Dependence assumption: Theoretical assumption about dependencies among possible network ties; determines the type of parameters in the model.
Exponential random graph models: a model for a social network, expressing a probability distribution of graphs with an exponential form: see also p*.
Homogeneity assumption: Assumption about which parameters to equate, to make the model identifiable.
Graph statistics: for homogeneous models these are counts of the configurations in the observed graph; more generally they may depend also on node-wise or dyad-wise covariates
Network Configuration: A small sub-graph that may be observed in the data and that is represented by parameters in the model: eg reciprocated ties, triangles.
Parameters: relate to specific network configurations that may be observed in the graph; a large positive parameter is interpreted as the presence of more of the configurations than might be expected from
chance (given the other effects in the model); a large negative parameter signifies the relative absence of the configuration.
p*: the term for exponential random graph models introduced by Wasserman and Pattison (1996).
Edge and dyad independence models:
Dyad independence: assumes that dyads are independent of one another; the model includes edge and
reciprocity parameters, and possibly also node or dyad attributes.
p1 models (Holland and Leinhardt): an early dyad independence model, including popularity and expansiveness effects.
p2 model: elaboration of p1 model, where popularity and expansiveness effects are random, and independent variables may be used to predict ties.
Simple random graphs, Bernoulli graphs, Erdos-Renyi graphs: assume that edges are independent of one another and are observed with a given probability.
Markov random graph models
alternating k-stars: a Markov parameter (and statistic) but used in the new higher order models; a particular combination of Markov k-star counts into the one statistic; equivalent to geometrically weighted degree counts; useful for modelling the degree distribution.
cyclic triad: a Markov graph configuration: in a directed network, ties ij, jk and ki are observed among actors i, j, and k.
degeneracy (or near-degeneracy): when a model implies that very few distinct graphs are probable, often only empty or complete graphs; degenerate models cannot be good models for social network data.
geometrically weighted degree counts: a statistic (and parameter) in higher order models: a sum of degree counts with geometrically decreasing weights; equivalent to alternating k-stars.
k-star: a Markov graph configuration: in a non-directed graph, k edges are expressed by the one actor.
k-in-star: a Markov graph configuration: in a directed graph, k arcs are directed to the one actor.
k-out-star: a Markov graph configuration: in a directed graph, k arcs are expressed by the one actor.
Markov dependence assumption: introduced by Frank and Strauss (1986), proposes that, conditional on the rest of the graph, two possible ties are dependent on one another when they share an actor.
mixed-star: a Markov graph configuration: a two path in a directed graph.
transitive triad: a Markov graph configuration: in a directed network, ties ij, jk and ik are observed among actors i, j, and k.
triangle: a Markov graph configuration: in a non-directed network, a clique of three actors, ties ij, jk and ik are observed among actors i, j, and k.
HIGHER ORDER MODEL
alternating independent-2-paths: a parameter (and statistic) in higher order models; a particular combination of k-independent-2-path counts into the one statistic; when this parameter is negative, together with a positive alternating k-triangle parameter, there is a tendency against 4-cycles in the network, unless those cycles include triangles (alternatively, the presence of many 2-paths between nodes is related to the formation of triangles.)
alternating k-triangles: a parameter (and statistic) in higher order models; a particular combination of ktriangle counts into the one statistic; expresses the tendency for many triangles to form together in the observed network; a positive parameter in the model suggests regions in the network of high triangulation, possibly core-periphery-type structures; a positive parameter, together with a negative alternating k-star parameter, suggests several smaller regions (possibly connected) of triangulation; equivalent to weighted shared partners.
Dyad-wise shared partners (dsp): a parameter (and statistic) in the higher order models; expresses the tendency in the observed network for dyads (whether tied or not) to have multiple shared partners; equivalent to alternating independent 2-paths.
Edge-wise shared partner distribution: Distribution of the number of dyads who are themselves related and who have a fixed number of shared partners.
Edge-wise shared partners (esp): a parameter (and statistic) in the higher order models; expresses the tendency in the observed network for tied nodes to have multiple shared partners; equivalent to
k-triangle: a configuration in higher order models; in a non-directed graph, the combination of k triangles, each sharing the one edge (the base of the k-triangle).
k-independent-2-paths: configurations in the higher order models; equivalent to k-triangles but without the base.
partial dependence assumption (Pattison & Robins, 2002): assumption for dependencies among possible ties created by the presence of other ties; permits models with higher order configurations than Markov configurations. For example, Xij and Xkl are conditionally dependent if xik = xjl = 1 or if xil = xjk = 1.
Monte Carlo Markov Chain maximum likelihood estimation (MCMCMLE): Method of estimation based on computer simulation; more principled than pseudolikelihood; produces reliable standard errors.
PNet: (Wang, Robins, & Pattison, 2005): Software that includes procedures for MCMCMLE for exponential random graph models – University of Melbourne, Australia.
Pseudo-likelihood estimation: an approximate method of estimation using logistic regression; does not
produce reliable standard errors; properties are not well understood.
Statnet: (Handcock, Hunter, Butts, Goodreau, & Morris, 2005). A software package using R, including procedures for MCMCMLE for exponential random graph models – University of Washington.
SIENA: (Boer, Huisman, Snijders, & Zeggelink, 2003). A procedure within the StOCNET software package that includes provisions for MCMCMLE for exponential random graph models – University of Groningen, the Netherlands. (http://stat.gamma.rug.nl/StOCNET)