The Small-Community Phenomenon in Networks
  We investigate several geometric models of network which simultaneously have some nice global properties, that the small diameter property, the small-community phenomenon, which is defined to capture the common experience that (almost) every one in our society belongs to some meaningful small communities by the authors (2011), and that under certain conditions on the parameters, the power law degree distribution, which significantly strengths the results given by van den Esker (2008), and Jordan (2010). The results above, together with our previous progress in Li and Peng (2011), build a mathematical foundation for the study of 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 concentration inequality by alternatively and iteratively applying both the sub- and super-martingale inequalities, which seems powerful, and which may have more potential applications. 
Angsheng Li, Pan Peng
math.PR