90 lines
3 KiB
TeX
90 lines
3 KiB
TeX
\documentclass[article,a4paper,11pt,openany,extrafontsizes]{memoir}
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\input{preamble}
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\usepackage[backend=biber,style=ieee,url=false,arxiv=abs]{biblatex}
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\addbibresource{proposal.bib}
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\tightlists%
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\begin{document}
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\maketitle
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% \subsection*{Title}
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% Topological Data Analysis of Time-dependent Networks
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\subsection*{Supervisors}
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Dr Heather Harrington (Mathematical Institute) and Dr Gesine Reinert
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(Department of Statistics)
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\subsection*{Description}
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Topological Data Analysis (TDA)~\cite{chazal_introduction_2017,
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oudot_persistence_2015, carlsson_topology_2009,
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edelsbrunner_computational_2010} is a family of techniques gaining
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an increasing importance in the analysis and visualization of
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high-dimensional data in machine learning applications.
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In this project, we will apply TDA techniques and persistent homology
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to time-dependent networks, in order to understand how the topological
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structure evolves over time in complex multilayer
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networks~\cite{kivela_multilayer_2014, porter_dynamical_2014}.
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There are two ways of obtaining time-dependent networks. Network data
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is available easily in many contexts: social networks and biological
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processes are two examples of systems evolving over time and that can
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be modelled as a graph. For instance, in social networks, links in ego
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networks have already been studied in the context of
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time-dependency~\cite{tabourier_predicting_2016}.
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The other large category is time series. It is possible to use a
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similarity measure to build a network from a set of time series taken
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from the same physical process. Although it could be applied to any
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set of time series, this has already been studied in the case of
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coupled oscillators (such as Kuramoto
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oscillators)~\cite{stolz_persistent_2017, schaub_graph_2016}. It is
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thus easy to find relevant datasets or to generate interesting data
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from physical simulations.
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It is then possible to apply existing TDA and persistent homology
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techniques to the networks, taking into account the temporal
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dimension. Certain methods have already been implemented in
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topological data analysis libraries~\cite{tierny_topology_2017,
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maria_gudhi_2014}, although they would have to be adapted to network
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data, and applied repeatedly to each time step. There is also a wide
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range of methods to explore, from the choice of the similarity
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measure, to the choice of filtration (in order to build a simplicial
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complex on the network), to the representation of topological
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structure. Each of these choices has a great influence on the final
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interpretation of the data, and may need to be adapted to each system.
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\subsection*{Prerequisite courses/knowledge}
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\begin{itemize}
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\item SM7 Probability and Statistics for Network Analysis
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\item Topological Data Analysis and Persistent
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Homology\footnote{\url{http://www.enseignement.polytechnique.fr/informatique/INF556/}}
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\end{itemize}
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\subsection*{Computing required?}
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Yes
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\subsection*{Data available?}
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Yes
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%%\nocite{*}
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% \bibliographystyle{ieeetr}
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% \bibliography{proposal}
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\printbibliography%
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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