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.
References
Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97
Atay FM, Jost J, Wende A (2004) Delays, connection topology, and synchronization of coupled chaotic maps. Phys Rev Lett 92(14):144101
Atay FM, Bıyıkoğlu T, Jost J (2006) Synchronization of networks with prescribed degree distributions. IEEE Trans Circuits Syst I 53(1):92–98
Atay FM, Bıyıkoğlu T, Jost J (2007) Network synchronization: spectral versus statistical properties. Phys D (to appear)
Banerjee A, Jost J (2007a) Laplacian spectrum and protein–protein interaction networks (preprint)
Banerjee A, Jost J (2007b) On the spectrum of the normalized graph Laplacian (preprint)
Banerjee A, Jost J (2007c) Graph spectra as a systematic tool in computational biology. Discrete Appl Math (submitted)
Barabási A-L, Albert RA (1999) Emergence of scaling in random networks. Science 286:509–512
Bolobás B (1998) Modern graph theory. Springer, Berlin
Bolobás B (2001) Random graphs. Cambridge University Press, London
Chung F (1997) Spectral graph theory. AMS, New York
Dorogovtsev SN, Mendes JFF (2003) Evolution of Networks. Oxford University Press, Oxford
Erdős P, Rényi A (1959) On random graphs. Public Math Debrecen 6:290–297
Godsil C, Royle G (2001) Algebraic graph theory. Springer, Berlin
Ipsen M, Mikhailov AS (2002) Evolutionary reconstruction of networks. Phys Rev E 66(4):046109
Jeong H, Tombor B, Albert R, Oltval ZN, Barabási AL (2000) The large-scale organization of metabolic networks. Nature 407(6804):651–654
Jost J (2007a) Mathematical methods in biology and neurobiology, monograph (to appear)
Jost J (2007b) Dynamical networks. In: Feng JF, Jost J, Qian MP (eds) Networks: from biology to theory. Springer, Berlin
Jost J, Joy MP (2001) Spectral properties and synchronization in coupled map lattices. Phys Rev E 65(1):16201–16209
Jost J, Joy MP (2002) Evolving networks with distance preferences. Phys Rev E 66:36126–36132
Merris R (1994) Laplacian matrices of graphs—a survey. Lin Alg Appl 198:143–176
Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network Motifs: simple building blocks of complex networks. Science 298(5594):824–827
Mohar B (1997) Some applications of Laplace eigenvalues of graphs. In: Hahn G, Sabidussi G (eds) Graph symmetry: algebraic methods and applications. Springer, Berlin, pp 227–277
Newman M (2003) The structure and function of complex networks. SIAM Rev 45:167–256
Ohno S (1970) Evolution by gene duplication. Springer, Berlin
Shen-Orr SS, Milo R, Mangan S, Alon U (2002) Network Motifs in the transcriptional regulation network of Escherichia coli. Nat Genet 31(1):64–68
Simon H (1955) On a class of skew distribution functions. Biometrika 42:425–440
Vázquez A (2002) Growing networks with local rules: Preferential attachment, clustering hierarchy and degree correlations. cond-mat/0211528
Wagner A (1994) Evolution of gene networks by gene duplications—a mathematical model and its implications on genome organization. Proc Nat Acad Sci USA 91(10):4387–4391
Watts DJ, Strogatz SH (1998) Collective dynamics of ‘Small-World’ networks. Nature 393(6684):440–442
White JG, Southgate E, Thomson JN, Brenner S (1986) The structure of the nervous-system of the Nematode Caenorhabditis-Elegans. Philos Trans R Soc Lond Ser B Biol Sci 314(1165):1–340
Wolfe KH, Shields DC (1997) Molecular evidence for an ancient duplication of the entire yeast genome. Nature 387(6634):708–713
Zhu P, Wilson R (2005) A study of graph spectra for comparing graphs. In: British machine vision conference 2005, BMVA, pp 679–688
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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