voter model

2 files changed, 5 insertions, 1 deletions

diff --git a/notes/formalisation.pdf b/notes/formalisation.pdf
index 9383d54..ae0c87f 100644
--- a/notes/formalisation.pdf
+++ b/notes/formalisation.pdf
diff --git a/notes/formalisation.tex b/notes/formalisation.tex
index acc58d2..66d34d6 100644
--- a/notes/formalisation.tex
+++ b/notes/formalisation.tex
@@ -134,9 +134,13 @@ $$\forall t< T-1, k, \ M^t_k \cdot \theta_i = b^{t+1}_k$$
 
 We can concatenate each equality in matrix form $M$, such that the rows of $M$ are the observed $blue$ nodes $M^t_k$ and the entries of $\vec b$ are the corresponding probabilities:
 
-$$M \cdot \theta = \vec b$$
+$$M \cdot \theta_i = \vec b$$
 
+Note that if $M$ had full column rank, then we could recover $\theta_i$ from $\vec b$. This is however an unreasonable assumption, even after having observed many cascades. We must explore which assumptions on $M$ allow us to recover $\theta_i$ from a small number of cascades. Further note that we do not immediately observe $\vec b$. Instead, we observe a bernoulli vector, whose j$^{th}$ entry is equal to $1$ with probability $b_j$ and $0$ otherwise. We will denote this vector $\vec w$, and denote by $\vec \epsilon$ the quantum noise vector such that $\vec w = \vec b + \vec \epsilon$. We can therefore reformulate the problem with our observables:
 
+$$M \cdot \theta_i + \vec \epsilon = \vec w$$
+
+We hope to show that we can exploit the sparsity of vector $\theta_i$ to apply common sparse recovery techniques. 
 
 \section{Independent Cascade Model}