The small-community phenomenon in networks

We investigate several geometric models of networks that simultaneously have some nice global properties, including the small-diameter property, the small-community phenomenon, which is defined to capture the common experience that (almost) everyone in society also belongs to some meaningful small communities, and the power law degree distribution, for which our result significantly strengthens those given in van den Esker (2008) and Jordan (2010). These results, together with our previous work in Li and Peng (2011), build a mathematical foundation for the study of both communities and the small-community phenomenon in various networks. In the proof of the power law degree distribution, we develop the method of alternating concentration analysis to build a concentration inequality by alternately and iteratively applying both the sub- and super-martingale inequalities, which seems to be a powerful technique with further potential applications.