Examining k-nearest neighbour networks: Superfamily phenomena and inversion

We examine the use of recurrence networks in studying non-linear deterministic dynamical systems. Specifically, we focus on the case of k-nearest neighbour networks, which have already been shown to contain meaningful (and more importantly, easily accessible) information about dynamics. Superfamily phenomena have previously been identified, although a complete explanation for its appearance was not provided. Local dimension of the attractor is presented as one possible determinant, discussing the ability of specific motifs to be embedded in various dimensions. In turn, the Lyapunov spectrum provides the link between attractor dimension and dynamics required. We also prove invertibility of k-nearest neighbour networks. A new metric is provided, under which the k-nearest neighbour and ϵ-recurrence construction methods produce identical networks. Hence, the already established ϵ-recurrence inversion algorithm applies equally to the k-nearest neighbour case, and inversion is proved. The change in metric necessarily distorts the shape of the reconstructed attractor, although topology is conserved.