  We study asymptotic 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. We consider coedge
ideals $I_G$ for $G \sim G(n, p)$ and establish asymptotic results about the
algebraic properties of $R/I_G$. Our main results involve fixing 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-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.
