  We consider the problem of counting 4-cycles ($C_4$) in an undirected graph
$G$ of $n$ vertices and $m$ edges (in bipartite graphs, 4-cycles are also often
referred to as $\textit{butterflies}$). There have been a number of previous
algorithms for this problem based on sorting the graph by degree and using
randomized hash tables. These are appealing in theory due to compact storage
and fast access on average. But, the performance of hash tables can degrade
unpredictably and are also vulnerable to adversarial input.
  We develop a new simpler algorithm for counting $C_4$ requiring
$O(m\bar\delta(G))$ time and $O(n)$ space, where $\bar \delta(G) \leq
O(\sqrt{m})$ is the $\textit{average degeneracy}$ parameter introduced by
Burkhardt, Faber & Harris (2020). It has several practical improvements over
previous algorithms; for example, it is fully deterministic, does not require
any sorting of the input graph, and uses only addition and array access in its
inner loops. To the best of our knowledge, all previous efficient algorithms
for $C_4$ counting have required $\Omega(m)$ space in addition to storing the
input graph.
  Our algorithm is very simple and easily adapted to count 4-cycles incident to
each vertex and edge. Empirical tests demonstrate that our array-based approach
is $4\times$ -- $7\times$ faster on average compared to popular hash table
implementations.
