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 kind of measurements do we dispose of in practice?

A diffusion process on a graph, which depends on the edges and edge weights in the graph, provides valuable information. By observing the sequence of nodes which become `infected' over time without knowledge of who has `influenced' whom, can we recover the graph on which this 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 to recover the edges of the graph from the observation of few influence cascades. We propose to show how sparse recovery can be applied to solve this recently introduced graph inference problem.

Recent research efforts in solving the graph inference problem have focused on constructing an effective algorithm which recovers a large majority of edges correctly from very few cascades. It has been 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 $m$ the number of nodes in the graph. In other words, the dependence of the number of cascades required to reconstruct the graph is (almost miraculously) logarithmic in the number of nodes of the graph. However, results in the sparse recovery literature lead us to believe that ${\cal O}(s \log m/s$ should be sufficient to recover the graph. In fact, we prove this lower bound in a very general sense. We also suggest a ${\cal O}(\Delta \log m)$-algorithm, which is almost tight. We also that the edge weights themselves can be estimated under the same assumptions.

Since the first few papers on link prediction in networks, the research community has made good progress in defining the Graph Inference problem more clearly and suggesting effective algorithms. 

Throughout this paper, we focus on the two main diffusion processes, outlined in the seminal work \cite{Kempe:03}: the independent cascade model (IC) and the linear threshold model.

\subsection{Related Work}

\paragraph{Past work}

The study of edge prediction in graph has been an active field of research for over a decade. \cite{GomezRodriguez:2010} was one of the first papers to study graph prediction from cascades. They introduce the {\scshape netinf} algorithm, which approximates the likelihood of cascades represented as a continuous process. The algorithm was later improved/modified in later work. Beside validation on generic networks, {\scshape netinf} is not known to have any theoretical recovery guarantees. \cite{Netrapalli:2012} studied solely the independent cascade model and obtained the first ${\cal O}(\Delta^2 \log m)$ guarantee on general networks. The algorithm is based around the same likelihood function we suggest, without the $\ell1$-norm penalty. However, the analysis depended strongly on a restrictive {\it correlation decay} assumption, which strongly restricts the number of new infections at every step. In this restricted setting, they show a complex lower bound, which is roughly $\Omega(\Delta \log (m/\Delta))$ lower bound for perfect support recovery with constant probability.

The work of \cite{Abrahao:13} study the same continuous-model framework as \cite{GomezRodriguez:2010} and obtain a ${\cal O}(\Delta^9 \log^2 \Delta \log m)$ support recovery algorithm. Their work also studies the information leveraged by different `parts' of the cascade, showing that a surprisingly important amount of information can be gleaned from the first `infections' of the cascade. We will reach a similar conclusion in section~\ref{sec:assumptions}. However, their model supposes a single-source model, where only one source is selected at random, which is less realistic in practice. Oftentimes, the `patient 0' is unknown to us, and a multi-source model intuitively models the more common situation of a delayed observation of the cascade.

The recent work of \cite{Daneshmand:2014} is very similar to our own, they consider a $\ell1$-norm penalty to their objective function, adapting standard results from sparse recovery to obtain a ${\cal O}(\Delta^3 \log m)$ algorithm under an irrepresentability condition. With stronger assumptions, they match the \cite{Netrapalli:2012} bound of ${\cal O}(\Delta^2 \log m)$, by exploiting a similar proof technique based around the KKT conditions of the objective function. Their work has the merit of studying a general framework of continuous functions. Similarly to \cite{Abrahao:13}, they place themselves in the restrictive single-source context. 

\paragraph{Our contributions}
Though our work follows closely the spirit of \cite{Netrapalli:2012} and \cite{Daneshmand:2014}, we believe we provide several significant improvements to their work. We study sparse recovery under less restrictive assumptions and obtain the first ${\cal O}(\Delta \log m)$ algorithm for graph inference from cascades. We also study the seemingly overlooked problem of weight recovery as well as the setting of the relaxed sparsity setting. Finally, we show these results are almost tight, by proving in section~\ref{sec:lowerbound} the first lower bound on the number of observations required to recover the edges and the edge weights of a graph in the general case. We study the case of the two best known diffusion processes for simplicity as outlined in \cite{Kempe:03}, but many arguments can be extended to more general diffusion processes.
