ERGM

ERGM, or Exponential Random Graph Model, is a family of statistical models used to analyze social networks by treating the probability of observing a particular network as a member of an exponential family. ERGMs aim to capture structural dependencies in networks, such as transitivity, homophily, and degree distributions, by relating observed patterns to a set of network statistics.

Mathematically, an ERGM specifies that the probability of a network y on n nodes is proportional to the exponential of a weighted sum of network statistics: P(Y = y) = exp(theta^T g(y)) / k(theta), where g(y) is a vector of statistics (for example, number of edges, number of triangles, or k-stars), theta is a corresponding parameter vector, and k(theta) is a normalizing constant that sums over all possible networks with n nodes. Inference involves estimating theta from observed network data. Because k(theta) is typically intractable to compute exactly for even modest n, estimation relies on methods such as Markov chain Monte Carlo maximum likelihood estimation (MCMC-MLE) or pseudo-likelihood approaches. Degeneracy, where the model assigns most probability to unrealistic networks, can arise if the statistics are poorly chosen or the model is mispecified. To mitigate this, practitioners utilize curved or geometrically weighted terms (e.g., GWESP) and careful model specification.

Applications of ERGMs span sociology, epidemiology, organizational studies, and biology, where researchers seek to understand the drivers of network structure. Software implementations, notably the ERGM framework in the R statnet suite, facilitate fitting and assessing these models. Limitations include computational demands, sensitivity to model specification, and challenges in interpreting highly complex parameterizations.