Rationalizing spatial exploration patterns of wild animals and humans through a temporal discounting framework.
ABSTRACT: Understanding the exploration patterns of foragers in the wild provides fundamental insight into animal behavior. Recent experimental evidence has demonstrated that path lengths (distances between consecutive turns) taken by foragers are well fitted by a power law distribution. Numerous theoretical contributions have posited that "Lévy random walks"-which can produce power law path length distributions-are optimal for memoryless agents searching a sparse reward landscape. It is unclear, however, whether such a strategy is efficient for cognitively complex agents, from wild animals to humans. Here, we developed a model to explain the emergence of apparent power law path length distributions in animals that can learn about their environments. In our model, the agent's goal during search is to build an internal model of the distribution of rewards in space that takes into account the cost of time to reach distant locations (i.e., temporally discounting rewards). For an agent with such a goal, we find that an optimal model of exploration in fact produces hyperbolic path lengths, which are well approximated by power laws. We then provide support for our model by showing that humans in a laboratory spatial exploration task search space systematically and modify their search patterns under a cost of time. In addition, we find that path length distributions in a large dataset obtained from free-ranging marine vertebrates are well described by our hyperbolic model. Thus, we provide a general theoretical framework for understanding spatial exploration patterns of cognitively complex foragers.
Project description:Many populations live in environments subject to frequent biotic and abiotic changes. Nonetheless, it is interesting to ask whether an evolving population's mean fitness can increase indefinitely, and potentially without any limit, even in a constant environment. A recent study showed that fitness trajectories of Escherichia coli populations over 50 000 generations were better described by a power-law model than by a hyperbolic model. According to the power-law model, the rate of fitness gain declines over time but fitness has no upper limit, whereas the hyperbolic model implies a hard limit. Here, we examine whether the previously estimated power-law model predicts the fitness trajectory for an additional 10 000 generations. To that end, we conducted more than 1100 new competitive fitness assays. Consistent with the previous study, the power-law model fits the new data better than the hyperbolic model. We also analysed the variability in fitness among populations, finding subtle, but significant, heterogeneity in mean fitness. Some, but not all, of this variation reflects differences in mutation rate that evolved over time. Taken together, our results imply that both adaptation and divergence can continue indefinitely--or at least for a long time--even in a constant environment.
Project description:Many empirical and descriptive models have been proposed since the beginning of the 20th century. In the present study, the power-law (Kennelly) and logarithmic (Péronnet-Thibault) models were compared with asymptotic models such as 2-parameter hyperbolic models (Hill and Scherrer), 3-parameter hyperbolic model (Morton), and exponential model (Hopkins). These empirical models were compared from the performance of 6 elite endurance runners (P. Nurmi, E. Zatopek, J. Väätäinen, L. Virén, S. Aouita, and H. Gebrselassie) who were world-record holders and/or Olympic winners and/or world or European champions. These elite runners were chosen because they participated several times in international competitions over a large range of distances (1500, 3000, 5000, and 10000 m) and three also participated in a marathon. The parameters of these models were compared and correlated. The less accurate models were the asymptotic 2-parameter hyperbolic models but the most accurate model was the asymptotic 3-parameter hyperbolic model proposed by Morton. The predictions of long-distance performances (maximal running speeds for 30 and 60 min and marathon) by extrapolation of the logarithmic and power-law models were more accurate than the predictions by extrapolation in all the asymptotic models. The overestimations of these long-distance performances by Morton's model were less important than the overestimations by the other asymptotic models.
Project description:BACKGROUND: Despite the common experience that interrupted sleep has a negative impact on waking function, the features of human sleep-wake architecture that best distinguish sleep continuity versus fragmentation remain elusive. In this regard, there is growing interest in characterizing sleep architecture using models of the temporal dynamics of sleep-wake stage transitions. In humans and other mammals, the state transitions defining sleep and wake bout durations have been described with exponential and power law models, respectively. However, sleep-wake stage distributions are often complex, and distinguishing between exponential and power law processes is not always straightforward. Although mono-exponential distributions are distinct from power law distributions, multi-exponential distributions may in fact resemble power laws by appearing linear on a log-log plot. METHODOLOGY/PRINCIPAL FINDINGS: To characterize the parameters that may allow these distributions to mimic one another, we systematically fitted multi-exponential-generated distributions with a power law model, and power law-generated distributions with multi-exponential models. We used the Kolmogorov-Smirnov method to investigate goodness of fit for the "incorrect" model over a range of parameters. The "zone of mimicry" of parameters that increased the risk of mistakenly accepting power law fitting resembled empiric time constants obtained in human sleep and wake bout distributions. CONCLUSIONS/SIGNIFICANCE: Recognizing this uncertainty in model distinction impacts interpretation of transition dynamics (self-organizing versus probabilistic), and the generation of predictive models for clinical classification of normal and pathological sleep architecture.
Project description:Modeling distributions of citations to scientific papers is crucial for understanding how science develops. However, there is a considerable empirical controversy on which statistical model fits the citation distributions best. This paper is concerned with rigorous empirical detection of power-law behaviour in the distribution of citations received by the most highly cited scientific papers. We have used a large, novel data set on citations to scientific papers published between 1998 and 2002 drawn from Scopus. The power-law model is compared with a number of alternative models using a likelihood ratio test. We have found that the power-law hypothesis is rejected for around half of the Scopus fields of science. For these fields of science, the Yule, power-law with exponential cut-off and log-normal distributions seem to fit the data better than the pure power-law model. On the other hand, when the power-law hypothesis is not rejected, it is usually empirically indistinguishable from most of the alternative models. The pure power-law model seems to be the best model only for the most highly cited papers in "Physics and Astronomy". Overall, our results seem to support theories implying that the most highly cited scientific papers follow the Yule, power-law with exponential cut-off or log-normal distribution. Our findings suggest also that power laws in citation distributions, when present, account only for a very small fraction of the published papers (less than 1 % for most of science fields) and that the power-law scaling parameter (exponent) is substantially higher (from around 3.2 to around 4.7) than found in the older literature.
Project description:The size distribution of neuronal avalanches in cortical networks has been reported to follow a power law distribution with exponent close to -1.5, which is a reflection of long-range spatial correlations in spontaneous neuronal activity. However, identifying power law scaling in empirical data can be difficult and sometimes controversial. In the present study, we tested the power law hypothesis for neuronal avalanches by using more stringent statistical analyses. In particular, we performed the following steps: (i) analysis of finite-size scaling to identify scale-free dynamics in neuronal avalanches, (ii) model parameter estimation to determine the specific exponent of the power law, and (iii) comparison of the power law to alternative model distributions. Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect). This scale-free dynamics suggests the power law as a model for the distribution of avalanche sizes. Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports. Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test. Both the power law distribution without and with exponential cut-off provided significantly better fits to the cluster size distributions in neuronal avalanches than the exponential, the lognormal and the gamma distribution. In summary, our findings strongly support the power law scaling in neuronal avalanches, providing further evidence for critical state dynamics in superficial layers of cortex.
Project description:Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter. We find these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology.
Project description:Efficient searching for resources such as food by animals is key to their survival. It has been proposed that diverse animals from insects to sharks and humans adopt searching patterns that resemble a simple Lévy random walk, which is theoretically optimal for 'blind foragers' to locate sparse, patchy resources. To test if such patterns are generated intrinsically, or arise via environmental interactions, we tracked free-moving <i>Drosophila</i> larvae with (and without) blocked synaptic activity in the brain, suboesophageal ganglion (SOG) and sensory neurons. In brain-blocked larvae, we found that extended substrate exploration emerges as multi-scale movement paths similar to truncated Lévy walks. Strikingly, power-law exponents of brain/SOG/sensory-blocked larvae averaged 1.96, close to a theoretical optimum (<i>µ</i> ? 2.0) for locating sparse resources. Thus, efficient spatial exploration can emerge from autonomous patterns in neural activity. Our results provide the strongest evidence so far for the intrinsic generation of Lévy-like movement patterns.
Project description:The putative scale-free nature of real-world networks has generated a lot of interest in the past 20 years: if networks from many different fields share a common structure, then perhaps this suggests some underlying ‘network law’. Testing the degree distribution of networks for power-law tails has been a topic of considerable discussion. Ad hoc statistical methodology has been used both to discredit power-laws as well as to support them. This paper proposes a statistical testing procedure that considers the complex issues in testing degree distributions in networks that result from observing a finite network, having dependent degree sequences and suffering from insufficient power. We focus on testing whether the tail of the empirical degrees behaves like the tail of a de Solla Price model, a two-parameter power-law distribution. We modify the well-known Kolmogorov–Smirnov test to achieve even sensitivity along the tail, considering the dependence between the empirical degrees under the null distribution, while guaranteeing sufficient power of the test. We apply the method to many empirical degree distributions. Our results show that power-law network degree distributions are not rare, classifying almost 65% of the tested networks as having a power-law tail with at least 80% power.
Project description:In this paper, we study the topological properties of the global supply chain network in terms of its degree distribution, clustering coefficient, degree-degree correlation, bow-tie structure, and community structure to test the efficient supply chain propositions proposed by E. J.S. Hearnshaw et al. The global supply chain data in the year 2017 are constructed by collecting various company data from the web site of Standard & Poor's Capital IQ platform. The in- and out-degree distributions are characterized by a power law of the form of ?in = 2.42 and ?out = 2.11. The clustering coefficient decays [Formula: see text] with an exponent ?k = 0.46. The nodal degree-degree correlations ?knn(k)? indicates the absence of assortativity. The bow-tie structure of giant weakly connected component (GWCC) reveals that the OUT component is the largest and consists 41.1% of all firms. The giant strong connected component (GSCC) is comprised of 16.4% of all firms. We observe that upstream or downstream firms are located a few steps away from the GSCC. Furthermore, we uncover the community structures of the network and characterize them according to their location and industry classification. We observe that the largest community consists of the consumer discretionary sector based mainly in the United States (US). These firms belong to the OUT component in the bow-tie structure of the global supply chain network. Finally, we confirm the validity of Hearnshaw et al.'s efficient supply chain propositions, namely Proposition S1 (short path length), Proposition S2 (power-law degree distribution), Proposition S3 (high clustering coefficient), Proposition S4 ("fit-gets-richer" growth mechanism), Proposition S5 (truncation of power-law degree distribution), and Proposition S7 (community structure with overlapping boundaries) regarding the global supply chain network. While the original propositions S1 just mentioned a short path length, we found the short path from the GSCC to IN and OUT by analyzing the bow-tie structure. Therefore, the short path length in the bow-tie structure is a conceptual addition to the original propositions of Hearnshaw.
Project description:High-throughput biology has contributed a wealth of data on chemicals, including natural products (NPs). Recently, attention was drawn to certain, predominantly synthetic, compounds that are responsible for disproportionate percentages of hits but are false actives. Spurious bioassay interference led to their designation as pan-assay interference compounds (PAINS). NPs lack comparable scrutiny, which this study aims to rectify. Systematic mining of 80+ years of the phytochemistry and biology literature, using the NAPRALERT database, revealed that only 39 compounds represent the NPs most reported by occurrence, activity, and distinct activity. Over 50% are not explained by phenomena known for synthetic libraries, and all had manifold ascribed bioactivities, designating them as invalid metabolic panaceas (IMPs). Cumulative distributions of ?200,000 NPs uncovered that NP research follows power-law characteristics typical for behavioral phenomena. Projection into occurrence-bioactivity-effort space produces the hyperbolic black hole of NPs, where IMPs populate the high-effort base.