810 lines
29 KiB
TeX
810 lines
29 KiB
TeX
\documentclass[a4paper,11pt,openany,extrafontsizes]{memoir}
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\input{preamble}
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\usepackage[firstpage]{draftwatermark}
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\begin{document}
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\pagestyle{plain}
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\tightlists%
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\begin{titlingpage}
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\begin{center}
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\vspace{1cm}
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\textsf{\Huge{University of Oxford}}\\
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\vspace{1cm}
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\includegraphics{branding/beltcrest.png}\\
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\vspace{2cm}
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\Huge{\thetitle}\\
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\vspace{2cm}
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\large{by\\[14pt]\theauthor\\[8pt]St Catherine's College}
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\vfill
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%% Inkscape L-system
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%% [C]++[C]++[C]++[C]++[C]
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%% B=DA++EA----CA[-DA----BA]++;C=+DA--EA[---BA--CA]+;D=-BA++CA[+++DA++EA]-;E=--DA++++BA[+EA++++CA]--CA;A=
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\begin{tikzpicture}
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\pgfdeclarelindenmayersystem{Penrose}{
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\symbol{M}{\pgflsystemdrawforward}
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\rule{M->OA++PA----NA[-OA----MA]++}
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}
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\draw[lindenmayer system={Penrose, axiom=[N]++[N]++[N]++[N]++[N],
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order=2, angle=36, step=4pt}]
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lindenmayer system;
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\end{tikzpicture}
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% \vspace{2.2cm}
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\vfill
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\large{A dissertation submitted in partial fulfilment of the degree of\\
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Master of Science in Applied Statistics}\\
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\vspace{.5cm}
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\large{\emph{Department of Statistics,\\ 24--29 St Giles, Oxford, OX1 3LB}}\\
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\vspace{1cm}
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\large{\thedate}
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\end{center}
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\end{titlingpage}
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%\chapterstyle{hangnum}
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%\chapterstyle{ell}
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%\chapterstyle{southall}
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\chapterstyle{wilsondob}
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\frontmatter
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\cleardoublepage%
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\chapter*{Declaration of authorship}
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\emph{This my own work (except where otherwise indicated).}\\[2cm]
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\begin{center}
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Date \hspace{.5\linewidth} Signature
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\end{center}
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\cleardoublepage%
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\begin{abstract}
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Abstract here
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\end{abstract}
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\cleardoublepage%
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\chapter*{Acknowledgements}%
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\label{cha:acknowledgements}
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Thank you!
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\cleardoublepage%
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\tableofcontents
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\listoffigures
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% \listoftables
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\clearpage
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\mainmatter%
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\chapter{Introduction}%
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\label{cha:introduction}
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\chapter{Graphs and Temporal Networks}%
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\label{cha:temporal-networks}
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\section{Definition and basic properties}%
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\label{sec:defin-basic-prop}
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In this section, we introduce the notion of temporal networks (or
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temporal graphs). This is a complex notion, with many concurrent
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definitions and interpretations.
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After clarifying the notations, we restate the standard definition of
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a non-temporal graph.
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\begin{notation}
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\begin{itemize}
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\item $\mathbb{N}$ is the set of non-negative natural numbers
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$0,1,2,\ldots$
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\item $\mathbb{N}^*$ is the set of positive integers $1,2,\ldots$
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\item $\mathbb{R}$ is the set of real numbers.
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$\mathbb{R}_+ = \{x\in\mathbb{R} \;|\; x\geq 0\}$, and
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$\mathbb{R}_+^* = \{x\in\mathbb{R} \;|\; x>0\}$.
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\end{itemize}
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\end{notation}
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\begin{defn}[Graph]
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A \emph{graph} is a couple $G = (V, E)$, where $V$ is a set of
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\emph{nodes} (or \emph{vertices}), and $E \subseteq V\times V$ is a
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set of \emph{edges}. A \emph{weighted graph} is defined by
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$G = (V, E, w)$, where $w : E\mapsto \mathbb{R}_+^*$ is called the
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\emph{weight function}.
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\end{defn}
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We also define some basic concepts that we will need later to build
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simplicial complexes on graphs.
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\begin{defn}[Clique]
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A \emph{clique} is a set of nodes where each pair is adjacent. That
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is, a clique $C$ of a graph $G = (V,E)$ is a subset of $V$ such that
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for all $i,j\in C, i \neq j \implies (i,j)\in E$. A clique is said
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to be \emph{maximal} if it cannot be augmented by any node, such
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that the resulting set of nodes is itself a clique.
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\end{defn}
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Temporal networks can be defined in the more general framework of
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\emph{multilayer networks}~\cite{kivela_multilayer_2014}. However,
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this definition is much too general for our simple applications, and
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we restrict ourselves to edge-centric time-varying
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graphs~\cite{casteigts_time-varying_2012}. In this model, the set of
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nodes is fixed, but edges can appear or disappear at different times.
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In this study, we restrict ourselves to discrete time stamps. Each
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interaction is taken to be instantaneous.
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%% TODO note about data collection, oversampling,
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%% duration of interactions
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\begin{defn}[Temporal network]
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A \emph{temporal network} is a tuple
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$G = (V, E, \mathcal{T}, \rho)$, where:
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\begin{itemize}
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\item $V$ is a set of nodes,
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\item $E\subseteq V\times V$ is a set of edges,
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\item $\mathbb{T}$ is the \emph{temporal domain} (often taken as
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$\mathbb{N}$ or any other countable set), and
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$\mathcal{T}\subseteq\mathbb{T}$ is the \emph{lifetime} of the
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network,
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\item $\rho: E\times\mathcal{T}\mapsto\{0,1\}$ is the \emph{presence
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function}, which determines whether an edge is present in the
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network at each time stamp.
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\end{itemize}
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The \emph{available times} of an edge are the set
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$\mathcal{I}(e) = \{t\in\mathcal{T}: \rho(e,t)=1\}$.
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\end{defn}
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Temporal networks can also have weighted edges. In this case, it is
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possible to have constant weights (edges can only appear or disappear
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over time, and always have the same weight), or time-varying
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weights. In the latter case, we can set the domain of the presence
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function to be $\mathbb{R}_+$ instead of $\{0,1\}$, where by
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convention a 0 weight corresponds to an absent edge.
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\begin{defn}[Additive and dismantling temporal
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networks]\label{defn:additive}
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A temporal network is said to be \emph{additive} if for all $e\in E$
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and $t\in\mathcal{T}$, if $\rho(e,t)=1$, then for all
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$t'>t, \rho(e, t') = 1$. An additive network can only gain edges
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over time.
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A temporal network is said to be \emph{dismantling} if for all
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$e\in E$ and $t\in\mathcal{T}$, if $\rho(e,t)=0$, then for all
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$t'>t, \rho(e, t') = 0$. An dismantling network can only lose edges
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over time.
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\end{defn}
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\section{Examples of applications}%
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\label{sec:exampl-appl}
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%% TODO
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\section{Network partitioning}%
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\label{sec:network-partitioning}
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%% TODO clarify, organise, references
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Temporal networks are a very active research subject, leading to
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multiple interesting problems. The additional time dimension adds a
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significant layer of complexity that cannot be adequately treated by
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the common methods on static graphs.
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Moreover, data collection can lead to large amount of noise in
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datasets. Combined with large dataset sized due to the huge number of
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data points for each node in the network, temporal graphs cannot be
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studied effectively in their raw form. Recent advances have been made
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to fit network models to rich but noisy
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data~\cite{newman_network_2018}, generally using some variation on the
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expectation-maximization (EM) algorithm.
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One solution that has been proposed to study such temporal data has
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been to \emph{partition} the time scale of the network into a sequence
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of smaller, static graphs, representing all the interactions during a
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short interval of time. The approach consists in subdividing the
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lifetime of the network in \emph{sliding windows} of a given length.
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We can then ``flatten'' the temporal network on each time interval,
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keeping all the edges that appear at least once (or adding their
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weights in the case of weighted networks).
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This partitioning is sensitive to two parameters: the length of each
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time interval, and their overlap. Of those, the former is the most
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important: it will define the \emph{resolution} of the study. If it is
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too small, too much noise will be taken into account; if it is too
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large, we will lose important information. There is a need to find a
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compromise, which will depend on the application and on the task
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performed on the network. In the case of a classification task to
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determine periodicity, it will be useful to adapt the resolution to
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the expected period: if we expect week-long periodicity, a resolution
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of one day seems reasonable.
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Once the network is partitioned, we can apply any statistical learning
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task on the sequence of static graphs. In this study, we will focus on
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classification of time steps. This can be used to detect periodicity,
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outliers, or even maximise temporal communities.
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%% TODO Talk about partitioning methods?
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\chapter{Topological Data Analysis and Persistent Homology}%
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\label{cha:tda-ph}
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%% TODO references
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\section{Basic constructions}%
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\label{sec:basic-constructions}
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\subsection{Homology}%
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\label{sec:homology}
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Our goal is to understand the topological structure of a metric
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space. For this, we can use \emph{homology}, which consists of
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associating a vector space $H_i(X)$ to a metric space $X$ and a
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dimension $i$. The dimension of $H_i(X)$ gives us the number of
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$i$-dimensional components in $X$: the dimension of $H_0(X)$ is the
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number of path-connected components in $X$, the dimension of $H_1(X)$
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is the number of holes in $X$, and the dimension of $H_2(X)$ is the
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number of voids.
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Crucially, these vector spaces are robust to continuous deformation of
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the underlying metric space (they are \emph{homotopy
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invariant}). However, computing the homology of an arbitrary metric
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space can be extremely difficult. It is necessary to approximate it in
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a structure that would be both combinatorial and topological in
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nature.
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\subsection{Simplicial complexes}%
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\label{sec:simplicial-complexes}
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To understand the topological structure of a metric space, we need a
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way to decompose it in smaller pieces that, when assembled, conserve
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the overall organisation of the space. For this, we use a structure
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called a \emph{simplicial complex}, which is a kind of
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higher-dimensional generalization of a graph.
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The building blocks of this representation is the \emph{simplex},
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which is the convex hull of an arbitrary set of points. Examples of
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simplices include single points, segments, triangles, and tetrahedrons
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(in dimensions 0, 1,, 2, and 3 respectively).
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\begin{defn}[Simplex]
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A \emph{$k$-dimensional simplex} $\sigma = [x_0,\ldots,x_k]$ is the
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convex hull of the set $\{x_0,\ldots,x_k\} \in \mathbb{R}^d$, where
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$x_0,\ldots,x_k$ are affinely independent. $x_0,\ldots,x_k$ are
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called the \emph{vertices} of $\sigma$, and the simplices defined by
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the subsets of $\{x_0,\ldots,x_k\}$ are called the \emph{faces} of
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$\sigma$.
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\end{defn}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[b]{.3\linewidth}
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\centering
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\begin{tikzpicture}
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\tikzstyle{point}=[circle,thick,draw=black,fill=blue!30,%
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inner sep=0pt,minimum size=15pt]
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\node (a)[point] at (0,0) {a};
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\end{tikzpicture}
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\caption{Single vertex}
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\end{subfigure}%
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%
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\begin{subfigure}[b]{.3\linewidth}
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\centering
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\begin{tikzpicture}
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\tikzstyle{point}=[circle,thick,draw=black,fill=blue!30,%
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inner sep=0pt,minimum size=15pt]
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\node (a)[point] at (0,0) {a};
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\node (b)[point] at (1.4,2) {b};
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\begin{scope}[on background layer]
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\draw[fill=blue!15] (a.center) -- (b.center) -- cycle;
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\end{scope}
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\end{tikzpicture}
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\caption{Segment}
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\end{subfigure}%
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%
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\begin{subfigure}[b]{.3\linewidth}
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\centering
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\begin{tikzpicture}
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\tikzstyle{point}=[circle,thick,draw=black,fill=blue!30,%
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inner sep=0pt,minimum size=15pt]
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\node (a)[point] at (0,0) {a};
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\node (b)[point] at (1.4,2) {b};
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\node (c)[point] at (2.8,0) {c};
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\begin{scope}[on background layer]
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\draw[fill=blue!15] (a.center) -- (b.center) -- (c.center) -- cycle;
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\end{scope}
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\end{tikzpicture}
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\caption{Triangle}
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\end{subfigure}%
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%
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\caption{Examples of simplices}%
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\label{fig:simplex}
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\end{figure}
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We then need a way to meaningfully combine these basic building blocks
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so that the resulting object can adequately reflect the topological
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structure of the metric space.
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\begin{defn}[Simplicial complex]
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A \emph{simplicial complex} is a collection $K$ of simplices such
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that:
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\begin{itemize}
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\item any face of a simplex of $K$ is a simplex of $K$
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\item the intersection of two simplices of $K$ is either the empty
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set, or a common face, or both.
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\end{itemize}
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\end{defn}
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\begin{figure}[ht]
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\centering
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\begin{tikzpicture}
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\tikzstyle{point}=[circle,thick,draw=black,fill=blue!30,%
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inner sep=0pt,minimum size=10pt]
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\node (a)[point] {};
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\node (b)[point,above right=1.4cm and 1cm of a] {};
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\node (c)[point,right=2cm of a] {};
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\node (d)[point,above right=.4cm and 2cm of b] {};
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\node (e)[point,above right=.4cm and 2cm of c] {};
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\node (f)[point,below right=.7cm and 1.3cm of c] {};
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\node (g)[point,right=2cm of d] {};
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\node (h)[point,below right=.4cm and 1.5cm of e] {};
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\begin{scope}[on background layer]
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\draw[fill=blue!15] (a.center) -- (b.center) -- (c.center) -- cycle;
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\draw (b) -- (d) -- (g);
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\draw (c.center) -- (e.center) -- (f.center) -- cycle;
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\draw (d) -- (e) -- (h);
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\end{scope}
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\node (1)[point,right=2cm of g] {};
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\node (2)[point,above right=.5cm and 1cm of 1] {};
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\node (3)[point,below right=.5cm and 1cm of 2] {};
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\node (4)[point,below left=1cm and .3cm of 3] {};
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\node (5)[point,below right=1cm and .3cm of 1] {};
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\node (6)[point,below left=1cm and .1cm of 5] {};
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\node (7)[point,below right=1cm and .1cm of 4] {};
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\node (8)[point,below right=.7cm and .7cm of 6] {};
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\begin{scope}[on background layer]
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\draw[fill=green!15] (1.center) -- (2.center) -- (3.center) -- (4.center) -- (5.center) -- cycle;
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\draw (1) -- (4) -- (2) -- (5) -- (3) -- (1);
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\draw[fill=blue!15] (6.center) -- (7.center) -- (8.center) -- cycle;
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\draw (5) -- (6) -- (4) -- (7);
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\end{scope}
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\end{tikzpicture}
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\caption{Example of a simplicial complex that has two connected
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components, two 3-simplices, and one 5-simplex.}%
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\label{fig:simplical-complex}
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\end{figure}
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The notion of simplicial complex is closely related to that of a
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hypergraph. One important distinction lies in the fact that a subset
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of a hyperedge is not necessarily a hyperedge itself.
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Using these definitions, we can define homology on simplicial
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complexes. %% TODO add reference for more details/do it myself?
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\subsection{Filtrations}%
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\label{sec:filtrations}
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%% TODO rewrite it using the Cech complex as an introductory example,
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%% to understand the problem with scale
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If we consider that a simplicial complex is a kind of
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``discretization'' of a subset of a metric space, we realise that
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there must be an issue of \emph{scale}. For our analysis to be
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invariant under small perturbations in the data, we need a way to find
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the optimal scale parameter to capture the adequate topological
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structure, without taking into account some small perturbations, nor
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ignoring some important smaller features.
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One possible solution to these problems is to consider all scales at
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once. This is the objective of \emph{filtered simplicial complexes}.
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\begin{defn}[Filtration]\label{defn:filt}
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A \emph{filtered simplicial complex}, or simply a \emph{filtration},
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$K$ is a sequence ${(K_i)}_{i\in I}$ of simplicial complexes such
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that:
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\begin{itemize}
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\item for any $i, j \in I$, if $i < j$ then $K_i \subseteq K_j$,
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\item $\bigcup_{i\in I} K_i = K$.
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\end{itemize}
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\end{defn}
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\section{Persistent Homology}%
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\label{sec:persistent-homology}
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We can now compute the homology for each step in a filtration. This
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leads to the notion of \emph{persistent
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homology}~\cite{carlsson_topology_2009,zomorodian_computing_2005},
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which gives all the information necessary to establish the topological
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structure of a metric space at multiple scales.
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\begin{defn}[Persistent homology]
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The \emph{$p$-th persistent homology} of a simplicial complex
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$K = {(K_i)}_{i\in I}$ is the pair
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$(\{H_p(K_i)\}_{i\in I}, \{f_{i,j}\}_{i,j\in I, i\leq j})$, where
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for all $i\leq j$, $f_{i,j} : H_p(K_i) \mapsto H_p(K_j)$ is induced
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by the inclusion map $K_i \mapsto K_j$.
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\end{defn}
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The functions $f_{i,j}$ allow one to link generators in each
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successive homology space in a filtration. Because each generator
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corresponds to a topological feature (connected component, hole, void,
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and so on, depending on the dimension $p$), we can determine whether
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it survives in the next step of the filtration. We can also determine
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when each feature is born and when it dies (if it dies at all). The
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couples of intervals (birth time, death time) depends on the choice of
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basis for each homology space $H_p(K_i)$. However, by the Fundamental
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Theorem of Persistent Homology~\cite{zomorodian_computing_2005}, we
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can choose basis vectors in each homology space such that the
|
|
collection of half-open intervals is well-defined and unique. This
|
|
construction is called a \emph{barcode}~\cite{carlsson_topology_2009}.
|
|
|
|
\section{Topological summaries: barcodes and persistence diagrams}%
|
|
\label{sec:topol-summ}
|
|
|
|
%% TODO need more context
|
|
|
|
To interpret the results of the persistent-homology computation, we
|
|
need to compare the output for a particular data set to a suitable
|
|
null model. For this, we need some kind of similarity measure between
|
|
barcodes and a way to evaluate the statistical significance of the
|
|
results.
|
|
|
|
One possible approach is to define a space in which we can project
|
|
barcodes and study their geometric properties. One such space is the
|
|
space of \emph{persistence
|
|
diagrams}~\cite{edelsbrunner_computational_2010}.
|
|
|
|
\begin{defn}[Multiset]
|
|
A \emph{multiset} $M$ is the couple $(A, m)$, where $A$ is the
|
|
\emph{underlying set} of $M$, formed by its distinct elements, and
|
|
$m : A\mapsto\mathbb{N}^*$ is the \emph{multiplicity function}
|
|
giving the number of occurrences of each element of $A$ in $M$.
|
|
\end{defn}
|
|
|
|
\begin{defn}[Persistence diagrams]
|
|
A \emph{persistence diagram} is the union of a finite multiset of
|
|
points in $\overline{\mathbb{R}}^2$ with the diagonal
|
|
$\Delta = \{(x,x) \;|\; x\in\mathbb{R}^2\}$, where every point of
|
|
$\Delta$ has infinite multiplicity.
|
|
\end{defn}
|
|
|
|
One adds the diagonal $\Delta$ for technical reasons. It is convenient
|
|
to compare persistence diagrams by using bijections between them, so
|
|
persistence diagrams must have the same cardinality.
|
|
|
|
In some cases, the diagonal in the persistence diagrams can also
|
|
facilitate comparisons between diagrams, as points near the diagonal
|
|
correspond to short-lived topological features, so they are likely to
|
|
be caused by small perturbations in the data.
|
|
|
|
One can build a persistence diagram from a barcode by taking the union
|
|
of the multiset of (birth, death) couples with the diagonal
|
|
$\Delta$. \autoref{fig:pipeline} summarises the entire pipeline.
|
|
|
|
\begin{figure}[ht]
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\tikzstyle{pipelinestep}=[rectangle,thick,draw=black,inner sep=5pt,minimum size=15pt]
|
|
\node (data)[pipelinestep] {Data};
|
|
\node (filt)[pipelinestep,right=1cm of data] {Filtered complex};
|
|
%% \node (barcode)[pipelinestep,right=1cm of filt] {Barcodes};
|
|
\node (dgm)[pipelinestep,right=1cm of filt] {Persistence diagram};
|
|
\node (interp)[pipelinestep,right=1cm of dgm] {Interpretation};
|
|
|
|
\draw[->] (data.east) -- (filt.west);
|
|
%% \draw[->] (filt.east) -- (barcode.west);
|
|
\draw[->] (filt.east) -- (dgm.west);
|
|
\draw[->] (dgm.east) -- (interp.west);
|
|
\end{tikzpicture}
|
|
|
|
\caption{Persistent homology pipeline}%
|
|
\label{fig:pipeline}
|
|
\end{figure}
|
|
|
|
One can define an operator $\dgm$ as the first two steps in the
|
|
pipeline. It constructs a persistence diagram from a subset of a
|
|
metric space, via persistent homology on a filtered complex.
|
|
|
|
We can now define several distances on the space of persistence
|
|
diagrams.
|
|
|
|
\begin{defn}[Wasserstein distance]
|
|
The \emph{$p$-th Wasserstein distance} between two diagrams $X$ and
|
|
$Y$ is
|
|
\[ W_p[d](X, Y) = \inf_{\phi:X\mapsto Y} \left[\sum_{x\in X} {d\left(x, \phi(x)\right)}^p\right] \]
|
|
for $p\in [1,\infty)$, and:
|
|
\[ W_\infty[d](X, Y) = \inf_{\phi:X\mapsto Y} \sup_{x\in X} d\left(x,
|
|
\phi(x)\right) \] for $p = \infty$, where $d$ is a distance on
|
|
$\mathbb{R}^2$ and $\phi$ ranges over all bijections from $X$ to
|
|
$Y$.
|
|
\end{defn}
|
|
|
|
\begin{defn}[Bottleneck distance]
|
|
The \emph{bottleneck distance} is defined as the infinite
|
|
Wasserstein distance where $d$ is the uniform norm:
|
|
$d_B = W_\infty[L_\infty]$.
|
|
\end{defn}
|
|
|
|
The bottleneck distance is symmetric, non-negative, and satisfies the
|
|
triangle inequality. However, it is not a true distance, as one can
|
|
come up with two distinct diagrams with bottleneck distance 0, even
|
|
on multisets that do not touch the diagonal $\Delta$.
|
|
|
|
\section{Stability}%
|
|
\label{sec:stability}
|
|
|
|
One of the most important aspects of topological data analysis is that
|
|
it is \emph{stable} with respect to small perturbations in the
|
|
data. More precisely, the second step of the pipeline
|
|
in~\autoref{fig:pipeline} is Lipschitz with respect to a suitable
|
|
metric on filtered complexes and the bottleneck distance on
|
|
persistence
|
|
diagrams~\cite{cohen-steiner_stability_2007,chazal_persistence_2014}. First,
|
|
we define a distance between subsets of a metric
|
|
space~\cite{oudot_persistence_2015}.
|
|
|
|
\begin{defn}[Hausdorff distance]
|
|
Let $X$ and $Y$ be subsets of a metric space $(E, d)$. The
|
|
\emph{Hausdorff distance} is defined by
|
|
\[ d_H(X,Y) = \max \left[ \sup_{x\in X} \inf_{y\in Y} d(x,y),
|
|
\sup_{y\in Y} \inf_{x\in X} d(x,y) \right]. \]
|
|
\end{defn}
|
|
|
|
We can now give an appropriate stability
|
|
property~\cite{cohen-steiner_stability_2007,chazal_persistence_2014}.
|
|
|
|
\begin{prop}
|
|
Let $X$ and $Y$ be subsets in a metric space. We have
|
|
\[ d_B(\dgm(X),\dgm(Y)) \leq d_H(X,Y). \]
|
|
\end{prop}
|
|
|
|
\section{Algorithms and implementations}%
|
|
\label{sec:algor-impl}
|
|
|
|
%% TODO
|
|
\cite{morozov_dionysus:_2018,bauer_ripser:_2018,reininghaus_dipha_2018,maria_gudhi_2014}
|
|
|
|
\section{Discussion}%
|
|
\label{sec:discussion}
|
|
|
|
%% TODO
|
|
|
|
%% information thrown away in filtrations and in PH
|
|
|
|
|
|
\chapter{Topological Data Analysis on Networks}%
|
|
\label{cha:topol-data-analys}
|
|
|
|
\section{Persistent homology for networks}%
|
|
\label{sec:pers-homol-netw}
|
|
|
|
We now consider the problem of applying persistent homology to network
|
|
data. An undirected network is already a simplicial complex of
|
|
dimension 1. However, this is not sufficient to capture enough
|
|
topological information; we need to introduce higher-dimensional
|
|
simplices. One method is to project the nodes of a network onto a
|
|
metric space~\cite{otter_roadmap_2017}, thereby transforming the
|
|
network data into a point-cloud data. For this, we need to compute the
|
|
distance between each pair of nodes in the network (e.g.\ with the
|
|
shortest-path distance). This also requires the network to be
|
|
connected. %% TODO defn of connected?
|
|
|
|
Another common method, for weighted networks, is called the
|
|
\emph{weight rank-clique filtration}
|
|
(WRCF)~\cite{petri_topological_2013}, which filters a network based
|
|
on weights. The procedure works as follows:
|
|
\begin{enumerate}
|
|
\item Consider the set of all nodes, without any edge, to be
|
|
filtration step~0.
|
|
\item Rank all edge weights in decreasing order $\{w_1,\ldots,w_n\}$.
|
|
\item At filtration step $t$, keep only the edges whose weights are
|
|
larger than or equal to $w_t$, thereby creating an unweighted graph.
|
|
\item Define the maximal cliques of the resulting graph to be
|
|
simplices.
|
|
\end{enumerate}
|
|
|
|
At each step of the filtration, we construct a simplicial complex
|
|
based on cliques; this is called a \emph{clique
|
|
complex}~\cite{zomorodian_tidy_2010}. The result of the algorithm is
|
|
itself a filtered simplicial complex (\autoref{defn:filt}), because a
|
|
subset of a clique is necessarily a clique itself, and the same is
|
|
true for the intersection of two cliques.
|
|
|
|
This leads to a first possibility for applying persistent homology to
|
|
temporal networks. It is possible to segment the lifetime of a network
|
|
into sliding windows, creating a time-independent graph on each window
|
|
by retaining only the edges available during the time interval. We can
|
|
then apply WRCF on each graph in the sequence, obtaining a filtered
|
|
complex for each window, to which we can then apply persistent
|
|
homology.
|
|
|
|
This method can quickly become very computationally expensive, as
|
|
finding all maximal cliques (e.g.\ using the Bron--Kerbosch algorithm)
|
|
is a complicated problem, with an optimal computational complexity of
|
|
$\mathcal{O}\big(3^{n/3}\big)$~\cite{tomita_worst-case_2006}. In
|
|
practice, one often restrict the search to cliques of dimension less
|
|
than or equl to a certain bound $d_M$. With this restriction, the new
|
|
simplicial complex is homologically equivalent to the original: they
|
|
have the same homology groups up to $H_{d_M-1}$.
|
|
|
|
%% TODO rewrite this paragraph
|
|
This method is sensitive to the choice of sliding windows on the time
|
|
scale. The width and the overlap of the windows can completely change
|
|
the networks created and their topological features. Too small a
|
|
window, and the network becomes too small to have any significant
|
|
topological properties, too large, and we lose important information
|
|
in the evolution of the network over time.
|
|
|
|
\section{Zigzag persistence}%
|
|
\label{sec:zigzag-persistence}
|
|
|
|
The standard algorithm to compute persistent homology
|
|
(see~\autoref{sec:persistent-homology}) relies on the fact that
|
|
filtrations (see~\autoref{defn:filt}) are nested sequences of
|
|
simplicial complexes:
|
|
\[ \cdots \subseteq K_{i-1} \subseteq K_i \subseteq K_{i+1} \subseteq
|
|
\cdots \]
|
|
|
|
One can now create an independent filtration (e.g.\ with WRCF) for
|
|
each time step. The issue is that the topological features will be
|
|
orthogonal to the time dimension.
|
|
|
|
Another possibility is to create a filtration along the time
|
|
dimension. The issue in this case is that the sequence is no longer
|
|
nested (except for additive or dismantling temporal networks,
|
|
see~\autoref{defn:additive}).
|
|
|
|
The solution to consider the time dimension is provided by
|
|
\emph{zigzag persistence}~\cite{carlsson_zigzag_2009}, which allows
|
|
one to compute persistence on alternating nested sequences:
|
|
\[ \cdots \supseteq K_{i-1} \subseteq K_i \supseteq K_{i+1} \subseteq
|
|
\cdots \]
|
|
|
|
This sequence can in turn be computed from a temporal network by
|
|
computing the union of each pair of consecutive time steps,
|
|
constructing an alternating sequence.
|
|
|
|
Zigzag persistence is a special case of the more general concept of
|
|
\emph{multi-parameter
|
|
persistence}~\cite{carlsson_theory_2009,dey_computing_2014}, where
|
|
filtrations can encompass multiple parameters.
|
|
|
|
%% Note about libraries implementing zigzag persistence: Dionysus
|
|
|
|
\chapter{Persistent Homology for Machine-Learning Applications}%
|
|
\label{cha:pers-homol-mach}
|
|
|
|
The output of persistent homology is not directly usable by most
|
|
statistical methods. For example, barcodes and persistence diagrams,
|
|
which are multisets of points in $\overline{\mathbb{R}}^2$, are not
|
|
elements of a metric space in which one can perform statistical
|
|
computations.
|
|
|
|
The distances between persistence diagrams defined
|
|
in~\autoref{sec:topol-summ} allow one to compare different
|
|
outputs. From a statistical perspective, it is possible to use a
|
|
generative model of simplicial complexes and to use a distance between
|
|
persistence diagrams to measure the similarity of our observations
|
|
with this null model~\cite{adler_persistent_2010}. This would
|
|
effectively define a metric space of persistence diagrams. It is even
|
|
possible to define some statistical summaries (means, medians,
|
|
confidence intervals) on these
|
|
spaces~\cite{turner_frechet_2014,munch_probabilistic_2015}.
|
|
|
|
%% TODO REFERENCES
|
|
|
|
The issue with this approach is that metric spaces do not offer enough
|
|
algebraic structure to be amenable to most machine-learning
|
|
techniques. One of the most recent development in the study of
|
|
topological summaries has been to find mappings between the space of
|
|
persistence diagrams and Banach spaces.
|
|
|
|
\section{Vectorization methods}%
|
|
\label{sec:vect-meth}
|
|
|
|
%% TODO
|
|
|
|
\subsection{Persistence landscapes}
|
|
|
|
Persistence landscapes~\cite{bubenik_statistical_2015} give a way to
|
|
project barcodes to a space where it is possible to add them
|
|
meaningfully. It is then possible to define means of persistence
|
|
diagrams, as well as other summary statistics.
|
|
|
|
The function mapping a persistence diagram to a persistence landscape
|
|
is \emph{injective}, but no explicit inverse exists to go back from a
|
|
persistence landscape to the corresponding persistence
|
|
diagram. Moreover, a mean of persistence landscapes does not
|
|
necessarily have a corresponding persistence diagram.
|
|
|
|
\begin{defn}[Persistence landscape]
|
|
The persistence landscape of a diagram $D = \{(b_i,d_i)\}_{i=1}^n$
|
|
is the set of functions $\lambda_k: \mathbb{R} \mapsto \mathbb{R}$,
|
|
for $k\in\mathbb{N}$, such that
|
|
\[ \lambda_k(x) = k\text{-th largest value of } \{f_{(b_i,
|
|
d_i)}(x)\}_{i=1}^n, \] (and $\lambda_k(x) = 0$ if the $k$-th
|
|
largest value does not exist), where $f_{(b,d)}$ is a
|
|
piecewise-linear function defined by:
|
|
\[ f_{(b,d)} =
|
|
\begin{cases}
|
|
0,& \text{if }x \notin (b,d),\\
|
|
x-b,& \text{if }x\in (b,\frac{b+d}{2}),\\
|
|
-x+d,& \text{if }x\in (\frac{b+d}{2},d)\,.
|
|
\end{cases}
|
|
\]
|
|
\end{defn}
|
|
|
|
Moreover, one can show that persistence landscapes are stable with
|
|
respect to the $L^p$ distance, and that the Wasserstein and bottleneck
|
|
distances are bounded by the $L^p$
|
|
distance~\cite{bubenik_statistical_2015}. We can thus view the
|
|
landscapes as elements of a Banach space in which we can perform the
|
|
statistical computations.
|
|
|
|
\subsection{Persistence images}
|
|
|
|
\cite{adams_persistence_2017}
|
|
|
|
\subsection{Tropical and arctic semirings}
|
|
|
|
\cite{kalisnik_tropical_2018}
|
|
|
|
\section{Kernel-based methods}%
|
|
\label{sec:kernel-based-methods}
|
|
|
|
\subsection{Persistent scale-space kernel}
|
|
|
|
\cite{reininghaus_stable_2015,kwitt_statistical_2015}
|
|
|
|
\subsection{Persistence weighted-Gaussian kernel}
|
|
|
|
\cite{kusano_kernel_2017}
|
|
|
|
\subsection{Sliced Wasserstein kernel}
|
|
|
|
\cite{carriere_sliced_2017}
|
|
|
|
\section{Comparison}%
|
|
\label{sec:comparison}
|
|
|
|
\chapter{Conclusions}%
|
|
\label{cha:conclusions}
|
|
|
|
|
|
|
|
|
|
\backmatter%
|
|
|
|
% \nocite{*}
|
|
\printbibliography%
|
|
|
|
\end{document}
|
|
|
|
|
|
|
|
%%% Local Variables:
|
|
%%% mode: latex
|
|
%%% TeX-master: t
|
|
%%% End:
|