Polising intro of model section

1 files changed, 34 insertions, 21 deletions

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index 4cef131..0f7eff7 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -1,31 +1,28 @@
-We consider a graph ${\cal G}= (V, E, \Theta)$, with $\Theta$ a set of parameters to the graph. Let $m\defeq |V|$. A \emph{cascade model} is a finite state space Markov process over $\{0, 1, \dots \}^V$. An \emph{influence cascade} is a realisation of the random process described by a cascade model.
+We consider a graph ${\cal G}= (V, E, \Theta)$, where $\Theta$ is $|V|\times
+|V|$ matrix of paremeters describing the edge weights of $\mathcal{G}$.
+A \emph{cascade model} is a Markov process over the finite state space $\{0, 1,
+\dots, K\}^V$. An \emph{influence cascade} is a realisation of the random
+process described by a cascade model.
 
-In practice, we restrict ourselves to discrete-time homogeneous cascade models. $\Theta$ usually represents the edge weights of ${\cal G}$. Each influence cascade begins at $t=0$ with a designated source `state' (e.g ``active''). All nodes in the source state at $t=0$ are called \emph{source nodes}.  A standard theoretical assumption we can make is that each node at $t=0$ has a constant probability $p_{\text{init}}$ of being in the source state. This is more realistic and less restrictive than ``single source'' assumption made in \cite{Daneshmand:2014} and \cite{Abrahao:13}.
+In practice, we restrict ourselves to discrete-time homogeneous cascade models.
+At $t=0$, each node in the graph has a constant probability $p_{init}$ of being
+in the ``active state''. The active nodes at $t=0$ are called the source nodes.
+Our probabilistic model for the source nodes is more realistic and less
+restrictive than the ``single source'' assumption made in \cite{Daneshmand:2014}
+and \cite{Abrahao:13}.
 
-Recovering the intrinsic graph parameters $\Theta$ from observed influence cascades is the central question of the present work and is important for the following two reasons:
-\begin{enumerate}
-    \item knowing the influence parameters allows for future prediction of
-        influence cascades.
-    \item when the set of edges $E$ is unknown, it can be recovered from the
-        influence parameters.
-\end{enumerate}
-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 for a large class of well-known cascade models.
 
 \subsection{Generalized Linear Cascade Models}
 
 Denoting by $X^t$ the state of the cascade at time step $t-1$, we interpret
-$X^t_j = 1$ for some node $j\in V$ as ``node $j$ exhibits the source nodes'
-behavior at time step $t+1$''.  We draw inspiration from \emph{generalized
-linear models} (GLM) to define a generalized linear cascade.
+$X^t_j = 1$ for some node $j\in V$ as ``node $j$ is active'', \emph{i.e}, ``$j$
+exhibits the source nodes' behavior at time step $t+1$''.  We draw inspiration
+from \emph{generalized linear models} (GLM) to define a generalized linear
+cascade.
 
-\begin{definition}
-    Let us denote by $\{\mathcal{F}_t, t\in\ints\}$ the natural filtration
-    induced by $\{X_t, t\in\ints\}$. A \emph{generalized linear cascade} is
-    characterized by the following equation:
+\begin{definition} Let us denote by $\{\mathcal{F}_t, t\in\ints\}$ the natural
+    filtration induced by $\{X_t, t\in\ints\}$. A \emph{generalized linear
+    cascade} is characterized by the following equation:
     \begin{displaymath}
         \P[X^{t+t}=x\,|\, \mathcal{F}_t] =
         \prod_{i\in V} f(\inprod{\theta_i}{X^{t}})^{x_i}
@@ -39,6 +36,10 @@ satisfies the Markov property:
 \begin{displaymath}
     \P[X^{t+1}=x\,|\mathcal{F}_t] = \P[X^{t+1}=x\,|\, X^t]
 \end{displaymath}
+Note also that $\E[X^{t+1}_i\,|\,X^t] = f(\inprod{\theta_i}{X^t})$. As such,
+$f$ can be interpreted as the inverse link function of our generalized linear
+cascade model.
+
 \subsection{Examples}
 \label{subsec:examples}
 
@@ -73,6 +74,18 @@ In the Voter Model, nodes can be either red or blue, where ``blue'' is the sourc
 
 \subsection{Maximum Likelihood Estimation}
 
+Recovering the intrinsic graph parameters $\Theta$ from observed influence
+cascades is the central question of the present work and is important for the
+following two reasons: \begin{enumerate} \item knowing the influence parameters
+            allows for future prediction of influence cascades.  \item when the
+        set of edges $E$ is unknown, it can be recovered from the influence
+parameters.  \end{enumerate} 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 for a large class of well-known
+cascade models.
+
 In both models, the unknown influence parameters are described by an $m\times
 m$ matrix $\Theta$. Furthermore, the information of the edges of $G$ is fully
 contained in $\Theta$: