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diff --git a/finale/mid_report.tex b/finale/mid_report.tex index d22a557..75f5f4a 100644 --- a/finale/mid_report.tex +++ b/finale/mid_report.tex @@ -155,29 +155,43 @@ authors' knowledge, novel in this context. \section{Model} -Weighted directed graph $G = (V, \Theta)$. $k=|V|$ is the number of nodes. -$\Theta\in\R_{+}^{k\times k}$. $\Theta$ implicitly defines the edge set $E$ of -the graph. +The GLC model is described over a directed graph $G = (V, \Theta)$. Denoting by +$k=|V|$ the number of nodes in the graph, $\Theta\in\R_{+}^{k\times k}$ is the +matrix of edge weights. Note that $\Theta$ implicitly defines the edge set $E$ of +the graph through the following equivalence: +\begin{displaymath} + (u,v)\in E\Leftrightarrow \Theta_{u,v} > 0,\quad + (u,v)\in V^2 +\end{displaymath} -Discrete time model, $t\in\N$. Nodes can be in one of two states: -\emph{susceptible}, \emph{infected} or \emph{immune}. $S_t:$ nodes susceptible -at the beginning of time step $t\in\N$. $I_t$ nodes infected at time step $t$. +The time is discretized and indexed by a variable $t\in\N$. The nodes can be in +one of three states: \emph{susceptible}, \emph{infected} or \emph{immune}. +Let us denote by $S_t$ the set of nodes susceptible at the beginning of time +step $t\in\N$ and by $I_t$ the set of nodes who become infected at this time +step. The following dynamics: \begin{displaymath} S_0 = V,\quad S_{t+1} = S_t \setminus I_t \end{displaymath} -Nodes which are no longer susceptible are immune. +expresses that the nodes infected at a time step are no longer susceptible +starting from the next time step (they become part of the immune nodes). -The dynamics of $I_t$ is described by a random Markovian process. Let us denote -by $x_t$ the indicator vector of $I_t$ at time step $t\in\N$. $x_0$ is drawn -from the \emph{source distribution} $p_s:\{0,1\}^n\to[0,1]$. For $t\geq 1$, -Markovian process: +The dynamics of $I_t$ are described by a random Markovian process. Let us +denote by $x_t$ the indicator vector of $I_t$ at time step $t\in\N$. $x_0$ is +drawn from a \emph{source distribution} $p_s:\{0,1\}^n\to[0,1]$. For $t\geq 1$, +we have: \begin{equation} \label{eq:markov} \forall i\in S_t,\quad \P\big(x_{i}^{t} = 1\,|\, x^{t-1}\big) = f(\bt_i\cdot x^{t-1}) \end{equation} -$\bt_i$ is the $i$th column of $\Theta$, $f:\R\to[0,1]$, plus independence for -$i$. A cascade continues until no more infected nodes. +where $\bt_i$ is the $i$th column of $\Theta$. The function $f:\R\to[0,1]$ can +be interpreted as the inverse link function of the model. Finally, the +transitions in \cref{eq:markov} occur independently for each $i$. A cascade +continues until no infected nodes remains. + +We refer the reader to \cite{pouget} for a more complete description of the +model and examples of common contagion models which can be interpreted as +specific instances of the GLC model. \section{Identifiability} |