Persistent homology of complex networks for dynamic state detection

Abstract

In this paper we develop a novel Topological Data Analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram-a two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.

Topological Analysis of Differential Effects of Ketamine and Propofol Anesthesia on Brain Dynamics

“…The exact significance of this is unclear, however it suggests that the TDA of OPNs captures dynamical relationships between the different states that TDA of the EPC, or the standard OPN analyses do not. The work by Myers et al [41] showed that TDA of OPNs can retrieve significant information about the system dynamics such as Lyapunov exponent even in the presence of noise and so future research aimed at understanding the applicability of these analyses to neural data look to be promising. If the similarities between the dynamics produced by ketamine and normal awareness are indicative of phenomenological consciousness, the similarities between normal awareness and propofol may be indicative of ketamine's specific effect on NMDA receptors that are not shared by propofol [66].…”

“…representations of the data [57,38], following the procedure discussed in [41]. Due to the difficulties associated with multi-dimensional OPNs [67], we applied this method to a uni-variate time series, constructed from the first temporal principle component of the data.…”

“…There is no agreed-upon method for choosing the optimal d and τ . Different researchers have suggested different criteria, including values that maximize the variance of the degree distribution [38], false-nearest-neighbours criteria [28], or the first local maxima in the permutation entropy [41]. One significant consideration is that the embedding dimension must be large enough that the state-space is sufficiently large to fully capture the range of patterns present in the data (otherwise the resulting OPN fails to capture all the dynamics), but not so large that every moment is unique (in which case, we reconstruct the time series as a path graph).…”

“…Previous work by Myers et al, [41] showed that persistence homology of OPNs can be informative about the dynamics driving a complex system. As with the EPC, the presence of cycles and higherorder structures in the discrete state-transition dynamics can shed light on emergent properties specific to the conditions.…”

“…We also used two slightly more complex TDA measures, the normalized persistence entropy [41] and the periodicity [47]. Analysis of variance found a significant difference between conditions (H=12.4, p=0.002), and post-hoc analysis found significant differences between the Propofol condition and the Ketamine condition (U = 63, p = 0.013) as well as between the Awake and Ketamine conditions (U = 65, p = 0.015).…”