# Why ERGM?

## To predict ties

• As function of individual covariates
• e.g. Are girls more popular than boys?
• As function of network structures
• e.g. If Adam is friends with Bill, and Bill is friends with Carl, what can we say about the chances of Adam and Carl being friends?

## To handle non-independence of observations

Suppose we…

• Want to predict dichotomous ties
• e.g. two countries at war
• Predictors are actor attributes
• e.g. government types and difference in defense budgets
• What’s wrong with estimating a logistic regression from an n x k matrix like this?
regime 1 regime 2 budget diff war?
dem dem $5e7 0 dem theo$1e9 0
dem dict \$2e9 1
• Even if one is not explicitly interested in how other ties affect the likelihood of a tie, network effects imply a correlation of residuals, so omitting them will produce biased estimates.
• e.g. If the U.S. and U.K. are both at war with Iraq, that fact must be considered for an unbiased estimate of the odds of the U.S. and U.K. being at war with each other.

Cranmer and Desmarais (2011) used ERGM to rebut conventional international relations wisdom on democracies going to war:

## To disentangle and quantify exogenous and endogenous effects

• Simultaneously model effect of network structures, actor attributes, and relational attributes

• Get estimates and uncertainty for each effect

# What do we get from an ERGM?

• Much like a logit: Change in the (log-odds) likelihood of a tie for a unit change in a predictor

• Predictors are network-level statistics, but we think about their changes locally

# How does it work?

• (See Pierre-Andre’s talk for details)

• Define joint-likelihood of all ties
• Observed set is expectation
• Choose set of ($$k$$) sufficient statistics ($$\Gamma$$)

• Use MCMC to find parameter values ($$\theta$$) for statistics that maximize the likelihood of the set of observed ties ($$Y_m$$)
• MCMC provides confidence intervals
• Maximize $$\theta$$:

$P(Y_m) = \dfrac{exp(-\sum_{j=1}^k\Gamma_{mj}\theta_j)}{\sum_{m=1}^Mexp(-\sum_{j=1}^k\Gamma_{mj}\theta_j)}$

• From parameter estimates, can estimate the probability of any edge:

$P(Y_{ij} | Y_{-ij}, \theta) = logistic \sum_{h=1}^{k}\theta_{h}\delta_{h}^{ij}(Y)$

That allows us to back out probabilities of interest, e.g.