Intro

3 files changed, 20 insertions, 6 deletions

diff --git a/paper/paper.tex b/paper/paper.tex
index 9eb0f06..6e94164 100644
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -100,6 +100,7 @@
 \input{sections/intro.tex}
 
 \section{Model}
+\label{sec:model}
 \input{sections/model.tex}
 
 \section{Results}
@@ -119,6 +120,7 @@
 % \input{sections/linear_threshold.tex}
 
 \section{Experiments}
+\label{sec:experiments}
 \input{sections/experiments.tex}
 
 \section{Discussion}
diff --git a/paper/sections/abstract.tex b/paper/sections/abstract.tex
index ad5b893..a07228f 100644
--- a/paper/sections/abstract.tex
+++ b/paper/sections/abstract.tex
@@ -1,10 +1,11 @@
 In the Graph Inference problem, one seeks to recover the edges of an unknown
 graph from the observations of influence cascades propagating over this graph.
 In this paper, we approach this problem from the sparse recovery perspective
-and provide an algorithm which recovers the graph edges with high probability
-provided that the number of measurements is $\Omega(s\log m)$ where $s$ is the
-maximum degree of the graph and $m$ is the number of nodes.
+and provide the first algorithm which recovers the graph's edges with high
+probability provided that the number of measurements is $\Omega(s\log m)$ where
+$s$ is the maximum degree of the graph and $m$ is the number of nodes.
 Furthermore, we show that our algorithm also recovers the edge weights (the
 parameters of the diffusion process) and is robust in the context of
 approximate sparsity. Finally we prove an almost matching lower bound of
-$\Omega(s\log\frac{m}{s})$.
+$\Omega(s\log\frac{m}{s})$ and validate our approach empirically on synthetic
+graphs.
diff --git a/paper/sections/intro.tex b/paper/sections/intro.tex
index 20b40f3..f4f2bd5 100644
--- a/paper/sections/intro.tex
+++ b/paper/sections/intro.tex
@@ -8,9 +8,14 @@ Tackling the graph inference problem means constructing a polynomial-time algori
  Throughout this paper, we focus on the three main discrete-time diffusion processes: the independent cascade model, the voter model, and the linear threshold model...
  \end{comment}
 
+Graphs have been extensively studied in their ability to propagate 
+
+Networks and social networks in particular have become an increasingly common
+subject of study in the recent years. 
+
 A diffusion process taking place over a graph provides valuable information
 about the presence and weights of its edges. \emph{Influence cascades} are a
-specific type of diffusions processes in which a particular infection behavior
+specific type of diffusion processes in which a particular infection behavior
 spreads over the nodes of the graph.  By only observing the ``infection times''
 of the nodes in the graph, one might hope to recover the underlying graph and
 the parameters of the cascade model. This problem is known in the literature as
@@ -53,7 +58,13 @@ The contributions of this paper are the following:
         observations.
 \end{itemize}
 
-The organization of the paper is as follows: ...
+The organization of the paper is as follows: we conclude the introduction by
+a survey of the related work. In Section~\ref{sec:model} we present our model
+of Generalized Linear Cascades and the associated sparse recovery formulation.
+Its theoretical guarantees are presented for various recovery settings in
+Section~\ref{sec:results}. The lower bound is presented in
+Section~\ref{sec:lowerbound}. Finally, we conclude with experiments in
+Section~\ref{sec:experiments}.
 
 \paragraph{Related Work}