Superlinear Scaling Research Articles

Is there more than one stickiness criterion?

Adhesion between an elastic body and a smooth, rigid substrate can lead to large tensile stresses between them. However, most macroscopic objects are microscopically rough, which strongly suppresses adhesion. A fierce debate has unfolded recently as to whether local or global parameters determine the crossover between small and large adhesion. Here, we report simulations revealing that the dependence of the pull-off force Fn on the surface energy γ does not only have two regimes of high and low adhesion but up to four regimes. They are related to contacts, which at the moment of rupture consist of (i) the last individual Hertzian-shaped contact, in which is linear in γ, (ii) a last meso-scale, individual patches with super-linear scaling, (iii) many isolated contact patches with extremely strong scaling, and (iv) a dominating largest contact patch, for which the pull-off stress is no longer negligible compared to the maximum, microscopic pull-off stress. Regime (iii) can be seen as a transition domain. It is located near the point where the surface energy is half the elastic energy per unit area in conformal contact. A criterion for the transition between regimes (i) and (ii) appears difficult to grasp.

Asymptotic burnout and homeostatic awakening: a possible solution to the Fermi paradox?

Previous studies show that city metrics having to do with growth, productivity and overall energy consumption scale superlinearly, attributing this to the social nature of cities. Superlinear scaling results in crises called ‘singularities’, where population and energy demand tend to infinity in a finite amount of time, which must be avoided by ever more frequent ‘resets’ or innovations that postpone the system's collapse. Here, we place the emergence of cities and planetary civilizations in the context of major evolutionary transitions. With this perspective, we hypothesize that once a planetary civilization transitions into a state that can be described as one virtually connected global city, it will face an ‘asymptotic burnout’, an ultimate crisis where the singularity-interval time scale becomes smaller than the time scale of innovation. If a civilization develops the capability to understand its own trajectory, it will have a window of time to affect a fundamental change to prioritize long-term homeostasis and well-being over unyielding growth—a consciously induced trajectory change or ‘homeostatic awakening’. We propose a new resolution to the Fermi paradox: civilizations either collapse from burnout or redirect themselves to prioritizing homeostasis, a state where cosmic expansion is no longer a goal, making them difficult to detect remotely.

A General Biphasic Bodyweight Model for Scaling Basal Metabolic Rate, Glomerular Filtration Rate, and Drug Clearance from Birth to Adulthood.

The objective of this study is to propose a unified, continuous, and bodyweight-only equation to quantify the changes of human basal metabolic rate (BMR), glomerular filtration rate (GFR), and drug clearance (CL) from infancy to adulthood. The BMR datasets were retrieved from a comprehensive historical database of male and female subjects (0.02 to 64 years). The CL datasets for 17 drugs and the GFR dataset were generated from published maturation and growth models with reported parameter values. A statistical approach was used to simulate the model-generated CL and GFR data for a hypothetical population with 26 age groups (from 0 to 20 years). A biphasic equation with two power-law functions of bodyweight was proposed and evaluated as a general model using nonlinear regression and dimensionless analysis. All datasets universally reveal biphasic curves with two distinct linear segments on log-log plots. The biphasic equation consists of two reciprocal allometric terms that asymptotically determine the overall curvature. The fitting results show a superlinear scaling phase (asymptotic exponent >1; ca. 1.5-3.5) and a sublinear scaling phase (asymptotic exponent <1; ca. 0.5-0.7), which are separated at the phase transition bodyweight ranging from 5 to 20 kg with a mean value of 10 kg (corresponding to 1 year of age). The dimensionless analysis generalizes and offers quantitative realization of the maturation and growth process. In conclusion, the proposed mixed-allometry equation is a generic model that quantitatively describes the phase transition in the human maturation process of diverse human functions.

Superlinear scaling of riverine biogeochemical function with watershed size

River networks regulate carbon and nutrient exchange between continents, atmosphere, and oceans. However, contributions of riverine processing are poorly constrained at continental scales. Scaling relationships of cumulative biogeochemical function with watershed size (allometric scaling) provide an approach for quantifying the contributions of fluvial networks in the Earth system. Here we show that allometric scaling of cumulative riverine function with watershed area ranges from linear to superlinear, with scaling exponents constrained by network shape, hydrological conditions, and biogeochemical process rates. Allometric scaling is superlinear for processes that are largely independent of substrate concentration (e.g., gross primary production) due to superlinear scaling of river network surface area with watershed area. Allometric scaling for typically substrate-limited processes (e.g., denitrification) is linear in river networks with high biogeochemical activity or low river discharge but becomes increasingly superlinear under lower biogeochemical activity or high discharge, conditions that are widely prevalent in river networks. The frequent occurrence of superlinear scaling indicates that biogeochemical activity in large rivers contributes disproportionately to the function of river networks in the Earth system.

Extended Dicke quantum battery with interatomic interactions and driving field

We investigate the charging process of quantum battery (QB) systems in an extended Dicke model with both atomic interactions and an external driving field. We focus on the effects of the atomic interaction and the external driving field on the charging performance of the QB and find that the maximum stored energy of the QB has a critical phenomenon. We analyze the critical behavior and obtain the analytical expression of the critical atomic interaction. The dependence of the maximum stored energy, the energy quantum fluctuations, and the maximum charging power on the number $N$ of the two-level systems are also discussed. In particular, for the maximum charging power, we obtain the quantum advantage of the QB, which approximately satisfies a superlinear scaling relation ${P}_{\text{max}}\ensuremath{\propto}{N}^{\ensuremath{\alpha}}$, where scaling exponent $\ensuremath{\alpha}$ varies with the number $N$ of the two-level systems. In the ultrastrong coupling regime, the atomic interaction can lead to faster battery charging, and the quantum advantage $\ensuremath{\alpha}=1.88$ can be achieved. While in the deep-strong coupling regime, the quantum advantage of the QB's maximum charging power is the same as that of the Dicke QB, i.e., $\ensuremath{\alpha}=1.5$.

A TensorFlow simulation framework for scientific computing of fluid flows on tensor processing units

A computational fluid dynamics (CFD) simulation framework for fluid-flow prediction is developed on the Tensor Processing Unit (TPU) platform. The TPU architecture is featured with accelerated dense matrix multiplication, large high bandwidth memory, and a fast inter-chip interconnect, making it attractive for high-performance scientific computing. The CFD framework solves the variable-density Navier-Stokes equation using a low-Mach approximation, and the governing equations are discretized by a finite-difference method on a collocated structured mesh. It uses the graph-based TensorFlow as the programming paradigm. The accuracy and performance of this framework is studied both numerically and analytically, specifically focusing on effects of TPU-native single precision floating point arithmetic. The algorithm and implementation are validated with canonical 2D and 3D Taylor-Green vortex simulations. To demonstrate the capability for simulating turbulent flows, simulations are conducted for two configurations, namely decaying homogeneous isotropic turbulence and a turbulent planar jet. Both simulations show good statistical agreement with reference solutions. The performance analysis shows a linear weak scaling and a superlinear strong scaling up to a full TPU v3 pod with 2048 cores.

Massively parallel linear-scaling Hartree-Fock exchange and hybrid exchange-correlation functionals with plane wave basis set accuracy.

We extend our linear-scaling approach for the calculation of Hartree-Fock exchange energy using localized in situ optimized orbitals [Dziedzic et al., J. Chem. Phys. 139, 214103 (2013)] to leverage massive parallelism. Our approach has been implemented in the onetep (Order-N Electronic Total Energy Package) density functional theory framework, which employs a basis of non-orthogonal generalized Wannier functions (NGWFs) to achieve linear scaling with system size while retaining controllable near-complete-basis-set accuracy. For the calculation of Hartree-Fock exchange, we use a resolution-of-identity approach, where an auxiliary basis set of truncated spherical waves is used to fit products of NGWFs. The fact that the electrostatic potential of spherical waves (SWs) is known analytically, combined with the use of a distance-based cutoff for exchange interactions, leads to a calculation cost that scales linearly with the system size. Our new implementation, which we describe in detail, combines distributed memory parallelism (using the message passing interface) with shared memory parallelism (OpenMP threads) to efficiently utilize numbers of central processing unit cores comparable to, or exceeding, the number of atoms in the system. We show how the use of multiple time-memory trade-offs substantially increases performance, enabling our approach to achieve superlinear strong parallel scaling in many cases and excellent, although sublinear, parallel scaling otherwise. We demonstrate that in scenarios with low available memory, which preclude or limit the use of time-memory trade-offs, the performance degradation of our algorithm is graceful. We show that, crucially, linear scaling with system size is maintained in all cases. We demonstrate the practicability of our approach by performing a set of fully converged production calculations with a hybrid functional on large imogolite nanotubes up to over 1400 atoms. We finish with a brief study of how the employed approximations (exchange cutoff and the quality of the SW basis) affect the calculation walltime and the accuracy of the obtained results.

Heterogeneous recruitment abilities to RNA polymerases generate nonlinear scaling of gene expression with cell volume

While most genes’ expression levels are proportional to cell volumes, some genes exhibit nonlinear scaling between their expression levels and cell volume. Therefore, their mRNA and protein concentrations change as the cell volume increases, which often have crucial biological functions such as cell-cycle regulation. However, the biophysical mechanism underlying the nonlinear scaling between gene expression and cell volume is still unclear. In this work, we show that the nonlinear scaling is a direct consequence of the heterogeneous recruitment abilities of promoters to RNA polymerases based on a gene expression model at the whole-cell level. Those genes with weaker (stronger) recruitment abilities than the average ability spontaneously exhibit superlinear (sublinear) scaling with cell volume. Analysis of the promoter sequences and the nonlinear scaling of Saccharomyces cerevisiae’s mRNA levels shows that motifs associated with transcription regulation are indeed enriched in genes exhibiting nonlinear scaling, in concert with our model.

Urban scaling of opioid analgesic sales in the United States.

Opioid misuse is a public health crisis in the United States. The origin of this crisis is associated with a sharp increase in opioid analgesic prescribing. We used the urban scaling framework to analyze opioid prescribing patterns in US commuting zones (CZs), i.e., groups of counties based on commuting patterns. The urban scaling framework postulates that a set of scaling relations can be used to predict health outcomes and behaviors in cities. We used data from the Drug Enforcement Administration's Automated Reports and Consolidated Ordering System (ARCOS) to calculate counts of oxycodone/hydrocodone pills distributed to 607 CZs in the continental US from 2006 to 2014. We estimated the scaling coefficient of opioid pill counts by regressing log(pills) on log(population) using a piecewise linear spline with a single knot at 82,363. Our results show that CZs with populations below the knot scaled superlinearly (β = 1.36), i.e., larger CZs had disproportionally larger pill counts compared to smaller CZs. On the other hand, CZs with populations above the knot scaled sublinearly (β = 0.92), i.e., larger CZs had disproportionally smaller pill counts compared to smaller CZs. This dual scaling pattern was consistent across US census regions. For CZs with population below the knot, the superlinear scaling of pills is consistent with the explanation that an increased number of successful matches between prescribers and users will lead to higher prescribing rates. The non-linear scaling behavior observed could be the result of a combination of factors, including stronger health care systems and prescribing regulation in largely populated commuting zones, as well as high availability of other opioids such as heroin in these commuting zones. Future research should explore potential mechanisms for the non-linearity of prescription opioid pills.

Scaling of urban income inequality in the USA.

Urban scaling analysis, the study of how aggregated urban features vary with the population of an urban area, provides a promising framework for discovering commonalities across cities and uncovering dynamics shared by cities across time and space. Here, we use the urban scaling framework to study an important, but under-explored feature in this community-income inequality. We propose a new method to study the scaling of income distributions by analysing total income scaling in population percentiles. We show that income in the least wealthy decile (10%) scales close to linearly with city population, while income in the most wealthy decile scale with a significantly superlinear exponent. In contrast to the superlinear scaling of total income with city population, this decile scaling illustrates that the benefits of larger cities are increasingly unequally distributed. For the poorest income deciles, cities have no positive effect over the null expectation of a linear increase. We repeat our analysis after adjusting income by housing cost, and find similar results. We then further analyse the shapes of income distributions. First, we find that mean, variance, skewness and kurtosis of income distributions all increase with city size. Second, the Kullback-Leibler divergence between a city's income distribution and that of the largest city decreases with city population, suggesting the overall shape of income distribution shifts with city population. As most urban scaling theories consider densifying interactions within cities as the fundamental process leading to the superlinear increase of many features, our results suggest this effect is only seen in the upper deciles of the cities. Our finding encourages future work to consider heterogeneous models of interactions to form a more coherent understanding of urban scaling.

Scaling of urban economic outputs: insights both from urban population size and population mobility

Urban scaling laws assume that the performance of a city largely relies on its urban population size. However, two cities with the same population size may have vastly different economic outputs, which reveals that factors apart from urban size (a measurement of intra-urban interactions) determine their economic outputs. Economic production is essentially the product of social interactions. Urban population size and interurban interactions reflected by population mobility were both considered to evaluate the scaling of urban economic outputs in this paper. We quantified the scaling relationship between urban economic outputs and interurban interactions, and compared it with the paradigm derived from urban size. Results showed that urban economic outputs scale with urban interactions across cities and present the same super-linear scaling regimes but with a greater scaling exponent. A deeper looking showed that interurban interactions definitely bring a more obvious super-linearity than population size but the scaling relationship between urban size and economic outputs is more robust. Urban population size has a greater impact on Gross Domestic Product (GDP), secondary and tertiary industries output products, total retail sales of consumer goods and total wage with 60– 65% relative contribution. For exports, interurban interactions and urban population size are almost equally important. We also proved that interactions between cities are significantly positively correlated with urban extra growth. These findings provide convictive evidence that in addition to population size, interurban interaction is also crucial for exploring the scaling of urban growth. Our results are enlightening to the study of mechanisms and evolutions of urban scaling that interurban interaction besides urban population size should both be a vital consideration to urban economic outputs.

A computational-graph partitioning method for training memory-constrained DNNs

Many state-of-the-art Deep Neural Networks (DNNs) have substantial memory requirements. Limited device memory becomes a bottleneck when training those models. We propose ParDNN, an automatic, generic, and non-intrusive partitioning strategy for DNNs that are represented as computational graphs. ParDNN decides a placement of DNN’s underlying computational graph operations across multiple devices so that the devices’ memory constraints are met and the training time is minimized. ParDNN is completely independent of the deep learning aspects of a DNN. It requires no modification neither at the model nor at the systems level implementation of its operation kernels. ParDNN partitions DNNs having billions of parameters and hundreds of thousands of operations in seconds to few minutes. Our experiments with TensorFlow on 16 GPUs demonstrate efficient training of 5 very large models while achieving superlinear scaling for both the batch size and training throughput. ParDNN either outperforms or qualitatively improves upon the related work.

How the geometry of cities determines urban scaling laws.

Urban scaling laws relate socio-economic, behavioural and physical variables to the population size of cities. They allow for a new paradigm of city planning and for an understanding of urban resilience and economics. The emergence of these power-law relations is still unclear. Improving our understanding of their origin will help us to better apply them in practical applications and further research their properties. In this work, we derive the basic exponents for spatially distributed variables from fundamental fractal geometric relations in cities. Sub-linear scaling arises as the ratio of the fractal dimension of the road network and of the distribution of the population embedded in three dimensions. Super-linear scaling emerges from human interactions that are constrained by the geometry of a city. We demonstrate the validity of the framework with data from 4750 European cities. We make several testable predictions, including the relation of average height of cities and population size, and the existence of a critical density above which growth changes from horizontal densification to three-dimensional growth.

The Urban Metabolism of Lima: Perspectives and Policy Indications for GHG Emission Reductions

The urban metabolism of Metropolitan Lima, the largest urban agglomeration in Peru, which has recently surpassed 10 million inhabitants, was analyzed in this study. This coastal megacity, located within the Rimac, Chillón, and Lurin watersheds constitutes the socioeconomic center of the country and is the hub of the main import and export routes. A multi-layer approach was used to explore material and energy flows in the Peruvian capital for a 10-year timeframe. Results show that in 2006 the GDP of the 49 districts that shape the metropolitan area of Lima was 105.2 billion USD-PPP, while in 2014 it reached about 200 billion USD-PPP. Based on this growth, we highlight that energy, electricity and water flows experienced a linear increase with respect to GDP, being the electricity consumption in years 2006, 2011 and 2014, 7295, 10,112, and 11,465 GWh, respectively. Regarding demographics, population growth ratios of GDP (650%), electricity consumption (400%), solid waste production (250%), and water (100%) confirm the results of super linear scaling found by Kennedy et al. (2015) for the other megacities. Finally, the increase in greenhouse gas (GHG) emissions was computed following an important shift in the primary energy sources to produce electricity. The most important change was linked to the shift from hydropower to natural gas, a trend that initiated in 2006. For instance, in 2001 79% of the total electricity production came from hydropower, whereas in 2014 69% was linked to natural gas. This shift produced an increase of GHG emissions of more than 200% in 2014 when compared to the electricity generation mix of 2001. Following these results, we strongly encourage policies for the decarbonization of the electricity production sector, as well as for mobility infrastructures, e.g., electric public and transport sector, with a progressive shift toward electric mobility.

Scaling of Urban Income Inequality in the USA

Urban scaling analysis, the study of how aggregated urban features vary with the population of an urban area, provides a promising framework for discovering commonalities across cities and uncovering dynamics shared by cities across time and space. Here, we use the urban scaling framework to study an important, but under-explored feature in this community — income inequality. We propose a new method to study the scaling of income distributions by analyzing total income scaling in population percentiles. We show that income in the least wealthy decile (10%) scales close to linearly with city population, while income in the most wealthy decile scale with a significantly superlinear exponent. In contrast to the superlinear scaling of total income with city population, this decile scaling illustrates that the benefits of larger cities are increasingly unequally distributed. For the poorest income deciles, cities have no positive effect over the null expectation of a linear increase. We repeat our analysis after adjusting income by housing cost, and find similar results. We then further analyze the shapes of income distributions. First, we find that mean, variance, skewness, and kurtosis of income distributions all increase with city size. Second, the Kullback–Leibler divergence between a city’s income distribution and that of the largest city decreases with city population, suggesting the overall shape of income distribution shifts with city population. As most urban scaling theories consider densifying interactions within cities as the fundamental process leading to the superlinear increase of many features, our results suggest this effect is only seen in the upper deciles of the cities. Our finding encourages future work to consider heterogeneous models of interactions to form a more coherent understanding of urban scaling.

Brilliant attosecond γ-ray emission and high-yield positron production from intense laser-irradiated nano-micro array

We investigate a novel scheme for brilliant attosecond γ-ray emission and high-yield positron production, which is accomplished with an ultra-intense laser pulse incident upon a Nano-Micro-array (NMA) with a substrate incorporated. This scheme is able to realize effectively electron acceleration and colliding geometry. Both the γ-ray flash and positron bunch are, then, generated with high conversion efficiency. At a laser intensity of 8 × 1023 W/cm2, ∼27% of the laser energy is transferred successfully into γ-rays and ∼0.7% of the laser energy into the positrons. As a consequence, ultra-short (∼440 as) and ultra-brilliant (∼1024 photons s−1 mm−2 mrad−2 per 0.1%BW at 15 MeV) γ-ray burst and high-yield (1.48 × 1011) and overdense (∼1022 cm−3) positron bunches are generated. We found a sub-linear scaling of laser-to-photon conversion efficiency (∝I00.75) and a superlinear scaling of laser-to-positron conversion efficiency (∝I02.5) with the laser intensity. Multi-dimensional particle-in-cell simulations show that particle (γ photon and positron) generation can be manipulated by the laser-focusing position and NMA's length and spacing. Optimal conditions for particle generation in NMAs are obtained, indicating that microwire arrays have the advantage over nanowire arrays in particle generation in the extreme laser fields. Furthermore, positron annihilation effects in the high-energy-density (HED) environment are discussed. The scheme using NMAs would provide effective avenues toward investigating attosecond nuclear science and HED physics with the coming 10 PW laser facilities.

Investigating duration and intensity of Covid-19 social-distancing strategies

The exponential character of the recent Covid-19 outbreak requires a change in strategy from containment to mitigation. Meanwhile, most countries apply social distancing with the objective to keep the number of critical cases below the capabilities of the health care system. Due to the novelty and rapid spread of the virus, an a priori assessment of this strategy was not possible. In this study, we present a model-based systems analysis to assess the effectiveness of social distancing measures in terms of intensity and duration of application. Results show a super-linear scaling between intensity (percent contact reduction) and required duration of application to have an added value (a lower number of fatalities). This holds true for an effective reproduction of R>1 and is reverted for R<1. If R is not reduced below 1, secondary effects of required long-term isolation are likely to unravel the added value of disease mitigation. If an extinction is not feasible, we recommend moderate social-distancing that is well balanced against capability limits of national health-care systems.

Diversity begets diversity in mammal species and human cultures

Across the planet the biogeographic distribution of human cultural diversity tends to correlate positively with biodiversity. In this paper we focus on the biogeographic distribution of mammal species and human cultural diversity. We show that not only are these forms of diversity similarly distributed in space, but they both scale superlinearly with environmental production. We develop theory that explains that as environmental productivity increases the ecological kinetics of diversity increases faster than expected because more complex environments are also more interactive. Using biogeographic databases of the global distributions of mammal species and human cultures we test a series of hypotheses derived from this theory and find support for each. For both mammals and cultures, we show that (1) both forms of diversity increase exponentially with ecological kinetics; (2) the kinetics of diversity is faster than the kinetics of productivity; (3) diversity scales superlinearly with environmental productivity; and (4) the kinetics of diversity is faster in increasingly productive environments. This biogeographic convergence is particularly striking because while the dynamics of biological and cultural evolution may be similar in principle the underlying mechanisms and time scales are very different. However, a common currency underlying all forms of diversity is ecological kinetics; the temperature-dependent fluxes of energy and biotic interactions that sustain all forms of life at all levels of organization. Diversity begets diversity in mammal species and human cultures because ecological kinetics drives superlinear scaling with environmental productivity.

Weighted hypersoft configuration model

Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions, we are dealing with configuration models. If the degree sequence is constrained exactly, the corresponding microcanonical ensemble of random graphs with a given degree sequence is the configuration model per se. If the degree sequence is constrained only on average, the corresponding grand-canonical ensemble of random graphs with a given expected degree sequence is the soft configuration model. If the degree sequence is not fixed at all but randomly drawn from a fixed distribution, the corresponding hypercanonical ensemble of random graphs with a given degree distribution is the hypersoft configuration model, a more adequate description of dynamic real-world networks in which degree sequences are never fixed but degree distributions often stay stable. Here, we introduce the hypersoft configuration model of weighted networks. The main contribution is a particular version of the model with power-law degree and strength distributions, and superlinear scaling of strengths with degrees, mimicking the properties of some real-world networks. As a byproduct, we generalize the notions of sparse graphons and their entropy to weighted networks.

A quantum extension of SVM-perf for training nonlinear SVMs in almost linear time

We propose a quantum algorithm for training nonlinear support vector machines (SVM) for feature space learning where classical input data is encoded in the amplitudes of quantum states. Based on the classical SVM-perf algorithm of Joachims \cite{joachims2006training}, our algorithm has a running time which scales linearly in the number of training examplesm(up to polylogarithmic factors) and applies to the standard soft-marginℓ1-SVM model. In contrast, while classical SVM-perf has demonstrated impressive performance on both linear and nonlinear SVMs, its efficiency is guaranteed only in certain cases: it achieves linearmscaling only for linear SVMs, where classification is performed in the original input data space, or for the special cases of low-rank or shift-invariant kernels. Similarly, previously proposed quantum algorithms either have super-linear scaling inm, or else apply to different SVM models such as the hard-margin or least squaresℓ2-SVM which lack certain desirable properties of the soft-marginℓ1-SVM model. We classically simulate our algorithm and give evidence that it can perform well in practice, and not only for asymptotically large data sets.