Under contraction conditions, an ESN trained on one-dimensional observations of an invertible dynamical system induces a C1 embedding with positive probability, and a linear readout makes the autonomous ESN topologically conjugate to the original system.
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TikZ-network is an open source software project for visualizing graphs and networks in LaTeX. It aims to provide a simple and easy tool to create, visualize and modify complex networks. The packaged is based on the PGF/TikZ languages for producing vector graphics from a geometric/algebraic description. Particular focus is made on the software usability and interoperability with other tools. Simple networks can be directly created within LaTeX, while more complex networks can be imported from external sources (e.g. igraph, networkx, QGIS, ...). Additionally, tikz-network supports visualization of multilayer networks in two and three dimensions. The software is available at: https://github.com/hackl/tikz-network.
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Embedding and Approximation Theorems for Echo State Networks
Under contraction conditions, an ESN trained on one-dimensional observations of an invertible dynamical system induces a C1 embedding with positive probability, and a linear readout makes the autonomous ESN topologically conjugate to the original system.