small changes

1 files changed, 2 insertions, 3 deletions

diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index 6fba6b2..cd0a5f8 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -20,8 +20,7 @@ a set $\mathcal{T}$ of observations. We will write $n\defeq |\mathcal{T}|$.
 \label{sec:main_theorem}
 
 In this section, we analyze the case where $\theta^*$ is exactly sparse. We
-write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. Our main theorem will rely
-on the now standard \emph{restricted eigenvalue condition}.
+write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. In our context, $S$ is the set of all nodes susceptible to influence our node $i$. In other words, if $\theta^*$ is exactly $s$-sparse, then node $i$ has at most $s$ parents. Our main theorem will rely on the now standard \emph{restricted eigenvalue condition}.
 
 \begin{definition}
     Let $\Sigma\in\mathcal{S}_m(\reals)$ be a real symmetric matrix and $S$ be
@@ -55,7 +54,7 @@ In the voter model, $\frac{f'}{f}(z) = \frac{1}{z}$ and $\frac{f'}{f}(1-z)
 1-\alpha$ for all $(i,j)\in E$. Similarly, in the Independent Cascade Model,
 $\frac{f'}{f}(z) = \frac{1}{1-e^z}$ and $\frac{f'}{1-f}(z) = -1$ and (LF) holds
 if $p_{i, j}\leq 1-\alpha$ for all $(i, j)\in E$. Remember that in this case we
-have $\Theta_{i,j} = \log(1-p_{i,j})$.
+have $\Theta_{i,j} = \log(1-p_{i,j})$. These assumptions are reasonable: if an edge has a weight very close to 0, then the ``infection'' will never happen along that edge for our set of observations and we can never hope to recover it.
 
 \begin{theorem}
 \label{thm:main}