Section 10.1.3, "Space, time and unitarity", page 267 of the book, page 285 of the PDF file:
This is the only textbook on loop quantum gravity and it can actually be partially understood by non-physicists. For example, in mid-90s I was trying to imagine how one could have discrete space at the quantum scale, but so that it still appears continuous at the macroscopic level. All variants I could come up with were pretty ugly. This book contains a very neat construction of that.
(Section 1.2.2. Quantum space: spin networks, starting from page 12 of the book, page 30 of the PDF file.)
Basically, when one measures any volume precisely enough, it should take a discrete value. Then some volumes (grains of space) are adjacent to each other, and the area of surface dividing two grains should also take a discrete value.
So the state is represented by a graph, the nodes are the grains of space labeled with discrete volume values, and the edges connect adjacent grains and are labeled with discrete surface area values.
This structure is called an "abstract spin network". The actual physical space is a quantum superposition of such spin networks.