7 research outputs found
Exactly solvable analogy of small-world networks
We present an exact description of a crossover between two different regimes
of simple analogies of small-world networks. Each of the sites chosen with a
probability from sites of an ordered system defined on a circle is
connected to all other sites selected in such a way. Every link is of a unit
length. Thus, while changes from 0 to 1, an averaged shortest distance
between a pair of sites changes from to .
We find the distribution of the shortest distances and obtain a
scaling form of . In spite of the simplicity of the models
under consideration, the results appear to be surprisingly close to those
obtained numerically for usual small-world networks.Comment: 4 pages with 3 postscript figure
Stochastic Aggregation: Rate Equations Approach
We investigate a class of stochastic aggregation processes involving two
types of clusters: active and passive. The mass distribution is obtained
analytically for several aggregation rates. When the aggregation rate is
constant, we find that the mass distribution of passive clusters decays
algebraically. Furthermore, the entire range of acceptable decay exponents is
possible. For aggregation rates proportional to the cluster masses, we find
that gelation is suppressed. In this case, the tail of the mass distribution
decays exponentially for large masses, and as a power law over an intermediate
size range.Comment: 7 page
Ising Model on Networks with an Arbitrary Distribution of Connections
We find the exact critical temperature of the nearest-neighbor
ferromagnetic Ising model on an `equilibrium' random graph with an arbitrary
degree distribution . We observe an anomalous behavior of the
magnetization, magnetic susceptibility and specific heat, when is
fat-tailed, or, loosely speaking, when the fourth moment of the distribution
diverges in infinite networks. When the second moment becomes divergent,
approaches infinity, the phase transition is of infinite order, and size effect
is anomalously strong.Comment: 5 page
Mesoscopics and fluctuations in networks
We describe fluctuations in finite-size networks with a complex distribution
of connections, . We show that the spectrum of fluctuations of the number
of vertices with a given degree is Poissonian. These mesoscopic fluctuations
are strong in the large-degree region, where ( is the
total number of vertices in a network), and are important in networks with
fat-tailed degree distributions.Comment: 3 pages, 1 figur