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diff --git a/paper/sections/experiments.tex b/paper/sections/experiments.tex index 9122e7d..e641992 100644 --- a/paper/sections/experiments.tex +++ b/paper/sections/experiments.tex @@ -22,8 +22,8 @@ (d) Sparse Kronecker ($\ell_2$-norm \emph{vs.} $n$ ) & (e) Non-sparse Kronecker ($\ell_2$-norm \emph{vs.} $n$) & (f) Watts-Strogatz (F$1$ \emph{vs.} $p_{\text{init}}$) \end{tabular} \captionof{figure}{Figures (a) and (b) report the F$1$-score in $\log$ scale -for 2 graphs as a function of the number of cascades $m$: (a) Barabasi-Albert graph, $300$ nodes, $16200$ edges. (b) -Watts-Strogatz graph, $300$ nodes, $4500$ edges. Figure (c) plots the Precision-Recall curve for various values for $\lambda$ ofr a Holme-Kim graph ($200$ nodes, $9772$ edges). Figures (d) and (e) report the +for 2 graphs as a function of the number of cascades $n$: (a) Barabasi-Albert graph, $300$ nodes, $16200$ edges. (b) +Watts-Strogatz graph, $300$ nodes, $4500$ edges. Figure (c) plots the Precision-Recall curve for various values of $\lambda$ for a Holme-Kim graph ($200$ nodes, $9772$ edges). Figures (d) and (e) report the $\ell_2$-norm $\|\hat \Theta - \Theta\|_2$ for a Kronecker graph which is: (d) exactly sparse (e) non-exactly sparse, as a function of the number of cascades $n$. Figure (f) plots the F$1$-score for the Watts-Strogatz graph as a function of $p_{init}$.} \label{fig:four_figs} \end{table*} @@ -42,10 +42,10 @@ We compare our \textsc{sparse mle} algorithm to 3 benchmarks: \textsc{greedy} an $$\hat \theta_i \in \arg \min_{\theta} \sum_{t \in {\cal T}} |f(\theta_i\cdot x^t) - x_i^{t+1}|^2 + \lambda \|\theta_i\|_1$$ \textsc{Lasso} has the merit of being both easier and faster to optimize numerically than the other convex-optimization based algorithms. It approximates the $\textsc{sparse mle}$ algorithm by making the assumption that the observations $x_i^{t+1}$ are of the form: $x_i^{t+1} = f(\theta_i\cdot x^t) + \epsilon$, where $\epsilon$ is random white noise. This is not valid in theory since $\epsilon$ \emph{depends on} $f(\theta_i\cdot x^t)$, however the approximation is validated in practice. -We did not benchmark against other known algorithms (\textsc{netrate} \cite{gomezbalduzzi:2011} and \textsc{first edge} \cite{Abrahao:13}) due to the discrete time assumption. These algorithms also suppose a single-source model, whereas \textsc{sparse mle}, \textsc{mle}, and \textsc{greedy} do not. Learning the graph in the case of a multi-source cascade model is harder (see Figure~\ref{fig:four_figs} (f)) but more realistic, since we rarely have access to ``patient 0'' in practice. +We did not benchmark against other known algorithms (\textsc{netrate} \cite{gomezbalduzzi:2011} and \textsc{first edge} \cite{Abrahao:13}) due to the discrete-time assumption. These algorithms also suppose a single-source model, whereas \textsc{sparse mle}, \textsc{mle}, and \textsc{greedy} do not. Learning the graph in the case of a multi-source cascade model is harder (see Figure~\ref{fig:four_figs} (f)) but more realistic, since we rarely have access to ``patient 0'' in practice. \paragraph{Graph Estimation} -In the case of the \textsc{lasso}, \textsc{mle} and \textsc{sparse mle} algorithms, we construct the edges of $\hat {\cal G} : \cup_{j \in V} \{i : \Theta_{ij} > 0.1\}$, \emph{i.e} by thresholding. Finally, we report the F1-score$=2 \text{precision}\cdot\text{recall}/(\text{precision}+\text{recall})$, which considers the number of true edges recovered by the algorithm over the total number of edges returned by the algorithm (\emph{precision}) with the number of true edges recovered by the algorithm over the total number of edges it should have recovered (\emph{recall}). +In the case of the \textsc{lasso}, \textsc{mle} and \textsc{sparse mle} algorithms, we construct the edges of $\hat {\cal G} : \cup_{j \in V} \{(i,j) : \Theta_{ij} > 0.1\}$, \emph{i.e} by thresholding. Finally, we report the F1-score$=2 \text{precision}\cdot\text{recall}/(\text{precision}+\text{recall})$, which considers \emph{(1)} the number of true edges recovered by the algorithm over the total number of edges returned by the algorithm (\emph{precision}) and \emph{(2)} the number of true edges recovered by the algorithm over the total number of edges it should have recovered (\emph{recall}). Over all experiments, \textsc{sparse mle} achieves higher rates of precision, recall, and F1-score. Interestingly, both \textsc{mle} and \textsc{sparse mle} perform exceptionally well on the Watts-Strogatz graph. \begin{comment} |