We  introduce concepts and tools at the intersection of information theory and network biology. We show that Shannon's information entropy, compressibility and algorithmic probability quantify different aspects of synthetic and biological networks at the intersection of local and global pattern detection  , including, the detection of the connectivity phase transition leading to the emergence of giant components in Erd\"os-R\'enyi random graphs, and the recovery of topological properties from numerical kinetic properties simulating gene expression arrays . We provide exact theoretical calculations, numerical approximations and error estimations of entropy, algorithmic probability and Kolmogorov complexity for different types of graphs characterizing their variant and invariant properties. We introduce formal definitions of complexity for both labelled and unlabelled graphs and prove that the Kolmogorov complexity of a labelled graph is a good approximation of the Kolmogorov complexity of the unlabelled graph  .
We survey and introduce concepts and tools at the intersection of information theory and network biology. We show that Shannon's information entropy, compressibility and algorithmic complexity quantify different aspects of synthetic and biological networks at the intersection of local and global pattern detection . This includes, for example , the detection of the connectivity phase transition leading to the emergence of giant components in Erdos-Renyi random graphs, and the recovery of topological properties from numerical kinetic properties simulating gene expression data . We provide exact theoretical calculations, numerical approximations and error estimations of entropy, algorithmic probability and Kolmogorov complexity for different types of graphs characterizing their variant and invariant properties. We introduce formal definitions of complexity for both labelled and unlabelled graphs and prove that the Kolmogorov complexity of a labelled graph is a good approximation of the Kolmogorov complexity of the unlabelled graph and thus a robust definition of graph complexity .
A
survey and
R
probability
complexity
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. This includes, for example
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including,
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Erd\"os-R\'enyi
Erdos-Renyi
R
arrays
data
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and thus a robust definition of graph complexity