3 research outputs found
Spectral plots and the representation and interpretation of biological data
It is basic question in biology and other fields to identify the char-
acteristic properties that on one hand are shared by structures from a
particular realm, like gene regulation, protein-protein interaction or neu- ral
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 repre- sentations, 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.Comment: 15 pages, 7 figure
Neighborhood properties of complex networks
A concept of neighborhood in complex networks is addressed based on the
criterion of the minimal number os steps to reach other vertices. This amounts
to, starting from a given network , generating a family of networks
such that, the vertices that are steps apart in
the original , are only 1 step apart in . The higher order
networks are generated using Boolean operations among the adjacency matrices
that represent . The families originated by the well known
linear and the Erd\"os-Renyi networks are found to be invariant, in the sense
that the spectra of are the same, up to finite size effects. A further
family originated from small world network is identified
q-Newton binomial: from Euler to Gauss
A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made
less mysterious by virtue of being generalized through the introduction of an
additional parameter