additional changes to model section

1 files changed, 11 insertions, 8 deletions

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index a645fcf..ae49830 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -1,6 +1,6 @@
 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.
 
-In practice, we restrict ourselves to discrete-time homogeneous cascade models. $\Theta$ usually represents the edge weights of ${\cal G}$. Finally, the state of each node can often be reduced to two classes: able to infect neighboring nodes at the next round, or unable to infect its neighbors.
+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}.
 
 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}
@@ -13,23 +13,26 @@ 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.
+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.
 
-\subsection{Generalized Linear 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 $j$ can infect other nodes at time step $t+1$''. By the Markov property of cascades, $$\mathbb{P}(X^t | X^0, X^1, \dots, X^{t-1}) = \mathbb{P}(X^t | X^{t-1})$$
-We consider the following \emph{generalized linear model} (GLM) framework for cascades.
+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 $t+1$''. By the Markov property of cascades, $$\mathbb{P}(X^t | X^0, X^1, \dots, X^{t-1}) = \mathbb{P}(X^t | X^{t-1})$$
+We draw inspiration from \emph{generalized linear models} (GLM) to define the following:
 
 \begin{definition}
-Let .... A \emph{generalized linear cascade} is:
+Let $f: \mathbb{R} \leftarrow \mathbb{R}$ be an inverse link function. Let ${\cal F}_{< t}$ be the filtration defined by $\{ X^0, X^1, \dots, X^{t-1} \}$. A \emph{generalized linear cascade} is defined as:
+$$\mathbb{P}(X^t = 1|{\cal F}_{< t}) = \exp(blabla$$
 \end{definition}
 
 \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}
+
 \subsubsection{Independent Cascade Model}
 
-In the independent cascade model, nodes can be either susceptible, active/infected or inactive. All nodes start either as susceptible or active. 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 $t+1$: nodes stay active for only one time step. The cascade continues until no active nodes remain.
+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.
 
 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}
@@ -46,7 +49,7 @@ 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
 
 \subsection{Maximum Likelihood Estimation}