53 research outputs found
Languages cool as they expand: Allometric scaling and the decreasing need for new words
We analyze the occurrence frequencies of over 15 million words recorded in millions of books published during the past two centuries in seven different languages. For all languages and chronological subsets of the data we confirm that two scaling regimes characterize the word frequency distributions, with only the more common words obeying the classic Zipf law. Using corpora of unprecedented size, we test the allometric scaling relation between the corpus size and the vocabulary size of growing languages to demonstrate a decreasing marginal need for new words, a feature that is likely related to the underlying correlations between words. We calculate the annual growth fluctuations of word use which has a decreasing trend as the corpus size increases, indicating a slowdown in linguistic evolution following language expansion. This ââcooling patternââ forms the basis of a third statistical regularity, which unlike the Zipf and the Heaps law, is dynamical in nature
Ordinary Percolation with Discontinuous Transitions
Percolation on a one-dimensional lattice and fractals such as the Sierpinski
gasket is typically considered to be trivial because they percolate only at
full bond density. By dressing up such lattices with small-world bonds, a novel
percolation transition with explosive cluster growth can emerge at a nontrivial
critical point. There, the usual order parameter, describing the probability of
any node to be part of the largest cluster, jumps instantly to a finite value.
Here, we provide a simple example of this transition in form of a small-world
network consisting of a one-dimensional lattice combined with a hierarchy of
long-range bonds that reveals many features of the transition in a
mathematically rigorous manner.Comment: RevTex, 5 pages, 4 eps-figs, and Mathematica Notebook as Supplement
included. Final version, with several corrections and improvements. For
related work, see http://www.physics.emory.edu/faculty/boettcher
Limited Urban Growth: London's Street Network Dynamics since the 18th Century
We investigate the growth dynamics of Greater London defined by the
administrative boundary of the Greater London Authority, based on the evolution
of its street network during the last two centuries. This is done by employing
a unique dataset, consisting of the planar graph representation of nine time
slices of Greater London's road network spanning 224 years, from 1786 to 2010.
Within this time-frame, we address the concept of the metropolitan area or city
in physical terms, in that urban evolution reveals observable transitions in
the distribution of relevant geometrical properties. Given that London has a
hard boundary enforced by its long-standing green belt, we show that its street
network dynamics can be described as a fractal space-filling phenomena up to a
capacitated limit, whence its growth can be predicted with a striking level of
accuracy. This observation is confirmed by the analytical calculation of key
topological properties of the planar graph, such as the topological growth of
the network and its average connectivity. This study thus represents an example
of a strong violation of Gibrat's law. In particular, we are able to show
analytically how London evolves from a more loop-like structure, typical of
planned cities, toward a more tree-like structure, typical of self-organized
cities. These observations are relevant to the discourse on sustainable urban
planning with respect to the control of urban sprawl in many large cities,
which have developed under the conditions of spatial constraints imposed by
green belts and hard urban boundaries.Comment: PlosOne, in publicatio
Modeling the scaling properties of human mobility
While the fat tailed jump size and the waiting time distributions
characterizing individual human trajectories strongly suggest the relevance of
the continuous time random walk (CTRW) models of human mobility, no one
seriously believes that human traces are truly random. Given the importance of
human mobility, from epidemic modeling to traffic prediction and urban
planning, we need quantitative models that can account for the statistical
characteristics of individual human trajectories. Here we use empirical data on
human mobility, captured by mobile phone traces, to show that the predictions
of the CTRW models are in systematic conflict with the empirical results. We
introduce two principles that govern human trajectories, allowing us to build a
statistically self-consistent microscopic model for individual human mobility.
The model not only accounts for the empirically observed scaling laws but also
allows us to analytically predict most of the pertinent scaling exponents
Impact of Single Links in Competitive Percolation -- How complex networks grow under competition
How a complex network is connected crucially impacts its dynamics and
function. Percolation, the transition to extensive connectedness upon gradual
addition of links, was long believed to be continuous but recent numerical
evidence on "explosive percolation" suggests that it might as well be
discontinuous if links compete for addition. Here we analyze the microscopic
mechanisms underlying discontinuous percolation processes and reveal a strong
impact of single link additions. We show that in generic competitive
percolation processes, including those displaying explosive percolation, single
links do not induce a discontinuous gap in the largest cluster size in the
thermodynamic limit. Nevertheless, our results highlight that for large finite
systems single links may still induce observable gaps because gap sizes scale
weakly algebraically with system size. Several essentially macroscopic clusters
coexist immediately before the transition, thus announcing discontinuous
percolation. These results explain how single links may drastically change
macroscopic connectivity in networks where links add competitively.Comment: non-final version, for final see Nature Physics homepag
A unified data representation theory for network visualization, ordering and coarse-graining
Representation of large data sets became a key question of many scientific
disciplines in the last decade. Several approaches for network visualization,
data ordering and coarse-graining accomplished this goal. However, there was no
underlying theoretical framework linking these problems. Here we show an
elegant, information theoretic data representation approach as a unified
solution of network visualization, data ordering and coarse-graining. The
optimal representation is the hardest to distinguish from the original data
matrix, measured by the relative entropy. The representation of network nodes
as probability distributions provides an efficient visualization method and, in
one dimension, an ordering of network nodes and edges. Coarse-grained
representations of the input network enable both efficient data compression and
hierarchical visualization to achieve high quality representations of larger
data sets. Our unified data representation theory will help the analysis of
extensive data sets, by revealing the large-scale structure of complex networks
in a comprehensible form.Comment: 13 pages, 5 figure
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