Analysis of nominations

To assess the hypotheses about nomination structure, we used exponential-family random graph models (ERGMs). This approach can be thought of as a kind of generalization of logistic regression to social networks–with the log-odds of a tie (here, a nomination) between two actors being dependent on a set of predictors of interest [21]. Those predictors may include characteristics of either or both nodes (e.g. their gender, class performance or outspokenness). However, it can also include structural factors involving the other ties in the network–e.g. the tendency for ties in a directed network to be mutual, or to form a triangle. When such structural terms are present, ties become conditionally dependent and estimation becomes more difficult, with Markov chain Monte Carlo-based methods the current state of the art for estimation [22]. Nevertheless, the coefficients may still be interpreted in terms of their contribution to the conditional log-odds of a tie, given all of the other ties in the network.

We specify two models, both of the general form:

where Y_ij represents the value of the tie from i to j, which equals 1 if i nominates j and 0 if they did not (we discuss missing data for these values in the SI). The quantity yijc represents the complement of y_ij, i.e. the state of all of the ties in the network other than y_ij. The δ vector represents the amount by which the model statistics change when y_ij is toggled from 0 to 1, and the θ vector represents the coefficients on these statistics.

The first model contains seven model statistics (δ₁ through δ₇) and the second model contains nine (δ₁ through δ₉):

δ ₁ = 1 for all dyads [the main effect or intercept];

δ ₂ = 1 if j nominates i, and 0 otherwise [mutuality];

δ ₃ = 1 if i is female and 0 otherwise [female nominator];

δ ₄ = 1 if both i and j are female and 0 otherwise [female-female bias];

δ ₅ = 1 if both i and j are male and 0 otherwise [male-male bias];

δ ₆ = 1 if i and j are in the same lab section [lab homophily];

δ ₇ = -1 if j has no nominations other than that from i [0-indegree];

δ ₈ = j’s final grade in the class [grade of nominee];

δ ₉ = 1 if j is outspoken, and 0 otherwise [outspokenness of nominee];

We use the R package network to process and store the data, and the R package ergm to estimate the θ coefficients for our two models for each survey wave [22,23]. The terms involving gender, grade, or outspokenness represent our core theoretical measures. We include mutuality since it is a basic phenomenon in directed networks (those where the relationship from i to j does not necessarily equal that from j to i), and we include lab homophily given that labs are a major structural element of the course. We include a unique propensity for individuals to have no nominations (called 0-indegree in network terminology) since this dramatically improved the fit of the model to the observed in-degree distribution, which is a condition for the statistical inference we later conduct (see S1 appendix for more information). Moreover, it is reasonable to expect that measures of renown such as that here would have more variation than expected by chance–that is, with more students who have either no or many nominations than otherwise expected. The δ on this term is negative given the unique condition that adding a tie reduces the statistic of interest (nodes without ties).