Characterizing the dynamics of unlabeled 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 unlabeled nodes, a case that has received relatively little attention so far. Situations in which node labeling cannot be tracked over time often emerge in practice due to technical challenges or privacy constraints. In unlabeled temporal networks, there is no one-to-one matching between a network snapshot and its adjacency matrix. Characterizing the dynamical properties of such unlabeled network trajectories is, therefore, nontrivial. Here, we exploit graph invariants to extend some recently proposed network-dynamical quantifiers of linear correlations and dynamical instability to the unlabeled setting. 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.