I'm reading the following paper on non-parametric standard errors and tests for network statistics - https://www.stats.ox.ac.uk/~snijders/Snijders_Borgatti.pdf by Snijders & Borgatti.

When comparing e.g. the density of two networks, we can calculate a t-statistic using the formula:

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Where Z1 and Z2 are the observed densities of each network, and SE1 and SE2 are the standard errors derived from bootstrapped samples of each network.

My question is, to generate a p-value for this t-statistic, what degrees of freedom should we be using (in the example they cite, network 1 has an N of 16 vertices, and network 2 an N of 15 vertices)? Is it analogous to the Student's t-test, so df = N1+N2-2 ?


1 Answer 1


If you believe that the number of edges coming from a vertex is reasonably approximated as normal, and the networks share maximum cardinality, then the $Z$s can be viewed as means of Gaussian random samples scaled by a constant. This constant, the total number of possible edges in the graph, is also implicitly present in the standard error estimates in the denominator and cancels out. Then the statistic is reduced to the same conditions as Student's t, and $t \sim T_{N_1 + N_2 -2}$ if you're willing to also willing to ignore the effect of dependence between observations (an edge correlates the counts for two vertices).

However, the laundry list assumptions above raises an eyebrow. Seeing as you're already comfortable with bootstrapping, I would advise considering a randomization test, which provides similar inference without the listed assumptions, at the cost of some power.


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