We consider a graph ${\cal G}= (V, E, \Theta)$, where $\Theta$ is a $|V|\times
|V|$ matrix of parameters describing the edge weights of $\mathcal{G}$.
Intuitively, $\Theta_{i,j}$ captures the ``influence'' of node $i$ on node $j$.
Let $m\defeq |V|$. For each node $j$, let $\theta_{j}$ be the $j^{th}$ column
vector of $\Theta$.  A {\color{red} discrete-time} \emph{Cascade model} is a
Markov process over a finite state space ${\{0, 1, \dots, K-1\}}^V$ with the
following properties:
\begin{enumerate}
    \item Conditioned on the previous time step, the transition events between
        two states in $\{0,1,\dots, K-1\}$ for each $i \in V$ are mutually
        independent.
\item Of the $K$ possible states, there exists a \emph{contagious state} such
    that all transition probabilities of the Markov process are either constant
    or can be expressed as a function of the graph parameters $\Theta$ and the
    set of ``contagious nodes'' at the previous time step.
  \item The initial probability over ${\{0, 1, \dots, K-1\}}^V$ is such that all
  nodes can eventually reach a \emph{contagious state} with non-zero
  probability. The ``contagious'' nodes at $t=0$ are called
    \emph{source nodes}.
\end{enumerate}

In other words, a cascade model describes a diffusion process where a set of
contagious nodes ``influence'' other nodes in the graph to become contagious. An
\emph{influence cascade} is a realisation of this random process, \emph{i.e.}
the successive states of the nodes in graph ${\cal G}$. Note that both the
``single source'' assumption made in~\cite{Daneshmand:2014} and
\cite{Abrahao:13} as well as the ``uniformly chosen source set'' assumption made
in~\cite{Netrapalli:2012} verify condition 3. {\color{red} why is it less
restrictive? explain}

In the context of Graph Inference,~\cite{Netrapalli:2012} focus
on the well-known discrete-time independent cascade model recalled below, which
\cite{Abrahao:13} and~\cite{Daneshmand:2014} generalize to continuous time. We
extend the independent cascade model in a different direction by considering
a more general class of transition probabilities while staying in the
discrete-time setting. We observe that despite their obvious differences, both
the independent cascade and the voter models make the graph inference problem
similar to the standard generalized linear model inference problem. In fact, we
define a class of diffusion processes for which this is true: the
\emph{Generalized Linear Cascade Models}. The linear threshold model is
a special case and is discussed in Section~\ref{sec:linear_threshold}.

\subsection{Generalized Linear Cascade Models}
\label{sec:GLC}

Let \emph{susceptible} denote any state which can become contagious at
the next time step with a non-zero probability. We draw inspiration from
generalized linear models to introduce Generalized Linear Cascades:

\begin{definition}
\label{def:glcm}
Let $X^t$ be the indicator variable of ``contagious nodes'' at time step $t$.
A \emph{generalized linear cascade model} is a cascade model such that for each
susceptible node $j$ in state $s$ at time step $t$, the probability of $j$
becoming ``contagious'' at time step $t+1$ conditioned on $X^t$ is a Bernoulli
variable of parameter $f(\theta_j \cdot X^t)$:
\begin{equation}
    \label{eq:glm}
    \mathbb{P}(X^{t+1}_j = 1|X^t)
    = f(\theta_j \cdot X^t)
\end{equation}
where $f: \reals \rightarrow [0,1]$
\end{definition}

In other words, each generalized linear cascade provides, for each node $j \in
V$ a series of measurements ${(X^t, X^{t+1}_j)}_{t \in {\cal T}_j}$ sampled from
a generalized linear model. Note also that $\E[X^{t+1}_i\,|\,X^t]
= f(\inprod{\theta_i}{X^t})$. As such, $f$ can be interpreted as the inverse
link function of our generalized linear cascade model.

% Denoting by $X^t$ the state of the cascade at time step $t$, we interpret
% $X^t_j = 1$ for node $j\in V$ as ``node $j$ is contagious'', \emph{i.e}, ``$j$
% exhibits the source nodes' behavior at time step $t$''.  We draw inspiration
% from \emph{generalized linear models} (GLM) to define a generalized linear
% cascade.

% \begin{definition}
%     \label{def:glcm}
%     Let us denote by $\{\mathcal{F}_t, t\in\ints\}$ the natural
%     filtration induced by $\{X_t, t\in\ints\}$. A \emph{generalized linear
%     cascade} is characterized by the following equation:
%     \begin{displaymath}
%         \P[X^{t+1}=x\,|\, \mathcal{F}_t] =
%         \prod_{i=1}^m f(\inprod{\theta_i}{X^{t}})^{x_i}
%         \big(1-f(\inprod{\theta_i}{X^{t}}\big)^{1-x_i}
%     \end{displaymath}
% where $f:\mathbb{R}\to[0,1]$ and $\theta_i$ is the $i$-th column of $\Theta$.
% \end{definition}

% It follows immediately from this definition that a generalized linear cascade
% satisfies the Markov property:
% \begin{displaymath}
%     \P[X^{t+1}=x\,|\mathcal{F}_t] = \P[X^{t+1}=x\,|\, X^t]
% \end{displaymath}
% Note also that $\E[X^{t+1}_i\,|\,X^t] = f(\inprod{\theta_i}{X^t})$. As such,
% $f$ can be interpreted as the inverse link function of our generalized linear
% cascade model.

\subsection{Examples}

\subsubsection{Independent Cascade Model}

In the independent cascade model, nodes can be either susceptible, contagious
or immune. At $t=0$, all source nodes are ``contagious'' and all remaining
nodes are ``susceptible''. At each time step $t$, for each edge $(i,j)$ where
$j$ is susceptible and $i$ is contagious, $i$ attempts to infect $j$ with
probability $p_{i,j}\in(0,1]$; the infection attempts are mutually independent.
If $i$ succeeds, $j$ will become contagious at time step $t+1$. Regardless of
$i$'s success, node $i$ will be immune at time $t+1$. In other words, nodes
stay contagious for only one time step. The cascade process terminates when no
contagious nodes remain.

If we denote by $X^t$ the indicator variable of the set of contagious nodes at
time step $t$, then if $j$ is susceptible at time step $t+1$, we have:
\begin{displaymath}
    \P\big[X^{t+1}_j = 1\,|\, X^{t}\big]
    = 1 - \prod_{i = 1}^m {(1 - p_{i,j})}^{X^t_i}.
\end{displaymath}
Defining $\Theta_{i,j} \defeq \log(1-p_{i,j})$, this can be rewritten as:
\begin{equation}\label{eq:ic}
    \tag{IC}
    \P\big[X^{t+1}_j = 1\,|\, X^{t}\big]
    = 1 - \prod_{i = 1}^m e^{\Theta_{i,j}X^t_i}
    = 1 - e^{\inprod{\theta_j}{X^t}}
\end{equation}

Therefore, the independent cascade model is a Generalized Linear Cascade model
with inverse link function $f : z \mapsto 1 - e^z$.

\subsubsection{The Linear Voter Model}

In the Linear Voter Model, nodes can be either \emph{red} or \emph{blue}.
Without loss of generality, we can suppose that the
\emph{blue} nodes are contagious. The parameters of the graph are normalized
such that $\forall i, \ \sum_j \Theta_{i,j} = 1$ and we assume that
$\Theta_{i,i}$ is always non-zero, meaning that all nodes have self-loops.
Each round, every node $j$ independently chooses
one of its neighbors with probability $\Theta_{i,j}$ and adopts their color. The
cascades stops at a fixed horizon time $T$ or if all nodes are of the same color.
If we denote by $X^t$ the indicator variable of the set of blue nodes at time
step $t$, then we have:
\begin{equation}
\mathbb{P}\left[X^{t+1}_j = 1 | X^t \right] = \sum_{i=1}^m \Theta_{i,j} X_i^t =
\inprod{\Theta_j}{X^t}
\tag{V}
\end{equation}

Thus, the linear voter model is a Generalized Linear Cascade model
with inverse link function $f: z \mapsto z$.

\subsubsection{Discretization of Continous Model}

Consider the continuous-time independent cascade model with exponential
transmission function (CICE) of~\cite{GomezRodriguez:2010, Abrahao:13,
Daneshmand:2014} where time is binned into intervals of length $\epsilon$. Let
$X^k$ be the set of nodes `infected' before or during the $k^{th}$ time
interval. Note that contrary to the discrete-time independent cascade model,
$X^k_i = 1 \implies X^{k+1}_i = 1$.  Let $\exp(p)$ be an
exponentially-distributed random variable of parameter $p$.  By the memoryless
property of the exponential, if we consider that $\forall i,\Theta_{i,j} = 0 if
(i,j) \notin E$, then if $X^k_j \neq 1$:
\begin{align*}
  \mathbb{P}(X^{k+1}_j = 1 | X^k) & = \mathbb{P}(\min_{i \in {\cal N}(j)}
  \exp(p_{i,j}) \leq \epsilon) \\
  & = \mathbb{P}(\exp( \sum_{i=1}^m \Theta_{i,j} X^t_i) \leq \epsilon) \\
  & = 1 - e^{- \epsilon \cdot \theta_j \cdot X^t}
\end{align*}
This formulation is consistent when $X^k_j = 1$ by considering the dummy
variables $\forall i, \Theta_{i,i} = +\infty$.  Therefore, the
$\epsilon-$binned-process of the continuous-time model with exponential
transmission function is a Generalized Linear Cascade model with inverse link
function $f:z\mapsto 1-e^{-\epsilon\cdot z}$.

\subsubsection{Logistic Cascades}

% \subsection{The Linear Threshold Model}

% In the Linear Threshold Model, each node $j\in V$ has a threshold $t_j$ from
% the interval $[0,1]$ and for each node, the sum of incoming weights is less
% than $1$: $\forall j\in V$, $\sum_{i=1}^m \Theta_{i,j} \leq 1$.

% Nodes have two states: susceptible or contagious. At each time step, each
% susceptible node $j$ becomes contagious if the sum of the incoming weights
% from contagious parents is greater than $j$'s threshold $t_j$. Nodes remain
% contagious until the end of the cascade, which is reached when no new
% susceptible nodes become contagious.

% As such, the source nodes are chosen, the process is entirely deterministic.
% Let $X^t$ be the indicator variable of the set of contagious nodes at time
% step $t-1$, then: \begin{equation} \nonumber X^{t+1}_j =
% \mathbbm{1}_{\sum_{i=1}^m \Theta_{i,j}X^t_i > t_j} =
% \mathbbm{1}_{\inprod{\theta_j}{X^t} > t_j} \end{equation} where we defined
% again $\theta_j\defeq (\Theta_{1,j},\ldots,\Theta_{m,j})$. In other words, for
% every step in the linear threshold model, we observe the following signal:

% \begin{equation} \label{eq:lt} \tag{LT} \mathbb{P} \left[X^{t+1}_j = 1 |
% X^t\right] = \text{sign} \left(\inprod{\theta_j}{X^t} - t_j \right)
% \end{equation} where ``sign'' is the function $\mathbbm{1}_{\cdot > 0}$.
\subsection{Maximum Likelihood Estimation}
\label{sec:mle}

\begin{figure}
  \includegraphics[scale=.4]{figures/drawing.pdf}
  \caption{Illustration of the sparse-recovery approach}
\end{figure}


Inferring the model parameter $\Theta$ from observed influence cascades is the
central question of the present work. Recovering the edges in $E$ from observed
influence cascades is a well-identified problem known as the \emph{Graph
Inference} problem. However, recovering the influence parameters is no less
important and has been seemingly overlooked so far. In this work we focus on
recovering $\Theta$, noting that the set of edges $E$ can then be recovered
through the following equivalence: $(i,j)\in E\Leftrightarrow  \Theta_{i,j}
\neq 0$

Given observations $(x^1,\ldots,x^n)$ of a cascade model, we can recover
$\Theta$ via Maximum Likelihood Estimation (MLE). Denoting by $\mathcal{L}$ the
log-likelihood function, we consider the following $\ell_1$-regularized MLE
problem:
\begin{displaymath}
    \hat{\Theta} \in \argmax_{\Theta} \frac{1}{n}
    \mathcal{L}(\Theta\,|\,x^1,\ldots,x^n) - \lambda\|\Theta\|_1
\end{displaymath}
where $\lambda$ is the regularization factor which helps preventing
overfitting and controls the sparsity of the solution.

The generalized linear cascade model is decomposable in the following sense:
given Definition~\ref{def:glcm}, the log-likelihood can be written as the sum
of $m$ terms, each term $i\in\{1,\ldots,m\}$ only depending on $\theta_i$.
Since this is equally true for $\|\Theta\|_1$, each column $\theta_i$ of
$\Theta$ can be estimated by a separate optimization program:
\begin{equation}\label{eq:pre-mle}
    \hat{\theta}_i \in \argmax_{\theta} \mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n)
    - \lambda\|\theta_i\|_1
\end{equation}
where we denote by ${\cal T}_i$ the time steps at which node $i$ is susceptible
and:
\begin{multline*}
    \mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n) = \frac{1}{|{\cal T}_i|}
    \sum_{t\in {\cal T}_i } x_i^{t+1}\log f(\inprod{\theta_i}{x^{t}}) \\ + (1 -
    x_i^{t+1})\log\big(1-f(\inprod{\theta_i}{x^t})\big)
\end{multline*}

In the case of the voter model, the measurements include all time steps until
we reach the time horizon $T$ or the graph coalesces to a single state. In the
case of the independent cascade model, the measurements include all time steps
until node $i$ becomes contagious, after which its behavior is deterministic.
Contrary to prior work, our results depend on the number of
measurements and not the number of cascades.

A sufficient condition for program~\eqref{eq:pre-mle} to be a convex program is
that $\mathcal{L}_i$ is a concave function. This will be the case if, for
example, $f$ and $(1-f)$ are both log-concave functions. Remember that
a twice-differentiable function $f$ is log concave iff. $f''f \leq f'^2$. It is
easy to verify this property for $f$ and $(1-f)$ in the Independent Cascade
Model and Voter Model.

{\color{red} TODO:~talk about the different constraints}