The social network structure of animal populations has major implications to survival, reproductive success, sexual selection, and pathogen transmission. Recent studies showed in various species that the structure of social networks and individuals’ positions in it are influenced by individual traits such as sex, age, and social rank, and can be heritable between generations. But as of yet, no general theory of social network structure exists that can explain the diversity of social networks observed in nature, and serve as a null model for detecting species and population-specific factors. Here we propose such a general model of social network structure. We consider the emergence of network structure as a result of two types of social bond formation: via social inheritance, in which newborns are likely to bond with maternal contacts, and via forming bonds randomly. We compare model output to data from several species, showing that it can generate networks with properties such as those observed in real social systems. Our model demonstrates that some of the observed properties of social networks, such as heritability of network position or assortative associations, can be understood as a consequence of social inheritance. Our results highlight the need to consider the dynamic processes that generate social structure in order to explain patterns of variation in social networks.