  We study matrix multiplication in the low-bandwidth model: There are $n$
computers, and we need to compute the product of two $n \times n$ matrices.
Initially computer $i$ knows row $i$ of each input matrix. In one communication
round each computer can send and receive one $O(\log n)$-bit message.
Eventually computer $i$ has to output row $i$ of the product matrix.
  We seek to understand the complexity of this problem in the uniformly sparse
case: each row and column of each input matrix has at most $d$ non-zeros and in
the product matrix we only need to know the values of at most $d$ elements in
each row or column. This is exactly the setting that we have, e.g., when we
apply matrix multiplication for triangle detection in graphs of maximum degree
$d$. We focus on the supported setting: the structure of the matrices is known
in advance; only the numerical values of nonzero elements are unknown.
  There is a trivial algorithm that solves the problem in $O(d^2)$ rounds, but
for a large $d$, better algorithms are known to exist; in the moderately dense
regime the problem can be solved in $O(dn^{1/3})$ communication rounds, and for
very large $d$, the dominant solution is the fast matrix multiplication
algorithm using $O(n^{1.158})$ communication rounds (for matrix multiplication
over fields and rings supporting fast matrix multiplication).
  In this work we show that it is possible to overcome quadratic barrier for
all values of $d$: we present an algorithm that solves the problem in
$O(d^{1.907})$ rounds for fields and rings supporting fast matrix
multiplication and $O(d^{1.927})$ rounds for semirings, independent of $n$.
