Relations between the set-complexity and the structure of graphs and their sub-graphs

Tomasz M Ignac,David J Galas,Nikita A Sakhanenko

doi:10.1186/1687-4153-2012-13

Abstract

We describe some new conceptual tools for the rigorous, mathematical description of the “set-complexity” of graphs. This set-complexity has been shown previously to be a useful measure for analyzing some biological networks, and in discussing biological information in a quantitative fashion. The advances described here allow us to define some significant relationships between the set-complexity measure and the structure of graphs, and of their component sub-graphs. We show here that modular graph structures tend to maximize the set-complexity of graphs. We point out the relationship between modularity and redundancy, and discuss the significance of set-complexity in this regard. We specifically discuss the relationship between complexity and entropy in the case of complete-bipartite graphs, and present a new method for constructing highly complex, binary graphs. These results can be extended to the case of ternary graphs, and to other multi-edge graphs, which are fundamentally more relevant to biological structures and systems. Finally, our results lead us to an approach for extracting high complexity modular graphs from large, noisy graphs with low information content. We illustrate this approach with two examples.

Highlights

Most physical, communications, social, and biological networks are usefully represented as graphs, with varying levels of complexity
Previous attempts to elucidate the fundamental concept of biological information have led to a proposed, general measure of complexity, or information content, based on Kolmogorov complexity [1,2], that resolves some of the perplexing paradoxes of biologically relevant meaning that arise in definitions of information and complexity [1]
A binary graph with the set-complexity score close to one exhibits a structure similar to an outcome of the transformation Fp applied to KN/2,N/2 with p ≈ 0.058. This result was generalized to graphs with M > 2, so-called multi-colored graphs. For this generalization we extended the definition of a complete bipartite graphs (CBG), and defined complete multi-partite graphs

Summary

Introduction

Communications, social, and biological networks are usefully represented as graphs, with varying levels of complexity. Our primary concern here is the representation and properties of biological networks, as reflected in the graphs used to represent these complex systems. Previous attempts to elucidate the fundamental concept of biological information have led to a proposed, general measure of complexity, or information content, based on Kolmogorov complexity [1,2], that resolves some of the perplexing paradoxes of biologically relevant meaning that arise in definitions of information and complexity [1]. We used this approach successfully in analyzing the information in gene interaction networks of yeast [3,4].

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Conclusion