  The centrality of a node within a network, however it is measured, is a vital
proxy for the importance or influence of that node, and the differences in node
centrality generate hierarchies and inequalities. If the network is evolving in
time, the influence of each node changes in time as well, and the corresponding
hierarchies are modified accordingly. However, there is still a lack of
systematic study into the ways in which the centrality of a node evolves when a
graph changes. In this paper we introduce a taxonomy of metrics of equality and
hierarchical mobility in networks that evolve in time. We propose an indicator
of equality based on the classical Gini Coefficient from economics, and we
quantify the hierarchical mobility of nodes, that is, how and to what extent
the centrality of a node and its neighbourhood change over time. These measures
are applied to a corpus of thirty time evolving network data sets from
different domains. We show that the proposed taxonomy measures can discriminate
between networks from different fields. We also investigate correlations
between different taxonomy measures, and demonstrate that some of them have
consistently strong correlations (or anti-correlations) across the entire
corpus. The mobility and equality measures developed here constitute a useful
toolbox for investigating the nature of network evolution, and also for
discriminating between different artificial models hypothesised to explain that
evolution.
