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2019-05
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Evans_Honors.pdf
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We study the asymptotic behavior of networks with discrete quadratic dynamics. While
single-map complex quadratic iterations have been studied over the past century, considering
ensembles of such functions, organized as coupled nodes in an oriented network, generates new,
interesting questions and applications to the life sciences. We extend results from single-node
dynamics to the more general case of networks, and present novel, network-speci c results.
We then consider two existing models from the dynamic networks literature: threshold-linear
networks and a reduced model of inhibitory neural clusters. We search for graph features which
lead to robust dynamics under minor perturbations within our model, as well as between the
three di erent models; in other words, we search for possible features of universality and the
conditions under which they hold. We create a classi cation system of large-scale networks. This
classi cation system is based on network dimensionality reduction (i.e. treating a large group
of nodes as a single node). Additionally, we present conditions under which reducing network
dimensionality is permittable. This has important implications for applications to the study
of natural networks (such as biological systems), which are often extremely large (composed of
many coupled nodes). Finally, we explore possible applications of the techniques used in these
three network models (complex quadratic networks, threshold-linear networks, and inhibitory
clustering neural networks) to other problems in the natural sciences: a chemical oscillator
model and a neural clustering model.
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