This talk concerns the stability and ergodic properties of skew products $T(x, y) =(f(x), g(x, y))$ in which $(f, X, \mu)$ is an ergodic map of compact metric space $X$ and $g : X \times Y \rightarrow Y$ is continuous. The set $X$ is the base, while $Y$ is the fiber. We consider the case where g is (non-uniformly) contracting. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the base space to the fiber space. Here, we consider the case that the contraction rates are non-uniform and hence specified by Lyapunov exponents and analogous quantities. We investigate the geometric structures of non-uniformly hyperbolic attractors of a certain class of skew products. We construct an open set of skew products over a linear expanding circle map such that any skew product belonging to this set admits a non-uniformly hyperbolic solenoidal attractor for which the following dichotomy is ascertained. This attractor is either a continuous invariant graph with nonempty interior or a thick bony attractor. Here, an attractor is, roughly speaking, a maximal attractor. Also, an attractor is thick if it has positive but not full Lebesgue measure. In our construction, the contraction in the fiber is non-uniform. Furthermore, we provide some related results on the ergodic properties of attracting graphs and stability results for such graphs under deterministic perturbations. In particular, we show that there exists an invariant ergodic physical measure whose support is contained in that attractor.