Theory of Interface: Category Theory, Directed Networks and Evolution of Biological Networks

TL;DR

This paper applies category theory to biological networks, revealing dual static and dynamic modes with a trade-off in centrality measures, and models their evolution to resemble real biological networks.

Contribution

It introduces a category-theoretic framework for understanding biological network modes and their trade-offs, along with an evolutionary model demonstrating emergence of real-world network features.

Findings

Trade-off between transport and coherence centralities in biological networks

Evolved networks exhibit features similar to real biological networks

Dynamic mode dominance leads to networks with realistic properties

Abstract

Biological networks have two modes. The first mode is static: a network is a passage on which something flows. The second mode is dynamic: a network is a pattern constructed by gluing functions of entities constituting the network. In this paper, first we discuss that these two modes can be associated with the category theoretic duality (adjunction) and derive a natural network structure (a path notion) for each mode by appealing to the category theoretic universality. The path notion corresponding to the static mode is just the usual directed path. The path notion for the dynamic mode is called lateral path which is the alternating path considered on the set of arcs. Their general functionalities in a network are transport and coherence, respectively. Second, we introduce a betweenness centrality of arcs for each mode and see how the two modes are embedded in various real biological network data. We find that there is a trade-off relationship between the two centralities: if the value of one is large then the value of the other is small. This can be seen as a kind of division of labor in a network into transport on the network and coherence of the network. Finally, we propose an optimization model of networks based on a quality function involving intensities of the two modes in order to see how networks with the above trade-off relationship can emerge through evolution. We show that the trade-off relationship can be observed in the evolved networks only when the dynamic mode is dominant in the quality function by numerical simulations. We also show that the evolved networks have features qualitatively similar to real biological networks by standard complex network analysis.