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| -rw-r--r-- | paper/sections/model.tex | 9 |
1 files changed, 4 insertions, 5 deletions
diff --git a/paper/sections/model.tex b/paper/sections/model.tex index 44236da..5934590 100644 --- a/paper/sections/model.tex +++ b/paper/sections/model.tex @@ -1,7 +1,7 @@ We consider a graph ${\cal G}= (V, E, \Theta)$, where $\Theta$ is a $|V|\times |V|$ matrix of parameters describing the edge weights of $\mathcal{G}$. Let $m\defeq |V|$. For each node $j$, let $\Theta_{j}$ be the $j^{th}$ column vector of $\Theta$. Intuitively, $\Theta_{i,j}$ captures the ``influence'' of node $i$ on node $j$. A \emph{Cascade model} is a Markov process over a finite state space $\{0, 1, \dots, K-1\}^V$ with the following properties: \begin{enumerate} \item Conditioned on the previous time step, the transition probabilities for each node $i \in V$ are mutually independent. -\item Of the $K$ possible states, there exists a \emph{contagious state} such that all transition probabilities of the Markov process can be expressed as a function of the graph parameters $\Theta$ and the set of ``contagious nodes'' at the previous time step. +\item Of the $K$ possible states, there exists a \emph{contagious state} such that all transition probabilities of the Markov process are either constant or can be expressed as a function of the graph parameters $\Theta$ and the set of ``contagious nodes'' at the previous time step. \item The initial probability over $\{0, 1, \dots, K-1\}^V$ of the Cascade Model is such that all nodes can eventually reach a \emph{contagious state} with non-zero probability. The ``contagious'' nodes at $t=0$ are called \emph{source nodes}. \end{enumerate} @@ -105,12 +105,11 @@ Despite their obvious differences, both models share a common similarity: condit \begin{definition} \label{def:glcm} -Let us denote by $\{\mathcal{F}_t, t\in\ints\}$ the natural filtration induced by the state of ${\cal G}$ up to time step t. A \emph{generalized linear cascade} is a cascade model characterized by the following equation: +A \emph{generalized linear cascade} is a cascade model characterized by the following equation: \begin{displaymath} - \P[X^{t+1}=x\,|\, \mathcal{F}_t] = - \prod_{i=1}^m f_{X^t_i,x_i}(\Theta_i \cdot X^t) +\forall j \in V, \; \P[X^{t+1}_j=x\,|\, X^t] = f(\Theta_j \cdot X^t) \end{displaymath} -where $f_{s,p}:\mathbb{R}\to[0,1]$ for each state pair $(s,p) \in \{0, \dots, K-1 \}$ +where $f:\mathbb{R}\to[0,1]$ \end{definition} |