What is the nature of language? How has it evolved in different species? Are there qualitative, well-defined classes of languages? Most studies of language evolution deal in a way or another with such theoretical contraption and explore the outcome of diverse forms of selection on the communication matrix that somewhat optimizes communication. This framework naturally introduces networks mediating the communicating agents, but no systematic analysis of the underlying landscape of possible language graphs has been developed. Here we present a detailed analysis of network properties on a generic model of a communication code, which reveals a rather complex and heterogeneous morphospace of language graphs. Additionally, we use curated data of English words to locate and evaluate real languages within this morphospace. Our findings indicate a surprisingly simple structure in human language unless particles with the ability of naming any other concept are introduced in the vocabulary. These results refine and for the first time complement with empirical data a lasting theoretical tradition around the framework of least effort language.