  Many complex networks depend upon biological entities for their preservation.
Such entities, from human cognition to evolution, must first encode and then
replicate those networks under marked resource constraints. Networks that
survive are those that are amenable to constrained encoding, or, in other
words, are compressible. But how compressible is a network? And what features
make one network more compressible than another? Here we answer these questions
by modeling networks as information sources before compressing them using
rate-distortion theory. Each network yields a unique rate-distortion curve,
which specifies the minimal amount of information that remains at a given scale
of description. A natural definition then emerges for the compressibility of a
network: the amount of information that can be removed via compression,
averaged across all scales. Analyzing an array of real and model networks, we
demonstrate that compressibility increases with two common network properties:
transitivity (or clustering) and degree heterogeneity. These results indicate
that hierarchical organization -- which is characterized by modular structure
and heterogeneous degrees -- facilitates compression in complex networks.
Generally, our framework sheds light on the interplay between a network's
structure and its capacity to be compressed, enabling investigations into the
role of compression in shaping real-world networks.
