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

Tomasz M Ignac, Nikita A Sakhanenko, David J Galas
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引用次数: 5

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.

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图及其子图的结构与集合复杂度的关系。
我们描述了一些新的概念工具,用于图的“集合复杂度”的严格的数学描述。这种集的复杂性已经被证明是分析某些生物网络和以定量方式讨论生物信息的有用度量。这里描述的进展使我们能够定义集复杂度度量与图及其组成子图的结构之间的一些重要关系。我们在这里展示了模图结构倾向于最大化图的集合复杂度。我们指出了模块化和冗余的关系,并讨论了集合复杂度在这方面的意义。我们特别讨论了完全二部图的复杂度与熵的关系,并提出了一种构造高度复杂二部图的新方法。这些结果可以推广到三元图的情况下,以及其他多边图,从根本上更相关的生物结构和系统。最后,我们的结果使我们能够从信息量低的大型噪声图中提取高复杂性模图。我们用两个例子来说明这种方法。
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