Abstract
It is basic question in biology and other fields to identify the characteristic properties that on one hand are shared by structures from a particular realm, like gene regulation, protein–protein interaction or neural networks or foodwebs, and that on the other hand distinguish them from other structures. We introduce and apply a general method, based on the spectrum of the normalized graph Laplacian, that yields representations, the spectral plots, that allow us to find and visualize such properties systematically. We present such visualizations for a wide range of biological networks and compare them with those for networks derived from theoretical schemes. The differences that we find are quite striking and suggest that the search for universal properties of biological networks should be complemented by an understanding of more specific features of biological organization principles at different scales.
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Notes
As always in mathematics, there is a notion of isomorphism: Graphs Γ1 and Γ2 are called isomorphic when there is a one-to-one map ρ between the vertices of Γ1 and Γ2 that preserves the neighborhood relationship, that is i∼ j precisely if ρ(i)∼ ρ(j). Isomorphic graphs are considered to be the same because they cannot be distinguished by their properties. In other words, when we speak about different graphs, we mean non-isomorphic ones.
In more precise terms, the degrees are Poisson distributed in the limit of an infinite graph size.
All networks are taken as undirected and unweighted. Thus, we suppress some potentially important aspects of the underlying data, but as our plots will show, we can still detect distinctive qualitative patterns. In fact, one can also compute the spectrum of directed and weighted networks, and doing that on our data will reveal further structures, but this is not explored in the present paper.
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Banerjee, A., Jost, J. Spectral plots and the representation and interpretation of biological data. Theory Biosci. 126, 15–21 (2007). https://doi.org/10.1007/s12064-007-0005-9
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DOI: https://doi.org/10.1007/s12064-007-0005-9