LQG is based on a smooth compact manifold M The configuration space is A the connections on M: by choice here we use SO(3) connections----the smooth rotation-valued 1-forms on M. The connections represent all the possible configurations of gravity, or curvature, on the manifold-----in other words the possible geometries. There is no fixed prior choice of metric. The quantum state space of the theory is a linear space L consisting of complex-valued functions on A. We are following the notation in the Rovelli-Upadhya LQG primer. A class of "cylindrical functions" is defined, spanning L, and using these functions an inner product is defined, so that we have a Hilbert space. Labeled networks enter here as a way of arriving at a basis for the Hilbert space. The set of "cylindrical functions" is highly redundant. They are very simple to define but there is a lot of overlap and the set is not linearly independent. To get a linearly independent spanning set of functions we have to be more methodical and selective. So to begin this thread I am going to describe a simplified version of labeled networks. Without loss of generality, the networks can be taken to be trivalent---three legs meeting at each node. A node where more than 3 legs meet can always be broken down into a kind of "traffic circle" of tee-joint, or trivalent, nodes.