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Diffstat (limited to 'paper/sections/model.tex')
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1 files changed, 8 insertions, 3 deletions
diff --git a/paper/sections/model.tex b/paper/sections/model.tex index 19d7506..fbcedf3 100644 --- a/paper/sections/model.tex +++ b/paper/sections/model.tex @@ -25,7 +25,8 @@ contagious nodes ``influence'' other nodes in the graph to become contagious. An the successive states of the nodes in graph ${\cal G}$. Note that both the ``single source'' assumption made in~\cite{Daneshmand:2014} and \cite{Abrahao:13} as well as the ``uniformly chosen source set'' assumption made -in~\cite{Netrapalli:2012} verify condition 3. +in~\cite{Netrapalli:2012} verify condition 3. {\color{red} why is it less +restrictive? explain} In the context of Graph Inference,~\cite{Netrapalli:2012} focus on the well-known discrete-time independent cascade model recalled below, which @@ -103,7 +104,7 @@ In the independent cascade model, nodes can be either susceptible, contagious or immune. At $t=0$, all source nodes are ``contagious'' and all remaining nodes are ``susceptible''. At each time step $t$, for each edge $(i,j)$ where $j$ is susceptible and $i$ is contagious, $i$ attempts to infect $j$ with -probability $p_{i,j}\in]0,1]$; the infection attempts are mutually independent. +probability $p_{i,j}\in(0,1]$; the infection attempts are mutually independent. If $i$ succeeds, $j$ will become contagious at time step $t+1$. Regardless of $i$'s success, node $i$ will be immune at time $t+1$. In other words, nodes stay contagious for only one time step. The cascade process terminates when no @@ -147,6 +148,9 @@ step $t$, then we have: Thus, the linear voter model is a Generalized Linear Cascade model with inverse link function $f: z \mapsto z$. +{\color{red} \subsubsection{Discretization of Continous Model} +TODO} + % \subsection{The Linear Threshold Model} % In the Linear Threshold Model, each node $j\in V$ has a threshold $t_j$ from @@ -171,6 +175,7 @@ with inverse link function $f: z \mapsto z$. % X^t\right] = \text{sign} \left(\inprod{\theta_j}{X^t} - t_j \right) % \end{equation} where ``sign'' is the function $\mathbbm{1}_{\cdot > 0}$. +{\color{red} Add drawing of math problem as in Edo's presentation} \subsection{Maximum Likelihood Estimation} @@ -227,4 +232,4 @@ a twice-differentiable function $f$ is log concave iff. $f''f \leq f'^2$. It is easy to verify this property for $f$ and $(1-f)$ in the Independent Cascade Model and Voter Model. -{\color{red} TODO: talk about the different constraints} +{\color{red} TODO:~talk about the different constraints} |