Truncated Power Law Research Articles

Local and global epidemic outbreaks in populations moving in inhomogeneous environments.

We study disease spreading in a system of agents moving in a space where the force of infection is not homogeneous. Agents are random walkers that additionally execute long-distance jumps, and the plane in which they move is divided into two regions where the force of infection takes different values. We show the onset of a local epidemic threshold and a global one and explain them in terms of mean-field approximations. We also elucidate the critical role of the agent velocity, jump probability, and density parameters in achieving the conditions for local and global outbreaks. Finally, we show that the results are independent of the specific microscopic rules adopted for agent motion, since a similar behavior is also observed for the distribution of agent velocity based on a truncated power law, which is a model often used to fit real data on motion patterns of animals and humans.

Motile male gametes of the araphid diatom Tabularia fasciculata search randomly for mates.

Sexuality in the marine araphid diatom Tabularia involves an unusual type of gamete, not only among diatoms but possibly in all of nature. The non-flagellated male gamete is free and vigorously motile, propelled by pseudopodia. However, the cues (if any) in their search for compatible female gametes and the general search patterns to locate them are unknown. We tracked and compared male gamete movements in the presence and absence of receptive female gametes. Path linearity of male movement was not affected by presence of female gametes. Male gametes did not move towards female gametes regardless of their proximity to each other, suggesting that the detection range for a compatible mate is very small compared to known algal examples (mostly spermatozoids) and that mate recognition requires (near) contact with a female gamete. We therefore investigated how male gametes move to bring insight into their search strategy and found that it was consistent with the predictions of a random-walk model with changes in direction coming from an even distribution. We further investigated the type of random walk by determining the best-fit distribution on the tail of the move length distribution and found it to be consistent with a truncated power law distribution with an exponent of 2.34. Although consistent with a Lévy walk search pattern, the range of move lengths in the tail was too narrow for Lévy properties to emerge and so would be best described as Brownian motion. This is somewhat surprising because female gametes were often outnumbered by male gametes, thus contrary to the assumption that a Brownian search mode may be most optimal with an abundant target resource. This is also the first mathematically analysed search pattern of a non-flagellated protistan gamete, supporting the notion that principles of Brownian motion have wide application in biology.

Nestedness in bipartite networks of Thuja plicata, Prunus laurocerasus and Buxus sempervirens and their pathogens

The trade with cultivated plants is one of the major pathways for the introduction of invasive species, pathogens included. Based on network analysis, the present study aimed the interaction between several species of cultivated woody perennials found in gardening outlets and nurseries trading with ornamental species and their documented pathogens. Focal species of the host list were Thuja plicata, Buxus sempervirens and Prunus laurocerasus, the selection being based on reported bestselling figures. Bipartite, qualitative, undirected networks were constructed to incorporate woody perennials as hosts and their documented pathogens. The tested network properties were: connectance, node degree distribution, web asymmetry and nestedness. Cluster analysis using Euclidian distance and niche overlap index of Pianka were employed as additional pattern description metrics. The main network containing 33 host species and 112 pathogens was characterized by truncated power law distribution fitting the observed degree distribution of hosts and power law distribution fitting the observed degree distribution of pathogens, low connectance (C = 0.12), intermediate web asymmetry (W = 0.54) and high significant nestedness (N = 0.94). The network containing three focal hosts showed significant lower nestedness (N = 0.54), higher asymmetry (W = 0.94) and higher connectance (C = 0.38). Cluster analysis revealed the separation of focal species distinctly, the majority of other hosts merging in one cluster. Due to the prevalence of specialized pathogens the niche breadth was narrow, with small overlap in resources’ partition (Pianka index = 0.31). Our results showed that a random assembly of hosts (woody ornamentals displayed for sale in retail centers and nurseries) could harbor pathogens which attached in a non-random manner, generating a characteristic pathosystem, with distinctive topology. The possible implications of the study consisted in a new insight in invasive spread and the inclusion of new pathogens in local pathogen communities using network analysis as a powerful investigation tool.

Non-Fickian transport through two-dimensional rough fractures: Assessment and prediction

Non-Fickian transport ubiquitously occurs across all scales within fractured geological media. Detailed characterization of non-Fickian transport through single fractures is thus critical for predicting the fate of solutes and other fluid-borne entities through fractured media. Our direct numerical simulations of solute transport through two-dimensional rough-walled fractures showed early arrival and heavy tailing in breakthrough curves (BTCs), which are salient characteristics of non-Fickian transport. Analyses for dispersion coefficients (DADE) using the standard advection-dispersion equation (ADE) led to errors which increased linearly with fracture heterogeneity. Estimated Taylor dispersion coefficients deviated from estimated DADE even at higher Peclet numbers. Alternatively, we used continuous time random walk (CTRW) model with truncated power law transition rate probability to characterize the non-Fickian transport. CTRW modeling markedly and consistently improved fits to the BTCs relative to those fitted with ADE solutions. The degree of deviation of transport from Fickian to non-Fickian is captured by the parameter β of the truncated power law. We found that β is proportional to fracture heterogeneity. We also found that the CTRW transport velocity can be predicted based on the flow velocity. Along with the ability to predict β, this is a major step toward prediction of transport through CTRW using measurable physical properties.

Influence of interfacial electron transfer on fluorescence blinking of quantum dots

The fluorescence blinking characteristics of the single CdSe/ZnS core/shell quantum dots (QDs) absorbed on the cover glass surface, indium-tin oxide (ITO) nanoparticles, and polymethyl methacrylate (PMMA) film surface are measured by a laser scanning confocal fluorescence microscopy. It is found that all the distributions of bright state duration time of QDs on the three different interfaces can be described by a truncated power law P(t)∝ t-αexp(-t/μ). The statistical on-time durations of single QDs absorbed on the ITO nanoparticles is shorter than on the glass. In addition, the on-time duration with single QDs absorbed on the PMMA is longer than on the others. These differences can be attributed to the diverse interfacial electron transfers between QD and different materials.

A complex network evolution model for network growth promoted by information transmission

In many real complex networks, information transmission occurs all the time. To study the effects of information transmission on the complex network evolution, we propose a new model for network growth promoted by the information transmission. The model includes three major steps: (i) New links attached to the nodes on the information transmission path, whose source point is chosen preferentially; (ii) the first link of the new node attached to the nodes in the local-world; (iii) other links of the new node attached to the nodes on the information transmission path, whose source point is the new node. The process of information transmission is simulated by self-avoiding random walk, and by considering the local information including its degree and distance; selective connection is established between the nodes on the information transmission path. Theoretical analysis and numerical simulation results show that the proposed model can not only reproduce small-world and scale-free network characteristics, but also indicate that shift power-law distribution and truncated power law function may form for different parameters which have some non-power-law features, such as exponential cutoff, and saturation for small variables. Moreover, in our model, the clustering coefficient is tunable without changing the degree distribution, and the model can also construct a network with assortative or disassortative mixed pattern.

An Empirical Study of a Chinese Online Social Network--Renren

Deeper knowledge of social networks' structure and temporal evolution enhances data mining for both research and education purposes. An empirical analysis of a Chinese social network, Renren, shows that it follows an exponentially truncated power law in degree distribution, and has a short average node distance.

Random walk analysis of ranging patterns of sympatric langurs in a complex resource landscape

The identification of random walk models to characterize the movement patterns of social groups of primates, and the behavioral processes that give rise to such movement patterns, remain open questions in movement ecology. Movement patterns characterized by a power-law tail with exponent between 1 and 3 (Lévy flight) occur when animals forage on scarce, randomly distributed resources. For primates and similar foragers with memory processes, movements resembling Lévy flights emerge when feeding trees (targets) are randomly distributed and the trunk size distribution of targets follows a power-law. We tested three competing random walk models to describe movement patterns of two langur species. We found a truncated power law to be the most suitable model. The power-law model was poorly supported by the data and hence we found no support for Lévy-flight-like behavior. Moreover, the spatial distribution of feeding trees and the probability distribution of feeding tree size differed from values suggested to result in Lévy-flight-like patterns. We identify intraspecific territoriality, foraging behavior, and the spatial and size distribution of food patches as plausible mechanisms that may have given rise to the observed movement patterns.

Truncated Power Law Analysis of Blinking SERS of Thiacyanine Molecules Adsorbed on Single Silver Nanoaggregates by Excitation at Various Wavelengths

From blinking surface-enhanced Raman scattering (SERS) of anionic thiacyanine adsorbed on single Ag nanoaggregates, the electromagnetic field and the molecular behavior in a nonemissive state were investigated by a truncated power law analysis. The power law that reproduces probability distribution of dark SERS events versus duration time was not truncated often by excitation at long wavelengths; otherwise it was truncated at the long tail. The truncation suggests a high energy barrier from nonemissive to emissive state and a short passage time of molecular random walk to overcome the energy barrier. The energy barrier in blinking SERS likely originates from a nanometer-ordered periodic optical trapping potential well, namely, electromagnetic field around a junction of the Ag nanoaggregate due to coupling of multipolar surface plasmon resonance, which is hardly induced by excitation at long wavelengths. This is consistent with the experimental excitation wavelength dependence of the truncation. At a low concentration of anionic thiacyanine, the power law was truncated at the short tail. The reason may be the short passage time of the molecule on the Ag surface adsorbing a small number of obstacles to reach the junction.

Power-Law Testing for Fault Attributes Distributions

This paper is devoted to statistical analysis of faults’ attributes. The distributions of lengths, widths of damage zones, displacements and thicknesses of fault cores are studied. Truncated power-law (TPL) is considered in comparison with commonly used simple power-law (PL) (or Pareto) distribution. The maximal likelihood and the confidence interval of the exponent for both PL and TPL are estimated by appropriate statistical methods. The Kolmogorov–Smirnov (KS) test and the likelihood ratio test with alternative non-nested hypothesis for exponential distribution are used to verify the statistical approximation. Furthermore, the advantage of TPL is proved by Bayesian information criterion. Our results suggest that a TPL is more suitable for describing fault attributes, and that its condition is satisfied for a wide range of fault scales. We propose that using truncated power laws in general might eliminate or relax the bias related to sampling strategy and the resolution of measurements (such as censoring, truncation, and cut effect) and; therefore, the most reliable range of data can be considered for the statistical approximation of fault attributes.

Ecological indicators for protection impact assessment at two scales in the Bozin and Marakhil protected area, Iran

Official protection can play a major role in biodiversity conservation, but it may have unintended effects on habitats at multiple scales. We evaluated the effects of this protection designation on forest area and structure at two spatial scales in Bozin and Marakhil protected area. We classified Landsat-5 TM images from 2001 and 2009, covering protected and unprotected areas and used for broad scale analysis, and IKONOS and GeoEye images from 2001 and 2009 for fine scale analysis. We used landscape structure metrics and the models that fit patch size distribution as the ecological indicators of protection impact. The protection effect was scale dependent, with significant habitat loss and fragmentation at the fine scale but not at the broad scale. Protection increased the number of large patches inside the park, as indicated by a shift from a truncated power law to a power law distribution of patch sizes. The habitat structural maintenance and aggregation inside the park, and fragmentation outside the park boundaries, signal a potential shift of human pressure from inside to directly outside of the park. This study indicates the need for spatial analyses on both sides of protection boundaries at multiple scales using complementary indicators to understand the impacts of protection on habitat structure.

Human-Mobility-Based Sensor Context-Aware Routing Protocol for Delay-Tolerant Data Gathering in Multi-Sink Cell-Phone-Based Sensor Networks

Ubiquitous use of cell phones encourages development of novel applications with sensors embedded in cell phones. The collection of information generated by these devices is a challenging task considering volatile topologies and energy-based scarce resources. Further, the data delivery to the sink is delay tolerant. Mobility of cell phones is opportunistically exploited for forwarding sensor generated data towards the sink. Human mobility model shows truncated power law distribution of flight length, pause time, and intercontact time. The power law behavior of inter-contact time often discourages routing of data using naive forwarding schemes. This work exploits the flight length and the pause time distributions of human mobility to design a better and efficient routing strategy. We propose a Human-Mobility-based Sensor Context-Aware Routing protocol (HMSCAR), which exploits human mobility patterns to smartly forward data towards the sink basically comprised of wi-fi hot spots or cellular base stations. The simulation results show that HMSCAR significantly outperforms the SCAR, SFR, and GRAD-MOB on the aspects of delivery ratio and time delay. A multi-sink scenario and single-copy replication scheme is assumed.

Scale effect and bimodality in the frequency distribution of species occupancy

Community patterns in species-by-site matrices provide valuable clues for inferring ecological processes at work. One such pattern is the occupancy frequency distribution (OFD) depicting the frequency distribution of row sums (i.e., occupancy) with a quarter OFDs of bimodal forms. Another pattern that also reflects the structure of row sums is the ranked species occupancy curve (RSOC), and has been shown to imply a 50% of bimodality in OFDs. The use of RSOCs has been advocated in literature over the OFD based on two conclusions from a 6-model inference using only 24 matrices: (i) RSOCs have two general forms, with half representing bimodal OFDs; (ii) there are no effects of spatial and study scales on RSOCs of different forms. Using a much more representative dataset of 289 matrices, I cast doubt on these two conclusions. A missing but dominant RSOC model (the truncated power law) is added. The number of species and the nestedness of the community differ significantly among matrices of different RSOC forms; however, the number of sites and the taxa in the studies do not differ among RSOC or OFD forms. The quarter OFDs of bimodal forms is reassured, with the least frequent occupancy consistent with Raunkiaer’s law of frequency. Importantly, a RSOC is mathematically transferrable to an OFD, with the derivative of the occupancy ranking curve being equal to the negative reciprocal of the occupancy frequency. Based on the type of the community (null versus interactive) and site environment (homogenous versus heterogeneous), four scenarios are needed to identify pre-inferring assemblage mechanisms. The results highlight the need for shifting research from the emphasis of marginal sums to the analysis of matrix structure for an in-depth understanding of the community assemblage patterns and mechanisms.

Understanding intra-urban trip patterns from taxi trajectory data

Intra-urban human mobility is investigated by means of taxi trajectory data that are collected in Shanghai, China, where taxis play an important role in urban transportation. From the taxi trajectories, approximately 1.5 million trips of anonymous customers are extracted on seven consecutive days. The globally spatio-temporal patterns of trips exhibit a significant daily regularity. Since each trip can be viewed as a displacement in the random walk model, the distributions of the distance and direction of the extracted trips are investigated in this research. The direction distribution shows an NEE–SWW-dominant direction, and the distance distribution can be well fitted by an exponentially truncated power law, with the scaling exponent β = 1.2 ± 0.15. The observed patterns are attributed to the geographical heterogeneity of the study area, which makes the spatial distribution of trajectory stops to be non-uniform. We thus construct a model that integrates both the geographical heterogeneity and distance decay effect, to interpret the observed patterns. Our Monte Carlo simulation results closely match to the observed patterns and thus validate the proposed model. According to the proposed model, in a single-core urban area, the geographical heterogeneity and distance decay effect improve each other when influencing human mobility patterns. Geographical heterogeneity leads to a faster observed decay, and the distance decay effect makes the spatial distribution of trips more concentrated.

The space density and X-ray luminosity function of non-magnetic cataclysmic variables

We combine two complete, X-ray flux-limited surveys, the ROSAT Bright Survey (RBS) and the ROSAT North Ecliptic Pole (NEP) survey, to measure the space density (\rho) and X-ray luminosity function (\Phi) of non-magnetic CVs. The combined survey has a flux limit of F_X \ga 1.1 \times 10^{-12} erg cm^{-2}s^{-1} over most of its solid angle of just over 2\pi, but is as deep as \simeq 10^{-14} erg cm^{-2}s^{-1} over a small area. The CV sample that we construct from these two surveys contains 20 non-magnetic systems. We carefully include all sources of statistical error in calculating \rho and \Phi by using Monte Carlo simulations; the most important uncertainty proves to be the often large errors in distances estimates. If we assume that the 20 CVs in the combined RBS and NEP survey sample are representative of the intrinsic population, the space density of non-magnetic CVs is 4^{+6}_{-2} \times 10^{-6} pc^{-3}. We discuss the difficulty in measuring \Phi in some detail---in order to account for biases in the measurement, we have to adopt a functional form for \Phi. Assuming that the X-ray luminosity function of non-magnetic CVs is a truncated power law, we constrain the power law index to -0.80 \pm 0.05. It seems likely that the two surveys have failed to detect a large, faint population of short-period CVs, and that the true space density may well be a factor of 2 or 3 larger than what we have measured; this is possible, even if we only allow for undetected CVs to have X-ray luminosities in the narrow range 28.7< log(L_X/erg\,s^{-1})<29.7. However, \rho as high as 2 \times 10^{-4} pc^{-3} would require that the majority of CVs has X-ray luminosities below L_X = 4 \times 10^{28} erg s^{-1} in the 0.5--2.0 keV band.

Effect of truncation on large deviations for heavy-tailed random vectors

This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be R d -valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the same is lost.

Inverse mapping of polarized optical emission from pulsars: basic formulation and determination of emission altitude

We present an inverse mapping approach to determining the emission height of the optical photons from pulsars, which is directly constrained by empirical data. The model discussed is for the case of the Crab pulsar. Our method, using the optical Stokes parameters, determines the most likely geometry for emission including magnetic field inclination angle ($\alpha$), observers line of sight angle ($\chi$) and emission height. We discuss the computational implementation of the approach, along with any physical assumptions made. We find that the most likely emission altitude is at 20% of the light cylinder radius above the stellar surface, in the open field region. We also present a general treatment of the expected polarisation from synchrotron source with a truncated power law spectrum of particles.

Chinese family name distributions in multiple scales

The distribution of Chinese family names is investigated based on data of the fifth national census of China in 2000, including 7329 Chinese family names and 1.28 billion people. The cumulative distribution function (CDF) of family name in multiple scales is presented and the correlation between distribution and scale is discussed. The fitting results show that CDFs at the scales of country, province, city and county all follow stretched exponential truncated power law, while the fitting parameters vary greatly. A characteristic index is proposed to measure the extent the distribution is stretched-exponential-like. The index increases as the scale goes down, indicating that the distribution at lower scale is more stretched-exponential-like, and the distribution at higher scale is more power-law-like. Such correlation between distribution and scale is verified by investigating how the distribution changes as the scale is expanded county by county. Furthermore, the correlation is qualitatively explained by artificially constructing multiple scales with a partition process.

A first principles derivation of animal group size distributions

Several empirical studies have shown that the animal group size distribution of many species can be well fit by power laws with exponential truncation. A striking empirical result due to Niwa is that the exponent in these power laws is one and the truncation is determined by the average group size experienced by an individual. This distribution is known as the logarithmic distribution. In this paper we provide first principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. In particular, we investigate two such models. Firstly, we look at a model in which groups merge whenever they meet but split with a constant probability per time step. This generates a distribution similar, but not identical to the logarithmic distribution. Secondly, we propose a model, based on preferential attachment, that produces the logarithmic distribution exactly. Our derivation helps explain why logarithmic distributions are so widely observed in nature. The derivation also allows us to link splitting and joining behavior to the exponent and truncation parameters in power laws.

A dual-frequency sub-arcsecond study of proto-planetary disks at mm wavelengths: first evidence for radial variations of the dust properties

(Abridged) We attempt to characterize the radial distribution of dust in disks around a sample of young stars from an observational point of view, and, when possible, in a model-independent way, by using parametric laws. We used the IRAM PdBI interferometer to provide very high angular resolution (down to 0.4" in some sources) observations of the continuum at 1.3 mm and 3 mm around a sample of T Tauri stars in the Taurus-Auriga region. The sample includes single and multiple systems, with a total of 23 individual disks. We used track-sharing observing mode to minimize the biases. We fitted these data with two kinds of models: a "truncated power law" model and a model presenting an exponential decay at the disk edge ("viscous" model). ect evidence for tidal truncation is found in the multiple systems. The temperature of the mm-emitting dust is constrained in a few systems. Unambiguous evidence for large grains is obtained by resolving out disks with very low values of the dust emissivity index Beta. In most disks that are sufficiently resolved at two different wavelengths, we find a radial dependence of Beta, which appears to increase from low values (as low as 0) at the center to about 1.7 -- 2 at the disk edge. The same behavior could apply to all studied disks. It introduces further ambiguities in interpreting the brightness profile, because the regions with apparent Beta = 0 can also be interpreted as being optically thick when their brightness temperature is high enough. Despite the added uncertainty on the dust absorption coefficient, the characteristic size of the disk appears to increase with a higher estimated star age. These results provide the first direct evidence of the radial dependence of the grain size in proto-planetary disks.