59 research outputs found

### Scaling and entropy in p-median facility location along a line

The p-median problem is a common model for optimal facility location. The
task is to place p facilities (e.g., warehouses or schools) in a
heterogeneously populated space such that the average distance from a person's
home to the nearest facility is minimized. Here we study the special case where
the population lives along a line (e.g., a road or a river). If facilities are
optimally placed, the length of the line segment served by a facility is
inversely proportional to the square root of the population density. This
scaling law is derived analytically and confirmed for concrete numerical
examples of three US Interstate highways and the Mississippi River. If facility
locations are permitted to deviate from the optimum, the number of possible
solutions increases dramatically. Using Monte Carlo simulations, we compute how
scaling is affected by an increase in the average distance to the nearest
facility. We find that the scaling exponents change and are most sensitive near
the optimum facility distribution.Comment: 7 pages, 6 figures, Physical Review E, in pres

### The Ising chain constrained to an even or odd number of positive spins

We investigate the statistical mechanics of the periodic one-dimensional
Ising chain when the number of positive spins is constrained to be either an
even or an odd number. We calculate the partition function using a
generalization of the transfer matrix method. On this basis, we derive the
exact magnetization, susceptibility, internal energy, heat capacity and
correlation function. We show that in general the constraints substantially
slow down convergence to the thermodynamic limit. By taking the thermodynamic
limit together with the limit of zero temperature and zero magnetic field, the
constraints lead to new scaling functions and different probability
distributions for the magnetization. We demonstrate how these results solve a
stochastic version of the one-dimensional voter model.Comment: 19 pages, 7 figures, to appear in Journal of Statistical Mechanic

### The geometry of percolation fronts in two-dimensional lattices with spatially varying densities

Percolation theory is usually applied to lattices with a uniform probability
p that a site is occupied or that a bond is closed. The more general case,
where p is a function of the position x, has received less attention. Previous
studies with long-range spatial variations in p(x) have only investigated cases
where p has a finite, non-zero gradient at the critical point p_c. Here we
extend the theory to two-dimensional cases in which the gradient can change
from zero to infinity. We present scaling laws for the width and length of the
hull (i.e. the boundary of the spanning cluster). We show that the scaling
exponents for the width and the length depend on the shape of p(x), but they
always have a constant ratio 4/3 so that the hull's fractal dimension D=7/4 is
invariant. On this basis, we derive and verify numerically an asymptotic
expression for the probability h(x) that a site at a given distance x from p_c
is on the hull.Comment: 13 pages, 7 figures, to appear in New Journal of Physic

### The topology of large Open Connectome networks for the human brain

The structural human connectome (i.e.\ the network of fiber connections in
the brain) can be analyzed at ever finer spatial resolution thanks to advances
in neuroimaging. Here we analyze several large data sets for the human brain
network made available by the Open Connectome Project. We apply statistical
model selection to characterize the degree distributions of graphs containing
up to $\simeq 10^6$ nodes and $\simeq 10^8$ edges. A three-parameter
generalized Weibull (also known as a stretched exponential) distribution is a
good fit to most of the observed degree distributions. For almost all networks,
simple power laws cannot fit the data, but in some cases there is statistical
support for power laws with an exponential cutoff. We also calculate the
topological (graph) dimension $D$ and the small-world coefficient $\sigma$ of
these networks. While $\sigma$ suggests a small-world topology, we found that
$D < 4$ showing that long-distance connections provide only a small correction
to the topology of the embedding three-dimensional space.Comment: 14 pages, 6 figures, accepted version in Scientific Report

### The spatial structure of networks

We study networks that connect points in geographic space, such as
transportation networks and the Internet. We find that there are strong
signatures in these networks of topography and use patterns, giving the
networks shapes that are quite distinct from one another and from
non-geographic networks. We offer an explanation of these differences in terms
of the costs and benefits of transportation and communication, and give a
simple model based on the Monte Carlo optimization of these costs and benefits
that reproduces well the qualitative features of the networks studied.Comment: 5 pages, 3 figure

### Optimal design of spatial distribution networks

We consider the problem of constructing public facilities, such as hospitals,
airports, or malls, in a country with a non-uniform population density, such
that the average distance from a person's home to the nearest facility is
minimized. Approximate analytic arguments suggest that the optimal distribution
of facilities should have a density that increases with population density, but
does so slower than linearly, as the two-thirds power. This result is confirmed
numerically for the particular case of the United States with recent population
data using two independent methods, one a straightforward regression analysis,
the other based on density dependent map projections. We also consider
strategies for linking the facilities to form a spatial network, such as a
network of flights between airports, so that the combined cost of maintenance
of and travel on the network is minimized. We show specific examples of such
optimal networks for the case of the United States.Comment: 6 pages, 5 figure

### Consensus time in a voter model with concealed and publicly expressed opinions

The voter model is a simple agent-based model to mimic opinion dynamics in
social networks: a randomly chosen agent adopts the opinion of a randomly
chosen neighbour. This process is repeated until a consensus emerges. Although
the basic voter model is theoretically intriguing, it misses an important
feature of real opinion dynamics: it does not distinguish between an agent's
publicly expressed opinion and her inner conviction. A person may not feel
comfortable declaring her conviction if her social circle appears to hold an
opposing view. Here we introduce the Concealed Voter Model where we add a
second, concealed layer of opinions to the public layer. If an agent's public
and concealed opinions disagree, she can reconcile them by either publicly
disclosing her previously secret point of view or by accepting her public
opinion as inner conviction. We study a complete graph of agents who can choose
from two opinions. We define a martingale $M$ that determines the probability
of all agents eventually agreeing on a particular opinion. By analyzing the
evolution of $M$ in the limit of a large number of agents, we derive the
leading-order terms for the mean and standard deviation of the consensus time
(i.e. the time needed until all opinions are identical). We thereby give a
precise prediction by how much concealed opinions slow down a consensus.Comment: 21 pages, 6 figures, to appear in J. Stat. Mech. Theory Ex

### Opinion formation models on a gradient

Statistical physicists have become interested in models of collective social
behavior such as opinion formation, where individuals change their inherently
preferred opinion if their friends disagree. Real preferences often depend on
regional cultural differences, which we model here as a spatial gradient $g$ in
the initial opinion. The gradient does not only add reality to the model. It
can also reveal that opinion clusters in two dimensions are typically in the
standard (i.e.\ independent) percolation universality class, thus settling a
recent controversy about a non-consensus model. However, using analytical and
numerical tools, we also present a model where the width of the transition
between opinions scales $\propto g^{-1/4}$, not $\propto g^{-4/7}$ as in
independent percolation, and the cluster size distribution is consistent with
first-order percolation.Comment: 12 pages, 8 figures, version accepted by PLoS ONE, online supplement
added as appendi

### Transition from connected to fragmented vegetation across an environmental gradient: scaling laws in ecotone geometry

A change in the environmental conditions across space—for example, altitude or latitude—can cause significant changes in the density of a vegetation type and, consequently, in spatial connectivity. We use spatially explicit simulations to study the transition from connected to fragmented vegetation. A static (gradient percolation) model is compared to dynamic (gradient contact process) models. Connectivity is characterized from the perspective of various species that use this vegetation type for habitat and differ in dispersal or migration range, that is, “step length” across the landscape. The boundary of connected vegetation delineated by a particular step length is termed the “ hull edge.” We found that for every step length and for every gradient, the hull edge is a fractal with dimension 7/4. The result is the same for different spatial models, suggesting that there are universal laws in ecotone geometry. To demonstrate that the model is applicable to real data, a hull edge of fractal dimension 7/4 is shown on a satellite image of a piñon‐juniper woodland on a hillside. We propose to use the hull edge to define the boundary of a vegetation type unambiguously. This offers a new tool for detecting a shift of the boundary due to a climate change

### Teaching Data Visualisation and Basic Map-Making Skills at a Liberal Arts College

U društvu koje sve više ovisi o podatcima komuniciranje podataka postalo je vitalno. Budući da se podatci često prenose grafički, mnogi fakulteti i sveučilišta nude tečajeve koji podučavaju principe vizualizacije podataka. U ovom se članku autor osvrće na svoje iskustvo podučavanja takvog kolegija na Liberal Arts Collegeu. Tečaj je upoznao studente s paketom paketa tidyverse za programski jezik R, uključujući paket za vizualizaciju podataka ggplot2 koji primjenjuje Wilkinsonovu (2005) "gramatiku grafike". Paketi sf i tmap korišteni su za proširenje mogućnosti paketa tidyverse za analizu geoprostornih podataka i izradu tematskih karata. Ovaj članak pruža uvid u učinkovite strategije podučavanja, uključujući predviđene ishode učenja, temeljni nastavni plan i program, odgojno-obrazovni proces i ocjenjivanje. Prikazani su i ogledni zadatci i vizualizacije kako bi se pokazalo kako su osnovne kartografske vještine prenesene studentima.Communicating data has become vital in an increasingly data-dependent society. Because data are often communicated graphically, many colleges and universities are offering courses that teach the principles of data visualisation. In this article, the author reflects on his experience of teaching such a course at a liberal arts college. The course introduced students to the tidyverse suite of packages for the programming language R, including the data visualisation package ggplot2, which implements Wilkinson’s (2005) ‘grammar of graphics’. The packages sf and tmap were used to extend the capabilities of the tidyverse packages for analysing geospatial data and producing thematic maps. This article provides insights into effective teaching strategies, including intended learning outcomes, the core syllabus, pedagogy and assessment. Sample tasks and visualisations are also presented to demonstrate how essential cartographic skills were imparted to students

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