# Finding Balance in Network of Friendships

I have a couple of data points on a large group of high school students:

A: Who their Friends are

B: Strength of Friendship for each friend (out of 5, 5 being best friends and 1 being acquaintances)

C: Grade Point Average for each student

To start, I want to confirm the "popular kids hangs out with other popular kids" phenomenon, I've aggregated the dataset and now I have all student and their friend count. What would be my next step? What kind of model should I be using to confirm the "popular kids" phenomenon. Next, I hypothesize that "high GPA kids hang out with other high GPA kids", which model should I use here?

Apologies if this is an easy question, I'm new to network analysis and not sure where to start.

• I would cluster first, then find mean and standard deviation of grades between various clusters. Here is how I would cluster: stats.stackexchange.com/questions/139490/… There are statistical tests to say "if mean is mu, standard deviation is sigma, and number of samples is n" is it more likely than not to come from the same distribution as mu_2, sigma_2 and n_2. – EngrStudent Dec 8 '16 at 3:09
• It's not clear how that would test the hypothesis, which is about tie formation between individuals. With clusters, it may happen the clusters themselves are assortative but all have the same mean, or that the clusters are not assortative but have different means due to some unrelated population variation. – samplesize1 Dec 8 '16 at 22:37

## 1 Answer

Both of these hypotheses relate to assortativity. Assortativity is a correlation metric that indicates whether nodes are likely to be connected to similar others (positive assortativity) or dissimilar others (negative assortativity).

Note that for your question about "popularity" you need to make a decision about how to measure popularity. The most straightforward metric would be the sum of incoming friendship weights. However, you have a few choices here -- check out the idea of network centrality.

Note that this does not directly provide for hypothesis testing in the statistical sense. Networks are difficult in this sense, because you generally only have a single network, so your statistical sample size is N=1. One common approach to developing a hypothesis test is to generate many random network with the same average degree as your empirical networks (and the same set of node attributes) and generate a null hypothesis distribution of the metric of interest, in this case assortativity. (That is, you generate many assortativity scores from your data, get the standard deviation, and determine a z-score for your empirical data point.)

For more advanced techniques, look into Exponential Random Graph Modeling (ERGM) which uses is comparable to multivariate regression and can test any set of potential tie predictors in addition to homophily.