Time series network induced subgraph distance as a metonym for dynamical invariants

The complex behaviour of dynamical systems can be discovered through the study of observed scalar time series via reconstruction of a suitable state space. Reconstruction can readily be achieved thanks to Takens embedding theorem and further insights about a system can be found by constructing complex networks based on the proximity to each other of these reconstructed states. An example of such a proximity network is the so-called phase space network which is a type of recurrence network. The structure and properties of these networks, including small induced subgraphs within them, describe the spatial distribution of reconstructed states in a way different from, but related to, correlation dimension. Beyond the makeup of the reconstructed states the time structure inherent in the time series is typically ignored in any subsequent analysis of proximity networks. Here, we construct proximity networks but retain information about the time structure of the reconstructed state space trajectory and hence assign time as an attribute of the nodes of the network. We show that metrics capturing the change in structure of induced subgraphs along the trajectory can qualitatively track behavioural changes of the system dynamics subject to a bifurcation parameter in a manner similar to time-averaging estimation methods of dynamical invariants. Moreover, we demonstrate that these new methods retain the capability of monitoring system change in the case of irregularly sampled data.