Computational Topology Techniques for Characterizing Time-Series Data

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and holes - could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems - e.g., the same note played on different musical instruments.Comment: 12 pages, 6 Figures, 1 Table, The Sixteenth International Symposium on Intelligent Data Analysis (IDA 2017

Rhythmic dynamics and synchronization via dimensionality reduction : application to human gait

Reliable characterization of locomotor dynamics of human walking is vital to understanding the neuromuscular control of human locomotion and disease diagnosis. However, the inherent oscillation and ubiquity of noise in such non-strictly periodic signals pose great challenges to current methodologies. To this end, we exploit the state-of-the-art technology in pattern recognition and, specifically, dimensionality reduction techniques, and propose to reconstruct and characterize the dynamics accurately on the cycle scale of the signal. This is achieved by deriving a low-dimensional representation of the cycles through global optimization, which effectively preserves the topology of the cycles that are embedded in a high-dimensional Euclidian space. Our approach demonstrates a clear advantage in capturing the intrinsic dynamics and probing the subtle synchronization patterns from uni/bivariate oscillatory signals over traditional methods. Application to human gait data for healthy subjects and diabetics reveals a significant difference in the dynamics of ankle movements and ankle-knee coordination, but not in knee movements. These results indicate that the impaired sensory feedback from the feet due to diabetes does not influence the knee movement in general, and that normal human walking is not critically dependent on the feedback from the peripheral nervous system

From brain to earth and climate systems: Small-world interaction networks or not?

We consider recent reports on small-world topologies of interaction networks derived from the dynamics of spatially extended systems that are investigated in diverse scientific fields such as neurosciences, geophysics, or meteorology. With numerical simulations that mimic typical experimental situations we have identified an important constraint when characterizing such networks: indications of a small-world topology can be expected solely due to the spatial sampling of the system along with commonly used time series analysis based approaches to network characterization

Community Structure Characterization

This entry discusses the problem of describing some communities identified in a complex network of interest, in a way allowing to interpret them. We suppose the community structure has already been detected through one of the many methods proposed in the literature. The question is then to know how to extract valuable information from this first result, in order to allow human interpretation. This requires subsequent processing, which we describe in the rest of this entry

Probing the topological properties of complex networks modeling short written texts

In recent years, graph theory has been widely employed to probe several language properties. More specifically, the so-called word adjacency model has been proven useful for tackling several practical problems, especially those relying on textual stylistic analysis. The most common approach to treat texts as networks has simply considered either large pieces of texts or entire books. This approach has certainly worked well -- many informative discoveries have been made this way -- but it raises an uncomfortable question: could there be important topological patterns in small pieces of texts? To address this problem, the topological properties of subtexts sampled from entire books was probed. Statistical analyzes performed on a dataset comprising 50 novels revealed that most of the traditional topological measurements are stable for short subtexts. When the performance of the authorship recognition task was analyzed, it was found that a proper sampling yields a discriminability similar to the one found with full texts. Surprisingly, the support vector machine classification based on the characterization of short texts outperformed the one performed with entire books. These findings suggest that a local topological analysis of large documents might improve its global characterization. Most importantly, it was verified, as a proof of principle, that short texts can be analyzed with the methods and concepts of complex networks. As a consequence, the techniques described here can be extended in a straightforward fashion to analyze texts as time-varying complex networks

Graph analysis of functional brain networks: practical issues in translational neuroscience

The brain can be regarded as a network: a connected system where nodes, or units, represent different specialized regions and links, or connections, represent communication pathways. From a functional perspective communication is coded by temporal dependence between the activities of different brain areas. In the last decade, the abstract representation of the brain as a graph has allowed to visualize functional brain networks and describe their non-trivial topological properties in a compact and objective way. Nowadays, the use of graph analysis in translational neuroscience has become essential to quantify brain dysfunctions in terms of aberrant reconfiguration of functional brain networks. Despite its evident impact, graph analysis of functional brain networks is not a simple toolbox that can be blindly applied to brain signals. On the one hand, it requires a know-how of all the methodological steps of the processing pipeline that manipulates the input brain signals and extract the functional network properties. On the other hand, a knowledge of the neural phenomenon under study is required to perform physiological-relevant analysis. The aim of this review is to provide practical indications to make sense of brain network analysis and contrast counterproductive attitudes