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diff --git a/paper/sections/intro.tex b/paper/sections/intro.tex index 4d44395..264476b 100644 --- a/paper/sections/intro.tex +++ b/paper/sections/intro.tex @@ -1,77 +1,75 @@ -\begin{comment} -A recent line of research has focused on applying advances in sparse recovery to -graph analysis. A graph can be interpreted as a signal that one seeks to -`compress' or `sketch' and then `recovered'. However, we could also consider the -situation where the graph is unknown to us, and we dispose of few measurements -to recover the signal. Which real-life processes allow us to `measure' the -graph? +%\begin{comment} +%A recent line of research has focused on applying advances in sparse recovery to +%graph analysis. A graph can be interpreted as a signal that one seeks to +%`compress' or `sketch' and then `recovered'. However, we could also consider the +%situation where the graph is unknown to us, and we dispose of few measurements +%to recover the signal. Which real-life processes allow us to `measure' the +%graph? -A diffusion process taking place on a graph can provide valuable information -about the existence of edges and their edge weights. By observing the sequence -of nodes which become `infected' over time without knowledge of who has -`infected' whom, can we recover the graph on which the process takes place? The -spread of a particular behavior through a network is known as an {\it Influence -Cascade}. In this context, the {\it Graph Inference}\ problem is the recovery of -the graph's edges from the observation of few influence cascades. {\color{red} -Cite references} We propose to show how sparse recovery can be applied to solve -this recently introduced graph inference problem. +%A diffusion process taking place on a graph can provide valuable information +%about the existence of edges and their edge weights. By observing the sequence +%of nodes which become `infected' over time without knowledge of who has +%`infected' whom, can we recover the graph on which the process takes place? The +%spread of a particular behavior through a network is known as an {\it Influence +%Cascade}. In this context, the {\it Network Inference}\ problem is the recovery +%of the graph's edges from the observation of few influence cascades. +%{\color{red} Cite references} We propose to show how sparse recovery can be +%applied to solve this recently introduced network inference problem. -{\color{red} Graph inference to Network inference} -Tackling the graph inference problem means constructing a polynomial-time -algorithm which recovers with high probability a large majority of edges -correctly from very few observations or {\it cascades}. Prior work shown that -the graph inference problem can be solved in ${\cal O}(poly(s) \log m)$ number -of observed cascades, where $s$ is the maximum degree and $m$ is the number of -nodes in the graph. Almost miraculously, the number of cascades required to -reconstruct the graph is logarithmic in the number of nodes of the graph. -Results in the sparse recovery literature lead us to believe that $\Omega(s \log -m/s)$ cascade observations should be sufficient to recover the graph. In fact, -we prove this lower bound in a very general sense: any non-adaptive graph -inference algorithm which succeeds in recovering the graph with constant -probability requires $\Omega(s \log m/s)$ observations. We show an almost tight -upper-bound by applying standard sparse recovery techniques and assumptions: -${\cal O}(s \log m)$ are sufficient to recover the graph. We show that the edge -weights themselves can also be recovered under the same assumptions. +%{\color{red} Graph inference to Network inference} +%Tackling the graph inference problem means constructing a polynomial-time +%algorithm which recovers with high probability a large majority of edges +%correctly from very few observations or {\it cascades}. Prior work shown that +%the graph inference problem can be solved in ${\cal O}(poly(s) \log m)$ number +%of observed cascades, where $s$ is the maximum degree and $m$ is the number of +%nodes in the graph. Almost miraculously, the number of cascades required to +%reconstruct the graph is logarithmic in the number of nodes of the graph. +%Results in the sparse recovery literature lead us to believe that $\Omega(s \log +%m/s)$ cascade observations should be sufficient to recover the graph. In fact, +%we prove this lower bound in a very general sense: any non-adaptive graph +%inference algorithm which succeeds in recovering the graph with constant +%probability requires $\Omega(s \log m/s)$ observations. We show an almost tight +%upper-bound by applying standard sparse recovery techniques and assumptions: +%${\cal O}(s \log m)$ are sufficient to recover the graph. We show that the edge +%weights themselves can also be recovered under the same assumptions. - Throughout this paper, we focus on the three main discrete-time diffusion - processes: the independent cascade model, the voter model, and the linear - threshold model\dots - \end{comment} + %Throughout this paper, we focus on the three main discrete-time diffusion + %processes: the independent cascade model, the voter model, and the linear + %threshold model\dots + %\end{comment} Graphs have been extensively studied for their propagative abilities: connectivity, routing, gossip algorithms, etc. -%One question is: can we -%understand and predict a diffusion process from the graph? Conversely, can we -%learn a graph from the diffusion process? A diffusion process taking place over a graph provides valuable information about the presence and weights of its edges. \emph{Influence cascades} are a specific type of diffusion processes in which a particular infectious behavior spreads over the nodes of the graph. By only observing the ``infection times'' of the nodes in the graph, one might hope to recover the underlying graph and the parameters of the cascade model. This problem is known in the literature as -the \emph{Graph Inference problem}. +the \emph{Network Inference problem}. -More precisely, solving the Graph Inference problem involves designing -an algorithm taking as input a set of observed cascades (realisations of the +More precisely, solving the Network Inference problem involves designing an +algorithm taking as input a set of observed cascades (realisations of the diffusion process) and recovers with high probability a large fraction of the -graph's edges. The goal is then to understand the relationship between the number -of observations, the probability of success, and the accuracy of the +graph's edges. The goal is then to understand the relationship between the +number of observations, the probability of success, and the accuracy of the reconstruction. -The Graph Inference problem can be decomposed and analyzed ``node-by-node''. +The Network Inference problem can be decomposed and analyzed ``node-by-node''. Thus, we will focus on a single node of degree $s$ and discuss how to identify its parents among the $m$ nodes of the graph. Prior work has shown that the -required number of observed cascades is $\O(poly(s)\log m)$. {\color{red} Add -reference} +required number of observed cascades is $\O(poly(s)\log m)$ +\cite{Netrapalli:2012, Abrahao:13}. -A more recent line of research has focused on applying advances in sparse -recovery to the graph inference problem. Indeed, the graph can be interpreted -as a ``sparse signal'' measured through influence cascades and then recovered. -The challenge is that influence cascade models typically lead to non-linear -inverse problems. The sparse recovery literature suggests that -$\Omega(s\log\frac{m}{s})$ cascade observations should be sufficient to recover -the graph.{\color{red} Add reference} However, the best known upper bound to -this day is $\O(s^2\log m)$.{\color{red} Add reference} +A more recent line of research~\cite{Daneshmand:2014} has focused on applying +advances in sparse recovery to the graph inference problem. Indeed, the graph +can be interpreted as a ``sparse signal'' measured through influence cascades +and then recovered. The challenge is that influence cascade models typically +lead to non-linear inverse problems. The sparse recovery literature suggests +that $\Omega(s\log\frac{m}{s})$ cascade observations should be sufficient to +recover the graph~\cite{donoho2006compressed, candes2006near}. However, the +best known upper bound to this day is $\O(s^2\log m)$~\cite{Netrapalli:2012, +Daneshmand:2014} The contributions of this paper are the following: @@ -82,9 +80,8 @@ The contributions of this paper are the following: encompasses the well-studied Independent Cascade Model and Voter Model. \item we give an algorithm which recovers the graph's edges using $\O(s\log m)$ cascades. Furthermore, we show that our algorithm is also able to - recover the edge weights (the parameters of the influence model), - a problem which has been seemingly overlooked so far. {\color{red} NOT - TRUE} + efficiently recover the edge weights (the parameters of the influence + model) up to an additive error term, \item we show that our algorithm is robust in cases where the signal to recover is approximately $s$-sparse by proving guarantees in the \emph{stable recovery} setting. @@ -123,11 +120,6 @@ support recovery algorithm, without the \emph{correlation decay} assumption. {\color{red} Du et.~al make a citation} -\begin{comment} -They assume a single-source model, where only one source is selected at random, -which is less realistic in practice since `patient 0' is rarely unknown to us. -\end{comment} - {\color{red} say they follow the same model as Gomez and abrahao} Closest to this work is a recent paper by \citet{Daneshmand:2014}, wherein the authors consider a $\ell_1$-regularized objective function. They adapt standard |