184 research outputs found
How to calculate the main characteristics of random uncorrelated networks
We present an analytic formalism describing structural properties of random
uncorrelated networks with arbitrary degree distributions. The formalism allows
to calculate the main network characteristics like: the position of the phase
transition at which a giant component first forms, the mean component size
below the phase transition, the size of the giant component and the average
path length above the phase transition. We apply the approach to classical
random graphs of Erdos and Renyi, single-scale networks with exponential degree
distributions and scale-free networks with arbitrary scaling exponents and
structural cut-offs. In all the cases we obtain a very good agreement between
results of numerical simulations and our analytical predictions.Comment: AIP conference proceedings format, 17 pages, 6 figure
Kauffman Boolean model in undirected scale free networks
We investigate analytically and numerically the critical line in undirected
random Boolean networks with arbitrary degree distributions, including
scale-free topology of connections . We show that in
infinite scale-free networks the transition between frozen and chaotic phase
occurs for . The observation is interesting for two reasons.
First, since most of critical phenomena in scale-free networks reveal their
non-trivial character for , the position of the critical line in
Kauffman model seems to be an important exception from the rule. Second, since
gene regulatory networks are characterized by scale-free topology with
, the observation that in finite-size networks the mentioned
transition moves towards smaller is an argument for Kauffman model as
a good starting point to model real systems. We also explain that the
unattainability of the critical line in numerical simulations of classical
random graphs is due to percolation phenomena
Supremacy distribution in evolving networks
We study a supremacy distribution in evolving Barabasi-Albert networks. The
supremacy of a node is defined as a total number of all nodes that
are younger than and can be connected to it by a directed path. For a
network with a characteristic parameter the supremacy of an
individual node increases with the network age as in an
appropriate scaling region. It follows that there is a relation between a node degree and its supremacy and the supremacy
distribution scales as . Analytic calculations basing on
a continuum theory of supremacy evolution and on a corresponding rate equation
have been confirmed by numerical simulations.Comment: 4 pages, 4 figure
Statistical mechanics of the international trade network
Analyzing real data on international trade covering the time interval
1950-2000, we show that in each year over the analyzed period the network is a
typical representative of the ensemble of maximally random weighted networks,
whose directed connections (bilateral trade volumes) are only characterized by
the product of the trading countries' GDPs. It means that time evolution of
this network may be considered as a continuous sequence of equilibrium states,
i.e. quasi-static process. This, in turn, allows one to apply the linear
response theory to make (and also verify) simple predictions about the network.
In particular, we show that bilateral trade fulfills fluctuation-response
theorem, which states that the average relative change in import (export)
between two countries is a sum of relative changes in their GDPs. Yearly
changes in trade volumes prove that the theorem is valid.Comment: 6 pages, 2 figure
Mean-field theory for clustering coefficients in Barabasi-Albert networks
We applied a mean field approach to study clustering coefficients in
Barabasi-Albert networks. We found that the local clustering in BA networks
depends on the node degree. Analytic results have been compared to extensive
numerical simulations finding a very good agreement for nodes with low degrees.
Clustering coefficient of a whole network calculated from our approach
perfectly fits numerical data.Comment: 8 pages, 3 figure
Ferromagnetic fluid as a model of social impact
The paper proposes a new model of spin dynamics which can be treated as a
model of sociological coupling between individuals. Our approach takes into
account two different human features: gregariousness and individuality. We will
show how they affect a psychological distance between individuals and how the
distance changes the opinion formation in a social group. Apart from its
sociological aplications the model displays the variety of other interesting
phenomena like self-organizing ferromagnetic state or a second order phase
transition and can be studied from different points of view, e.g. as a model of
ferromagnetic fluid, complex evolving network or multiplicative random process.Comment: 8 pages, 5 figure
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