Bird songs are acoustic communication signals primarily used in male-male aggression and in male-female attraction. These are often monotonous patterns composed of a few phrases, yet some birds have extremely complex songs with a large phrase repertoire, organized in non-random fashion with discernible patterns. Since structure is typically associated with function, the structures of complex bird songs provide important clues to the evolution of animal communication systems. Here we propose an efficient network-based approach to explore structural design principles of complex bird songs, in which the song networks--transition relationships among different phrases and the related structural measures--are employed. We demonstrate how this approach works with an example using California Thrasher songs, which are sequences of highly varied phrases delivered in succession over several minutes. These songs display two distinct features: a large phrase repertoire with a 'small-world' architecture, in which subsets of phrases are highly grouped and linked with a short average path length; and a balanced transition diversity amongst phrases, in which deterministic and non-deterministic transition patterns are moderately mixed. We explore the robustness of this approach with variations in sample size and the amount of noise. Our approach enables a more quantitative study of global and local structural properties of complex bird songs than has been possible to date.