  This paper proposes that common measures for network transitivity, based on
the enumeration of transitive triples, do not reflect the theoretical
statements about transitivity they aim to describe. These statements are often
formulated as comparative conditional probabilities, but these are not directly
reflected by simple functions of enumerations. We think that a better approach
is obtained by considering the linear regression coefficient of ties
$i\,\rightarrow\,j$ on the number of two-paths
$i\,\rightarrow\,k\rightarrow\,j$ for the $(n-2)$ possible intermediate nodes
$k$. Two measures of transitivity based on correlation coefficients between the
existence of a tie and the existence, or the number, of two-paths are
developed, and called "Transitivity Phi" and "Transitivity Correlation". Some
desirable properties for these measures are studied and compared to existing
clustering coefficients, in both random (Erd\"os-Renyi) and in stylized
networks (windmills). Furthermore, it is shown that under the condition of zero
Transitivity Correlation, the total number of transitive triples is determined
by four underlying features of any directed graph: size, density, reciprocity,
and the covariance between indegrees and outdegrees. Also, it is demonstrated
that plotting conditional probability of ties, given the number of two-paths,
provides valuable insights into empirical regularities and irregularities of
transitivity patterns.
