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SESSION III
Metrics and Models
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Sensitivity Analysis of Social Network Data and Methods: Some
Preliminary Results
Stanley Wasserman and Douglas SteinTey
University of Illinois at ChampaignUrbana
Abstract
Four social network indices are examined in depth degree centraliza
tion, betweenness centralization,transitivity, and efficiency. This study uses
Monte Carlo techniques and standard random graph distributions to gen
erate graphs. We seek to establish the ~rnlmAwnrk for ~ =~n~rn1 Her ^f
resistance of network statistics.
~~ 4~D~ ~ ~~ · ~ ~~ 4% ^~4 Cal 45~1~1 ~ ~1~\J1 ~ w !
Social network analysis has been used for the past seventy years to advance research in the
social and behavioral sciences. Major breakthroughs over the past ten years, both substantive and
methodological, have allowed this paradigm to greatly expand its usefulness, especially in comm7,ni
cation, broadly defined (including internet research), organizational science, and epidemiology. The
focus of this research is on the application of the paradigm to policy issues' especially those arising
governmentally, and the study and expansion of standard methodology to these very important
research questions. There are several major methodological questions under study:
 · What are the effects, and more importantly, the implications, of measurement error on
social networks?
~ = _
· What can be done to control for the lack of independence of interaction measurements
taken on the respondents (a common problem with egocentered networks)?
· Are there better methods for analyzing longitudinal networks, their composition and struc
ture, than generalized linear models?
Social network analysis is discussed in detail in Wasserman and Faust (1994); further, theo
retical concerns are highlighted in Monge and Contractor (2003). Applications of the paradigm to
many substantive disciplines, as well as discussion of how networks have advanced these disciplines,
can be fo''nd in Wasserman and Galaskiewicz (1994). Recent methodological developments are
described in Carrington, Scott, and Wasserman (2003), written to update Wasserman and Faust
(1994) by presenting advances made during the 1990's after the publication of Wasserman and
Faust (1994~. We will assume that most of the material Dresented in Wassermn.n n.nr1 Fit. (1 qqaN
is known to the reader.
~ ~ ~ ~ ~ ,
Our research program has a variety of components. We first focus attention on two areas:
This research was supported by grants from the National Science Foundation (#ITS9980189), and the Once
of Natural Research (#N0001~0210877~. email addresses: stanwassOuiuc . ecu; steinleyOs . psych. uiuc . ecu. Affiil
iations: Department of Psychology, Department of Statistics, and The Beckman Institute for Advanced Science and
Technology.
DYNAMIC SOCIAL NETWORK MODELING ED CYSTS
197
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SENSITIVITY ANALYSIS
2
· Robustness and resistance of network statistics and measures
· Study of the effects of measurement error (in a variety of different forms) on standard
analyses
Four important statistics are: measures of connectivity (quantifying how much actors are
"tied together"), actor centrality, counts of bridges, numbers and frequencies of cohesive subgroups,
and the efficiency of a network.. We will took carefully at how these descriptive network statistics
can be used in substantive context, and comment on their mathematical definitions, their statistical
properties, their errorproneness, and their general resistance to sampling designs.
This presentation uses standard distribution theory for graphs and directed graphs (thereby
allowing for parameter estimation, unlike nonconventional computational models with probabilistic
assumptions). We sample from these distributions, and study the effects of modifications of distri
bution assumptions on the variability of standard statistics. Such studies are but a first step in a
complete robustness and resistance theory for social networks.
Introduction
We begin with a graph, single set of nodes At, and a set of lines. Here, we will only be
interested in graphs, not digraphs. It is common to use this mathematical concept to represent a
social network, a set of n actors and a collection of r social relations that specify how these actors
are related to one another.
Here, we let r = 1, focusing just on networks with single, nonclirected relations, and assuming
that relational ties take on just two values.
We let ~ = {1, 2, . . ., g) denote the set of actors, and X denote a particular relation defined
on the actors. Specifically, X is a set of ordered pairs recording the presence or absence of relational
ties between pairs of actors. This social relation can be represented by a 9 x 9 matrix X, with
elements
J lif~i,j)~X,
O otherwise
We will use a variety of graph characteristics and statistics throughout this presentation
Distribution theory
The first step for any probabilistic model of a network is to construct a dependence graph.
Such a device allows us to distinguish among the many possible graph probability distributions,
which can often be characterized by considering which relational ties are assumed to be statistically
independent.
We define a dependence graph (as it applies to network relational variables) and then show
how it can distinguish among basic graph distributions (such as those described in Wasserman and
Faust, 1994, chapter 13~. This dependence graph is also the starting point for the His mersley
Clifford Theorem (Besag, 1974), which posits a very general probability distribution for these
network random variables using the postulated dependence graph. The exact form of the depen
dence graph depends on the nature of the substantive hypotheses about the social network under
study; we briefly discuss several such hypotheses.
198
DYNAMIC SOCIAL NETWOR~MODELING^D ISIS
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SENSITIVITY ANALYSIS
Theory
3
Any observed single relational network may be regarded as a realization x = Fiji of a random
twoway binary array X = ~Xij]. In general, the entries of the array X cannot be assumed to be
independent; consequently, it is helpful to specify a dependence structure for the random variables
{Xij) as originally suggested by Ffank and Strauss (1986).
The dependence structure for these random var~abies is determined by the dependence graph
D of the random array X. Z) is itself a graph whose nodes are elements of the index set {(i, j); i, j ~
J/, i ~ j} for the random variables in X, and whose edges signify pairs of the random variables
that are assumed to be conditionally dependent (given the values of all other random variables).
More formally, a dependence graph for a univariate social network has node set
JVD = {(i, j); i, j ~ An, i ~ j).
The edges of D are given by
ED = {((i, j), (k, 1~), where Xij and Xk~ are not conditionally independent).
This specific dependence graph is a version of an independence graph, as it is termed in the graphical
modelling literature (for example, Lauritzen, 1996; Robins, 1997); see Robins (1998) for an extended
discussion of the application of graphical modelling techniques to social network models.
We also note that all of these concepts and definitions can be extended to multirelational
networks (Robins and Pattison, 2002) without too much difficulty. A recent, thorough review of
these ideas can be found in the trio of papers (Wasserman and Robins, 2002; Robins and Pattison,
2002; and Koehly and Pattison, 2002) written for the Carrington, Scott, and Wasserman (2003)
volume.
Distributions
As Ffank and Strauss (1986) observed for univariate graphs and associated tw~way binary
arrays, several wellknown classes of distributions for random graphs may be specified in terms
of the structure of the dependence graph. Pattison and Wasserman (2001) and Wasserman and
Pattison (2000) note that there are three major classes Bernoulli graphs and conditional uniform
graph distributions, Dyadic dependence distributions, and p*. Other probabilistic graph models
are described by Bollobas (1985), although the primary focus in the mathematics literature is
on asymptotic behavior of various graph statistics as the size of the node set increases (whereas
typically in social network analysis we wish to analyze social networks on a fixed node set).
The assumption of conditional independence for all pairs of random variables representing
distinct relational ties (that is, Xij and Xk~ are independent whenever i id k and j id 1) leads to the
class of Bernoulli graphs (Ffank and Nowicki, 1993~. The dependence graph for such a distribution
has no edges; it is empty. A Bernoulli graph assumes complete independence of relational ties; the
probability that the tie i ~ j is present is Pij. If Pij = 0.5 for all ties, the distribution is often
referred to as the uniform random (di)graph distribution, U. All (di)gTaphs are equally likely to
occur; hence the uniform probability aspect of the distribution. A more general Bernoulli graph
distribution fixes the Pij at P; each edge can be viewed as the outcome of a biased coin toss, with
probability P of a "success".
The uniform distribution U conditions on no graph properties, while the uniform distribution
AL, statistically conditions on the number L of edges in the graph. All Idiographs with L = I
DYNAMIC SOCKS NETWO=MODEL~G ED TRYSTS
199
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SENSITIVITY ANALYSIS
4
lines (arcs) are equally likely; Idiographs with L 74 1 lines (arcs) have probability 0 There are
many other conditional uniform distributions, including the classic OMAN distribution which
fixes the counts of the dyed states and assumes that all digraphs with the specified dyed census are
equally likely, and Ul{Xi+}, {X+j} which fixes the outdegrees and indegrees. Many such conditional
uniform distributions are described in Chapter 13 of Wasserman end Faust (1994~. Some of these
distributions have simple dependence graphs; for example, the Taxi+) distribution, which fixes
only the outdegrees, has a dependence graph with edge set ED = f((i,j)7(i' k)), for all j 76 k for
every i).
The assumption of conditional dependence of Xij and Xk' If and only if {k, ti = {j, if leads
to the class of dyed dependence models (see Wasserman, 1987; Wasserman and Pattison, 2000),
the second family of graph distributions mentioned above. These "multinomial dyed" distributions
assume all dyads are statistically independent, but the states of any specific dyed are not. It
postulates substantively interesting parameterizations for the probabilities of the various dyed
states. The depenclence graph for such distributions has an edge set with edges connecting only the
two random variables withing each dyed: ED = {~(i, j), (j, i)), for all ~ 76 j). This class of models
was termed pi by Holland and Leinhardt (1977, 1981), and has a long history (see Chapters 15
and 16 of Wasserman and Faust 1994~. Although for some parameterizations it Is easy to fit, its
assumption of independence across dyads is not terribly realistic.
*
p
For an observed network, which we consider to be a realization x of a random array X, we
assume the existence of a dependence graph ID for the random array X.
There are, of course, general dependence graphs, with arbitrary edge sets. Such dependence
graphs yield a very general probability distribution for a(di~graph, which we have termed p*. Such
distributions belong to a very general exponential family; they are often referred to as exponential
random graph models (perhaps a misnomer, since nearly all distributions can be rearranged into
an exponential form).
One very general dependence graph, for which this distribution was first developed, assumes
conditional independence of Xij and Xk~ if and only if {i, j) n In, t) = 0. This type of dependency
resembles a Markov spatial process, so these dependencies were defined as a Markov graph by Frank
and Strauss (1986). This p* family of distributions has horn ~xt.~nd.?~1 in manor war. 'n.nr1 ~.~t.im~t.
of its parameters scrutinized.
~ _ ,, my ~ ~ ~~ ~~ v., v _~
, ~
The HammersleyClifford theorem (Besag, 1974) establishes that a probability model for
X depends only on the cliques of the dependence graph Z). As mentioned, application of the
Ha~nmersleyClifford theorem yields a characterization of Pr(X = x) in the form of an exponential
family of distributions, as discussed in detail, in for example, Wasserman and Robins (2002~.
For our sensitivity analyses, we will postulate various, albeit simple, dependence graphs, and
adopt the associated probability distribution, in order to study the effects of node and line removals
on graph statistics.
Network statistics
An important concern in social network studies is the connectivity of a network. Connec
tivity measures how much actors are tied together within a network. For example, using sexual
networks and disease simply for illustrative purposes, actors are unlikely to contract HIV via sexual
200
DY7IAMIC SOCIAL NETWORK MODELING ED ANALYSIS
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SENSITIVITY ANALYSIS
intercourse if they are not tied to others (i.e., they are isolates). These isolates can, in fact, help
to deter the spread of the disease. Conversely, networks with greater connectivity are more likely
to include actors who will contract the disease due to more prevalent sexual interactions. Con
nectivity is also important to terrorist networks because it determines how much information can
be 'transmitted' throughout a network. Lack of connectivity (i.e., isolation) can harm individuals.
Although connectivity in general is a very important consideration connectivity is dependent on
time. In other words, relationships are continually beginning and ending, in a persistent state of
flux. Such continual change influences the spread of information among a set of actors. There are
many different connectivity measures; here, we focus on balance (frequency of transitive triads).
Centrality is also an important network measurement for most network relations. bolter,
Rothenberg, and Coyle (1995) suggest that the centrality of network members to determine the
'gatekeepers' of the network those who function as important links within the network, linking
together otherwise unconnected nodes (Ffeeman, 1980~. Laumann, Gagnon, Michael, and Michaels
(1994) define these central members as the 'core group' of a network. The number of core members
within a network can substantially affect the spread of information throughout a network. A simple
measure of actor degree (the number actors adjacent to a 'focal' actor) can determine which actors
serve as core members. Structural characteristics of actors are also very important. In studies on
terrorism, degree centrality and prestige indices would aid in finding individuals who 'determine'
the sending and receiving of information. They also can help 'locate' important subgroups such as
core and periphery networks, which have important consequences for individuals in these subgroups.
Measures of centralization combine actorlevel indices, and are straightforward to investigate.
~otter, et al. (1995) and Laumann, et al. (1994) also discuss the importance of bridges
to the connectivity of a network. A bridge is an individual (or group of individuals) that spans
two disconnected subgroups. Such network measures can help to identify those people who are
crucial to the connectedness of a terrorist network (usually referred to as cutpoints, see Wasserman
and Faust, 1994, Ch. 43. For example, actor betweenness centrality indices could held t.n icl`?nt.iRr
t, I l ~
cuose network members who link actors to some type of influential subnetwork. In this situation,
the bridge member functions to 'monitor' the flow of support/information/resources within the
network.
Bienenstock and Bonacich (2002) and Borgatti (2002) recently introduced a new measure to
describe the efficiency of a network. Efficiency measures the average distance between all pairs of
nodes in a network. When all possible pairs of nodes are connected, the efficiency of the network
is equal to unity; when all of the nodes are isolates the efficiency of the network is equal to zero.
Borgatti (2002) indicates that efficiency may be the key measure to consider when deciding which
nodes to delete from the network in order to maximize the impact of the deletion. This especially
seems to be the case when the resources used for node deletion are limited.
Sensitivity analyses
. . ,
Many analyses of standard social network data sets involve summarizing the relational data
with a substantivelymeaningful, carefully chosen set of network statistics. Many researchers do not
go the "model route", and simply focus attention on this small set of statistics, perhaps including
these statistics as explanatory variables in linear models with actor measurements as responses.
One hope in such studies is that the chosen statistics are not only meaningful, but aIe "good"
statistically. But this hope begs us to ask the question: "How robust and resistant are typical,
network (multirelational) statistics"?
DYNAMIC SOCKS NETWO~MODE~G ED ^=YSIS
201
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202
SENSITIVITY ANALYSIS
6
Thus, our concern here is the robustness of network statistics, such as those theoretically
relevant to the substantive applications of interest here, and to the presence or absence of particular
relational ties or actors. In statistics, robustness is defined as how well a parameter estimate
behaves when basic assumptions (particularly distributional assumptions) are violated. Resistance
is usually defined as how well statistics behave when data become "messy", full of outliers and
"ugly" observations. Since most social network studies are not parametric, resistance is perhaps
the more important concern. Shouldn't a network analyst know how well his or her analysis is
going to "hold up" or "behave" if some modeling assumptions change, or if some of the data are
"ugly" or removed?
We term the study of resistance of network statistics sensitivity analysis, as it is often called
in the statistical modelling literature. It should be obvious that sensitivity analyses of standard
network statistics are desperately needed. For example, focus on centrality as one of these network
statistics  one of the methodological tools used by the majority of network analysts, and, as
mentioned earlier, a substantivelymeaningful way of looking at most types of networks. It is
straightforward to study how various centrality measurements change when certain ties in a relation
are altered. This study can be done systematically find scientifically and should yield valuable
information into the resistance of centrality indices. Statisticians have been designing studies such
as these (to investigate the resistance and robustness of, say, parameter estimates) for many years.
Is it not time to do the same for network analysis? All of the alterations to the relation will be
made in the context of policy networks (especially those that are egocentered, longitudinal, and
multirelational) so that the findings will indeed reflect what could occur in the real world.
Monte Cario Technique
In order to study the resistance of various statistics, "artificial" sociomatrices for a nondi
rected relation were generated. We were concerned with three key aspects of the sociomatrix.
1. First, the number of actors (nodes) is considered. For this preliminary study, two network
sizes, 10 and 25, were considered.
2. Second, the density of the sociomatrices was varied. The values of the density were deter
mined by the maximum number of dyads (9X(~l)) possible for each sociomatrix. Starting at zero,
these values were allowed to vary in steps of .ltmax(dyads)~. For example, for a network with 10
actors, max~dyads) = 45. Therefore, the stepsize is 5 (.1~45), after rounding). So, for a network
with 10 actors, the number of ties considered were 0, 5, 10, 15, 20, 2o, 30, 35, 40, and 45.
3. All matrices were generated randomly from CAL. For a fixed 9, the appropriate number
of ties were randomly assigned to the lower triangles. Following the lower triangular assignment,
the sociomatrices were symmetrized.
4. Finally, 1,000 random sociomatrices were generated for each condition.
Results
Degree Centralization
The characteristics of degree centralization do not depend on the size of the network, but
rather the density of the network. Furthermore, degree centralization is not monotonically related
to the density of the network. In addition, when density is fixed, the observed value of degree
centralization is extremely discrete. For example, with 10 actors and a density of 0.44, only five
different values of degree centralization were observed. Table 1 provides the results for degree
DYIJAMIC SOCIAL NETWORK MODELING AND ANALYSIS
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SENSITIVITY ANALYSIS
7
centralization when 10 actors are considered (the pattern of results for 25 actors was very similar).
Extreme discreteness of of degree centralization is observed; specifically, it Is seen that no more
Table 1: Mean and Variances of DistributionsDegree Centralization
# of Actors # of Lines Mean Degree Centralization Variance # Mass points
10 5 0.2037 0.0058 3
10 0.2615 0.0091 5
15 0.2874 0.0101 5
20 0.3022 0.0100 5
25 0.2926 0.0086 4
30 0.2668 0.0072 3
35 0.2243 0.0046 2
40 0.1386 0.0000 1
than five probability mass points are observed for any given density (eight were the most observed
when 25 actors were considered). Both network sizes indicate the same nonmonotonic relationship
between density and degree centralization (Figure 1 indicates that the underlying distribution
governing this relationship may be approximated by a binomial distribution). In any case, this may
indicate that the density of the network, not the size, has the greatest eject on degree centralization.
Figure 1. Degree centrality by density for 10 actors (1,000 sociomatrices)
0.35
0.3
0.25
co 0.2
c'
ID
553.1 5
0.1
ans
o
_
/, I ~ ' '
~ on  
\
\\
~ I · ~
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Density
DYNAMIC SOCIAL NETWORK MODE ED ISIS
203
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SENSITIVITY ANALYSIS
Table 2:
and Variances of Dis;tribilt.ion~R~t.w~nn=~ (gentry ligation
of Actors # of Lines Mean Betweenness Centralization Variance ~ Mass points
10 5 0.0708 0.0020 33
10 0.2280 0.0077 833
15 0.2109 0.0072 960
20 0.1666 0.0049 911
25 0.1143 0.0024 661
30 0.0640 0.0007 545
35 0.0302 0.0001 581
40 0.0084 0.0000 66
Betweenness Cer~tralizatiorz
8
Like degree centralization, betweenness centralization is not dependent upon the size of the
network but the density of the network. For both network sizes, the betweenness measure has a
peak at a density between 0.1 and 0.2 and then proceeds to sharply decline as density increases.
Table 2 indicates the results when 10 actors were examined (the same pattern was observed when
the 25 actor sociomatrices were examined). A key difference between degree centralization and
betweenness centralization is that the latter is not nearly as discrete. Even though several values
of betweenness were observed, the small variance indicates that the values were not extremely
different. Figure 2 indicates a skewed relationship between density and betweenness. Although the
same skewness is present at both sizes of sociomatrices examined, the peals tends to move towards
the yaxis as size increases. Figure 2 indicates that this relationship might be approximated by an
Fdistribution.
Balance
Balance is unlike the other two network statistics. As would be guessed, balance increases as
density increases. A peculiar trait of balance is that for several small values of density it is close to
zero. The exponential increase does not tend to occur until density is about 0.3 (this is observable
in Table 3~. Figure 3 further indicates the exponential relationship between balance and density
within the sociomatrices (this relationship is the same for both network sizes) This phenomenon
was observed for both network sizes studied.
Efficiency
Efficiency behaves in a similiar same manner as balance. As the density of the network
increases, efficiency increases as well (this pattern of behavior was observed for both network sizes
studied). This type of behavior is clear from the definition of efficiency. However, what is striking
is the rapid, inverse exponential increase. This increase begins almost immediately as density
increases from zero. Figure 4 indicates the initial inverse exponential relationship that eventually
becomes a linear relationship when density is approximately 0.40. The only difference between the
204
DYNAMIC SOCKS HETWO~MODEL~G~D TRYSTS
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SENSITIVITY ANALYSIS
Figure 2. Group betweenness by density for 10 actors (1,000 sociomatrices)
0.25
0.2
in
89.15
Q
m
° 0.1
C'
0.05
 /
o
0.1 0.2
0.4 0.5 0.6 0.7
Density
0.8 0.9 1
Table 3: Mean and Variances of DistributionsBalance
.
# of Actors # of Lines Mean Balance Variance
10
~ Mass points
10
15
20
25
30
35
40
9
0.0006
0.0085
0.0319
0.0808
0.1619
0.2847
0.4608
0.6965
0.0000
0.0001
0.0002
0.0003
0.0004
0.0004
0.0003
0.0001
2
4
10
15
18
16
15
9
networks of 10 actors and the networks of 25 actors is the initial inverse exponential relationship is
smoother in the larger sociomatrices.
Discussion
This paper begins building the framework for the analysis of resistance and sensitivity in social
networks. This is accomplished through the generation of several thousand random sociomatrices
with known distributions. Calculating graph statistics for the networks with known structure
reveals several key features inherent in social networks.
DYNAMIC SOCIAL NT:TWORK MODELING AND ANALYSIS
205
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SENSITIVITY ANALYSIS
Figure S. Balance by density for 10 actors (1,000 sociomatrices)
1
of
0.8
0.7
0.6
80.5
co
to
0.4
0.3
0.2
0.1
I r ~ I I ~ I A

O 1_
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 1
Density
10
First, regardless of the network statistic, the measurement is not linearly related to the
centrality of the network. Second, none of the measurements axe related to centrality in the same
manner or to the same degree. These two characteristics pose several interesting questions for
future research.
Primarily, the interaction between the different graph statistics is crucial In determining
overall sensitivity of a network. These interactions will also have an impact on cleterm~ng the
configuration that will lead to the optimum amount of resistance or vulnerability of a network. In
addition, by establishing regular features to be elected in networks of various size and density, it
will be possible to determine if a given observed network is abnormal. This observed abnormality
Table 4: Mean and Variances of DistributionsEfficiency
# of Actors # of Lines Mean EfficiencG ~
.
10 5 0.1621 0.0000 2
10 0.4413 0.0001 4
15 0.6115 0.0002 10
20 0.7087 0.0003 15
25 0.7757 0.0004 18
30 0.8332 0.0004 16
35 0.8889 0.0003 15
40 0.9444 0.0001 9
206
DYNAMIC SOCIAL NETWORK MODELING AND ANALYSIS
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SENSITIVITY ANALYSIS
Figure 4. =~~ ~ ~~ t~ 1 l ~ ^ ,' ^^^
~mclency ny density tor lo actors (l'OOO sociomatrices)
1
o.s
0.8
0.7
0.6
c'
·O 0.5
0.4
0.3
0.2
0.1

/ . I '




O ~ . . . . . . . . .
0 0.1 0.2 0.3 0.4






0.5 0.6 0.7 0.8 0.9 1
Density
11
may indicate different possibilities such as: nonobserved actors, nonobserved links, or networks
bred on different underlying distributions.
Furthermore, the nonuniformity of the network statistics, as related to centrality, indicates
the need to study other graph statistics in more detail. Finally, ad of the work presented here will
be generalized to directed graphs and additional statistics will be examined (instars, outstare, etc.).
In addition to this generalization, analytical derivations of the simulation results will be pursued.
This approach will possibly lead to approximating phenomena in the field of social networks with
underlying statistical distributions in the saline manner that Is used in several other fields.
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DYNAMIC SOCIAL NETWORK MODFL~G~D CYSTS