Characterising the dynamics of unlabelled temporal networks

Networks model the architecture backbone of complex systems. The backbone itself can change over time leading to what is called `temporal networks'. Interpreting temporal networks as trajectories in graph space of a latent graph dynamics has recently enabled the extension of concepts and tools from dynamical systems and time series to networks. Here we address temporal networks with unlabelled nodes, a case that has received relatively little attention so far. Situations in which node labelling cannot be tracked over time often emerge in practice due to technical challenges, or privacy constraints. In unlabelled temporal networks there is no one-to-one matching between a network snapshot and its adjacency matrix. Characterizing the dynamical properties of such unlabelled network trajectories is nontrivial. We here exploit graph invariants to extend to the unlabelled setting network-dynamical quantifiers of linear correlations and dynamical instability. In particular, we focus on autocorrelation functions and the sensitive dependence on initial conditions. We show with synthetic graph dynamics that the measures are capable of recovering and estimating these dynamical fingerprints even when node labels are unavailable. We also validate the methods for some empirical temporal networks with removed node labels.