  In this paper, we study new batch-dynamic algorithms for the $k$-clique
counting problem, which are dynamic algorithms where the updates are batches of
edge insertions and deletions. We study this problem in the parallel setting,
where the goal is to obtain algorithms with low (polylogarithmic) depth. Our
first result is a new parallel batch-dynamic triangle counting algorithm with
$O(\Delta\sqrt{\Delta+m})$ amortized work and $O(\log^* (\Delta+m))$ depth with
high probability, and $O(\Delta+m)$ space for a batch of $\Delta$ edge
insertions or deletions. Our second result is an algebraic algorithm based on
parallel fast matrix multiplication. Assuming that a parallel fast matrix
multiplication algorithm exists with parallel matrix multiplication constant
$\omega_p$, the same algorithm solves dynamic $k$-clique counting with
$O\left(\min\left(\Delta m^{\frac{(2k - 1)\omega_p}{3(\omega_p + 1)}},
(\Delta+m)^{\frac{2(k + 1)\omega_p}{3(\omega_p + 1)}}\right)\right)$ amortized
work and $O(\log (\Delta+m))$ depth with high probability, and
$O\left((\Delta+m)^{\frac{2(k + 1)\omega_p}{3(\omega_p + 1)}}\right)$ space.
Using a recently developed parallel $k$-clique counting algorithm, we also
obtain a simple batch-dynamic algorithm for $k$-clique counting on graphs with
arboricity $\alpha$ running in $O(\Delta(m+\Delta)\alpha^{k-4})$ expected work
and $O(\log^{k-2} n)$ depth with high probability, and $O(m + \Delta)$ space.
Finally, we present a multicore CPU implementation of our parallel
batch-dynamic triangle counting algorithm. On a 72-core machine with two-way
hyper-threading, our implementation achieves 36.54--74.73x parallel speedup,
and in certain cases achieves significant speedups over existing parallel
algorithms for the problem, which are not theoretically-efficient.
