remodeling results section

1 files changed, 12 insertions, 4 deletions

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index ae49830..6acfade 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -13,7 +13,7 @@ We note that recovering the edges in $E$ from observed influence cascades is
 a well-identified problem known as the \emph{Graph Inference} problem. However,
 recovering the influence parameters is no less important and has been seemingly
 overlooked so far. The machinery developped in this work addresses both
-problems, in the context of generalized linear cascade models, defined below. We show that this definition encompasses a large class of well-known cascade models.
+problems for a large class of well-known cascade models.
 
 \subsection{Generalized Linear Cascade Models}
 
@@ -28,11 +28,11 @@ $$\mathbb{P}(X^t = 1|{\cal F}_{< t}) = \exp(blabla$$
 \subsection{Examples}
 \label{subsec:examples}
 
-In this section, we show that both the Independent Cascade Model and the Voter model are Generalized Linear Cascades. Discussion about the Linear Threshold model has been deferred to Section~\ref{sec:linear_threshold}
+In this section, we show that both the Independent Cascade Model and the Voter model are Generalized Linear Cascades. We discuss the Linear Threshold model in Section~\ref{sec:linear_threshold}
 
 \subsubsection{Independent Cascade Model}
 
-In the independent cascade model, nodes can be either susceptible, active/infected or inactive. At $t=0$, all source nodes are ``active'' and all remaining nodes are ``susceptible''. At each time step $t$, for each edge $(i,j)$ where $j$ is susceptible and $i$ is active, $i$ attempts to infect $j$ with probability $p_{i,j}\in[0,1]$. If $i$ succeeds, $j$ will become active at time step $t+1$. Regardless of $i$'s success, node $i$ will be inactive at time $t+1$. In other words, nodes stay active for only one time step. The cascade process terminates When no active nodes remain.
+In the independent cascade model, nodes can be either susceptible, active or inactive. At $t=0$, all source nodes are ``active'' and all remaining nodes are ``susceptible''. At each time step $t$, for each edge $(i,j)$ where $j$ is susceptible and $i$ is active, $i$ attempts to infect $j$ with probability $p_{i,j}\in[0,1]$. If $i$ succeeds, $j$ will become active at time step $t+1$. Regardless of $i$'s success, node $i$ will be inactive at time $t+1$. In other words, nodes stay active for only one time step. The cascade process terminates when no active nodes remain.
 
 If we denote by $X^t$ the indicator variable of the set of active nodes at time step $t-1$, then if $j$ is susceptible at time step $t-1$, we have:
 \begin{displaymath}
@@ -49,7 +49,11 @@ Defining $\Theta_{i,j} \defeq \log(1-p_{i,j})$, this can be rewritten as:
 
 \subsubsection{The Voter Model}
 
-In the Voter Model, nodes have two states
+In the Voter Model, nodes can be either red or blue, where ``blue'' is the source state. The parameters of the graph are normalized such that $\forall i, \ \sum_j \Theta_{i,j} = 1$. Each round, every node $j$ 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. If we denote by $X^t$ the indicator variable of the set of blue nodes at time step $t-1$, then we have:
+\begin{equation}
+\mathbb{P}\left[X^{t+1}_j = 1 | X^t \right] = \sum_{i=1}^m \Theta_{i,j} X_i^t = \inprod{\theta_j}{X^t}
+\tag{V}
+\end{equation}
 
 \subsection{Maximum Likelihood Estimation}
 
@@ -95,3 +99,7 @@ the first step at which $j$ becomes active, we can rewrite the  MLE program
     \sum_{t=1}^{n_j-1}\log\big(1-f(\inprod{\theta_j}{x^t})\big)\\
 \log f(\inprod{\theta_j}{x^{n_j}})+ \lambda\|\theta_j\|
 \end{multline}
+
+
+
+