  We consider the problem of embedding a dynamic network, to obtain
time-evolving vector representations of each node, which can then be used to
describe changes in behaviour of individual nodes, communities, or the entire
graph. Given this open-ended remit, we argue that two types of stability in the
spatio-temporal positioning of nodes are desirable: to assign the same
position, up to noise, to nodes behaving similarly at a given time
(cross-sectional stability) and a constant position, up to noise, to a single
node behaving similarly across different times (longitudinal stability).
Similarity in behaviour is defined formally using notions of exchangeability
under a dynamic latent position network model. By showing how this model can be
recast as a multilayer random dot product graph, we demonstrate that unfolded
adjacency spectral embedding satisfies both stability conditions. We also show
how two alternative methods, omnibus and independent spectral embedding,
alternately lack one or the other form of stability.
