2,315 research outputs found
Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?
We study the Laplacian operator of an uncorrelated random network and, as an
application, consider hopping processes (diffusion, random walks, signal
propagation, etc.) on networks. We develop a strict approach to these problems.
We derive an exact closed set of integral equations, which provide the averages
of the Laplacian operator's resolvent. This enables us to describe the
propagation of a signal and random walks on the network. We show that the
determining parameter in this problem is the minimum degree of vertices
in the network and that the high-degree part of the degree distribution is not
that essential. The position of the lower edge of the Laplacian spectrum
appears to be the same as in the regular Bethe lattice with the
coordination number . Namely, if , and
if . In both these cases the density of eigenvalues
as , but the limiting behaviors near
are very different. In terms of a distance from a starting vertex,
the hopping propagator is a steady moving Gaussian, broadening with time. This
picture qualitatively coincides with that for a regular Bethe lattice. Our
analytical results include the spectral density near
and the long-time asymptotics of the autocorrelator and the
propagator.Comment: 25 pages, 4 figure
p-forms on d-spherical tessellations
The spectral properties of p-forms on the fundamental domains of regular
tesselations of the d-dimensional sphere are discussed. The degeneracies for
all ranks, p, are organised into a double Poincare series which is explicitly
determined. In the particular case of coexact forms of rank (d-1)/2, for odd d,
it is shown that the heat--kernel expansion terminates with the constant term,
which equals (-1)^{p+1}/2 and that the boundary terms also vanish, all as
expected. As an example of the double domain construction, it is shown that the
degeneracies on the sphere are given by adding the absolute and relative
degeneracies on the hemisphere, again as anticipated. The eta invariant on a
fundamental domain is computed to be irrational. The spectral counting function
is calculated and the accumulated degeneracy give exactly. A generalised
Weyl-Polya conjecture for p-forms is suggested and verified.Comment: 23 pages. Section on the counting function adde
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