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Diffstat (limited to 'paper/sections')
| -rw-r--r-- | paper/sections/experiments.tex | 6 | ||||
| -rw-r--r-- | paper/sections/results.tex | 13 |
2 files changed, 10 insertions, 9 deletions
diff --git a/paper/sections/experiments.tex b/paper/sections/experiments.tex index d13472e..80f95fa 100644 --- a/paper/sections/experiments.tex +++ b/paper/sections/experiments.tex @@ -33,7 +33,8 @@ In this section, we validate empirically the results and assumptions of Section~ \paragraph{Experimental setup} We evaluate the performance of the algorithms on synthetic graphs, chosen for their similarity to real social networks. We therefore consider a Watts-Strogatz graph ($300$ nodes, $4500$ edges) \cite{watts:1998}, a Barabasi-Albert graph ($300$ nodes, $16200$ edges) \cite{barabasi:2001}, a Holme-Kim power law graph ($200$ nodes, $9772$ edges) \cite{Holme:2002}, and the recently introduced Kronecker graph ($256$ nodes, $10000$ edges) \cite{Leskovec:2010}. Undirected graphs are converted to directed graphs by doubling the edges. -For every reported data point, we sample edge weights and generate $n$ cascades from the Independent Cascade model for $n \in \{100, 500, 1000, 2000, 5000\}$. We compare for each algorithm the estimated graph $\hat {\cal G}$ with ${\cal G}$. The initial probability of a node being a source is fixed to $0.05$, i.e. an average of $15$ nodes source nodes per cascades for all experiments, except for Figure~\label{fig:four_figs} (f). All edge weights are chosen uniformly in the interval $[0.2, 0.7]$, except when testing for approximately sparse graphs (see paragraph on robustness). Adjusting for the variance of our experiments, all data points are reported with at most a $\pm 1$ error margin. The parameter $\lambda$ is chosen to be of the order ${\cal O}(\sqrt{\log m / (\alpha n)})$. +For every reported data point, we sample edge weights and generate $n$ cascades +from the (IC) model for $n \in \{100, 500, 1000, 2000, 5000\}$. We compare for each algorithm the estimated graph $\hat {\cal G}$ with ${\cal G}$. The initial probability of a node being a source is fixed to $0.05$, i.e. an average of $15$ nodes source nodes per cascades for all experiments, except for Figure~\label{fig:four_figs} (f). All edge weights are chosen uniformly in the interval $[0.2, 0.7]$, except when testing for approximately sparse graphs (see paragraph on robustness). Adjusting for the variance of our experiments, all data points are reported with at most a $\pm 1$ error margin. The parameter $\lambda$ is chosen to be of the order ${\cal O}(\sqrt{\log m / (\alpha n)})$. \paragraph{Benchmarks} @@ -44,8 +45,7 @@ $$\hat \theta_i \in \arg \min_{\theta} \sum_{t \in {\cal T}} |f(\theta_i\cdot x^ 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. The true positives are the edges which appear both in ${\cal G}$ and $\hat {\cal G}$ and the true negatives are the edges which appear in neither. 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 $\hat {\cal G} : \cup_{j \in V} \{i : \Theta_{ij} > 0.1\}$, \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}). 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} diff --git a/paper/sections/results.tex b/paper/sections/results.tex index c39f9da..5d63cbd 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -130,9 +130,10 @@ $\hat{\theta}_\lambda$ is the solution of \eqref{eq:pre-mle}: To prove Theorem~\ref{thm:main}, we apply Lemma~\ref{lem:negahban} with $\tau_{\mathcal{L}}=0$. Since $\mathcal{L}$ is twice differentiable and convex, -assumption \eqref{eq:rc} is implied by the restricted eigenvalue condition -\eqref{eq:re}. The upper bound on the $\ell_{\infty}$ norm of -$\nabla\mathcal{L}(\theta^*)$ is given by Lemma~\ref{lem:ub}. +assumption \eqref{eq:rc} with $\kappa_{\mathcal{L}}=\frac{\gamma}{2}$ is +implied by the (RE) condition \eqref{eq:re}. The upper bound +on the $\ell_{\infty}$ norm of $\nabla\mathcal{L}(\theta^*)$ is given by +Lemma~\ref{lem:ub}. \begin{lemma} \label{lem:ub} @@ -335,7 +336,8 @@ We will need the following additional assumptions on the inverse link function $ \left|\frac{f''}{f}\right|\right) \leq\frac{1}{\alpha} \end{equation} -whenever $f(\inprod{\theta^*}{x})\notin\{0,1\}$. +whenever $f(\inprod{\theta^*}{x})\notin\{0,1\}$. These conditions are once +again non restrictive in the (IC) model and (V) model. \begin{proposition} \label{prop:fi} @@ -391,8 +393,7 @@ m)$. \paragraph{(RE) vs Irrepresentability Condition} \cite{Daneshmand:2014} rely on an `incoherence' condition on the hessian of the -likelihood function. Their condition is equivalent to the more commonly called -{\it (S,s)-irrepresentability} condition: +likelihood function also known as the {\it (S,s)-irrepresentability} condition: \begin{comment} \begin{definition} |