21,781 research outputs found
Degree Distribution of Competition-Induced Preferential Attachment Graphs
We introduce a family of one-dimensional geometric growth models, constructed
iteratively by locally optimizing the tradeoffs between two competing metrics,
and show that this family is equivalent to a family of preferential attachment
random graph models with upper cutoffs. This is the first explanation of how
preferential attachment can arise from a more basic underlying mechanism of
local competition. We rigorously determine the degree distribution for the
family of random graph models, showing that it obeys a power law up to a finite
threshold and decays exponentially above this threshold.
We also rigorously analyze a generalized version of our graph process, with
two natural parameters, one corresponding to the cutoff and the other a
``fertility'' parameter. We prove that the general model has a power-law degree
distribution up to a cutoff, and establish monotonicity of the power as a
function of the two parameters. Limiting cases of the general model include the
standard preferential attachment model without cutoff and the uniform
attachment model.Comment: 24 pages, one figure. To appear in the journal: Combinatorics,
Probability and Computing. Note, this is a long version, with complete
proofs, of the paper "Competition-Induced Preferential Attachment"
(cond-mat/0402268
Evolutionary Model of the Personal Income Distribution
The aim of this work is to establish the personal income distribution from
the elementary constituents of a free market; products of a representative good
and agents forming the economic network. The economy is treated as a
self-organized system. Based on the idea that the dynamics of an economy is
governed by slow modes, the model suggests that for short time intervals a
fixed ratio of total labour income (capital income) to net income exists
(Cobb-Douglas relation). Explicitly derived is Gibrat's law from an
evolutionary market dynamics of short term fluctuations. The total private
income distribution is shown to consist of four main parts. From capital income
of private firms the income distribution contains a lognormal distribution for
small and a Pareto tail for large incomes. Labour income contributes an
exponential distribution. Also included is the income from a social insurance
system, approximated by a Gaussian peak. The evolutionary model is able to
reproduce the stylized facts of the income distribution, shown by a comparison
with empirical data of a high resolution income distribution. The theory
suggests that in a free market competition between products is ultimately the
origin of the uneven income distribution
Preferential attachment in growing spatial networks
We obtain the degree distribution for a class of growing network models on
flat and curved spaces. These models evolve by preferential attachment weighted
by a function of the distance between nodes. The degree distribution of these
models is similar to the one of the fitness model of Bianconi and Barabasi,
with a fitness distribution dependent on the metric and the density of nodes.
We show that curvature singularities in these spaces can give rise to
asymptotic Bose-Einstein condensation, but transient condensation can be
observed also in smooth hyperbolic spaces with strong curvature. We provide
numerical results for spaces of constant curvature (sphere, flat and hyperbolic
space) and we discuss the conditions for the breakdown of this approach and the
critical points of the transition to distance-dominated attachment. Finally we
discuss the distribution of link lengths.Comment: 9 pages, 12 figures, revtex, final versio
Evolutionary Model of the Growth and Size of Firms
The key idea of this model is that firms are the result of an evolutionary
process. Based on demand and supply considerations the evolutionary model
presented here derives explicitly Gibrat's law of proportionate effects as the
result of the competition between products. Applying a preferential attachment
mechanism for firms the theory allows to establish the size distribution of
products and firms. Also established are the growth rate and price distribution
of consumer goods. Taking into account the characteristic property of human
activities to occur in bursts, the model allows also an explanation of the
size-variance relationship of the growth rate distribution of products and
firms. Further the product life cycle, the learning (experience) curve and the
market size in terms of the mean number of firms that can survive in a market
are derived. The model also suggests the existence of an invariant of a market
as the ratio of total profit to total revenue. The relationship between a
neo-classic and an evolutionary view of a market is discussed. The comparison
with empirical investigations suggests that the theory is able to describe the
main stylized facts concerning the size and growth of firms
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