A Stochastic Model for Directed Graphs with Transition Rates Determined by
Reciprocity
Stanley S. Wasserman
Sociological Methodology, Vol. 11. (1980), pp. 392-412.
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A STOCHASTIC MODEL
FOR DIRECTED GRAPHS
WITH TRANSITION RATES
DETERMINED BY RECIPROCITY
Stanley S. Wasserman
UNIVERSITY OF MINNESOTA
This chapter addresses the problem of describing mathematically the evolution of a binary directed graph, or social network, over time. We develop a new statistical method for the
analysis of social networks and discuss its usefulness for sociologists
Partial support was provided by the National Science Foundation, Grant
SOC 73-05489 to Carnegie-Mellon University. Revisions were made while the
author was a Postdoctoral Training Fellow of the Social Science Research Council.
This chapter is an abbreviated version of Technical Report 305, University of
Minnesota, School of Statistics. I wish to thank Joseph Galaskiewicz for comments
on the manuscript.
STOCHASTIC MODEL FOR DIRECTED GRAPHS
and others studying networks of individuals, organizations, or
groups.
Interest in substantive sociological network theory is at an
all-time high level-more and more researchers have been using the
social network paradigm to study social structure. The number of
sophisticated mathematical models for directed graphs is increasing
at a pace equal to the rise in interest. Mathematicians and statisticians are seeing the application of their research to social networks.
But even with the growing popularity of networks, many of
the methods used by network analysts are very elementary, and
some are possibly obsolete. This situation has both positive and
negative features. O n the positive side, because most of the quantitative aspects of the field are relatively new, much of the data
analysis is exploratory in nature, and simple well-known methods
are often sufficient to provide both sociological insights and directions for future research. O n the negative side, the elementary
methods typically involve univariate statistics such as density
measures, correlation coefficients, and the like, and researchers have
not been able to benefit from carefully developed stochastic models
and modern methods of multivariate statistical analysis. To paraphrase Burt (1978), a discipline of applied network analysis needs to
be defined, emphasized, and used to bridge the gap between the
substantive sociological theorists and the mathematical and statistical modelers.
This chapter was presented at a conference on Stochastic
Process Models for Social Structure in December 1977. The current
gap between sociologists and mathematicians was apparent at the
conference-although everyone recognized the need for dynamic
stochastic analysis of networks, the new models described in the
papers given by statisticians were either so mathematical that they
were not accessible to most participants or they involved mathematical and statistical assumptions that were gross simplifications.
In order to tap the current interest in stochastic modeling, it is
necessary both to develop a new methodology based on a stochastic
modeling framework that is reasonably free of mathematical difficulties and to demonstrate how the methodology could be used to
answer fundamental sociological questions. This chapter is a timid
start in that direction.
STANLEY S. WASSERMAN
Social groups are acknowledged to change over time, an
evolution that alters the links that exist between group members.
However, there has been little effort directed at the construction of
realistic models for this evolution. Undirected graphs and multivalued graphs have been effectively modeled stochastically, but the
fitting of these models to binary directed graphs yields little insight
into the binary processes under investigation. Indeed, the most
important features of these processes are the on/off nature and the
directedness of the arcs in the structural graph. Ignoring these two
qualities in model construction would be a fundamental error.
We shall describe a simple stochastic model for the process
of change in a binary digraph and propose several strategies for
estimating the parameters of the model. The appropriate strategy
for a given situation depends on the number of times the group is
observed in its evolution. An application of this model to several
data sets (Wasserman, 1977) has yielded some interesting insights
into group processes.
This chapter uses a general mathematical language for
presentation of results, since the model has application in other
areas, such as communication, transportation, and the natural
sciences (see Wasserman, 1978). Sociological applications are discussed in the last section. The mathematics have been simplified,
and all proofs have been omitted. Those interested can consult
Wasserman (1977) or the technical report mentioned in the opening
footnote for details.
The stochastic model described here is quite simple. But it is
unfair to assess the merits of such a model by its complexity or lack
thereof. Much can be learned from simple models. The information
to be gained may yield valuable insights into the process in question. The ideas presented in this chapter may allow researchers to
postulate and analyze more complicated models.
I. INTRODUCTION TO THE PROBLEM
Consider a directed graph or digraph that structurally represents the state of some process, sociological or otherwise, at time t. A
digraph is a set V = {v,, u p , . . . , vg) of nodes and a set L =
{I,, I,, . . . , I,) of directed arcs connecting pairs of nodes. We let
STOCHASTIC MODEL FOR DIRECTED GRAPHS
395
li = vjvk be the directed line running from node vj to node vk and
further stipulate that the digraph be binary; that is, if two distinct
arcs exist such that li = vjvk and li, = ujvk, then li = li8.In addition,
arcs li = vjvi do not exist, ruling out the existence of loops in the
digraph. We commonly let D, represent a digraph with g nodes.
Let X(t) be the adjacency matrix representing the state of D,at
time t. Specifically, X(t) = [Xij(t)], where
Xii(t) =
if vivi E L at time t
otherwise
The time parameter t is assumed to be continuous, t 2 0.
The matrices x, w, y, z, . . . are single states of the continuous time stochastic process X(t). The process has a state space S of
all possible (g x g) binary-valued matrices with zero diagonal2~b-I)in number, making S quite large.
The problem that this chapter considers is the formulation
and evaluation of a stochastic model for X(t), where the transitions
between states of S depend on the state of the digraph at time t. We
accomplish this by assuming X(t) to be a continuous-time Markov
chain and by allowing the infinitesimal transition rates to be specific
functions of the elernents of X(t). This solution to the problem was
first suggested by Holland and Leinhardt (1977a) and has been
further elaborated on by Holland and Leinhardt (1977b) and by
Wasserman (1977, 1978). Several models using these assumptions
are discussed in depth by Wasserman (1977), including the model
incorporating only reciprocity, which is the subject of this discussion.
In Section 11, we discuss the Holland-Leinhardt modeling
framework and outline its assumptions. We present a simple model
for digraphs based on reciprocity and describe how this particular
parameterization substantially reduces the size of the state space. In
Section 111, we compute moments and equilibrium distribution for
the stochastic process arising from the reciprocity model.
The parameterization chosen for the reciprocity model has
four parameters that can vary from digraph to digraph. In the latter
sections of this chapter we estimate these parameters. In Section IV,
we consider whether an observed set of observations of a digraph is
representable as a continuous-time Markov chain and give a simple
396
STANLEY
S. WASSERMAN
procedure for computing reasonable parameter estimates. Maximum-likelihood estimation, a computationally more difficult procedure, is discussed in Section V. We conclude with some comments
for social networkers.
II. T H E M O D E L I N G F R A M E W O R K
The Holland-Leinhardt framework consists of two assumptions regarding the stochastic nature of the arcs X i j ( t ) . The first is
that X ( t ) is a Markov chain:
ASSUMPTION
1. Markov
X ( t ) is a standard Markov chain with jnite state space S and
Pxy(t,h) = P { X ( t
+ h ) = yl X ( t ) = x)
(1)
as the probability transition matrix.
+
Second, we assume that for small intervals of time ( t , t
h ) , the
changes in the arcs of a digraph are statistically independent:
ASSUMPTION
2. Conditional Change Independence
Assumption 2 is crucial. It implies that the probability of
any two arcs changing simultaneously is essentially zero. In a small
interval of time, only two changes can occur for a single arc: Arcs
present at time t may disappear at time t
h and vice versa. Thus
we represent the probability of arc changes as
+
Note that A, the change rate function, depends on the state of the
digraph at time t and on t itself.
Let q,,(t) be the injnitesimal transition rates of X ( t ) , the
STOCHASTIC MODEL FOR DIRECTED GRAPHS
digraph process. We have
Aij(x,t)
9xy(t) =
I0
I
if y and x differ only in the
(i,j)th element
if y and x differ by more than one
element
(4a)
and
as terms of Q(t), the matrix of infinitesimal transition rates. Wasserman (1977) discusses some characteristics of S and proves that if
a simple condition is satisfied, specifically a restriction to nonzero
change rates Aii(x,t), then the digraph process has an equilibrium
distribution.
III. THE RECIPROCITY MODEL AND THE DYAD PROCESS
One can postulate various functional forms for the change
rate function A, defined in Equation (3), and generate many models
incorporating Assumptions (1) and (2) of the modeling framework.
We propose one such model in this section with a parameterization
that produces a state space D of only four states. The model is for
reciprocity where the tendency over time for the arc vivi to exist
depends only on the presence or absence of arc upi.
We shall consider the (9,) dyads, or 2-subgraphs, of D,, which
by assumption are independent and identically distributed. After
computing moments and equilibrium distribution of the dyad
process, we briefly describe its probability transition matrix.
We dichotomize the change rate function (3) into two
functions to allow for both types of arc changes. Define
398
STANLEY S. WASSERMAN
for small h. Note that the rates of the process are now time-homogeneous and, consequently, the digraph process is stationary in time.
For the reciprocity model, we assume that
There are g(g - 1) pairs of the change intensities (6a) and (6b) such
that the pair for ( i , j ) depends only on the pair for (j,i).
T he
parameters ho and XI are measures of the "overall" rate of change
for an arc, and p0 and p, measure the "importance" of a reciprocated arc. Common sense suggests that empirically
and
since, in the absence of a reciprocated arc, there should be a greater
A,,). Moreover, we
tendency for arcs to appear than disappear (A,
assume that in the presence of a reciprocated arc (1) the tendency
for an arc to appear from vi to vj should increase (po 0) and
(2) the tendency for the arc from vi to vj to disappear should
decrease (pl
0). Positivity of hl
p1 requires that p,
-Al.
Inequalities (7a) and (7b) are more likely to be true in friendship
networks where the tendency over time is toward mutuality. These
parameters are discussed further in Section VI.
Let
<
<
+
>
>
be the dyad for the pair of nodes (i,j ) . The state space D of the dyad
Dij(t) contains four states as illustrated in Figure 1. The Q intensity
matrix for the dyad process is shown in Table 1.
The parameterization (Equations 6a and 6b) of the reciprocity model yields a set of dyad processes {Dij(t)) that are independent. The entire X(t) digraph process can be represented as ( 4 )
independent dyad processes consisting of the symmetrically positioned pairs of off-diagonal elements of X(t). Moreover, the {Dij(t))
STOCHASTIC MODEL FOR DIRECTED GRAPHS
Figure 1. States contained in D
w .
4
I
2
(a) Mutual Relation (M)
(b) Asymmetric Relation ( A l )
*
J
2
(c) Asymmetric Relation (AO)
.
2
(d) Null Relation (N)
are identically distributed, continuous-time, four-state Markov
chains with state space D and intensity matrix Q.
We now compute the moments of the dyad process-that is,
the probability, on average, that the arc vivj is present or absent and
that vivj and viai are present at time t. Let
TABLE 1 Q Matrix for the Dyad Process State
( 0 )
(1,0)
(0,l)
(1,l)
-2Ao
A,
A,
0
A0
-(A0
+ A1 + PO)
0
A1
+ PI
A@
0
-
0
A1
+ 1 + PO
+ PI
0
A0 + Po
A,
yo
-2@1 + P I )
+
400
STANLEY S. WASSERMAN
be the vector of first and second moments of the {Dij(t)).Also define
The moments m(t) are computed by solving the differential equation
The solution of ( l l ) , subject to a set of initial conditions at time
to, 1s
where n(t) is the solution of the homogeneous equation
Calculation of the integral (Equation 12) yields a very
complicated solution for m(t) that gives little insight into the nature
of the process. However, the equilibrium distribution of the dyad
process, the probabilities that a dyad is in one of the four states as
t + co, is simple to calculate and easy to comprehend. There are
three ways the equilibrium probabilities can be found: letting
t + co in Expression (12); setting the system of differential equations in (1 l ) to zero and solving for m(oo); or showing that the dyad
process is reversible, allowing the equilibrium probabilities to be
simply found from the reversibility equations. Mathematical details
are given in Wasserman (1977).
Let (q{Dij(t);Dij(t h ) ) ) be the elements of the Q matrix
for the dyad process given in Table 1. We define
+
rM(t)= P{Dij(t) = (1,l))
(mutual)
7iAl(t) = P{Dij(t) = (1,O))
(asymmetric)
7iAO(t)= P{Dij(t) = (0,l))
(asymmetric)
' i ~ ~ (=t )P{Dii(t) = (0,O))
(null)
(14)
STOCHASTIC MODEL FOR DIRECTED GRAPHS
40 1
as the elements of ~ ( t )the
, vector of probabilities of the four states of
the dyad process. The equilibrium probabilities are
These probabilities are simple functions of the model parameters
and can be estimated by computing them with parameter estimates.
Comparing Equation (15) with the empirical distributions gives
information on how close the network and the dyad process are to
equilibrium.
The probability transition matrix for the dyad process is a
very complicated expression. We find it by examining the eigenvalues and eigenvectors of the (4 x 4) infinitesimal generator Q and
then compute etQ. The reader is referred to Wasserman (1977) for
the details.
IV. EMBEDDABILITY
To estimate the four parameters of the reciprocity model
and determine whether the model provides a good description of the
evolution of a directed graph, we must use empirical observations
on the process. We use the general notion of embeddability, the
determination of whether an obserued probability transition matrix
could have arisen from a continuous-time Markov chain. We continue the discussion in the next section, where we estimate the
model parameters.
Suppose that we have observations on some process z(t)
denoted by r,(t), z2(t), . . . , zK(t).Assume that the process has finite
state space Z, with states labeled 1, 2 , . . . , N. For example, if
r(t) = Dii(t),the dyad process, we have K = (3), Z = D, and N = 4.
402
STANLEY S. WASSERMAN
These observations {z(t)) are collected at times t = to, tl, . . . , tn,
where the t, are distinct increasing positive numbers.
If n is knite and nonzero, we define the empirical probabili~
1 5 n, with elements [pii(tl - tk)]
transition matrix P(t, - tk),0 5 k
as follows (where t = tl - tk):
<
where Tj(t) is the number of 2's in state i at time t, and in state j
at time tl. These estimates are maximum-likelihood estimates of the
elements of the probability transition matrix of a stationary,
discrete-time Markov chain (Anderson and Goodman, 1957). For
moderate n, there is a substantial number of estimated transition
matrices that can be used to test the suitability of a continuous-time
Markov chain for the data.
Embeddability was first posed as a problem by Elfving
(1937) and discussed by Kingman (1962), but only recently has the
problem been rigorously solved by Singer and Spilerman (1974).
The embedding problem, as formulated by Singer and Spilerman
(1976), is:
Find simple test criteria on the elements of an observed
stochastic matrix @(t),0
t
cc,which will guarantee that it
car, be written in the form P(t) = etB for some Q w i t h elements
for i # j
qij 2 0
< <
A
50
for i = 1,2, . . . , N
C qii = 0
f o r i = 1 , 2 , . . . ,N
qii
(17)
i
The data-analytic problem is to find the subclass of all Q matrices
with structure (17) that could have given rise to the observed ^P
matrix or matrices.
This subclass of Q matrices is
Q = {all matrices Q with structure (17) such that (tl - t k ) Q
(18)
= log F(t, - t,) for 0
k
1 5 n)
< <
403
STOCHASTIC MODEL FOR DIRECTED GRAPHS
Note that Qmay contain more than one element even when n = 1
because the logarithm function of a matrix is a "one to many"
function. Moreover, when n
1, the collection of Q matrices
compatible with a subset of the empirical ^P matrices may not
coincide with the Q matrices compatible with a different subset.
This situation is caused by the negation of the Markov assumption
of time-homogeneous transition rates.
Although there are many necessary conditions for Q to be
nonempty, no simple sufficiency criteria exist. One must verify that
the observed ^P matrix satisfies the necesary conditions; and even
then, there may not be a Q matrix compatible with the observed
transition matrix.
1f ^P has positive, distinct, real eigenvalues, however, there is
a unique Q matrix for a given P. The scalar logarithm function is
multiple-valued only when it has complex arguments and is undefined with negative arguments. The following theorem, due to
Sylvester, and also given by Singer and Spilerman (1976), states the
result for this special instance.
>
A
THEOREM.
SYLVESTER'S
If P is an ( N X N ) matrix w i t h
is single-valued
distinct eigenvalues hl, h2, . . . , AN, and f f
in a neighborhood of each of the eigenvalues, then
( a )
For proof see either Gantmacher (1960) or Sylvester (1883).
The application of this theory to the reciprocity model is
straightforward. Suppose, for example, that we assume the model is
operating and we have observations on the digraph at time to and t,.
.,We reduce X(to) and X(tl) to two sets of (q) dyads and form
P(tl - to), a 4 x 4 matrix of dyad transitions. If this empirical probability transition matrix has four positive, real, distinct eigenvalues, we can calculate the unique empirical Q matrix for these
observations. Since we know the functional form of Q (Table I), we
can easily solve for the parameters A,, A,, po, p1 by setting the
theoretical elements of Q equal to the observed, calculated values.
Wasserman (1977) gives conditions on the elements of ^P
A
404
STANLEY S. WASSERMAN
which ensure that the eigenvalues of ^P are real, distinct, and
positive. The conditions generally hold if the diagonal elements of ^P
are large, relative to the off-diagonal elements. This "diagonaldominant" situation is likely to occur when the two observation
time points to and tl are close. Taking observations close in time
allows only a few transitions to other states so that the diagonal
elements remain near unity.
Occasionally log ^P yields a Q that does not have structure
(17); that is, the off-diagonal elements of
may be negative. If so,
then one may force the negative elements to zero, adjust the remaining terms to maintain the zero-row.,sum, and obtain approximate parameter estimates. O r a different P can be examined if there
are more than two observations on the process. This strategy of
estimating Q by computing logarithms of empirical transition
matrices is flexible, but it yields good estimates. We discuss the
usefulness of this technique in the next section.
V. PARAMETER ESTIMATION
In this section we examine estimation of the parameters of
the reciprocity model. There are three possible sampling schemes:
continuous record; single observation; and two or more observations.
Continuous Record
When one has a continuous record of all changes in each of
the ( 9 , ) dyads for a time interval (t,,,~),estimation is relatively easy
and need not be discussed. Billingsley (1961) discusses the theory for
general Markov processes. One assumes that the Q matrix is a
function of a vector of parameters 8 and reduces the continuous
record { D i j ( t ) ,to 5 t 5 T ) for all dyads to observations on the
discrete jumps of the process and the total waiting time in each
state. Billingsley estimates 8 by maximum likelihood and discusses
likelihood ratio tests.
Single Observation
When one has a finite number of observations on a continuous-time Markov chain, estimation is rather difficult. Darwin
(1956) recognizes this and states that the complex form of the
STOCHASTIC MODEL FOR DIRECTED GRAPHS
405
likelihood, with observations on the process taken once or at regular
intervals, almost prohibits the use of maximum-likelihood estimation. Keiding (1974, 1975) considers sampling of the process at
equidistant time points to, ti, tZj,. . . , tni, but only derives asymptotic results for n -+ c ~ .
We now consider estimation of the four parameters of the
reciprocity model, A,, A,, po, pl, assuming that we have a single
observation on the entire digraph. A single adjacency matrix is the
data set most frequently collected by sociologists studying social
networks. We discuss the likelihood function of the parameters,
given this single adjacency matrix, and estimate two meaningful
functions of the parameters.
Suppose we have a sole observation of the stationary X(t)
process, which we label x. The matrix x contains (8) independent
observations of the dyad process Dij(t) with state space D. We are
interested in the numbers of each type of dyad. Since the labeling of
the nodes in the directed graph is arbitrary, we cannot distinguish a
(1,O) asymmetric from a (0,l) asymmetric. Consequently, the information in x can be summarized by three statistics:
xiixji = number of mutuals
M(t) =
i>i
A(t) =
x
(1 - xir)xii + xii(l - xii)] = number of asymrnetrics i >i N(t) =
x
(1 - xii)(l - xfi) = number of nulls i>i +
+
where M(t)
A(t)
N(t) = (92).
Define L(0 I x) as the likelihood function of the parameters
given the single matrix x, where 0 = (Ao,A,, po, pl)'. L is merely a
multinomial likelihood:
L(O 1 X) =
7iM(t)xtlxl*~Al(t)"'l(l-~l,)
all dyads
i>l
x
(21)
~~~(t)(l-x~,)x~%7i~(t)(~-x,~)(~-x,,)
406
STANLEY S. WASSERMAN
where the T'S, defined in Equation (14), are the probabilities of the
four states of the dyad process. The log likelihood can be written
succinctly as
+
where aA(t) = aAl(t) aAO(t).Indeed, M(t), A(t), and N(t) are
sufficient statistics.
The expressions for a M ,a*, and aNare linear combinations
of the elements of the m(t) vector of moments given in (12) and
hence are quite complicated expressions. For computational ease,
we shall use the steady-state values of a M ,T*, and aN,defined in
(15). Let
so that
The log likelihood (22) is
log L(0 I x) = M(t)log(a/2b) - N(t)log(2b/c)
-
($)log(a/2b
+ 6/26 + 1)
(26)
where A(t) = (92) - M(t) - N(t).
The likelihood function (26) depends on four unknown
parameters 8, but it contains only two "pieces of information": M(t)
and N(t). Hence we can estimate only two functions of the parameters, which are the change ratios
8 , is the ratio of the probabilities of change, in a small time interval,
in the presence of a reciprocated arc and O 2 is the ratio in the absence
of a reciprocated arc. The maximum-likelihood estimates of
(see Wasserman, 1977) are
Q2
and
407
STOCHASTIC MODEL FOR DIRECTED GRAPHS
and are easily computed from the data.
There are two other ratios that are more interesting than fll
and f12. These are
These ratios, K~ and K,, directly emphasize the importance of a
reciprocated arc: K~ ( K ~is) the effect of a reciprocated arc on the
change from a nonchoice to a choice (choice to a nonchoice). We
1 and K~
1. But these "reciprocity" ratios
suspect that K~
cannot be estimated via maximum likelihood with only one observation on the digraph. (See Wasserman, 1977, for proof of this fact.)
Note that the ratio of the change ratios fll and 8, is an odds
ratio. We have
>
<
the increase in the odds of a new arc vivj coming into existence
during the interval (t, t
h ) due to the existence of ujui. With no
reciprocity effect, log (f11/02) = 0; if log (01/f12) 0, a positive
reciprocity effect is present. So even though fll and f12 considered
separately are not very informative, their ratio is quite interesting.
+
>
Two or More Observations
We now examine situations where one has a data set containing several observations on the directed graph. First consider
408
STANLEY S. WASSERMAN
>
n = 2, and let X(tl) and X(t2), t2
tl, be the two observations on
the process. We examine each of the pairs [Xii(tl), Xii(tl)] and
[Xij(tz),Xii(t2)] in turn and form a 4 x 4 contingency table with
rows corresponding to the dyad state at time tl and columns corresponding to the t2 state. There are (8) "counts" in this table. (See
Bishop, Fienberg, and Holland, 1975, for discussion of representation
of data from Markov models as contingency tables.)
We let T denote such a table, with entries (tkl),where
k = 1 = null (0,O)
k = 2 = asymmetric (1,O)
k = 3 = asymmetric (0,l)
(32)
k = 4 = mutual (1,l)
and similarly for the subscript I.
The likelihood function when n = 2 is
L[e I x(t1) = x, ~ ( t , )=
= [ ~ , ( t , ) ~ (TA(tl)A(tl)
~~)
TN(tl)~(tl)~
where the [pkl(tZ- tl)] are the elements of the probability transition matrix for the dyad process.
Suppose n
2. We now have observations on the digraph
process X(tl), X(t2), . . . , X(tn). We form (n - 1) matrices Tm,
m = 1, 2 , . . . , (n - I), with elements (tklm),where Tm gives the
transitions of the dyads at time tm to the dyads at time tm+,. The
process is stationary, so that the probability transition matrix for the
dyads depends only on tm+, - t,. The likelihood function is
>
When we have equidistant sampling, such that t, - t, =
t, - t2 = . . . - tn - tnPl = t, then the likelihood (34) simplifies to
STOCHASTIC MODEL FOR DIRECTED GRAPHS
We can "pool" transitions across time points if we are certain of
time homogeneity, so that we have an effective sample of ( n - 1) (9,)
dyads.
To estimate the four parameters 8, we can differentiate the
logarithm of ( 3 3 ) or ( 3 4 )with respect to each of the four parameters,
setting the resulting derivatives to zero, to obtain a system of four
equations in four unknowns. But the PkI'sare nonlinear, being sums
of exponentials. In this situation, maximum-likelihood estimation is
not only unreliable, but the solutions can be obtained only approximately.
We may study the embeddability of the data in hand,
however, and develop a new strategy. If n = 2, we compute the
(4 x 4 ) empirical probability transition matrix and, using the
., rules
outlined in the previous section, find an empirical Qmatrix, Q. The
elements of Q are simple functions of the parameters Ao, A,, po, pl;
consequently, reliable estimates of the parameters are easily obtained. If n
2, we have several Q matrices; we compute several
plausible estimates of 0 and study each. We can use these estimates
as starting values for a Newton-Raphson iterative solution to the
likelihood equations, or we can explore the likelihood function in
the vicinity of these points, as is done in Wasserman (1977).
>
A
VI. SOCIOLOGICAL CONSIDERATIONS
To conclude this chapter, we wish to mention two ways in
which the model can be utilized by social network researchers. First,
the reciprocity model is a device to study the "structural tendency"
toward reciprocity; second, it serves as a benchmark, a standard to
which empirical data can be contrasted to uncover higher levels of
structural tendencies that might be present.
The utility of the model is that by specifying the probabilities that dyads change state, it allows the theorist to postulate how
the current structure of a group with a fixed amount of reciprocity
influences the future structure. We can study tendencies toward
410
STANLEY S. WASSERMAN
integration or symmetry by examining whether null relations become asymmetric relations, asymmetric relations become symmetric, mutual relations, and whether mutual relations exhibit a large
degree of stability. We can also examine tendencies away from
symmetry by postulating that arcs will disappear over time, that
dyads are likely to go through transitions such as M + A + Nmutual to asymmetric to null. Whatever the overall direction of
movement, the four parameters of the model not only measure the
direction but also state the overall effect of reciprocated arcs on the
movement.
The parameters ho and A, allow us to assess whether asymmetric ties tend to be formed (A,) and whether asymmetric ties tend
to disappear (A,) over time, in the absence of the reciprocated arc
vjvi.The parameters must be positive, and their relative magnitude
reflects these two tendencies. With movement toward symmetry, ho
should exceed hl, since the number of linkages should increase over
time. In contrast, movements away from symmetry imply that Xo
should be less than A,.
The parameter sums Xo
p, and hl
p1 measure the
chance that linkages between actors i and j form, or disappear, in
the presence of the reciprocated linkage. Thus p, and p, directly
measure the effect of reciprocated arcs on tendencies toward and
away from symmetry. Large values of A,
po indicate movement
from asymmetric toward mutual relations, while small values of
A, + p1 imply that mutual relationships are unlikely to lose linkages.
The second major use of the reciprocity model is as a
benchmark-a null model against which networks with complicated
structure can be compared. A study of residuals from the model and
its fit will allow researchers to isolate more complicated structural
tendencies, such as popularity, isolation, and expansiveness. Because
of its simplicity, the reciprocity model will rarely fit a data set so
well that a sociologist can confidently conclude that reciprocity
effects are adequate to describe the structure of a given network.
However, the model is valuable in that it allows both the opportunity to rule out reciprocity as the sole structural tendency and the
chance to find other tendencies in the model's lack of fit.
There is a strong tradition of the model-as-benchmark in
sociometric analysis. The Davis-Holland-Leinhardt transitivity
+
+
+
STOCHASTIC MODEL FOR DIRECTED GRAPHS
41 1
studies are based on the same philosophy. After adjusting a triad
census for effects of mutuality and asymmetry (Holland and Leinhardt, 1975) and, more recently, distributions of indegrees and
outdegrees (Holland and Leinhardt, 1979), a researcher can study
how far the census is away from its expected value to determine how
much transitivity remains. The ideal state of no intransitivity is a
benchmark, with small values of T indicating adherence to the ideal.
This modeling strategy is likely to increase in future sociometric
research, as researchers realize that we are limited to these simple
models. The construction of realistic, "all-inclusive" models is just
too difficult mathematically.
This research is limited in other ways. The reciprocity
model incorporates Markov assumptions that may not be appropriate for specific types of networks. Moreover, the transition rates
are postulated as constants over time, and dyads are assumed to be
independent and subject to the same homogeneous rates. Even so,
the history of Markov modeling in sociology, beginning with
Kemeny and Snell (1960, 1962) and continuing with Coleman
(1965), is rich and has given researchers many insights into sociological processes. A Markov model of reciprocity in social networks
should prove to be a useful tool for social networkers.
REFERENCES
ANDERSON, T. W., AND GOODMAN, L. A.
1957 "Statistical inference about Markov chains." Annals of Mathematical Statistics 28:89-110.
BILLINGSLEY, P.
1961 Statistical Inferences for Markov Processes. Chicago: University of
Chicago Press.
BISHOP, Y. M. M., FIENBERG, S. E., AND HOLLAND, P. W.
1975 Discrete Multivariate Analysis: Theory and Practice. Cambridge,
Mass.: MIT Press.
BURT, R. S.
1978 "Applied network analysis: An overview." Sociological Methods
and Research 7:123-130.
COLEMAN, J. S.
1965 Introduction to Mathematical Sociology. Glencoe, Ill.: Free Press.
DARWIN, J. H.
1956 "The behaviour of an estimator for a simple birth and death
process." Biometrika 43:23-31.
412 STANLEY S. WASSERMAN
ELFVING, G.
1937 "Zur Theorie der Markoffschen Ketten." Acta Social Science
Fennicae n., Series A.2, 8: 1-1 7.
GANTMACHER, F. R.
1960 The Theory of Matrices. Vol. 1. New York: Chelsea.
HOLLAND, P. W., AND LEINHARDT, S.
1975 "Local structure in social networks." In D. R. Heise (Ed.), Sociological Methodology 1976. San Francisco: Jossey-Bass. 1977a "A dynamic model for social networks." Journal ofMathematica1 Sociologv 5:5-20.
1977b "Social structure as a network process." Zeitschrift fur Soziologie
6:386-402.
1979 "Structural sociometry." In P. W. Holland and S. Leinhardt
(Eds.), Social Networks: Surveys, Advances and Commentary. New
York: Academic Press.
KEIDING, N.
1974 "Estimation in the birth process." Biometrika 61:71-80.
1975 "Maximum likelihood estimation in the birth-and-death process model." Annals of Statistics 3:363-372.
KEMENY, J. G., AND SNELL, J. L.
1960 Finite Markov Chains. Princeton: Van Nostrand.
1962 Mathematical Models in the Social Sciences. Waltham, Mass.: Blaisdell.
KINGMAN, J. F. C.
1962 "The imbedding problem for finite Markov chains." Zeitschrift
fur Wahrscheinlichkeitstheorie 1: 14-24.
SINGER, B., AND SPILERMAN, S.
1974 "Social mobility models for heterogeneous populations." In
H. Costner (Ed.), Sociological Methodology, 1973- 1974. San
Francisco: Jossey-Bass.
1976 "The representation of social processes as Markov models."
American Journal of Sociology 82: 1-54.
SYLVESTER, J. J.
1883 "On the equation to the secular inequalities in the planetary
theory." Philosophical Magazine 16:267-269.
WASSERMAN, S. S.
1977 "Stochastic models for directed graphs." Unpublished doctoral
dissertation. Department of Statistics, ~HarvardUniversity.
1978 "Models for binary directed graphs and their applications."
Advances in Applied Probability 10:803-8 18.