diff --git a/dissertation/dissertation.tex b/dissertation/dissertation.tex index abdafac..b38f5c8 100644 --- a/dissertation/dissertation.tex +++ b/dissertation/dissertation.tex @@ -416,9 +416,52 @@ convention a zero weight corresponds to an absent edge. 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}% \label{sec:pers-homol-netw}