  It is known that, if we take a countable model of Zermelo--Fraenkel set
theory ZFC and "undirect" the membership relation (that is, make a graph by
joining $x$ to $y$ if either $x\in y$ or $y\in x$), we obtain the
Erd\H{o}s--R\'enyi random graph. The crucial axiom in the proof of this is the
Axiom of Foundation, so it is natural to wonder what happens if we delete this
axiom, or replace it by an alternative (such as Aczel's Anti-Foundation Axiom).
The resulting graph may fail to be simple, it may have loops (if $x\in x$ for
some $x$) or multiple edges (if $x\in y$ and $y\in x$ for some $x,y$). We show
that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the
"random loopy graph" (which is $\aleph_0$-categorical and homogeneous), but if
we keep multiple edges, the resulting graph is not $\aleph_0$-categorical, but
has infinitely many $1$-types. Moreover, if we keep only loops and double edges
and discard single edges, the resulting graph contains countably many connected
components isomorphic to any given finite connected graph with loops.
