30 research outputs found
Organization of complex networks without multiple connections
We find a new structural feature of equilibrium complex random networks
without multiple and self-connections. We show that if the number of
connections is sufficiently high, these networks contain a core of highly
interconnected vertices. The number of vertices in this core varies in the
range between and , where is the number of
vertices in a network. At the birth point of the core, we obtain the
size-dependent cut-off of the distribution of the number of connections and
find that its position differs from earlier estimates.Comment: 5 pages, 2 figure
Multifractal properties of growing networks
We introduce a new family of models for growing networks. In these networks
new edges are attached preferentially to vertices with higher number of
connections, and new vertices are created by already existing ones, inheriting
part of their parent's connections. We show that combination of these two
features produces multifractal degree distributions, where degree is the number
of connections of a vertex. An exact multifractal distribution is found for a
nontrivial model of this class. The distribution tends to a power-law one, , in the infinite network limit.
Nevertheless, for finite networks's sizes, because of multifractality, attempts
to interpret the distribution as a scale-free would result in an ambiguous
value of the exponent .Comment: 7 pages epltex, 1 figur
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
Giant strongly connected component of directed networks
We describe how to calculate the sizes of all giant connected components of a
directed graph, including the {\em strongly} connected one. Just to the class
of directed networks, in particular, belongs the World Wide Web. The results
are obtained for graphs with statistically uncorrelated vertices and an
arbitrary joint in,out-degree distribution . We show that if
does not factorize, the relative size of the giant strongly
connected component deviates from the product of the relative sizes of the
giant in- and out-components. The calculations of the relative sizes of all the
giant components are demonstrated using the simplest examples. We explain that
the giant strongly connected component may be less resilient to random damage
than the giant weakly connected one.Comment: 4 pages revtex, 4 figure