Entropy of Some Models of Sparse Random Graphs With Vertex-Names

Consider the setting of sparse graphs on N vertices, where the vertices have distinct "names", which are strings of length O(log N) from a fixed finite alphabet. For many natural probability models, the entropy grows as cN log N for some model-dependent rate constant c. The mathematical content of this paper is the (often easy) calculation of c for a variety of models, in particular for various standard random graph models adapted to this setting. Our broader purpose is to publicize this particular setting as a natural setting for future theoretical study of data compression for graphs, and (more speculatively) for discussion of unorganized versus organized complexity.