From 8a611d4cc0e979d62ee29a75d757bfd2b3ff0cdb Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Fri, 6 Feb 2015 16:25:28 -0500
Subject: Cosmetic changes

---
 paper/sections/experiments.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

(limited to 'paper/sections/experiments.tex')

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.
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