  Researchers developing implementations of distributed graph analytic
algorithms require graph generators that yield graphs sharing the challenging
characteristics of real-world graphs (small-world, scale-free, heavy-tailed
degree distribution) with efficiently calculable ground-truth solutions to the
desired output. Reproducibility for current generators used in benchmarking are
somewhat lacking in this respect due to their randomness: the output of a
desired graph analytic can only be compared to expected values and not exact
ground truth. Nonstochastic Kronecker product graphs meet these design criteria
for several graph analytics. Here we show that many flavors of triangle
participation can be cheaply calculated while generating a Kronecker product
graph. Given two medium-sized scale-free graphs with adjacency matrices $A$ and
$B$, their Kronecker product graph has adjacency matrix $C = A \otimes B$. Such
graphs are highly compressible: $|{\cal E}|$ edges are represented in ${\cal
O}(|{\cal E}|^{1/2})$ memory and can be built in a distributed setting from
small data structures, making them easy to share in compressed form. Many
interesting graph calculations have worst-case complexity bounds ${\cal
O}(|{\cal E}|^p)$ and often these are reduced to ${\cal O}(|{\cal E}|^{p/2})$
for Kronecker product graphs, when a Kronecker formula can be derived yielding
the sought calculation on $C$ in terms of related calculations on $A$ and $B$.
We focus on deriving formulas for triangle participation at vertices, ${\bf
t}_C$, a vector storing the number of triangles that every vertex is involved
in, and triangle participation at edges, $\Delta_C$, a sparse matrix storing
the number of triangles at every edge.

Discrete Mathematics;Social and Information Networks;Combinatorics