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Late twentieth-century developments in statistical models for social networks reﬂect an increasing theoretical focus in the social and behavioral sciences on the interdependence of social actors in dynamic, network-based social settings. As a result, a growing importance has been accorded the problem of modeling the dynamic and complex interdependencies among network ties and the actions of the individuals whom they link. The early focus of statistical network modeling on the mathematical and statistical properties of Bernoulli and dyad-independent random graph distributions has been replaced by eﬀorts to construct theoretically and empirically plausible parametric models for structural network phenomena and their changes over time. These models in this research paper are based on graphs where the nodes are individual actors, which are ‘linked’ by lines representing relational ties.

## 1. Statistical Models For Social Networks

### 1.1 Notation

In most of the literature on statistical models for networks, the set N = {1, 2,…, n} of network nodes is regarded as ﬁxed and the edges, or ties, between nodes are assumed to be random. The tie linking node i to node j (i, j ϵ N) may be denoted by the random variable X_{ij}. In the simplest case of binary-valued random variables, X_{ij }takes the value 1 if the tie is present and 0 otherwise. Other cases of interest include:

(a) valued networks where X_{ij} is assumed to take values in the set {0, 1,…, C – 1};

(b) multiple relational or multivariate networks, where the variable X_{ijk }represents the possible tie of type k from node i to node j (with k ϵ R = {1, 2,…, r}, a ﬁxed set of types of tie); and

(c) time-dependent networks, where X_{ijt }represents the tie from node i to node j at time t.

In each case, network ties may be directed (e.g., X_{ij }and X_{ji }are distinct random variables) or nondirected (e.g., X_{ij }and X_{ji }are not distinguished). The n n array X = [X_{ij}] of random variables can be regarded as the adjacency matrix of a random (directed) graph on N; the state space of all possible realizations of these arrays may be denoted by Ω_{n}. The array x = [x_{ij}] denotes a realization of X, with x_{ij} = 1 if there is an observed tie from node i to node j, and x_{ij} = 0 otherwise. In some cases, models may also refer to variables measured on actor attributes: for instance, the mth attribute Z_{i}^{[m]} of actor i gives rise to a random vector Z ^{[m] }with realization z^{[m]}.

The adjacency matrix of an illustrative nondirected binary network is shown in Table 1. The data are from a study of interorganizational networks amongst mental health agencies conducted by Morrisey et al. (1994). The nodes of the network are 37 mental health agencies in one of the US cities in the Morrissey study; an edge links two agencies if they engage in regular and mutual exchange of information. The graphical representation of these data is shown in Fig. 1. We use this network to illustrate several of the distributions and models described below.

### 1.2 Bernoulli Graphs

A simple statistical model for a (directed) graph assumes a Bernoulli distribution, in which each edge,

or tie, is statistically independent of all others and governed by a theoretical probability P_{ij}. In addition to edge independence, simpliﬁed versions also assume equal probabilities across ties; other versions allow the probabilities to depend on structural parameters (Frank and Nowicki 1993). These distributions have often been used as models (Bollobas 1985), but are of questionable utility due to the independence assumption. Bernoulli graph distributions are described at length by Erdos and Renyi (1960) and Frank (1981).

### 1.3 Dyadic Structure In Networks

Statistical models for social network phenomena have been developed from their edge-independent beginnings in a number of major ways. In one important series of developments, the p model introduced by Holland and Leinhardt (1981), and developed by Fienberg and Wasserman (1981), recognized the theoretical and empirical importance of dyadic structure in social networks, that is, of the interdependence of the variables X_{ij} and X_{ji}. This class of Bernoulli dyad distributions and their generalization to valued, multivariate, and time-dependent forms, gave parametric expression to ideas of reciprocity and exchange in dyads and their development over time. The model assumes that each dyad (X_{ij}, X_{ji}) is independent of every other and, in a commonly constrained form, speciﬁes:

where θ is a density parameter; ρ is a reciprocity parameter; the parameters α_{i} and β reﬂect individual diﬀerences in expansiveness and popularity; and λ_{ij} ensures that probabilities for each dyad sum to 1. This model is log-linear.

An important elaboration of the p model permits the dependence of density, popularity and expansiveness parameters on a partition of the nodes into subsets that can either be speciﬁed a priori (e.g., from hypotheses about social position) or can be identiﬁed a posteriori from observed patterns of node interrelationships (e.g., Wasserman and Faust 1994). The resulting stochastic blockmodels represent hypotheses about the interdependence of social positions and the patterning of network ties (as originally elaborated in the notion of blockmodel by White et al. 1976). A potentially valuable variant of these models regards the allocation of nodes to blocks as a latent classiﬁcation; Snijders and Nowicki (1997) have provided Bayesian estimation methods for the case of two latent classes. Another valuable extension of the p_{1} model is the so-called p_{2} model (Lazega and van Duijn 1997). Like the p_{1} model, the p_{2} model assumes dyad independence, but it adds the possibility of arbitrary node and dyad covariates for the parameters of the p_{1} model, and models the unexplained parts of the popularity and expansiveness parameters as random rather than ﬁxed eﬀects.

### 1.4 Null Models For Networks

Despite the useful parameterization of dyadic eﬀects associated with the p model and its generalizations, the assumption of dyadic independence has attracted sustained criticism. Thus, another series of developments has been motivated by the problem of assessing the degree and nature of departures from simple structural assumptions like dyadic independence. Following a general approach exempliﬁed by early work of Rapoport (e.g., Rapoport 1949), a number of conditional uniform random graph distributions were introduced as null models for exploring the structural features of social networks (see Wasserman and Faust 1994 for a historical account). These distributions, denoted by U|Q, are deﬁned over subsets Q of the state space Ω_{n} of directed graphs, and assign equal probability to each member of Q. The subset Q is usually chosen to have some speciﬁed set of properties (e.g., a ﬁxed number of mutual, asymmetric, and null dyads, as in the U|MAN distribution of Holland and Leinhardt 1975). When Q is equal to Ω_{n} the distribution is referred to as the uniform (di)graph distribution, and is equivalent to a Bernoulli distribution with homogeneous tie probabilities. Enumeration of the members of Q and simulation of U|Q is often straightforward (e.g., see Wasserman and Pattison, in press), although certain cases, such as the distribution that is conditional on the indegree (∑x_{ij}) and outdegreee (∑_{ij}) of each node i in the net- work, require more complicated approaches (Snijders 1991, Wasserman 1977).

A typical application of these distributions is to assess whether the occurrence of certain higher-order (e.g., triadic) features in an observed network is unusual, given the assumption that the data arose from a uniform distribution that is conditional on plausible lower-order (e.g., dyadic) features (e.g., Holland and Leinhardt 1975). For example, the network displayed in Table 1 has 84 edges, and 333 and 69 triads with exactly 2 and 3 edges, respectively. If Q denotes the subset of complete Ω with 84 edges, the expected number of triads of each kind is 250.1 and 50.8, and the observed values lie at the 0.003 and 0.005 percentiles of the u|x_{++} = 84 distribution, respectively. This general approach has also been developed for the analysis of multivariate social networks. The best- known example of this approach is probably the quadratic assignment procedure or model (QAP) for networks (Hubert and Baker 1978). In this case, the association between two graphs deﬁned on the same set of nodes is assessed using a uniform multivariate graph distribution that is conditional on the unlabelled graph structure of each univariate graph. In a more general framework for the evaluation of structural properties in multivariate graphs, Pattison et al. (in press) also introduced various multigraph distributions associated with stochastic blockmodel hypotheses.

### 1.5 Extra-Dyadic Local Structure In Networks

A signiﬁcant step in the development of parametric statistical models for social networks was taken by Frank and Strauss (1986) with the introduction of the class of Markov random graphs. This class of models permitted the parameterization of extra-dyadic local structural forms and so allowed a more explicit link between some important network conceptualizations and statistical network models. Frank and Strauss (1986) adapted research originally developed for modeling spatial dependencies among observations to graphs. They observed that the Hammersley-Cliﬀord the oremprovidesa general probability distribution for X from a speciﬁcation of which pairs (X_{ij}, X_{kl}) of edge random variables are conditionally dependent, given the values of all other random variables.

Deﬁne a dependence graph D with node set N(D) = {(X_{ij}: i, j ϵ N, i ≠ j} and edge set E(D) = {(X_{ij}, X_{kl}): X_{ij }and X_{kl }are assumed to be conditionally dependent, given the rest of X}. Frank and Strauss used D to obtain a model for Pr(X = x), denoted p* by Wasserman and Pattison (1996), in terms of parameters and substructures corresponding to cliques of D. The model has the form

where:

(a) the summation is over all cliques P of D (with a clique of D deﬁned as a nonempty subset P of N(D) such that |P| = 1 or (X_{ij}, X_{kl}) ϵ E (D) for all X_{ij}, X_{kl} ϵ P);

(b) z_{p}(x) = x_{i}Π_{j} x_{ij} is the (observed) network statistic corresponding to the clique ρ of D; and

(c) c = Σ_{x} exp {Σ_{p}α_{p}z_{p}(x) is a normalizing quantity.

speciﬁcation of a dependence graph raises a deep theoretical issue for network analysis: what constitutes an appropriate set of dependence assumptions? Frank and Strauss (1986) proposed a Markov assumption, in which (X_{ij}, X_{kl}) ϵ E(D) whenever {i, j} ∩ {k, l} ≠ φ. This assumption implies that the occurrence of a network tie from one node to another is conditionally dependent on the presence or absence of other ties in a local neighborhood of the tie. A Markovian local neighborhood for X_{ij }comprises all possible ties involving i and/or j. Any network tie is seen both as dependent on its neighboring ties and as participating in the neighborhood of other ties, so the resulting model embodies the assumption that social networks have a local self-organizing quality. In the case of Markovian neighborhoods, cliques in the dependence graph correspond to triadic and star-like network conﬁgurations. A homogeneity assumption, that model parameters do not depend on the speciﬁc identities of nodes but only on the network conﬁgurations to which they correspond, yields a model that expresses the probability of the network in terms of propensities for subgraphs of triadic and star-like structures to occur.

For example, a homogeneous Markov model for the network of Table 1 with parameters corresponding to edges, 2-stars, and triangles appears to be a substantial improvement in ﬁt over a Bernoulli model with just an edge parameter, since the mean absolute residual for ﬁtted edge probabilities drops from .220 to .175. Pseudolikelihood estimates for model parameters (Wasserman and Pattison 1996) are –3.31, 0.056, and 0.616 for edges, 2-stars, and triangles, respectively, suggesting that graphs with a large number of triangles are relatively more likely in the assumed homogeneous Markov p* distribution than those with few.

p* random graph models permit the parameterization of many important ideas about local structure in univariate social networks, including transitivity, local clustering, degree variability, and centralization. Valued (Robins et al. 1999) and multivariate (Pattison and Wasserman 1999) generalizations also lead to parameterizations of substantively interesting multi- relational concepts, such as those associated with balance and clusterability, generalized transitivity and exchange, and the strength of weak ties.

### 1.6 Prospects For Structural Models

The p* family of random graph models poses many interesting questions, both substantive and statistical. A major empirical question is the suﬃciency of these local structural descriptions: do they suﬃce, or do they need to accommodate other characterizations of local dependence, more global considerations, other types of constraints (e.g., temporal and spatial), or variability in local structural dependencies? On the statistical side, major questions are raised by the complexity and rich parameterization of models that are deemed to be theoretically plausible. For instance, does a Markov chain Monte Carlo maximum likelihood approach to parameter estimation oﬀer a viable alternative to approximate methods such as pseudolikelihood estimation (Strauss and Ikeda 1990)? How can these distributions best be simulated, and what are the convergence properties of various approaches, such as the Metropolis-Hastings algorithm?

## 2. Dynamic Models

Empirically and theoretically defensible parametric models for networks constitute one challenge for statistical modeling, but an arguably more signiﬁcant challenge is to develop models for the emergence of network phenomena, including the evolution of networks and the unfolding of individual actions (e.g., voting, attitude change, decision-making) and interpersonal transactions (e.g., patterns of communication or interpersonal exchange) in the context of longer standing relational ties. Early attempts to model the evolution of networks in either discrete or continuous time assumed dyad independence and Markov processes in time.

A step towards continuous time Markov chain models for network evolution that relaxes the assumption of dyad independence has been taken by Snijders (1996). These models incorporate both random change and change arising from individuals’ attempts to optimize features of their local network environment (such as the extent to which reciprocity and balance prevail) and use the computer package SIENA for ﬁtting. A tension function is proposed to represent the function that actors wish to minimize and Robbins-Munro procedures are used to estimate free parameters. The approach illustrates the potentially valuable role of simulation techniques for models that make empirically plausible assumptions; clearly, such methods provide a promising focus for future development.

The value of simulation has also been demonstrated by Watts and Strogatz (1998) and Watts (1999) in their analysis of the so-called small-world phenomenon (the tendency for individuals to be linked by short acquaintanceship network paths). They considered a probabilistic model for network change in which the probability of a new tie was determined, in part, by existing patterns of neighboring ties (a clustering component) and, in part, by a small random component. They used simulations to establish that only a small random component was needed for the simulated network to be likely to possess the small-world property.

## 3. Models Bridging Analytic Levels

A recurring theme among social theorists is the need for models that integrate the dynamic, interdependent, interpersonal, and cultural contexts in which social action occurs. The network modeling challenge posed by such framing of the nature of social processes is the development of models that represent interdependencies across several analytic categories (e.g., individuals, relational ties, settings, groups).

Snijder’s (1996) model for network change represents one step in this direction; the class of social inﬂuence models represents another (Friedkin 1998). These models are exempliﬁed by the network eﬀect model for the mth attribute variable:

where α is a network eﬀects parameter, y is a matrix of exogenous attribute variables with corresponding parameter vector β, and ε is a vector of residuals.

A further step in the direction of modeling interdependent node attributes and network ties has been to generalize the dependence graph approach to p* network models (Sect. 1.5) so as to include both node attribute variables (Z ^{[m]}) and relational variables (X ). This step is facilitated if directed dependencies are permitted in the dependence graph (i.e., dependencies in which one random variable is assumed to be conditionally dependent on a second, but the second not on the ﬁrst). Robins et al. (in press) drew on graphical modeling techniques to construct models with directed dependencies among network ties and individual attributes. If the dependencies are directed from network ties to node attributes, models for social inﬂuence are obtained; with dependencies oriented from node attributes to network ties, social selection models result. Indeed, this generalization introduces substantial ﬂexibility in model construction, with potential applications to joint inﬂuence and selection models, discrete time models for network change, network path models, and group-level attribute models.

The dependence approach of Sect. 1.5 has also been used to construct principled models for more general multiway relational data structures representing interdependent analytic categories. Examples to date have included bipartite (Faust and Skvoretz 1999) and k-partite (Pattison et al. 1998) graphs and illustrate how this general distribution has the capacity for a systematic and coherent approach to a variety of major network modeling challenges.

## 4. Conclusion

In addition to some of the speciﬁc future prospects already noted, a more general expectation is that the current classes of models are likely to be the foundation of substantial empirical and model-building activity in the early twenty-ﬁrst century. From their early beginnings, statistical approaches to network modeling have now reached a point where sustained interaction is likely to be empirical research that allows the development of greatly improved models for processes occurring on networks, with important applications in many ﬁelds of social and behavioral science, including epidemiology, organizational studies, social psychology, and politics.

For more information on these approaches to social networks analysis, please see Wasserman and Faust (1994) and Wasserman and Pattison (1996, in press).

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