  We study homological properties of random quadratic monomial ideals in a
polynomial ring $R = {\mathbb K}[x_1, \dots x_n]$, utilizing methods from the
Erd\"{o}s-R\'{e}nyi model of random graphs. Here for a graph $G \sim G(n, p)$
we consider the `coedge' ideal $I_G$ corresponding to the missing edges of $G$,
and study Betti numbers of $R/I_G$ as $n$ tends to infinity. Our main results
involve setting the edge probability $p = p(n)$ so that asymptotically almost
surely the Krull dimension of $R/I_G$ is fixed. Under these conditions we
establish various properties regarding the Betti table of $R/I_G$, including
sharp bounds on regularity and projective dimension, and distribution of
nonzero normalized Betti numbers. These results extend work of Erman and Yang,
who studied such ideals in the context of conjectured phenomena in the
nonvanishing of asymptotic syzygies. Along the way we establish results
regarding subcomplexes of random clique complexes as well as notions of
higher-dimensional vertex $k$-connectivity that may be of independent interest.
