When Pam Cronrath's husband Bill died last year, after nearly 60 years of marriage, she knew what she wanted to do, but not exactly how. I promised him a super wake, she told the BBC. What she didn't expect was that keeping the promise would lead her into the world of holograms, technology more commonly associated with celebrities than memorial services in rural America. A self-confessed tech enthusiast, she says her outlook was shaped by a career that stretched back to the early days of the internet. Several years ago, while speaking at a medical conference, she watched a doctor appear as a full-body hologram broadcast live across the United States.

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.