**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

**Suggested Citation:**"Session III: Metrics and Models Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results." National Research Council. 2003.

*Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers*. Washington, DC: The National Academies Press. doi: 10.17226/10735.

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SESSION III Metrics and Models

Sensitivity Analysis of Social Network Data and Methods: Some Preliminary Results Stanley Wasserman and Douglas SteinTey University of Illinois at Champaign-Urbana 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 (#ITS99-80189), and the Once of Natural Research (#N0001~02-1-0877~. 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

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 error-proneness, 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

SENSITIVITY ANALYSIS Theory 3 Any observed single relational network may be regarded as a realization x = Fiji of a random two-way 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 JV-D = {(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 well-known 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

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 Hammersley-Clifford 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~nmersley-Clifford 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

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 actor-level 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 sub-network. 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 substantively-meaningful, 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

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 substantively-meaningful 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

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 Distributions-Degree 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

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 y-axis as size increases. Figure 2 indicates that this relationship might be approximated by an F-distribution. 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

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 Distributions-Balance . # 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

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 Distributions-Efficiency # 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

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: non-observed actors, non-observed links, or networks bred on different underlying distributions. Furthermore, the non-uniformity 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. References Besag, J.E. (1974~. Spatial interaction and the statistical analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, Series B. 36, 192-236. Bienenstock, E. J., and Bonacich, P. (2002). Balancing efficiency and vulnerability in social networks. Paper presented at the National Academy of Science Workshop on Social Network Analysis, Washington, D.C. (November, 2002). Bollobas, B. (1985~. Random Graphs. London: Academic Press. Borgati, S. P. (2002~. The key player problem. Paper presented at the National Academy of Science Workshop on Social Network Analysis, Washington, D. C. (November, 20029. ~ · 1 TO ~ ~ . . ~ . ~ ~-arrmgron, r.J.' Scott, J., and Wasserman, 5. (eds.) (2003). Models and Methods in Social Network Analysis. New York: Cambridge University Press. Frank, O., and Nowicki, K. (1993~: Exploratory statistical analysis of networks. In J. Gimbel, J. W. Kennedy, and L. V. Quintas (eds.), Quo Vadis Graph Theory? A Source Book for Challenges and Directions. Amsterdam: North-Holland. (also Annals of Discrete Mathematics. DYNAMIC SOCL4L NETWORK MOD~:~WG AD CYSTS 207

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