Geographical Coarse Graining of Complex Networks

We perform a renormalization-grouplike numerical analysis of geographically embedded complex networks on a two-dimensional square lattice. At each step of the coarse-graining procedure, the four vertices on each $2\times 2$ square box are merged to a single vertex, resulting in a coarse-grained system of smaller size. Repetition of the process leads to the observation that the coarse-graining procedure does not alter the qualitative characteristics of the original scale-free network, which opens the possibility of subtracting a smaller network from the original network without destroying the important structural properties. The implication of the result is also suggested in the context of the recent study of the human brain functional network.