Test for triadic closure and triadic protection in temporal relational event data

Abstract

Temporal relational events are evidence of dynamically evolving social networks. The timing of the creation and dissolving of enduring ties, such as friendships or alliances, often depend on a large variety of factors. Particularly, the presence of the so-called triadic or transitive effects suggests a certain maturity of the underlying social process and is an important feature of various social relationships. Various models have been proposed to capture various determinants of such temporal relational events. The main obstacle for widely using these models in practice is their computational complexity, especially for modern, online recorded data. The aim of this paper is to propose a simple test for the presence of triadic effects in relational event data. We propose a joint test for triadic closure and triadic protection of ties, based on a combination of a method-of-moments estimator and a Hotelling’s T2 test. Such test is computationally fast and statistically near-efficient, and we show how the test is particularly insightful for the analysis of two studies involving relational event data.

Appendices

Appendices

1.1 Starting values for Newton–Raphson

Suppose that $\varDelta t$ is the time interval between two observations. Then, the probability that links disappear during time $\varDelta t$ is given by

$$\begin{aligned} P(T\le \varDelta t) = 1 - \mathrm {e}^{\mu _0 \varDelta t}. \end{aligned}$$

The empirical probability can be expressed as the ratio between the number of extinguished links ($s_{10}$) and the total number of links ($s_1$):

$$\begin{aligned} {\hat{P}}(T\le \varDelta t) = \frac{s_{10}}{s_1}. \end{aligned}$$

By equating the empirical and theoretical probability, we obtain a rough estimate for parameter $\mu$

$$\begin{aligned} \mu _0^{(0)} = - \frac{\ln \left( 1-\frac{s_{10}}{s_1}\right) }{\varDelta t}. \end{aligned}$$

Respectively, we can show that

$$\begin{aligned} \lambda _0^{(0)} = - \frac{\ln \left( 1-\frac{s_{01}}{s_0}\right) }{\varDelta t}, \end{aligned}$$

where $s_{01}$ denotes the number of appeared links and $s_0$ the total number of non-links.