Examining k-nearest neighbour networks: Superfamily phenomena and inversion

Abstract

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.

Three-dimensional reconstruction of single-cell chromosome structure using recurrence plots

“… 7 requires additional information such as the frequencies of pairs or read counts, hence the RPR method is suitable for Hi-C data with a few read counts. Second, the proposed method is based on mathematical proofs 13 14 and seems to work well with larger datasets. For example, the RPR method functioned well, even for a large-scale dataset with over 10,000 points 3 , whereas the example with Lense’s method handled only up to 1,000 points 7 .…”

“…However, we have chosen to focus on the RPR method we previously proposed in 2008 11 because (i) it works, even if the original time series is multivariate, and (ii) we have proven a theorem 13 that the metric space recovered using our prior method 11 is equivalent to the original Euclidean metric under mild conditions. In addition, the RPR method is known to be rather robust, even if we change the definition of the closeness 14 . Therefore, we used the method to reconstruct the 3D chromosome structure from Hi-C data.…”

Persistent homology of complex networks for dynamic state detection

“…This is built by adding a vertex to represent each x i ∈ χ, and for each x i , adding an edge to the k closest points x j ∈ χ. This construction, and in particular the investigation of motifs in the resulting graph, has been extensively studied [16][17][18].…”

Twenty years after “The Impact of Chaos”: what have been achieved and what should be answered

“…In addition, we need to ensure theoretically, by proofs, that methods for nonlinear dynamics work appropriately. For this sake, ordinal patterns [69], symbolic dynamics [67] and recurrence plots [53,[96][97][98] seem to have some advantages judging from the recent literature. For example, to identify directional couplings and slow driving forces from point processes, we need to combine two extensions of Takens' theorems: one by Stark [36] and the other by Huke and Broomhead [39].…”