Information-theoretic-based measures have been useful in quantifying network complexity. Here we briefly survey and contrast (algorithmic) information-theoretic methods which have been used to characterize graphs and networks. We illustrate the strengths and limitations of Shannon’s entropy, lossless compressibility and algorithmic complexity when used to identify aspects and properties of complex networks. We review the fragility of computable measures on the one hand and the invariant properties of algorithmic measures on the other demonstrating how current approaches to algorithmic complexity are misguided and suffer of similar limitations than traditional statistical approaches such as Shannon entropy. Finally, we review some current definitions of algorithmic complexity which are used in analyzing labelled and unlabelled graphs. This analysis opens up several new opportunities to advance beyond traditional measures.