Mathematically, graphs defy a systematic and complete classification, and empirically, the graphs representing networks come in a bewildering multitude. We have developed some tools (8; 9; 710) that at least allow for a rough classification of graphs that reflects the difference in the empirical domains from which network data are produced and that does not depend on sophisticated visualization tools.
As such, a graph is a rather simple formal structure. It consists of nodes or vertices that are connected by edges or links. These nodes then represent the elements of a network (and we shall often not distinguish between the network and its underlying graph), and the edges represent relations between them. These could be chemical interactions as in intracellular networks of genes, proteins, or metabolites, synaptic connections between neurons, physical links in infrastructural networks, links between Internet pages, co-occurrences between words in sentences or on text pages, email contacts between people, co-authorships between scientists, and so on. This structure then can be expected to be somehow adapted to the function of the network, by evolution, self-organization, or design. In turn, any dynamics supported by the network will be constrained by this underlying structure.
Our approach is based on associating certain mathematical objects—which ultimately just yield some numbers—to a graph which reflect its structural properties and which in particular encode the constraints on the dynamics that it can support. The mathematical objects will be an operator, the graph Laplacian (a discrete analogue of the Laplace operator in real analysis), and its eigenfunctions, and the numbers alluded to will be the eigenvalues of that operator.
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L.A. Adamic and N. Glance, The political blogosphere and the 2004 US election: Divided they blog, in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005)Google Scholar
R. Guimera et al, Self-similar community structure in a network of human interactions, Physical Review E 68, 2003, 065103(R)CrossRefGoogle Scholar
M. Horton, H. Stark, A. Terras, What are zeta functions of graphs and what are they good for? In Quantum graphs and their applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 415, 2006, 173–189Google Scholar
M. Ipsen, A.S. Mikhailov, Evolutionary reconstruction of networks, Phys. Rev. E 66(4), 046109, 2002Google Scholar
H. Jeong et al, The large-scale organization of metabolic networks, Nature 407, 2000, 651–654CrossRefGoogle Scholar
J. Jost, Mathematical methods in biology and neurobiology, monograph, to appearGoogle Scholar
J. Jost, in: J.F. Feng, J. Jost, M.P. Qian (eds.), Networks: From Biology to Theory, 35–62, Springer, Berlin, 2007CrossRefzbMATHGoogle Scholar
A. Vazquez et al., Modelling of protein interaction networks, ComPlexUs 1, 2003, 38–44Google Scholar
A. Wagner, How the global structure of protein interaction networks evolves, Proc. Roy. Soc. B 270, 2003, 457–466CrossRefGoogle Scholar
A. Wagner, Evolution of gene networks by gene duplications — A mathematical model and its implications on genome organization, Proc. Nat. Acad. Sciences USA 91(10), 1994, 4387–4391CrossRefzbMATHGoogle Scholar