Information Content of Colored Motifs in Complex Networks

Abstract

We study complex networks in which the nodes of the network are tagged with different colors depending on the functionality of the nodes (colored graphs), using information theory applied to the distribution of motifs in such networks. We find that colored motifs can be viewed as the building blocks of the networks (much more so than the uncolored structural motifs can be) and that the relative frequency with which these motifs appear in the network can be used to define the information content of the network. This information is defined in such a way that a network with random coloration (but keeping the relative number of nodes with different colors the same) has zero color information content. Thus, colored motif information captures the exceptionality of coloring in the motifs that is maintained via selection. We study the motif information content of the C. elegans brain as well as the evolution of colored motif information in networks that reflect the interaction between instructions in genomes of digital life organisms. While we find that colored motif information appears to capture essential functionality in the C. elegans brain (where the color assignment of nodes is straightforward) it is not obvious whether the colored motif information content always increases during evolution, as would be expected from a measure that captures network complexity. For a single choice of color assignment of instructions in the digital life form Avida, we find rather that colored motif information content increases or decreases during evolution, depending on how the genomes are organized, and therefore could be an interesting tool to dissect genomic rearrangements.

Correlation of automorphism group size and topological properties with program-size complexity evaluations of graphs and complex networks

“…An interesting perspective regarding such graphs is to investigate the complexity [3,4] or information content of a graph [5]. While Shannon information theory [5][6][7] and counting symmetries [8,9] have been applied to measure information content/complexity of graphs, little has been done, by contrast, to demonstrate the utility of Kolmogorov complexity as a numerical tool for graph and real-world network investigations. Some theoretical connections between graphs and algorithmic randomness have been explored (e.g.…”

Dissecting molecular network structures using a network subgraph approach

“…In other words, we adopt the notion that N-node subgraphs are the basic building blocks, but do not necessarily occur frequently in a network. Adami et al (2011) studied undirected colored graphs (where nodes are labeled with different colors) and showed that the relative frequency of the colored motifs can be used to define the information content of the network. Here, we consider subgraphs that are directed graphs which could contain cycles.…”

Discovering Colored Network Motifs

“…There has been some work on motif detection using colors only in nodes [9], but not with colored edges, even if it was acknowledged that incorporating connection types would lead to richer results [6]. The work by Adami et al [5] considered the case of colors only in nodes and uses an entropy-based measurement to determine significance. Quian et al [10] also use colors solely in nodes and swap colors in the network, that is, the randomized networks are topological equivalent to the original, but with a different permutation of the colors in the nodes.…”

“…For example, in metabolic networks, we can distinguish between two sets of nodes: reactions and chemical compounds [4]. By ignoring these labels we may be missing important patterns, and it has been shown that by associating different colors to the Pedro Ribeiro • Fernando Silva CRACS & INESC-TEC, DCC-FCUP, Universidade do Porto, Portugal e-mail: pribeiro@dcc.fc.up.pt, fds@dcc.fc.up.pt nodes, their information content is richer [5]. The same can be said about edges, and previous experiments have shown that we would gain if it was possible to distinguish between different types of connections in biological networks [6].…”