The Small World Phenomenon in Hybrid Power Law Graphs

The small world phenomenon, that consistently occurs in numerous existing networks, refers to two similar but different properties — small average distance and the clustering effect. We consider a hybrid graph model that incorporates both properties by combining a global graph and a local graph. The global graph is modeled by a random graph with a power law degree distribution, while the local graph has specified local connectivity. We will prove that the hybrid graph has average distance and diameter close to that of random graphs with the same degree distribution (under certain mild conditions). We also give a simple decomposition algorithm which, for any given (real) graph, identifies the global edges and extracts the local graph (which is uniquely determined depending only on the local connectivity). We can then apply our theoretical results for analyzing real graphs, provided the parameters of the hybrid model can be appropriately chosen.