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| -rw-r--r-- | paper/sections/model.tex | 22 |
1 files changed, 11 insertions, 11 deletions
diff --git a/paper/sections/model.tex b/paper/sections/model.tex index 5e55c7a..cd7485d 100644 --- a/paper/sections/model.tex +++ b/paper/sections/model.tex @@ -51,7 +51,7 @@ In other words, each generalized linear cascade provides a series for each node % $f$ can be interpreted as the inverse link function of our generalized linear % cascade model. -\subsection{examples} +\subsection{Examples} \subsubsection{Independent Cascade Model} @@ -78,11 +78,11 @@ Defining $\Theta_{i,j} \defeq \log(1-p_{i,j})$, this can be rewritten as: = 1 - e^{\inprod{\theta_j}{X^t}} \end{equation} -Therefore, the independent cascade model is a Generalized Linear Cascade model with link function $f : z \mapsto 1 - e^z$ +Therefore, the independent cascade model is a Generalized Linear Cascade model with canonical link function $f : z \mapsto 1 - e^z$ \subsubsection{The Linear Voter Model} -In the Linear Voter Model, nodes can be either \emph{red} or \emph{blue}. Both states verify the properties of a contagious state, but for ease of presentation, we will suppose that the contagious nodes are the \emph{blue}. The parameters of the graph are normalized such that $\forall i, +In the Linear Voter Model, nodes can be either \emph{red} or \emph{blue}. Both states are symetric, but without loss of generality, we can suppose that the \emph{blue} nodes are contagious and the \emph{red} nodes are susceptible. The parameters of the graph are normalized such that $\forall i, \ \sum_j \Theta_{i,j} = 1$. Each round, every node $j$ independently chooses one of its neighbors with probability $\Theta_{ij}$ and adopts their color. The cascades stops at a fixed horizon time T or if all nodes are of the same color. @@ -93,7 +93,7 @@ step $t$, then we have: \tag{V} \end{equation} -Therefore, the independent cascade model is a Generalized Linear Cascade model with link function $f: z \mapsto z$ +Therefore, the independent cascade model is a Generalized Linear Cascade model with canonical link function $f: z \mapsto z$ % \subsection{The Linear Threshold Model} @@ -137,7 +137,7 @@ $\Theta$ via Maximum Likelihood Estimation (MLE). Denoting by $\mathcal{L}$ the log-likelihood function, we consider the following $\ell_1$-regularized MLE problem: \begin{displaymath} - \hat{\Theta} \in \argmax_{\Theta} \mathcal{L}(\Theta\,|\,x^1,\ldots,x^n) + \hat{\Theta} \in \argmax_{\Theta} \frac{1}{n} \mathcal{L}(\Theta\,|\,x^1,\ldots,x^n) + \lambda\|\Theta\|_1 \end{displaymath} where $\lambda$ is the regularization factor which helps at preventing @@ -149,16 +149,16 @@ of $m$ terms, each term $i\in\{1,\ldots,m\}$ only depending on $\theta_i$. Since this is equally true for $\|\Theta\|_1$, each column $\theta_i$ of $\Theta$ can be estimated by a separate optimization program: \begin{equation}\label{eq:pre-mle} - \hat{\theta}_i \in \argmax_{\theta}\frac{1}{n_i}\mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n) + \hat{\theta}_i \in \argmax_{\theta} \frac{1}{n}\mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n) + \lambda\|\theta_i\|_1 \end{equation} -where we denote by $n_i$ the first step at which node $i$ becomes contagious and -where: +where we denote by ${\cal T}_i$ the time steps at which node $i$ is susceptible and: \begin{multline} - \mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n) = \frac{1}{n_i} - \sum_{t=1}^{n_i-1}\log\big(1-f(\inprod{\theta_i}{x^t})\big)\\ -+\log f(\inprod{\theta_i}{x^{n_i}})+ \lambda\|\theta_i\|_1 + \mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n) = \frac{1}{|{\cal T}_i|} \sum_{t\in {\cal T}_i } x_i^{t+1}\log f(\inprod{\theta_i}{x^{t}}) + \dots \nonumber \\ + (1 - x_i^{t+1})\log\big(1-f(\inprod{\theta_i}{x^t})\big) + \lambda\|\theta_i\|_1 \nonumber \end{multline} +Note that in the case of the voter model, TODO horizon time. In the case of the independent cascade model... + TODO: discuss conditions on $f$ under which this program is convex. For LC and VM it is convex. |