diff options
Diffstat (limited to 'finale/sections/model.tex')
| -rw-r--r-- | finale/sections/model.tex | 17 |
1 files changed, 10 insertions, 7 deletions
diff --git a/finale/sections/model.tex b/finale/sections/model.tex index 3fe316a..4917bfb 100644 --- a/finale/sections/model.tex +++ b/finale/sections/model.tex @@ -1,7 +1,7 @@ -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: +The GLC model is described over a directed weighted 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 @@ -30,9 +30,12 @@ we have: 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. As noted in \cite{pouget} many -commonly studied contagion models can be cast as specific instances of the GLC -model. +continues until no infected nodes remains. + +As noted in \cite{pouget} many commonly studied contagion models can be cast as +specific instances of the GLC model. In particular the Independent Cascade +model of \cite{Kempe:03} which we will use as a running example corresponds to +the specific case where $f(z) = 1 - e^{-z}$. \Cref{eq:markov} and a source distribution $p_s$ together completely specify the probability distribution of a cascade $\mathbf{x} = (x_t)_{t\geq 0}$ given |