We study complex networks in which the nodes of the network are tagged withdifferent colors depending on the functionality of the nodes (colored graphs),using information theory applied to the distribution of motifs in suchnetworks. We find that colored motifs can be viewed as the building blocks ofthe networks (much more so than the uncolored structural motifs can be) andthat the relative frequency with which these motifs appear in the network canbe used to define the information content of the network. This information isdefined in such a way that a network with random coloration (but keeping therelative number of nodes with different colors the same) has zero colorinformation content. Thus, colored motif information captures theexceptionality of coloring in the motifs that is maintained via selection. Westudy the motif information content of the C. elegans brain as well as theevolution of colored motif information in networks that reflect the interactionbetween instructions in genomes of digital life organisms. While we find thatcolored motif information appears to capture essential functionality in the C.elegans brain (where the color assignment of nodes is straightforward) it isnot obvious whether the colored motif information content always increasesduring evolution, as would be expected from a measure that captures networkcomplexity. For a single choice of color assignment of instructions in thedigital life form Avida, we find rather that colored motif information contentincreases or decreases during evolution, depending on how the genomes areorganized, and therefore could be an interesting tool to dissect genomicrearrangements.
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