\subsection{The Restricted Eigenvalue Condition}

Proving the restricted eigenvalue assumption for correlated measurements is non-trivial. Under reasonable assumptions on the graph parameters, we can show a very crude ${\cal O}(N)$-lower bound for $\gamma_n$ by exploiting only the first set of measurements, where only the source nodes are active. Note that even though we waste a lot of information, we obtain similar asymptotic behavior than previous work.

\subsection{The Irrepresentability Condition}

If our objective is to recover only the support of the graph and to do it exactly , then the irrepresentability condition\footnote{{\it incoherence} in the text} used in \cite{Daneshmand:2014} is believed to be necessary. 

However, assuming we only wish to recover all edges above a certain threshold, bounding the $\ell2$-error allows us to recover the full graph above a certain minimum threshold. Furthermore, it is intuitive that the irrepresentability condition is stronger than our suggested {\bf(RE)} assumption. In fact, by adapting slightly the following result from \cite{vandegeer:2009}, we can show that a `strong' irrepresentability condition directly {\it implies} the {\bf(RE)} condition for $\ell2$-recovery

\begin{proposition}
If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then the restricted eigenvalue condition holds with constant: $(1 - 3\epsilon)^2 \lambda_{\min}n/s$
\end{proposition}