general relativity - How is gravity proportional to space-time curvature in the rubber-sheet analogy?

In General Relativity, Einstein established that gravity is due to the curvature produced by objects in space.

We all know that gravity is proportional to mass. The picture Einstein painted looks like this: The bigger the object, the bigger the curvature and hence the stronger gravity.

Let's say that the material inside the Earth is taken out and Earth is made less dense. Now the gravitational pull will be decreased according to Newton's formula due to reduction in mass.

But as per Einstein's picture the dimensions of curvature will be same, as the size of Earth is the same still and the gravity of Earth should not be altered.

I've pictured the scenario something like this: http://youtu.be/0HnaLnUdYvs?t=275

Please clear up my confusion.