VA-Index: Quantifying Assortativity Patterns in Networks with Multidimensional Nodal Attributes

Network connections have been shown to be correlated with structural or external attributes of the network vertices in a variety of cases. Given the prevalence of this phenomenon network scientists have developed metrics to quantify its extent. In particular, the assortativity coefficient is used to capture the level of correlation between a single-dimensional attribute (categorical or scalar) of the network nodes and the observed connections, i.e., the edges. Nevertheless, in many cases a multi-dimensional, i.e., vector feature of the nodes is of interest. Similar attributes can describe complex behavioral patterns (e.g., mobility) of the network entities. To date little attention has been given to this setting and there has not been a general and formal treatment of this problem. In this study we develop a metric, the vector assortativity index (VA-index for short), based on network randomization and (empirical) statistical hypothesis testing that is able to quantify the assortativity patterns of a network with respect to a vector attribute. Our extensive experimental results on synthetic network data show that the VA-index outperforms a baseline extension of the assortativity coefficient, which has been used in the literature to cope with similar cases. Furthermore, the VA-index can be calibrated (in terms of parameters) fairly easy, while its benefits increase with the (co-)variance of the vector elements, where the baseline systematically over(under)estimate the true mixing patterns of the network.