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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.