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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 + $G = (V, E, w)$, where $w : E\mapsto \mathbb{R}_+$ is called the \emph{weight function}. \end{defn} @@ -147,6 +147,8 @@ convention a zero weight corresponds to an absent edge. \section{Examples of applications}% \label{sec:exampl-appl} +%% TODO + \section{Network partitioning}% \label{sec:network-partitioning} @@ -224,9 +226,9 @@ conserve the overall organisation of the space. For this, we use a structure called a \emph{simplicial complex}, which is a kind of higher-dimensional generalization of graphs. -The building blocks of this representation will be \emph{simplices}, +The building blocks of this representation will be \emph{simplexes}, which are simply the convex hull of an arbitrary set of -points. Examples of simplices include single points, segments, +points. Examples of simplexes include single points, segments, triangles, and tetrahedrons (in dimensions 0, 1,, 2, and 3 respectively). @@ -234,7 +236,7 @@ respectively). The \emph{$k$-dimensional simplex} $\sigma = [x_0,\ldots,x_k]$ is the convex hull of the set $\{x_0,\ldots,x_k\} \in \mathbb{R}^d$, where $x_0,\ldots,x_k$ are affinely independent. $x_0,\ldots,x_k$ - are called the \emph{vertices} of $\sigma$, and the simplices + are called the \emph{vertices} of $\sigma$, and the simplexes defined by the subsets of $\{x_0,\ldots,x_k\}$ are called the \emph{faces} of $\sigma$. \end{defn} @@ -282,7 +284,7 @@ respectively). \caption{Triangle} \end{subfigure}% % - \caption{Examples of simplices}% + \caption{Examples of simplexes}% \label{fig:simplex} \end{figure} @@ -292,11 +294,11 @@ so that the resulting object can adequately reflect the topological structure of the metric space. \begin{defn}[Simplicial complex] - A \emph{simplicial complex} is a collection $K$ of simplices such + A \emph{simplicial complex} is a collection $K$ of simplexes such that: \begin{itemize} \item any face of a simplex of $K$ is a simplex of $K$ - \item the intersection of two simplices of $K$ is either the empty + \item the intersection of two simplexes of $K$ is either the empty set or a common face or both. \end{itemize} \end{defn} @@ -339,7 +341,7 @@ structure of the metric space. \end{scope} \end{tikzpicture} \caption{Example of a simplicial complex, with two connected - components, two 3-simplices, and one 5-simplex.}% + components, two 3-simplexes, and one 5-simplex.}% \label{fig:simplical-complex} \end{figure} @@ -364,7 +366,7 @@ important smaller features. %% TODO rewrite using the Cech filtration as an example? The ideal solution to these problems is to consider all scales at -once: this is the objective of \emph{filtered simplical complexes}. +once: this is the objective of \emph{filtered simplicial complexes}. \begin{defn}[Filtration] A \emph{filtered simplicial complex}, or simply a \emph{filtration}, @@ -411,7 +413,7 @@ construction is called a \emph{barcode}. In order 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 a similarity +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. @@ -498,11 +500,11 @@ 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 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 +simplexes. The first possible method is to project the network on a 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 +distance between each pair of nodes in the network (with the shortest +path distance for instance). This also requires the network to be connected. Another usual method for weighted networks is called the \emph{weight @@ -515,7 +517,7 @@ filters the network based on weights. The procedure works as follows: \item At filtration step $t$, keep only the edges whose weights are less than $w_t$, thus creating an unweighted graph. \item Define the maximal cliques of the resulting graph to be - simplices. + simplexes. \end{enumerate} At each step of the filtration, we construct a simplicial complex @@ -573,7 +575,7 @@ compute persistence on alternating nested sequences: This sequence can in turn be computed from a temporal network by computing the union of each pair of consecutive time steps, -constructing a alternating sequence. +constructing an alternating sequence. Zigzag persistence is a special case of the more general concept of \emph{multi-parameter @@ -665,7 +667,7 @@ statistical computations. \cite{reininghaus_stable_2015,kwitt_statistical_2015} -\subsection{Persistence weighted gaussian kernel} +\subsection{Persistence weighted Gaussian kernel} \cite{kusano_kernel_2017}