  The k-Clique problem is a fundamental combinatorial problem that plays a
prominent role in classical as well as in parameterized complexity theory. It
is among the most well-known NP-complete and W[1]-complete problems. Moreover,
its average-case complexity analysis has created a long thread of research
already since the 1970s. Here, we continue this line of research by studying
the dependence of the average-case complexity of the k-Clique problem on the
parameter k. To this end, we define two natural parameterized analogs of
efficient average-case algorithms. We then show that k-Clique admits both
analogues for Erd\H{o}s-R\'{e}nyi random graphs of arbitrary density. We also
show that k-Clique is unlikely to admit neither of these analogs for some
specific computable input distribution.
