We construct a curve graph consisting of non-peripheral, simple closed curves for surfaces of infinite type. We show that it is connected and often hyperbolic. We use the non-peripheral curve graph to show that a large set of big mapping class groups have at most quadratic divergence. A section application is that we prove a large set of big mapping class groups have infinite coarse rank. This talk is based on joint work with Kasra Rafi and Assaf Bar-Natan. If time permits, we will discuss recent work joint with Alex Squires; we show that geodesics in non-peripheral curve graphs also satisfy the bounded geodesic image theorem.