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@ -75,11 +75,128 @@ Thank you!
\label{cha:introduction} \label{cha:introduction}
\chapter{Graphs and Temporal Networks}%
\label{cha:temporal-networks}
\section{Definition and basic properties}%
\label{sec:defin-basic-prop}
In this section, we will introduce the notion of temporal networks or
graphs. This is a complex notion, with many concurrent definitions and
interpretations. First, we restate the standard definition of a
non-temporal, static graph.
\begin{defn}[Graph]
A \emph{graph} is a couple $G = (V, E)$, where $V$ is a finite set
of \emph{nodes} (or \emph{vertices}), and $E \subseteq V\times V$ is
a set of \emph{edges}. A \emph{weighted graph} is defined by
$G = (V, E, w)$, where $w : E\mapsto \mathbb{R}_+$ is fcalled the
\emph{weight function}.
\end{defn}
We also define some basic concepts that will be needed later on to
build simplicial complexes on graphs.
\begin{defn}[Clique]
A \emph{clique} is a set of nodes where each pair is connected. That
is, a clique $C$ of a graph $G = (V,E)$ is a subset of $V$ such that
$\forall i,j\in C, i \neq j \implies (i,j)\in E$. A clique is said
to be \emph{maximal} if it cannot be augmented by any node.
\end{defn}
Temporal networks are defined in the more general framework of
\emph{multilayer networks}~\cite{kivela_multilayer_2014}. However,
this definition is much too general for our simple applications, and
we restrict ourselves to edge-centric time-varying
graphs~\cite{casteigts_time-varying_2012}. In this model, the set of
nodes is fixed and doesn't change over time, whereas edges can appear
or disappear at different timestamps.
\begin{defn}[Temporal network]
A \emph{temporal network} (or graph) is a tuple
$G = (V, E, \mathcal{T}, \rho)$, where:
\begin{itemize}
\item $V$ is a finite set of nodes,
\item $E\subseteq V\times V$ is a set of edges,
\item $\mathbb{T}$ is the \emph{temporal domain} (often taken as
$\mathbb{N}$ or $\mathbb{R}_+$), and
$\mathcal{T}\subseteq\mathbb{T}$ is the \emph{lifetime} of the
network,
\item $\rho: E\times\mathcal{T}\mapsto\{0,1\}$ is the \emph{presence
function}, which determines whether an edge is present in the
network at each timestamp.
\end{itemize}
The \emph{available dates} of an edge are the set
$\mathcal{I}(e) = \{t\in\mathcal{T}: \rho(e,t)=1\}$.
\end{defn}
Temporal networks can also have weighted edges. In this case, it is
possible to have constant weights (edges can only appear or disappear
over time, and always have the same weight), or time-varying
weights. In the latter case, we can set the domain of the presence
function to be $\mathbb{R}_+$ instead of $\{0,1\}$, where by
convention a zero weight corresponds to an absent edge.
\begin{defn}[Additive temporal network]
A temporal network is said to be \emph{additive} if for all $e\in E$
and $t\in\mathcal{T}$, if $\rho(e,t)=1$, then
$\forall t'>t, \rho(e, t') = 1$. Edges can only be added to the
network, never removed.
\end{defn}
\section{Examples of applications}%
\label{sec:exampl-appl}
\section{Network partitioning}%
\label{sec:network-partitioning}
Temporal networks are a very active research subject, leading to
multiple interesting problems. The additional time dimension adds a
significant layer of complexity that cannot be adequately treated by
the common methods on static graphs.
Moreover, data collection can lead to large amount of noise in
datasets. Combined with large dataset sized due to the huge number of
data points for each node in the network, temporal graphs cannot be
studied effectively in their raw form. Recent advances have been made
to fit network models to rich but noisy
data~\cite{newman_network_2018}, generally using some variation on the
expectation-maximization (EM) algorithm.
One solution that has been proposed to study such temporal data has
been to \emph{partition} the time scale of the network into a sequence
of smaller, static graphs, representing all the interactions during a
short interval of time. The approach consists in subdividing the
lifetime of the network in \emph{sliding windows} of a given length.
We can then ``flatten'' the temporal network on each time interval,
keeping all the edges that appear at least once (or adding their
weights in the case of weighted networks).
This partitioning is sensitive to two parameters: the length of each
time interval, and their overlap. Of those, the former is the most
important: it will define the \emph{resolution} of the study. If it is
too small, too much noise will be taken into account; if it is too
large, we will lose important information. There is a need to find a
compromise, which will depend on the application and on the task
performed on the network. In the case of a classification task to
determine periodicity, it will be useful to adapt the resolution to
the expected period: if we expect week-long periodicity, a resolution
of one day seems reasonable.
Once the network is partitioned, we can apply any statistical learning
task on the sequence of static graphs. In this study, we will focus on
classification of time steps. This can be used to detect periodicity,
outliers, or even maximise temporal communities.
%% TODO Talk about partitioning methods?
\chapter{Topological Data Analysis and Persistent Homology}% \chapter{Topological Data Analysis and Persistent Homology}%
\label{cha:tda-ph} \label{cha:tda-ph}
\section{Homology}% \section{Basic constructions}
\label{sec:basic-constructions}
\subsection{Homology}%
\label{sec:homology} \label{sec:homology}
Our goal is to understand the topological structure of a metric Our goal is to understand the topological structure of a metric
@ -98,7 +215,7 @@ space can be extremely difficult. It is necessary to approximate it in
a structure that would be both combinatorial and topological in a structure that would be both combinatorial and topological in
nature. nature.
\section{Simplicial Complexes}% \subsection{Simplicial Complexes}%
\label{sec:simplicial-complexes} \label{sec:simplicial-complexes}
In order to understand the topological structure of a metric space, we In order to understand the topological structure of a metric space, we
@ -235,7 +352,7 @@ of a hyperedge is not necessarily a hyperedge itself.
Using these definitions, we can define homology on simplicial Using these definitions, we can define homology on simplicial
complexes. %% TODO add reference for more details/do it myself? complexes. %% TODO add reference for more details/do it myself?
\section{Filtrations}% \subsection{Filtrations}%
\label{sec:filtrations} \label{sec:filtrations}
If we consider that a simplicial complex is a kind of If we consider that a simplicial complex is a kind of
@ -307,7 +424,7 @@ space.
\begin{defn}[Persistence diagrams] \begin{defn}[Persistence diagrams]
A \emph{persistence diagram} is the union of a finite multiset of A \emph{persistence diagram} is the union of a finite multiset of
points in $\bar{\mathbb{R}}^2$ zith the diagonal points in $\overline{\mathbb{R}}^2$ zith the diagonal
$\Delta = \{(x,x) \;|\; x\in\mathbb{R}^2\}$, where every point of $\Delta = \{(x,x) \;|\; x\in\mathbb{R}^2\}$, where every point of
$\Delta$ has infinite multiplicity. $\Delta$ has infinite multiplicity.
\end{defn} \end{defn}
@ -347,122 +464,12 @@ diagonal $\Delta$.
\section{Stability}% \section{Stability}%
\label{sec:stability} \label{sec:stability}
\section{Algorithms and implementations}%
\label{sec:algor-impl}
\chapter{Temporal Networks}% \chapter{Topological Data Analysis on Networks}%
\label{cha:temporal-networks} \label{cha:topol-data-analys}
\section{Definition and basic properties}%
\label{sec:defin-basic-prop}
In this section, we will introduce the notion of temporal networks or
graphs. This is a complex notion, with many concurrent definitions and
interpretations. First, we restate the standard definition of a
non-temporal, static graph.
\begin{defn}[Graph]
A \emph{graph} is a couple $G = (V, E)$, where $V$ is a finite set
of \emph{nodes} (or \emph{vertices}), and $E \subseteq V\times V$ is
a set of \emph{edges}. A \emph{weighted graph} is defined by
$G = (V, E, w)$, where $w : E\mapsto \mathbb{R}_+$ is fcalled the
\emph{weight function}.
\end{defn}
We also define some basic concepts that will be needed later on to
build simplicial complexes on graphs.
\begin{defn}[Clique]
A \emph{clique} is a set of nodes where each pair is connected. That
is, a clique $C$ of a graph $G = (V,E)$ is a subset of $V$ such that
$\forall i,j\in C, i \neq j \implies (i,j)\in E$. A clique is said
to be \emph{maximal} if it cannot be augmented by any node.
\end{defn}
Temporal networks are defined in the more general framework of
\emph{multilayer networks}~\cite{kivela_multilayer_2014}. However,
this definition is much too general for our simple applications, and
we restrict ourselves to edge-centric time-varying
graphs~\cite{casteigts_time-varying_2012}. In this model, the set of
nodes is fixed and doesn't change over time, whereas edges can appear
or disappear at different timestamps.
\begin{defn}[Temporal network]
A \emph{temporal network} (or graph) is a tuple
$G = (V, E, \mathcal{T}, \rho)$, where:
\begin{itemize}
\item $V$ is a finite set of nodes,
\item $E\subseteq V\times V$ is a set of edges,
\item $\mathbb{T}$ is the \emph{temporal domain} (often taken as
$\mathbb{N}$ or $\mathbb{R}_+$), and
$\mathcal{T}\subseteq\mathbb{T}$ is the \emph{lifetime} of the
network,
\item $\rho: E\times\mathcal{T}\mapsto\{0,1\}$ is the \emph{presence
function}, which determines whether an edge is present in the
network at each timestamp.
\end{itemize}
The \emph{available dates} of an edge are the set
$\mathcal{I}(e) = \{t\in\mathcal{T}: \rho(e,t)=1\}$.
\end{defn}
Temporal networks can also have weighted edges. In this case, it is
possible to have constant weights (edges can only appear or disappear
over time, and always have the same weight), or time-varying
weights. In the latter case, we can set the domain of the presence
function to be $\mathbb{R}_+$ instead of $\{0,1\}$, where by
convention a zero weight corresponds to an absent edge.
\begin{defn}[Additive temporal network]
A temporal network is said to be \emph{additive} if for all $e\in E$
and $t\in\mathcal{T}$, if $\rho(e,t)=1$, then
$\forall t'>t, \rho(e, t') = 1$. Edges can only be added to the
network, never removed.
\end{defn}
\section{Examples of applications}%
\label{sec:exampl-appl}
\section{Network partitioning}%
\label{sec:network-partitioning}
Temporal networks are a very active research subject, leading to
multiple interesting problems. The additional time dimension adds a
significant layer of complexity that cannot be adequately treated by
the common methods on static graphs.
Moreover, data collection can lead to large amount of noise in
datasets. Combined with large dataset sized due to the huge number of
data points for each node in the network, temporal graphs cannot be
studied effectively in their raw form. Recent advances have been made
to fit network models to rich but noisy
data~\cite{newman_network_2018}, generally using some variation on the
expectation-maximization (EM) algorithm.
One solution that has been proposed to study such temporal data has
been to \emph{partition} the time scale of the network into a sequence
of smaller, static graphs, representing all the interactions during a
short interval of time. The approach consists in subdividing the
lifetime of the network in \emph{sliding windows} of a given length.
We can then ``flatten'' the temporal network on each time interval,
keeping all the edges that appear at least once (or adding their
weights in the case of weighted networks).
This partitioning is sensitive to two parameters: the length of each
time interval, and their overlap. Of those, the former is the most
important: it will define the \emph{resolution} of the study. If it is
too small, too much noise will be taken into account; if it is too
large, we will lose important information. There is a need to find a
compromise, which will depend on the application and on the task
performed on the network. In the case of a classification task to
determine periodicity, it will be useful to adapt the resolution to
the expected period: if we expect week-long periodicity, a resolution
of one day seems reasonable.
Once the network is partitioned, we can apply any statistical learning
task on the sequence of static graphs. In this study, we will focus on
classification of time steps. This can be used to detect periodicity,
outliers, or even maximise temporal communities.
%% TODO Talk about partitioning methods?
\section{Persistent homology for networks}% \section{Persistent homology for networks}%
\label{sec:pers-homol-netw} \label{sec:pers-homol-netw}