674 research outputs found
k-Generalized Statistics in Personal Income Distribution
Starting from the generalized exponential function
, with
, proposed in Ref. [G. Kaniadakis, Physica A \textbf{296},
405 (2001)], the survival function ,
where , , and , is
considered in order to analyze the data on personal income distribution for
Germany, Italy, and the United Kingdom. The above defined distribution is a
continuous one-parameter deformation of the stretched exponential function
\textemdash to which reduces as
approaches zero\textemdash behaving in very different way in the and
regions. Its bulk is very close to the stretched exponential one,
whereas its tail decays following the power-law
. This makes the
-generalized function particularly suitable to describe simultaneously
the income distribution among both the richest part and the vast majority of
the population, generally fitting different curves. An excellent agreement is
found between our theoretical model and the observational data on personal
income over their entire range.Comment: Latex2e v1.6; 14 pages with 12 figures; for inclusion in the APFA5
Proceeding
Zipf's law in Nuclear Multifragmentation and Percolation Theory
We investigate the average sizes of the largest fragments in nuclear
multifragmentation events near the critical point of the nuclear matter phase
diagram. We perform analytic calculations employing Poisson statistics as well
as Monte Carlo simulations of the percolation type. We find that previous
claims of manifestations of Zipf's Law in the rank-ordered fragment size
distributions are not born out in our result, neither in finite nor infinite
systems. Instead, we find that Zipf-Mandelbrot distributions are needed to
describe the results, and we show how one can derive them in the infinite size
limit. However, we agree with previous authors that the investigation of
rank-ordered fragment size distributions is an alternative way to look for the
critical point in the nuclear matter diagram.Comment: 8 pages, 11 figures, submitted to PR
Universal scaling in sports ranking
Ranking is a ubiquitous phenomenon in the human society. By clicking the web
pages of Forbes, you may find all kinds of rankings, such as world's most
powerful people, world's richest people, top-paid tennis stars, and so on and
so forth. Herewith, we study a specific kind, sports ranking systems in which
players' scores and prize money are calculated based on their performances in
attending various tournaments. A typical example is tennis. It is found that
the distributions of both scores and prize money follow universal power laws,
with exponents nearly identical for most sports fields. In order to understand
the origin of this universal scaling we focus on the tennis ranking systems. By
checking the data we find that, for any pair of players, the probability that
the higher-ranked player will top the lower-ranked opponent is proportional to
the rank difference between the pair. Such a dependence can be well fitted to a
sigmoidal function. By using this feature, we propose a simple toy model which
can simulate the competition of players in different tournaments. The
simulations yield results consistent with the empirical findings. Extensive
studies indicate the model is robust with respect to the modifications of the
minor parts.Comment: 8 pages, 7 figure
Bidding process in online auctions and winning strategy:rate equation approach
Online auctions have expanded rapidly over the last decade and have become a
fascinating new type of business or commercial transaction in this digital era.
Here we introduce a master equation for the bidding process that takes place in
online auctions. We find that the number of distinct bidders who bid times,
called the -frequent bidder, up to the -th bidding progresses as
. The successfully transmitted bidding rate by the
-frequent bidder is obtained as , independent of
for large . This theoretical prediction is in agreement with empirical data.
These results imply that bidding at the last moment is a rational and effective
strategy to win in an eBay auction.Comment: 4 pages, 6 figure
Power-law distributions from additive preferential redistributions
We introduce a non-growth model that generates the power-law distribution
with the Zipf exponent. There are N elements, each of which is characterized by
a quantity, and at each time step these quantities are redistributed through
binary random interactions with a simple additive preferential rule, while the
sum of quantities is conserved. The situation described by this model is
similar to those of closed -particle systems when conservative two-body
collisions are only allowed. We obtain stationary distributions of these
quantities both analytically and numerically while varying parameters of the
model, and find that the model exhibits the scaling behavior for some parameter
ranges. Unlike well-known growth models, this alternative mechanism generates
the power-law distribution when the growth is not expected and the dynamics of
the system is based on interactions between elements. This model can be applied
to some examples such as personal wealths, city sizes, and the generation of
scale-free networks when only rewiring is allowed.Comment: 12 pages, 4 figures; Changed some expressions and notations; Added
more explanations and changed the order of presentation in Sec.III while
results are the sam
Scaling laws of strategic behaviour and size heterogeneity in agent dynamics
The dynamics of many socioeconomic systems is determined by the decision
making process of agents. The decision process depends on agent's
characteristics, such as preferences, risk aversion, behavioral biases, etc..
In addition, in some systems the size of agents can be highly heterogeneous
leading to very different impacts of agents on the system dynamics. The large
size of some agents poses challenging problems to agents who want to control
their impact, either by forcing the system in a given direction or by hiding
their intentionality. Here we consider the financial market as a model system,
and we study empirically how agents strategically adjust the properties of
large orders in order to meet their preference and minimize their impact. We
quantify this strategic behavior by detecting scaling relations of allometric
nature between the variables characterizing the trading activity of different
institutions. We observe power law distributions in the investment time
horizon, in the number of transactions needed to execute a large order and in
the traded value exchanged by large institutions and we show that heterogeneity
of agents is a key ingredient for the emergence of some aggregate properties
characterizing this complex system.Comment: 6 pages, 3 figure
Predicted and Verified Deviations from Zipf's law in Ecology of Competing Products
Zipf's power-law distribution is a generic empirical statistical regularity
found in many complex systems. However, rather than universality with a single
power-law exponent (equal to 1 for Zipf's law), there are many reported
deviations that remain unexplained. A recently developed theory finds that the
interplay between (i) one of the most universal ingredients, namely stochastic
proportional growth, and (ii) birth and death processes, leads to a generic
power-law distribution with an exponent that depends on the characteristics of
each ingredient. Here, we report the first complete empirical test of the
theory and its application, based on the empirical analysis of the dynamics of
market shares in the product market. We estimate directly the average growth
rate of market shares and its standard deviation, the birth rates and the
"death" (hazard) rate of products. We find that temporal variations and product
differences of the observed power-law exponents can be fully captured by the
theory with no adjustable parameters. Our results can be generalized to many
systems for which the statistical properties revealed by power law exponents
are directly linked to the underlying generating mechanism
Generalized (m,k)-Zipf law for fractional Brownian motion-like time series with or without effect of an additional linear trend
We have translated fractional Brownian motion (FBM) signals into a text based
on two ''letters'', as if the signal fluctuations correspond to a constant
stepsize random walk. We have applied the Zipf method to extract the
exponent relating the word frequency and its rank on a log-log plot. We have
studied the variation of the Zipf exponent(s) giving the relationship between
the frequency of occurrence of words of length made of such two letters:
is varying as a power law in terms of . We have also searched how
the exponent of the Zipf law is influenced by a linear trend and the
resulting effect of its slope. We can distinguish finite size effects, and
results depending whether the starting FBM is persistent or not, i.e. depending
on the FBM Hurst exponent . It seems then numerically proven that the Zipf
exponent of a persistent signal is more influenced by the trend than that of an
antipersistent signal. It appears that the conjectured law
only holds near . We have also introduced considerations based on the
notion of a {\it time dependent Zipf law} along the signal.Comment: 24 pages, 12 figures; to appear in Int. J. Modern Phys
Proportionate vs disproportionate distribution of wealth of two individuals in a tempered Paretian ensemble
We study the distribution P(\omega) of the random variable \omega = x_1/(x_1
+ x_2), where x_1 and x_2 are the wealths of two individuals selected at random
from the same tempered Paretian ensemble characterized by the distribution
\Psi(x) \sim \phi(x)/x^{1 + \alpha}, where \alpha > 0 is the Pareto index and
is the cut-off function. We consider two forms of \phi(x): a bounded
function \phi(x) = 1 for L \leq x \leq H, and zero otherwise, and a smooth
exponential function \phi(x) = \exp(-L/x - x/H). In both cases \Psi(x) has
moments of arbitrary order.
We show that, for \alpha > 1, P(\omega) always has a unimodal form and is
peaked at \omega = 1/2, so that most probably x_1 \approx x_2. For 0 < \alpha <
1 we observe a more complicated behavior which depends on the value of \delta =
L/H. In particular, for \delta < \delta_c - a certain threshold value -
P(\omega) has a three-modal (for a bounded \phi(x)) and a bimodal M-shape (for
an exponential \phi(x)) form which signifies that in such ensembles the wealths
x_1 and x_2 are disproportionately different.Comment: 9 pages, 8 figures, to appear in Physica
Emergence of Zipf's Law in the Evolution of Communication
Zipf's law seems to be ubiquitous in human languages and appears to be a
universal property of complex communicating systems. Following the early
proposal made by Zipf concerning the presence of a tension between the efforts
of speaker and hearer in a communication system, we introduce evolution by
means of a variational approach to the problem based on Kullback's Minimum
Discrimination of Information Principle. Therefore, using a formalism fully
embedded in the framework of information theory, we demonstrate that Zipf's law
is the only expected outcome of an evolving, communicative system under a
rigorous definition of the communicative tension described by Zipf.Comment: 7 pages, 2 figure
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