Community Structures Are Definable in Networks, and Universal in Real World

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.