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Dissertation: bibliography

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Dimitri Lozeve 2018-07-30 12:04:17 +01:00
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@ -265,9 +265,10 @@ once: this is the objective of \emph{filtered simplical complexes}.
\label{sec:persistent-homology}
We can now compute the homology for each step in a filtration. This
leads to the notion of \emph{persistent homology}, which gives us all
the information necessary to establish the topological structure of
the metric space at multiple scales.
leads to the notion of \emph{persistent
homology}~\cite{carlsson_topology_2009,zomorodian_computing_2005},
which gives us all the information necessary to establish the
topological structure of the metric space at multiple scales.
\begin{defn}[Persistent homology]
The \emph{$p$-th persistent homology} of a simplicial complex
@ -378,11 +379,12 @@ build simplicial complexes on graphs.
\end{defn}
Temporal networks are defined in the more general framework of
\emph{multilayer networks}. However, this definition is much too
general for our simple applications, and we restrict ourselves to
edge-centric time-varying graphs. In this model, the set of nodes is
fixed and doesn't change over time, whereas edges can appear or
disappear at different timestamps.
\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
@ -470,14 +472,15 @@ data. An undirected network is already a simplicial complex of
dimension 1. However, this will not be sufficient to capture enough
topological information: we need to introduce higher-dimensional
simplices. The first possible method is to project the network on a
metric space, thus 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 (via shortest path distance for instance). This
also requires the network to be connected.
metric space~\cite{otter_roadmap_2017}, thus 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 (via shortest path
distance for instance). This also requires the network to be
connected.
Another usual method for weighted networks is called the \emph{weight
rank clique filtration} (WRCF), which filters the network based on
weights. The procedure works as follows:
rank clique filtration} (WRCF)~\cite{petri_topological_2013}, which
filters the network based on weights. The procedure works as follows:
\begin{enumerate}
\item Set the set of all nodes, without any edge, as filtration
step~0.