  Given an unknown $n \times n$ matrix $A$ having non-negative entries, the
\emph{inner product} (IP) oracle takes as inputs a specified row (or a column)
of $A$ and a vector $v \in \mathbb{R}^{n}$, and returns their inner product. A
derivative of IP is the induced degree query in an unknown graph $G=(V(G),
E(G))$ that takes a vertex $u \in V(G)$ and a subset $S \subseteq V(G)$ as
input and reports the number of neighbors of $u$ that are present in $S$. The
goal of this paper is to understand the strength of the inner product oracle.
Our results in that direction are as follows: (I) IP oracle can solve bilinear
form estimation, i.e., estimate the value of ${\bf x}^{T}A\bf{y}$ given two
vectors ${\bf x},\, {\bf y} \in \mathbb{R}^{n}$ with non-negative entries and
can sample almost uniformly entries of a matrix with non-negative entries; (ii)
We tackle for the first time weighted edge estimation and weighted sampling of
edges that follow as an application to the bilinear form estimation and almost
uniform sampling problems, respectively; (iii) induced degree query, a
derivative of IP can solve edge estimation and an almost uniform edge sampling
in induced subgraphs. To the best of our knowledge, these are the first set of
Oracle-based query complexity results for induced subgraphs. We show that
IP/induced degree queries over the whole graph can simulate local queries in
any induced subgraph; (iv) Apart from the above, we also show that IP can solve
several problems related to matrix, like testing if the matrix is diagonal,
symmetric, doubly stochastic, etc.
