  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.
