  Community detecting is one of the main approaches to understanding networks
\cite{For2010}.
  However it has been a longstanding challenge to give a definition for
community structures of networks. Here we found that community structures are
definable in networks, and are universal in real world. We proposed the notions
of entropy- and conductance-community structure ratios. It was shown that the
definitions of the modularity proposed in \cite{NG2004}, and our entropy- and
conductance-community structures are equivalent in defining community
structures of networks, that randomness in the ER model \cite{ER1960} and
preferential attachment in the PA \cite{Bar1999} model are not mechanisms of
community structures of networks, and that the existence of community
structures is a universal phenomenon in real networks. Our results demonstrate
that community structure is a universal phenomenon in the real world that is
definable, solving the challenge of definition of community structures in
networks. This progress provides a foundation for a structural theory of
networks.
