7. Testing Transitivity in Digraphs

Abstract

The problem of testing the transitivity of a relationship observed in a digraph, taking as many nontransitivity related irregularities as possible into account, is studied. Two test quantities are used: (1) the proportion of transitive triples out of all nonvacuously transitive triples, and (2) the density difference (the difference between mean local transitivity density and overall edge density). The null distribution used is the rather complex uniform distribution on digraphs conditional on the indegrees and outdegrees. A simulation study is made in order to estimate critical values of the tests for different significance levels. When all vertices have the same indegree and outdegree, the occurrence of transitive triples is rather infrequent in most conditional uniform graphs; this is reflected by low critical values of the transitivity-related test statistics. When both the indegree and outdegree sequences are skewed in the same direction, there is a small number of vertices with large indegrees and outdegrees. This results in a clustering structure, in which transitive triples occur frequently; in such conditional uniform graphs, the critical values of the test statistics are rather high. The powers of the tests are estimated against the Bernoulli transitive triple model, which assumes a simple random graph distribution in which the transitivity is high. The test based on density difference has the highest power in many cases. The tests are applied to a large set of classroom sociograms, and in this situation it is also found that uniform randomness is rejected in favor of transitivity most frequently when the test based on the density difference is used. However, the vast majority of these sociograms are so far from the uniform distribution that the null hypothesis of uniform randomness is rejected regardless of which test is used. Nevertheless, the results imply that the density difference is the best detector of transitivity related to the measures examined.