\documentclass{article}
\usepackage{fullpage, amsmath, amssymb}

\title{Extensions}

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\section{Gram matrix}

In the ICML paper, we showed that the graph could be recovered as long as the
expected gram matrix $\mathcal{G}$ of observations had its restricted
eigenvalues lower bounded away from $0$.  Recall that the gram matrix $G$ is
given by $X X^T$, where $X$ is an $n\times m$ matrix, where $n$ is the number of
measurements and $m$ is the number of nodes and if $X_i$ is a column vector
indicating the nodes infected at step $i$, then the rows of $X$ are given by
$X_i^T$. It follows that $$\mathcal{G} = \mathbb{E}[\sum_i X_i X_i^T]$$

Note that the indices of the sum  are themselves random variables!  Furthermore,
recall the definition of a restricted eigenvalue as $\|X_{S^c}\|_1 \leq 3
\|X_S\|_1$.

\subsection{Voter model}

In the voter model, $\mathbb{P}[X_j^{t+1} = 1 | X^t] = \sum_{i=1}^m A_{i,j}
X^t_i$, where $A$ is the weighted adjacency matrix of the graph. Furthermore,
the model runs indefinitely until time $T$, a hyperparameter of the model.
Therefore, $\mathbb{E}[X_{n+1} | X_n] = A^T X_n$ and $\mathcal{G} = \sum_i A^i
\mathbb{E}[X_0 X_0^T] (A^T)^i$.  In the case of the single-source model,
$\mathbb{E}[X_0X_0^T] = \frac{1}{m} I_m$ and it follows that $$\mathcal{G} =
\frac{1}{m} \sum_{i=1}^T A^i (A^T)^i$$

\section{Submodularity of Generalized Linear Cascades}

For which link functions is the resulting influence function submodular? We know
it to be true for the IC and LT model, what about the Logistic cascade model?

The answer is no. If we take the example of three nodes $(A, B, C)$ with the
possiblity to influence a remaining node D with respective edge weights: $a$,
$b$, and $c$, then we see that the following equality must be verified:
$$2\sigma(a+b) \geq \sigma(a+b+c) + \sigma(a)$$

We see that for $a=.5$, $,b=.5$, and $c=1$, the equality is violated.
Interestingly however is that if we let the scale parameter of the sigmoid go to
infinity, we recover the LT model which is submodular. In fact, by slowly
increasing the scale parameter, it becomes harder to find values of $a, b, c$
which violate the inequality. We therefore have a parameterized function which
is `close to submodular'.
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