\documentclass[10pt]{article}

\usepackage{amsmath, amssymb, amsthm, bbm, fullpage}
\date{}
\author{}
\newtheorem{claim}{Claim}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}

\title{Supplementary Material \\ {\small Inferring Graphs from Cascades: A Sparse Recovery Framework}}

\begin{document}

\maketitle


\subsection*{Learnability and Optimization}

\paragraph{Notation}

Let $G = (V, E)$ be a weighted directed graph. Without loss of generality, we can assign a weight $p_{u,v} \in [0,1]$ to every possible edge $(u,v) \in V^2$. Let $m$ be the number of non-zero edges of $G$. Let $\sigma_G(S)$ be the influence of the set $S \subseteq V$ in graph $G$ under the independent cascade model. For $k \in \mathbb{N}^*$, let $O_k \in \arg\max_{|S| \leq k} \sigma_G(S)$ and $\sigma_{G}^* = \sigma_G(O_k)$ where we omit the $k$ when it is unambiguous.

\begin{claim}
\label{cla:oracle}
For two graphs $G$ and $G'$ such that $V = V'$. If for all $(u,v) \in V^2$, $p_{u,v} \geq p'_{u,v} \geq p_{u,v} - \frac{1}{\alpha m}$, then $\forall S \subseteq V, \sigma_{G}(S) \geq \sigma_{G'}(S) \geq (1 - 1/\alpha) \sigma_{G}(S)$
\end{claim}

\begin{proof}
Let $S \subseteq V$ be any seed set. From \cite{Kempe:03}, by sampling unweighted subgraph $H \sim G$, where $(u,v)$ is an edge in $H$ with probability $p_{u,v}$, we can write:

\begin{equation}\sigma_G(S) = \sum_{v\in V} \sigma_G(S)[v]  = \sum_{v \in V} \mathbb{E}_{H \sim G} \left[ \mathbbm{1}_{v \text{ reachable from S in H}}\right] 
\end{equation}

where $\sigma_G(S)[v]$ is the influence of $S$ on $v\in V$ in $G$. A similar equation can be written for $H' \sim G'$. Note that if $v \in S$, then $\sigma_G(S)[v] = \sigma_G'(S)[v]$ trivially. Suppose $v \in V \backslash S$. We can couple the sampling of $H'$ and $H$ such that $H'$ is always a subgraph of $H$ and such that that any edge $(u,v)$ is in $H$ but not in $H'$ with probability $p_{u,v} - p'_{u,v} \leq \frac{1}{\alpha m}$. Since any non-zero edge of $H'$ is always an edge in $H$, we have $\sigma_{G}(S)[v] \geq \sigma_G'(S)[v]$. The probability that $H$ and $H'$ have the same non-zero edge set is $(1 -\frac{1}{\alpha m})^m \geq 1 - 1/\alpha$, with equality when $m=1$. By adopting the convention $0/0 = 1$, 

$$\frac{\sigma_{G'}(S)[v]}{\sigma_{G}(S)[v]} = \mathbb{E} \left[ \frac{\mathbbm{1}_{v \text{ reachable from S in H'}}}{\mathbbm{1}_{v \text{ reachable from S in H}}}\right] \geq \left(1 - \frac{1}{\alpha}\right) \mathbb{E} \left[ \frac{\mathbbm{1}_{v \text{ reachable from S in H'}}}{\mathbbm{1}_{v \text{ reachable from S in H}}} \middle| H = H'\right] = (1 - 1/\alpha)$$

It follows that $\sigma_{G'}(S)[v] \geq (1 -1/\alpha) \sigma_{G}(S)[v]$.
\end{proof}

 We can use Claim~\ref{cla:oracle} to find a constant factor approximation algorithm to maximising influence on $G$ by maximising influence on $G'$:

\begin{proposition}
\label{prop:approx_optim}
Suppose we have an unknown graph $G$ and a known graph $G'$ such that $V = V'$ and for all $(u,v) \in V^2, |p_{u,v} - p_{u,v}'| \leq  \frac{1}{\alpha m}$. Then for all $k \in \mathbb{N}^*$, we can find a set $\hat O_k$ such that $\sigma_{G}(\hat O_k) \geq (1 - e^{\frac{2}{\alpha} - 1}) \sigma^*_{G}$
\end{proposition}

\begin{proof}
For every edge $(u,v) \in V^2$, let $\hat p = p'_{u,v} - \frac{1}{\alpha m}$. We are now in the conditions of Claim~\ref{cla:oracle} with $\alpha \leftarrow \alpha/2$. We return the set $\hat O_k$ obtained by greedy maximisation on $\hat G$. It is a classic exercise to show then that $\sigma_G(\hat O_k) \geq 1 - e^{\frac{2}{\alpha} - 1}$ (see Pset 1, CS284r).
\end{proof}

A small note on the approximation factor: it is only $>0$ for $\alpha > 2$. Note that $\alpha \geq 7 \implies 1 - e^{\frac{2}{\alpha} - 1} \geq \frac{1}{2}$ and that it converges to $(1 - 1/e)$ which is the best possible polynomial-time approximation ratio to influence maximisation unless $P = NP$. Also note that in the limit of large $m$, $(1 -\frac{1}{\alpha m})^m \rightarrow \exp(-1/\alpha)$ and the approximation ratio goes to $1 - \exp(-\exp(-2/\alpha))$.

\subsection*{Obtaining $\| p - p^*\|_{\infty} \leq 1/2m$}

Note that with $n \geq C \frac{s^2}{\gamma \alpha} \log m$ measurements for $C$ an absolute constant, we have for every node $i$, $$\|\theta_i - \theta_i^*\|_2 \leq \frac{6}{\gamma}\sqrt{\frac{s \log m}{\alpha n^{1-\delta}}}$$ with probability $1 - 2e^{-n^\delta \log m}$

Suppose we want this for every node, then by using the first measurements of every cascade where a node isn't infected at $t=0$, which occurs with probability $1 - \frac{1}{p_{\text{init}}}$ (we'll ignore this for now), we have by union bound, for $n \geq C \frac{\max(s)^2}{\gamma \alpha} \log m$ measurements:
$$\|\theta - \theta^*\|_{2} \leq \frac{6}{\gamma}\sqrt{\frac{|E| \log m}{\alpha n^{1-\delta}}}$$ 
with probability $1 - 2\exp^{-(n^{\delta} -1) \log m}$ (by union bound). Note that we could also have reasoned with the $\|\cdot\|_{\infty}$-norm. In which case, we have for $\Delta:= \max(s)$, the same number of measurements and the same probability:
$$\|\theta - \theta^*\|_{\infty} \leq \frac{6}{\gamma}\sqrt{\frac{\Delta\log m}{\alpha n^{1-\delta}}}$$

Suppose $n^{1-\delta} \geq \frac{36 C \Delta |E|}{\gamma^2 \alpha} \log m$, then with high probability:
$$|\theta_{ji} - \theta^*_{ji}| \leq \| \theta - \theta^* \|_{\infty} \leq \frac{1}{\sqrt{|E|C}}$$

We can now show that $f_{\theta^*} \geq \frac{1}{\sqrt{C}} f_{\theta}$ in the independent cascade model.




\subsection*{(RE) with high probability}

\subsubsection*{(RE) for the gram matrix}

Recall the expression of the Hessian:
\begin{equation}
\nonumber
    \nabla^2\mathcal{L}(\theta^*)
    = \frac{1}{|\mathcal{T}|}\sum_{t\in\mathcal{T}}x^t(x^t)^T
    \bigg[x_i^{t+1}\frac{f''f-f'^2}{f^2}(\theta^* \cdot x^t)\\
    -(1-x_i^{t+1})\frac{f''(1-f) + f'^2}{(1-f)^2}(\theta^* \cdot x^t)\bigg]
\end{equation}

This can be rewritten as $X J X^T$ where $J$ is a diagonal matrix with coefficients either $g_1$ or $g_2$. Under certain regularity conditions for $f$ (such as strong log-concavity), we can consider only the gram matrix: $\Delta^T X^T J X \Delta \geq \gamma_1 \Delta^T X^T X \Delta$

Denote $p_i$ the probability that node $i$ is infected at the beginning of the cascade. Let $\mathbb{P}(i)$ be the probability that $i$ is infected at one point during a cascade. Let $\mathbb{P}(S)$ for any set $S \subseteq N$ be the probability that $S$ is infected (all at once) at one point during a cascade. The diagonal of the Gram Matrix can be written as:
\begin{equation}
\nonumber
c_j := \mathbb{P}(i) = p_i - \sum_{S \subseteq {\cal N}(i)} (-1)^{|S|} \mathbb{P}(S) \prod_{j \in S} w_{ji}  =  p_i + \sum_{j \in N(i)} w_{ji}c_j + \dots
\end{equation}
Note that we have the following bounds for $c_j$:

\begin{align}
c_j &\leq p_i + \sum_{j \in S} w_{ji} c_j \tag{PageRank}\\
c_j &\geq p_i + \max_{j \in S} w_{ji} c_j \tag{?}
\end{align}

\paragraph{Simple claim}

If we assume that every node has an independent probability $p_i \in ]0,1[$ to be a seed node, then by taking the first measuremnent of every cascade, we get a matrix of the form

$[[p_1, p_1 p_2, \dots],[p_1 p_2, p_2, p_2 p_3, \dots] \dots]$

which can be written as $D \times (A + (I - D))$ where $D = diag(p_i)$ and $A_{i,j} = p_j$ for all $i$.

Since $\det(D(A + (I-D))) \geq det(D)det(A) + det(D)det(I-D) \geq \min(p_i (1-p_i))^m > 0$ and we have our $\gamma > 0$


\subsection*{Relaxed (RE) with high probability}

\subsection*{Lower Bound}


\bibliography{./sparse}
\bibliographystyle{plain}

\end{document}