cause they provide evidence of the existence of relational properties or social laws.
3. Modeling social structure
The definition that we have provided for social structure does not yield a simple, static pattern that we can point to and call social structure. It does permit social behavior to be structured in a regular fashion. It implies that a structuring proc
ess of ongoing rearrangement is a permanent aspect of social behavior. It has direct implications for any attempt to construct a mathematical model of social structure. A model of social struc
ture, according to our definition, must take into account the network of relational ties between individuals, the dynamic and stochastic nature of change in the arrangement of the ties, the contin
uous nature of change over time and the impact of current structure on future structure. In this section we discuss each of these requirements.
3.1 Network
In mathematics, networks are generalizations of directed graphs in which the edges and nodes may have values. The term social network as usually employed rarely exploits this full generality Typically, it refers to a set of individuals, the nodes, and a relation which is represented by
arrows connecting nodes (individuals). This is precisely what we shall mean by a network. Leav
ing the issue of weighted edges and valued nodes for another time, we shall use network and graph interchangeably. Note that this notion of a net
work carries with it no a priori assumption of relational properties. Consequently, the notion of an elementary relation that we introduced earlier and that of a network or graph introduced here are one and the same. We will use the matrix notation, X , to indicate a particular network:
Xy = 1 if i j, Xy = 0 otherwise; x- = 0 by convention.
One of the appealing features of thinking about and representing social relations in terms of a social network is that networks emphasize patterns of arrangements or configurations of ties. In other words, they draw attention to structure. Empiri
cal networks have been explored by social scientists since the early work of MORENO (1934) for precisely this reason. If we hypothesize that social relations possess certain qualities then we should be able to distinguish the types of config
urations that do not involve contradictions in the implications of these properties and by enumer
ating occurrences in an empirical network, we can obtain information relevant to accepting or rejecting the hypothesis. There are no particular limits to the kinds of configurations that can be studied in this manner. Constructing a precise hypothesis test is not trivial but tools have been developed to facilitate this process. (See, for example, HOLLAND and LEINHARDT, 1975a,
1975b, 1976, WASSERMAN 1977a).
It is important to understand what is at issue here. The social network is an abstracted and simplified version of a complex phenomenon.
For example, we abstract the nature of the inter
personal sentiment felt, expressed, or demonstrated by one person for another with a single directed line connecting two points. We throw subtleties and nuances aside and concentrate solely on the presence or absence of a relational tie. By exam
ining the patterns that these abstracted and simpli
fied ties make, we infer what properties the social relation possesses. Once we ascertain these prop
erties we can summarize the social structure by listing them. Since in an empirical situation individuals are connected to one another by many different types of relations, the amount by which even a few properties of social relations
392 Zeitschrift für Soziologie, Jg. 6, Heft 4, Oktober 1977, S. 3 8 6 -4 0 2 condition the occurrence of relational configura
tions may be vast.
A mathematical model of social structure must represent this network. The relation defined on the population is synonomous with the social network schematizing the ties between individuals Since there is no essential difference between the set theoretic or graph theoretic representations of a relation (or the algebraic matrix representa
tion), whatever properties hold for the relation, R, must hold for the graph, X. Thus, while we can think of structure in set theoretic terms as aggregations of individuals, we can also visualize structure in the sense of a diagram which schema
tizes the relation.
3.2 Dynamics o f Structure
An observation on an empirical network provides us with a snapshot of social structure, a picture of a social relation defined on a population at an instant in time. But the rules of structure operate over time in a dynamic fashion altering the probability of a tie occurring between a pair of individuals as a function of the tie’s overall con
sistency with the implication of the relation’s properties. Exogenous factors such as life cycle effects, social and geographical mobility, physical constraints or qualities of the individuals may also impact on the dynamic features of relational structure. But even if these did not intervene a fundamental feature of the laws is operation over time. The network that we observe empirically is a consequence of the network existing im
mediately preceding the observation of the net
work, its arrangement, and the properties of the relations. Whatever the role of exogenous factors, it is at a level distinct from and not contrary to the role of the social laws.
When we examine snapshots of structure we engage in a study of statics, which is, after all, a special case of dynamics. If we do so with the under
standing that we are examining an aspect of a process which has an unobserved past and future then we at least comprehend the nature of our problem although analytic tools with which to proceed may not be readily forthcoming. But when we presume that what we see is the social structure then we put ourselves on very weak ground. If social structure is dynamic we can have
little idea of the stage the process is in from one observation. Has it been ongoing for some time?
Is it just begun? Is it stationary? Will it change rapidly? We usually can’t say. If it is changing rapidly and we say that the observed arrangement is permanent we will be wrong. If we act on it we may well prove foolish. In a dynamic situation the only time that the study of statics makes sense is when equilibrium can be assumed. But the danger here is that a dynamic process in equilibrium does not mean an absence of change.
Thus, a mathematical model of social structure as we conceive of it, must be able to represent the network of relations dynamically. It must be a model which acknowledges that the process of structuring occurs over time.
3.3 Continuous time
How should the model envisage change occurring?
Are changes continuous or discrete? The way we have conceived of it, the properties of relations are always present and, consequently, they operate to foster change in configurational arrangements all the time. Opportunities for change do not arise on some regular basis, there is no cadence of rearrangement. Social change is not only a fundamental feature of social life ever present, it is ever occurring. When we obtain data on an empirical social network, such as socio
metric data, we effectively discretize this process.
But this is a convenience which simplifies data collection; it is not a necessity. To maintain adherence to the notion of structural properties operating regularly and without regard to explicit periodicities the process should develop in continuous time.
3.4 Stochastic
Must change occur? We have said that change is fundamental and ever present but is it certain? In the case of elementary kinship relations, the likelihood of change in a relation is zero, it can
not occur. At the other extreme, in friendship, where the tie is the result of a discretionary choice, fluctuations from one moment to the next are not at all unusual.
We do not mean to imply that structural laws
P. W. Holland, S. Leinhardt: Social Structure as a Network Process 393 based on relational properties will necessitate
modifications in relational arrangements when their implications are contradicted or that they guarantee stability when contradictions do not arise.
They have an impact in the sense that when they ope
rate, the conditional probability of a tie differs from the unconditional probability of a tie. Thus change is made more or less likely, it is not determined or made necessary. The more impor
tant the property, the greater the difference in the probabilities. In other words, structural laws are essentially stochastic.
The distinction between a stochastic and a deterministic representation of social structure is critical. Earlier, when we referred to a relation being symmetric it meant that aRb if and only if bRa. For a symmetric social relation we now say that if aRb then bRa is made more likely. Another way of thinking about this is through the empiri
cal notion of tendency. We could say that the social relation tends to be symmetric in that when aRb is observed bRa tends also to be observed. In other words, in an empirical situa
tion, if R tends to be symmetric and we observe aRb we may observe bRa. If the tendency toward symmetry is strong, the contrary situation will be rare. If the tendency is not strong, i.e., the signal is weak, empirical verification of a presumed sym
metry may be difficult to obtain.
If the existence of social structure means that certain arrangements of relations are made likely but not necessary and that these increased likelihoods will manifest themselves over time then we need a model which can capture both the stochastic and time dynamic features simul
taneously. But we must also remember what it is that is changing. If we talk of social structure in terms of relations which we identify with a net
work then what changes in time is the entire network, not a specific configuration or a particu
lar tie but the entire set of ties, the ordered pairs, that are the moment to moment definition of the relation. The random variable here is the entire network. It is the network that transits over time from one arrangement, one network, to another. An effective model must adhere to this notion o f a process on the network.
4. A framework for modeling social structure We formalize our notion of social structure by identifying it with a probability distribution over
a set of graphs. In the absence of social structure, all graphs are equally likely. Structural laws exist when some graphs are more likely than others.
Describing the probability distribution which accords with empirical reality provides us with an
“explanation” of that reality. We began our re
search collabroation by searching for evidence of structure (HOLLAND and LEINHARDT 1970).
Having convinced ourselves that structure exists (HOLLAND and LEINHARDT 1975b), we have proceeded to specify a general model for social structure, a continuous, matrix-valued stochastic process, and have begun its study.
4.1 Evidence o f structure
In our past research, we focused on the construc
tion of methods for determining whether empiri
cal social networks evidenced structure. The problem, as we saw it, was to create a sufficiently conditional model of a random graph and to use this model as a contrast for empirical graphs. By sufficiently conditional we mean to condition the probability distribution over all graphs for various features already known to influence structure but which operated at nodal or pair levels. In other words, we wanted to ask the question of whether these known factors were sufficient to explain the observed arrangements of relational networks.
The method we developed was a general null hypothesis test procedure. A graph is reduced to 16 summary statistics — counts of the triads.
These empirical frequencies are contrasted with computed expected frequencies for a graph that is similar to the empirical graph except that it is random. We were able to achieve a high level of conditioning, i.e., our random graphs were hardly random in the usual sense. They were highly structured save for the operation of rela
tional properties we were studying in the empi
rical graph.
One of these relational properties was transitivity.
Examination of hundreds of empirical graphs of interpersonal affect led us to conclude that the extent to which transitive configurations occurred
394 Zeitschrift für Soziologie, Jg. 6, Heft 4, Oktober 1977, S. 3 8 6-402 could not be explained on the basis of lower level
relational properties. This was important to us for two reasons. First, many sociological and social psychological propositions about social behavior presuppose transitivity. Second, transi
tivity is a quality of triples. That is, its presence can only be verified by examining triples of ordered pairs of nodes in which three different nodes participate. This, for us, is what is meant by social, i.e., something beyond ego and an im
mediate other. Consequently, it is for us the most elementary level at which questions of social structure become interesting.
The apparatus that we constructed to determine if interesting structure existed in social networks is quite elaborate. Transitivity is not the only relational property that can be tested nor is it the only one we are interested in. It is just one of the more obviously important properties. The ap
paratus as currently operationalized can detect evidence of any configuration that can be con
tained in a triad. Modifications enable studies of higher order configurations (HOLLAND and LEINHARDT 1975a, 1975b).
4.2 The network process
To model the dynamic, probabilistic nature of social structure we use a stochastic process. This is not the only approach that can be taken. For example, HUNTER’S (1974) model is dynamic but deterministic. Others (e.g., KATZ and PROC
TOR 1959, and SORENSEN and HALLINAN 1974) have developed stochastic models but these differ in important respects from ours.
The model we present here has been described in detail elsewhere (HOLLAND and LEINHARDT 1977) and has recently been studied extensively by WASSERMAN (1977b). Consequently, in this paper we will concentrate on developing the model as a general mathematical framework for discussing social structure and social change.
There are two mathematical notions that underly our model. The first is that of a multigraph and the second is that of a finite-state continuous
time stochastic process.
At any time t we will identify the structure of a
social system as a multigraph. Briefly, a multi
graph is defined by a set of individuals (or nodes) denoted by i, j, k , . . . , and a family of binary relations, R ., R 2, . . . Rf, . . . defined on the individuals. The actual relations used to describe the social system depend on the interest of the investigator and the type of social system under study. In this development we will take the { Rj } as given. Rather than using the notation of relations it is a little simpler to identify a relation, Rr , with its sociomatrix Xr (or, as it is called in graph theory, its adjacency matrix). Xr is defined as the matrix whose (i, j)-entry is Xyr = 1 if iRrj (i.e., if i stands in the relation Rr to j), otherwise Xjjr =0). The collection G of all the sociomatrices X j, . , Xr, . . . is the multigraph of the social system at a given point in time.
The notion of a stochastic process is essential in our model for describing social change and its effect on structure. We conceptualize social change as the lawful evolution of the social network as it takes on one state at one moment and another at the next. Thus, the social structure can be conceived of as transiting from one state in the state space to another over time. Nothing prevents it from returning to states previously occupied, nothing pevents it from remaining in the state it currently occupies. Exactly which state it will occupy is a probabilistic function of time.
A probablistic system that changes from state to state as time passes is a continuous-time stochastic process. For us, it is the entire multigraph that changes over time. We denote the social system (i.e., the multigraph) at time t by G(t) and each of the constituent sociomatrices by Xr (t). Each of the Xr (t) may be thought of as a binary-matrix valued stochastic process and the entire collection of these, G(t), is a multigraph-valued stochastic process. Of course, being a random function of time does not necessarily mean that structure is random in the sense that any state in the state space is equally likely. If social structure exists in the sense that social relations possess properties, then certain of the states in the state space be
come more likely than others. In others words, we anticipate that a social network will transit through the state space in an orderly fashion progressing towards arrangements that minimize configurational contradictions with relational properties.
P. W. Holland, S. Leinhardt: Social Structure as a Network Process 395 In our model this orderliness is brought in
through formulas for the change intensities.
These formulas determine the probability that a relation between a pair of individuals will change within some small time period.
A fundamental assumption of the model is that the change intensity, the probability of a change in the relational status of a pair of individuals some small time in the future, depends on the state of the entire network at that moment. This defines the change intensities (G(t), t) by
P {Xijr(t+h)= 1—Xijr (t)IG(t)}
= hXijr (G(t), t) + 0(h) a s h ^ O (1) Several assumptions are necessary for mathemati
cal tractability. We assume that no two edges of the multigraph change simultaneously. One way to think about this is to imagine that the infini
tesimally small periods of time in the future are so small that they almost discretize time. In other words, when a change occurs it occurs instanta
neously — so quickly that no other change is occur
ring at the same time and no other change is affected by it. In other words, the changes are conditionally independent given the current structure G(t). We also assume that instantaneous states do not occur, i.e., when a state is reached the network remains in it for some exponential
ly long period of time.
By positing specific functional forms for the change intensities we may propose particular models of social structure. This is one natural way of establishing a regression framework for the study of structural processes. Different func
tions for the change intensities represent different hypotheses regarding what is and what is not an important influence on the structure of a social system. The procedure is completely general. A model may bring in qualities of the nodes or in
dividuals such as sex, race, age, etc., properties of the pairs, of the triples and higher order configu
rations including general features such as connec
tivity, etc.
A necessary simplification to make the study of the process’s transitions sufficient is Markovity.
The Markov condition asserts that knowledge of the future development of the process is com
pletely determined by knowledge of the current
state. We have said earlier that current structure influences future structure and Markovity is one way of assuring that current structure not only matters, it is all that matters. Thus we model social structure as a system of finite-state, binary-matrix-valued, continuous-time Markov chains. But does this make sense? Is it reasonable to think that structural development has no memory and that the history of how an arrange
ment was achieved implies nothing for the future development of the structure? While we admit this is a simplifying assumption, we are willing to argue that it is not an unreasonable assumption.
We are assuming that the motivation for evolution in social structure derives from relational proper
ties. Structure materializes as a consequence of these properties. Initial situations are rearranged over time as relations are added and lost within the network. What happens next, in the immediate future, depends on relevant aspects of the individuals in the network and on the extent of structure already achieved. Does it matter what the structure was in the past? If symmetry is an issue, does it matter if asymmetry occurred in the past if symmetry exists now? We hazard the guess that in reality past structure exerts little effect when compared with present structure, so little that the assumption of Markovity, an ab
sence of memory, is a reasonable approximation.
The basic stochastic process G(t) is called the network process. It is the process of transitions from multigraph to multigraph which differ by one edge only. Social change, according to this model, evolves by edge changes occuring con
tinuously in time, conditioned by the properties of the relations effecting the structure of the entire multigraph and any other hypothesized, relevant, quantifiable feature of the social net
work at each moment in time.
By using a stochastic process to model social structure we do not restrict the arrangement of ties in a social network to any particular form.
Social change retains a natural quality. Each sociomatrix can be thought of as an observation on the particular process as it exists in an empiri
cal social system. Statistical inference can then be employed and the sociomatrices become data from which the parameters of the structural process, the properties of relations and other re
levant factors, can be estimated.
396 Zeitschrift für Soziologie, Jg. 6, Heft 4, Oktober 1977, S. 386 -4 0 2 The actual procedure involves specifying equations
for the change intensities. Elaborate models that depend on nodal, pair, higher-order configura
tional and temporal qualities may be specified.
However, estimation in many cases is not straight
forward and it pays to study some simple proc
esses. Examples appear in HOLLAND and LEIN- HARDT (1977), and further results occur in WASSERMAN (1977).
To illustrate what is involved consider a model for sociometric choice that involves only one relation.
(This illustration is drawn from HOLLAND and LEINHARDT 1977, section 3.1). Since there is only one relation we denote the network process by X(t) and specific realizations by x(t). In this case, we can construct a model which has parame
ters for individual, pair and triadic properties such as individual expansiveness, differential popularity, choice reciprocity, and transitivity, as follows.
First observe that we can express any Xjj (x) as Ay (x) = (1 -Xjj) X0lj (x)+XjjXj y (x) (2) where x denotes all of x except xy. Xgjj (x) is the value of Xy (x) when xy = 0 and Xjy (x) is the value of Xq jJ (x) when xy = 1. X0y (x) is the inten
sity of change from a non-choice to a choice while Xiij(x) is the intensity of a change from a choice into a non-choice. Models are specified by stating a form for Xgy and X^y. The specific forms we consider here are:
Aoij (x) = A0 + e0 Xj+ + tt0 x+j + r0 (Sxjk xkj)
Xlij ( x ) = Al + € i Xj+ + 7Ty X+j +M1 Xji
+ T1 kJ j ( 1 - xik) xjk + T2kJ. xki (1 - x kj) (4) the parameters X0 and X1 in (3) und (4) give the overall rate of change. For example, if Xq is high and Xj is low then there will be a general, overall tendency for the number of choices to increase over time and to persist once made. The parameters 6q and €\ are for individual expansiveness as reflect
ed in x i+. The parameter eq measures the intensity of a change between i and j from a nonchoice to a choice, given the numer of choices i is already making. The parameter €\ is the intensity of a change between i and j from a choice to a non
choice given the same information. The para
meters 7Tq and 7ii are similar to 6q and ej except that they are related to individual popularity as measured by x+j. The parameters and ß\ give the intensities for changes that depend on the state of Xjj (i.e., the effect of reciprocity). The parameters r 0, Tj and 72 reflect various aspects of transitivity that we may wish to investigate.
The parameter tq reflects a tendency for a non
choice that creates many intransitivities to change into a choice. The parameters r j and t i reflect the two ways in which a choice that creates many intransitivities may tend to change into a non
choice.
As illustrated above, we can construct a model that includes any accountable effect. These could be quartet and other higher-order effects or, heterogeneous populations, such as males and females, if the parameters depend on i and j, the individual nodes.
Estimation of the parameters yields measures of the effects of the respective factors. Their inter
pretation is not unlike that of a regression equation in which parameters measure the strength of relationship between response and carrier variables.
Here, however, we are concerned with the impact of structural laws on the process by which social networks tend, over time, to acquire recognizable and socially meaningful arrangements.
By conceiving of social structure as the dynamic consequence of ordering properties possessed by social relations we are able to develop a formaliza
tion in terms of a probability distribution on a set of multigraphs. The specific model we propose for studying a social structure is only meant as a first step. It obviously oversimplifies what we know about social systems but we believe that these simplifications do not reduce the system to an absurdity. Furthermore the gain in mathematical tractability is vast. Indeed, applications to data may require even further simplifications until better observational tools are available.
We mentioned earlier that social structure often acquires a quality of persistence in arrangement.
Once a group forms, change in relational struc
ture is likely to be rapid as ties are made and relationships tested, and cemented or broken.
After some period of time even in informal groups of friends the relational structure often
P. W. Holland, S. Leinhardt: Social Structure as a Network Process 397 acquires an apparent stability. While some change
or flux continues indefinitely, if the group mem
bers do not change, many of the ties become relatively permanent. It is safe to say that when single observations are taken on the social network, as with a sociometric test, the investigator usually implicitly assumes the network is stable in time.
While the issue of network stability in a particular social system is an empirical question, conceiving of the social structure as a network process yields a direct and natural way of conceptualizing this state and studying individual sociomatrices.
This derives from the fact that a finite-state Markov process in the limit possesses a stationary distribution (assuming regularity conditions). If we have reason to believe that the system has achieved stability in the sense that the various structural and non-structural properties of the relations, individuals and environment are in equilibrium then we can assume that the single sociogram is an observation on the network’s stationary distribution. We may then study the structural tendencies of the system by making inferences about parameters of the underlying stationary distribution. Consequently, the model permits us to study both the dynamic process using longitudinal observations on the network and its stationary arrangement using a single observation. 5
5. A theoretical synthesis
We initiated this discussion by observing that in sociology social structure was a common though inconsistently defined term. This point is not original with us. So many have concerned them
selves with it that we need not. Instead, we refer the interested reader to NADEL (1957: 2—3), and BLAU (1975). In the context of such diversity it is inevitable that we could find a point of view not unsimilar to our own. Indeed, much of what appears in BLAU’s edited volume on social structure (1975) can be used in support of our general approach. For example, GOODE in his commentary on HOMAN’s and MERTON’S work cites WITTGENSTEIN’S focus on networks as the essential feature of structure. “Pay atten
tion to the network”, GOODE paraphrases, “the geometry of its arrangement, and not the charac
teristics of the thing that net describes.” (BLAU 1975: 74) For GOODE the things arranged in the network and whose geometry we should
study are social ties. Similar points of view are expressed by BLAU, COLEMAN and HOMANS (BLAU 1975) among others.
While there are some commonalities between our approach and those of other social scientists we certainly do not mean to argue that we have achieved a synthesis of all contemporary efforts.
In fact, when pushed, we must admit that much of this seeming commonality is just that, a super
ficial rather than essential feature. While the terms geometry or articulation of relations may be used by sociologists when they discuss social structure, most often the terms appear as metaphors and are meant to sensitize the investigator rather than to call forth an explicit model. This metaphoric useage is in clear contrast to the analytic frame
work we have attempted to construct. Our pur
pose has been to build procedures for empirically testing hypotheses about social structure which represent precise and parsimonious statements about relationships among structural variables.
To do this we have had to rely on statistical procedures and mathematical models, tools that differ substantially from the verbal arguments common in research on social networks. To be fair, other network researchers often are interested in networks as independent variables; a purpose quite different from ours. Their concern focuses on the effect of various relational arrangements on other variables such as diffusion of information, voting behavior, problem solving effectiveness, etc. What our approach offers such studies is the possibility of new and more exact structural variables with which to investigate behavior which is thought to be conditioned by social structure.
While we do not claim close ties between the approach we have described and more traditional views, there are two anthropologists, S.F. NADEL and A.R. RADCLIFFE-BROWN, whose concerns have been similar to ours. Both stressed the need to understand regularities in social structure and both represented social structure as a network.
While we did not aim to achieve it, to some degree our approach synthesizes ideas presented earlier by NADEL and RADCLIFFE-BROWN. In this section we contrast their views with ours.
5.1 Nadel: Structures o f roles
In his book, The Theory o f Social Structure
398 Zeitschrift für Soziologie, Jg. 6, Heft 4, Oktober 1977, S. 38 6 -4 0 2 (1957)4 5, NADEL develops a perspective which
is remarkably similar to ours. NADEL focuses on what he sees as several fundamental aspects of structure: abstraction, pattern and invariance. He develops a “theory” of social structure that envisages relationships between roles, the relation
ships are abstractions of behavior that individuals engage in by virtue of the roles they fill. These rela
tionships establish a pattern that can be conceived of in terms of a structure. These patterns have a quality of invariance over time and while indi
viduals may move into and out of particular roles, this underlying pattern of the society’s social struc
ture, remains essentially consistent over time. In what follows we paraphrase and quote from NA
DEL to flesh-out this brief synopsis.
NADEL identifies structure with orderly arrange
ment. It is an abstract quality implicit in the way that relations position pairs of components relative to one another. He defines structure as follows: “Structure indicates an ordered arrange
ment of parts, which can be treated as transposable, being relatively invariant, while the parts them
selves are variable.” (p. 8) People within a society are the parts. They act towards one another in lawful, consistent ways. These actions demonstrate the existence of relationships. NADEL’s notion of abstraction involves a simplification of this action, an identification of an essential compo
nent devoid of irrelevant and inconsistent aspects.
Relationships are abstractions of concrete behavior in the sense that they represent the es
sential character of the behavioral act and im
plicitly indicate linkages between the individuals who are interacting. To quote, ‘Thus, in identi
fying any relationship we already abstract5 from the qualitatively varying modes of behavior an invariant relational aspect — the linkage between people they signify. ” (p. 10.) It is in arrangements of relationships that structure is manifest: “It is clear, on the other hand, that all relationships, through the linkage or mutuality they signify, serve to ‘position’, ‘order’ or ‘arrange’ the human material of societies.” (p. 11). For NADEL relationships are aspects of roles, each role being a multigraph with various relationships linking it to other roles. Thus, societies are characterized
4 All page references to NADEL are to this source.
5 Author’s emphasis.
by structures of roles which possess enduring organizing properties for its members. “Though relationships and roles (more precisely, relation
ships in virtue of roles) ‘arrange’ and ‘order’ the human beings who make up the society, the col
lection of existing relationships must itself be an orderly one . . .. ” (p. 11.) Social structure, thus,
“ . . . must correspond to something like an overall system, network or pattern.” (p. 12) In other words, social structure, the society’s collec
tion of social ties, must be the result of an orderly, lawful, aggregation process. The individual ties, the relationships between roles, come to be ar
rayed in a regular fashion, into a network that has a recognizable, transposable organizational invariance: “By ‘network’ . . . I mean the inter
locking of relationships whereby the interaction in one determines those occurring in the others . . . Let me stress that I am using the term in a . . . technical sense. For I do not merely wish to indicate the ‘links’ between persons; this is adequately done by the word relationship. Rather, I wish to indicate the further linkage of the links themselves and the important consequence that, what happens so-to-speak between one pair of
‘knots’, must affect what happens between the adjacent ones. It is in order to illustrate this in
terrelatedness or interlocking of the relationships .. . that we require an additional term, and ‘net
work’ seems the most appropriate.” (p. 16—17) Thus, NADEL describes social structure as a system of relationships in the sense that their organization imposes a structure on them. In this manner the structure of the network or its tendency towards its invariant abstracted pattern exerts an influence on each individual linkage.
NADEL was near to capturing the lawfulness with which we model the structuring of social relations over time. He also acknowledged that social structure had dynamic qualities. “We might say that roles and relationships are being con
stantly redistributed through the population; or we might say that people move in and out of roles and relationships, so that these too go and come, cease and begin again.” (p. 132) The con
figurations of roles, however, have patterns that are invariant. “Thus, in spite of the explicit admission of the time dimension we are still entitled to speak of structure. We need only concede that it has dynamic properties, containing within it the displacement and replacement.”
(p. 132) NADEL’s dynamics are those of indi
P. W. Holland, S. Leinhardt: Social Structure as a Network Process 399 viduals, not of the structure itself. Individuals
move in and out of the system of roles. Some
times roles go unfilled, sometimes individuals move from one role to another and their relational ties change as a consequence. But even when a role is vacant there is an underlying, unobserved structure that supports the unobserved role struc
ture during its inactive period and which becomes manifest at the moment an actor fills the role.
The social network continues to be formed and reformed but in an orderly way. The process of development exhibits time invariance even though at specific moments the apparent structure and the underlying structure do not exactly coincide.
Obviously, we differ from NADEL in many respects. Our concept of dynamics emphasizes change and structural evolution whereas NADEL’s emphasizes invariance. For us the observed net
work is the multigraph; for NADEL the network is an unobserved but systematically organized abstraction. For NADEL relational ties between individuals are determined through their respective roles; for us the linkage between two individuals is a stochastic function of the properties of rela
tions. For NADEL the social structure is mani
fested in arrangements of roles; for us social structure arranges individuals. What is invariant for NADEL are the role structures whereas we see relational properties as comparatively in
variant.
5.2 Radcliffe-Brown: Structure in empirical ties NADEL emphasizes abstracting relationships from behavioral acts and associating these abstractions with roles rather than individuals. RADCLIFFE- BROWN (1940)6, in contrast, identifies social structure with the extant demonstrable ties linking individuals in a population. “.. . (D)irect observation does reveal to us that .. . human beings are connected by a complex network of social relations. I use the term ‘social structure’
to denote this network of actually existing rela
tions.” (p. 2 ) Thus, the study of social structure, which RADCLIFFE-BROWN saw as “most fundamental” to social anthropology, is for him the study of empirical social networks.
6 All page references to RADCLIFFE-BROWN are to this source.
RADCLIFFE-BROWN is explicit about the im
portance of social structure in everyday activity.
Indeed, he asserts that individual behavioral acts derive not simply from the volition of individuals but as a consequence of the social structure, the arrangement of the relations in the population.
“So also the social phenomena which we observe in any human society” , the wirtes, “are not the im
mediate result of the nature of human beings, but are the result of the social structure by which they are united.” (p. 3 ) The “uniting” of individuals into a society is manifested in the net
work of social relations. RADCLIFFE-BROWN’s use of the term relation is not simply equivalent to the term tie: “A particular social relation be
tween two persons . . . exists only as part of a wide network of social relations, involving many other persons, and it is this network which I regard as the object of our investigations.” (p.
3 )
RADCLIFFE-BROWN never develops a cohesive specification of configurational patterns of rela
tions. Instead, he directs the study of social structure towards inventories of structures, a research process he identifies with morphology,
“.. . consisting in the definition, comparisons and classification of diverse structural systems . . . ” (p. 6). This scheme is never specifically detailed although examples suggest that he is concerned here with the identification of multi
graphs of socially siginificant extant relations.
Though he provides no precise orientation to particular patterns RADCLIFFE-BROWN does make clear why these inventories of extant social networks are critically important to social scientists. “The problem here,” he observes, “is how do structural systems persist? What are the mechanisms which maintain a network of social relations in existence, and how do they work?”
(p. 6 ) In our terms, these questions refer to the social laws, the properties of relations that persist and yield similar though not necessary identical arrangements in different populations and at different moments. This last element, the dynamics of structure, is visualized by RADCLIFFE- BROWN in a form closer to our appreciation than to NADEL’s: “. .. (T)he social life constantly renews the social structure. Thus the actual relations of persons and groups of persons change from year to year, or even from day to day. New members come into a community by birth or
