  We consider the following general hidden hubs model: an $n \times n$ random
matrix $A$ with a subset $S$ of $k$ special rows (hubs): entries in rows
outside $S$ are generated from the probability distribution $p_0 \sim
N(0,\sigma_0^2)$; for each row in $S$, some $k$ of its entries are generated
from $p_1 \sim N(0,\sigma_1^2)$, $\sigma_1>\sigma_0$, and the rest of the
entries from $p_0$. The problem is to identify the high-degree hubs
efficiently. This model includes and significantly generalizes the planted
Gaussian Submatrix Model, where the special entries are all in a $k \times k$
submatrix. There are two well-known barriers: if $k\geq c\sqrt{n\ln n}$, just
the row sums are sufficient to find $S$ in the general model. For the submatrix
problem, this can be improved by a $\sqrt{\ln n}$ factor to $k \ge c\sqrt{n}$
by spectral methods or combinatorial methods. In the variant with $p_0=\pm 1$
(with probability $1/2$ each) and $p_1\equiv 1$, neither barrier has been
broken.
  We give a polynomial-time algorithm to identify all the hidden hubs with high
probability for $k \ge n^{0.5-\delta}$ for some $\delta >0$, when
$\sigma_1^2>2\sigma_0^2$. The algorithm extends to the setting where planted
entries might have different variances each at least as large as $\sigma_1^2$.
We also show a nearly matching lower bound: for $\sigma_1^2 \le 2\sigma_0^2$,
there is no polynomial-time Statistical Query algorithm for distinguishing
between a matrix whose entries are all from $N(0,\sigma_0^2)$ and a matrix with
$k=n^{0.5-\delta}$ hidden hubs for any $\delta >0$. The lower bound as well as
the algorithm are related to whether the chi-squared distance of the two
distributions diverges. At the critical value $\sigma_1^2=2\sigma_0^2$, we show
that the general hidden hubs problem can be solved for $k\geq c\sqrt n(\ln
n)^{1/4}$, improving on the naive row sum-based method.
