Can Power-Law Scaling and Neuronal Avalanches Arise from Stochastic Dynamics?

The presence of self-organized criticality in biology is often evidenced by a power-law scaling of event size distributions, which can be measured by linear regression on logarithmic axes. We show here that such a procedure does not necessarily mean that the system exhibits self-organized criticality. We first provide an analysis of multisite local field potential (LFP) recordings of brain activity and show that event size distributions defined as negative LFP peaks can be close to power-law distributions. However, this result is not robust to change in detection threshold, or when tested using more rigorous statistical analyses such as the Kolmogorov–Smirnov test. Similar power-law scaling is observed for surrogate signals, suggesting that power-law scaling may be a generic property of thresholded stochastic processes. We next investigate this problem analytically, and show that, indeed, stochastic processes can produce spurious power-law scaling without the presence of underlying self-organized criticality. However, this power-law is only apparent in logarithmic representations, and does not survive more rigorous analysis such as the Kolmogorov–Smirnov test. The same analysis was also performed on an artificial network known to display self-organized criticality. In this case, both the graphical representations and the rigorous statistical analysis reveal with no ambiguity that the avalanche size is distributed as a power-law. We conclude that logarithmic representations can lead to spurious power-law scaling induced by the stochastic nature of the phenomenon. This apparent power-law scaling does not constitute a proof of self-organized criticality, which should be demonstrated by more stringent statistical tests.     

Power-law scaling in dimension-to-biomass relationship of fish schools

Motivated by the finding that there is some biological universality in the relationship between school geometry and school biomass of various pelagic fishes in various conditions, I here establish a scaling law for school dimensions: the school diameter increases as a power-law function of school biomass. The power-law exponent is extracted through the data collapse, and is close to 35. This value of the exponent implies that the mean packing density decreases as the school biomass increases, and the packing structure displays a mass-fractal dimension of 53. By exploiting an analogy between school geometry and polymer chain statistics, I examine the behavioral algorithm governing the swollen conformation of large-sized schools of pelagics, and I explain the value of the exponent.     

Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing

The expected number of new infections per day per infectious person during an epidemic has been found to exhibit power-law scaling with respect to the susceptible fraction of the population. This is in contrast to the linear scaling assumed in traditional epidemiologic modeling. Based on simulated epidemic dynamics in synthetic populations representing Los Angeles, Chicago, and Portland, we find city-dependent scaling exponents in the range of 1.7-2.06. This scaling arises from variations in the strength, duration, and number of contacts per person. Implementation of power-law scaling of the new infection rate is quite simple for SIR, SEIR, and histogram-based epidemic models. Treatment of the effects of the social contact structure through this power-law formulation leads to significantly lower predictions of final epidemic size than the traditional linear formulation.     

Power-law versus exponential distributions of animal group sizes

There has been some confusion concerning the animal group size: an exponential distribution was deduced by maximizing the entropy; lognormal distributions were practically used; as power-law decay with exponent 3/2 was proposed in physical analogy to aerosol condensation. Here I show that the animal group-size distribution follows a power-law decay with exponent 1, and is truncated at a cut-off size which is the expected size of the groups an arbitrary individual engages in. An elementary model of animal aggregation based on binary splitting and coalescing on contingent encounter is presented. The model predicted size distribution holds for various data from pelagic fishes and mammalian herbivores in the wild.     

Impacts of fish aggregation devices on size structures of skipjack tuna Katsuwonus pelamis

Tuna purse seine fisheries target fish aggregated in schools, including free schools that are formed naturally based on fish biology and aggregations associated with natural and/or artificial drifting objects. Using data collected from skipjack tuna (Katsuwonus pelamis) fisheries, we evaluated differences in size structures between drifting-floating-object-associated schools and unassociated schools. We developed a generalized linear model to remove impacts of environmental variables on skipjack size composition. This study indicates that the drifting-floating-object-associated schools tended to have significantly wider size ranges than the unassociated schools. This suggests that unassociated schools were likely formed based on similarity in sizes among individuals within a school while drifting-floating-object-associated schools were probably composed of individuals of large size ranges and their formation was not based on the “size selection” rule. We concluded that the unassociated schools and the drifting-floating-object-associated schools were formed through different mechanisms, and drifting floating objects could aggregate unassociated schools of different size structures. Thus, a large scale of deployment of man-made floating objects might disrupt the spatial aggregation pattern of fish that otherwise tended to school based on their sizes in the absence of floating objects.     

School size affects individual feeding success in three-spined sticklebacks (Gastevosteus aculeatus L.)

The effect of school size on the feeding success of individual three-spined sticklebacks was studied. We found that the proportion of fish feeding on benthic prey increased with school size and that fish in large schools tended to start feeding sooner than fish in small schools. The total number of strikes also increased in larger schools. Despite this evidence for a foraging benefit associated with school membership we propose that improved feeding returns do not alone explain stickleback schooling.     

Modeling fractal structure of city-size distributions using correlation functions

Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Using the idea from general fractals and scaling, I propose a dual competition hypothesis of city development to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another, but represent different processes of urban evolution, respectively. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is the Pareto effect indicating city number increase (external complexity), and the other the Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences.     

Emergence of Scale-Free Close-Knit Friendship Structure in Online Social Networks

Although the structural properties of online social networks have attracted much attention, the properties of the close-knit friendship structures remain an important question. Here, we mainly focus on how these mesoscale structures are affected by the local and global structural properties. Analyzing the data of four large-scale online social networks reveals several common structural properties. It is found that not only the local structures given by the indegree, outdegree, and reciprocal degree distributions follow a similar scaling behavior, the mesoscale structures represented by the distributions of close-knit friendship structures also exhibit a similar scaling law. The degree correlation is very weak over a wide range of the degrees. We propose a simple directed network model that captures the observed properties. The model incorporates two mechanisms: reciprocation and preferential attachment. Through rate equation analysis of our model, the local-scale and mesoscale structural properties are derived. In the local-scale, the same scaling behavior of indegree and outdegree distributions stems from indegree and outdegree of nodes both growing as the same function of the introduction time, and the reciprocal degree distribution also shows the same power-law due to the linear relationship between the reciprocal degree and in/outdegree of nodes. In the mesoscale, the distributions of four closed triples representing close-knit friendship structures are found to exhibit identical power-laws, a behavior attributed to the negligible degree correlations. Intriguingly, all the power-law exponents of the distributions in the local-scale and mesoscale depend only on one global parameter, the mean in/outdegree, while both the mean in/outdegree and the reciprocity together determine the ratio of the reciprocal degree of a node to its in/outdegree. Structural properties of numerical simulated networks are analyzed and compared with each of the four real networks. This work helps understand the interplay between structures on different scales in online social networks.     

Statistical analyses support power law distributions found in neuronal avalanches

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.     

Difference on reproductive trait of skipjack tuna <Emphasis Type="Italic">Katsuonus pelamis</Emphasis> female between schools (free vs FAD school) in the tropical western and central Pacific Ocean

In order to clarify the difference on reproductive traits of female skipjack tuna between fish aggregation device (FAD) and free-swimming (Free) school, we examined the relative condition factor and sex ratio, minimum size at first maturity and spawning fraction in tropical western and central Pacific Ocean. The relative condition factor in developing (FAD: 1.00 ± 0.06, Free: 1.05 ± 0.07) and spawning capable phase (FAD: 1.01 ± 0.06, Free: 1.04 ± 0.06) was higher in the free-swimming school than those in the fish aggregation device school (p < 0.01). The proportion of mature fish was significant difference between each schools (p < 0.05). The sex ratios in the schools were not biased (p > 0.05). Observed minimum size at first maturity were 40.0 cm FL in FAD and 42.5 cm FL in free-swimming school. Spawning capable fish were observed year around in FAD and for seven months in free-swimming school. The estimated total spawning fractions based on the postovulatory follicle method were 0.50 and 0.49 in the FAD and free-swimming schools, respectively. These results suggested that differences in the school type did not affect inherent spawning characteristics (i.e. sex ratio, minimum size at first maturity, or spawning season) and may shortly affect the oogenesis throughout the food availability.     

An assessment of preferential attachment as a mechanism for human sexual network formation

Recent research into the properties of human sexual-contact networks has suggested that the degree distribution of the contact graph exhibits power-law scaling. One notable property of this power-law scaling is that the epidemic threshold for the population disappears when the scaling exponent rho is in the range 2 < rho < or = 3. This property is of fundamental significance for the control of sexually transmitted diseases (STDs) such as HIV/AIDS since it implies that an STD can persist regardless of its transmissibility. A stochastic process, known as preferential attachment, that yields one form of power-law scaling has been suggested to underlie the scaling of sexual degree distributions. The limiting distribution of this preferential attachment process is the Yule distribution, which we fit using maximum likelihood to local network data from samples of three populations: (i) the Rakai district, Uganda; (ii) Sweden; and (iii) the USA. For all local networks but one, our interval estimates of the scaling parameters are in the range where epidemic thresholds exist. The estimate of the exponent for male networks in the USA is close to 3, but the preferential attachment model is a very poor fit to these data. We conclude that the epidemic thresholds implied by this model exist in both single-sex and two-sex epidemic model formulations. A strong conclusion that we derive from these results is that public health interventions aimed at reducing the transmissibility of STD pathogens, such as implementing condom use or high-activity anti-retroviral therapy, have the potential to bring a population below the epidemic transition, even in populations exhibiting large degrees of behavioural heterogeneity.     

Emergence of Oblong School Shape: Models and Empirical Data of Fish

The main benefit of the oblong shape of schools of fish is supposed to be the protection against predation. Models of self‐organised travelling groups have shown that this shape may arise as a side effect of the avoidance of collisions with group members. These models were developed for schools of fish in open water, whereas the oblong shape of schools of real fish has mostly been observed in schools in tanks. Therefore, it is not known how school shape in a tank originates neither in models nor in real fish. To find out what causes this shape, we use the combination of a theoretical and an empirical study. We test the predictions produced by our earlier models regarding the effect of school size on the school shape both in a model of self‐organised schooling in a tank and empirically. Empirically, we study the 3D positions of all individuals in the schools of 10–60 real mullets (Chelon labrosus). We calculate for each individual its distance to its nearest neighbour and its velocity and we measure per school its length and width. The relation between school shape and size in the model and in the real mullets supports our prediction and thus supports the hypothesis that school shape may be emergent from the avoidance of collisions during coordinated travelling.     

The effect of school size on the structure and dynamics of minnow schools

Varying the number of fish in schools of minnows (Phoxinus phoxinus) affects their three-dimensional structure and internal dynamics. Previous authors have suggested that single pairs of fish can be considered a school, but internal organization and structure of two-fish schools are quantitatively different from those for larger schools. Time series analyses show that correlations between fishes' instantaneous velocities increase with school size and as interfish distances decrease. Average cross-correlations of fishes' velocities indicate that leader/follower relationships are common in two-fish schools, but they are not seen for schools with more fish in them. Pairs of fish tend to swim one behind another and on the same level, but larger schools take on a more three-dimensional appearance.     

Further evidence for size-assortative schooling in sticklebacks

Using brook ( Culaea inconstans ) and 10-spined ( Pungitius pungitius ) sticklebacks we examined body-size related schooling behaviour. Small and large sticklebacks were allowed to choose between two test schools, of small and of large individuals, with and without a piscivorous fish ( Oncorhynchus mykiss ) visible. Sticklebacks of both species preferred the company of fish of matching body size: small fish associating with a school of small fish, large fish with a school of large fish. While no interspecific differences were found in responses to school selection, body size and predator presence did affect selection of school-type. In both species, small fish tended to show a stronger preference for matching schools. The preference was enhanced in small fish with presence of a predator, but not in large fish. These observations are further evidence for assortative schooling in sticklebacks.     

Three dimensional structure and morphology of pelagic fish schools

In fish resource assessment, it is very important to know about the behaviour and form of fish schools. This paper describes the three‐dimensional (3D) morphology and internal structure of pelagic schools observed using vertical‐scanning multibeam sonar. The acoustic data were collected in waters off Venezuela, Senegal, and Mexico. The data were used to derive metrics of school location, density, shape and internal structure from a total of 668 schools: 257 from Mexico, 343 from Venezuela and 68 from Senegal. In general, school shapes were amoeboid‐like (e.g. spheres or ellipsoids) and not simply geometric. Also, the fish were patchily‐distributed, forming both nuclei (groups) and vacuoles (empty spaces) within the schools. The results support the hypothesis of ‘auto‐organization of the fish inside schools’, meaning that the distribution of fish within a school arises from the elementary reaction of the individuals.     

Scaling exponents and rank-size distributions as indicators of landscape character and change

I used a rank-size distribution model and its associated scaling exponents to describe the organization of Yellowstone National Park before and after the 1988 fire season using the statistical distribution of patch sizes. Rank-size distributions indicate the relative effect of patch size on landscape structure, and whether the size of patches differs from what is expected from the model. Scaling exponents describe the distribution and magnitude of change in patch size, and may indicate the effect of fire on ecological processes including succession and resource distribution. The results of my analysis suggest that fires during the 1988 fire-season substantially affected the distribution of patch size in Yellowstone National Park, where large patches have a disproportionate effect on landscape character. For example, patches ≥100 ha occupy a majority of the area even though they represent a minority number of patches in the landscape. Additionally, rank-size distributions indicate fractal properties existed over several orders of magnitude, signifying that processes acting at one level of the landscape hierarchy may be acting similarly at other levels of the hierarchy. This has implications for linking the scaling properties of patch size with other scale-based phenomena including allometry. Finally, the distribution of patch sizes in conjunction with fire-return interval may be useful in assessing the likelihood of landscape-level disturbances.     

Frequency distributions from birth,death, and creation processes

The time-dependent frequency distribution of groups of individuals versus group size was investigated within a continuum approximation, assuming a simplified individual growth, death and creation model. The analogy of the system to a physical fluid exhibiting both convection and diffusion was exploited in obtaining various solutions to the distribution equation. A general solution was approximated through the application of a Green's function. More specific exact solutions were also found to be useful. The solutions were continually checked against the continuum approximation through extensive simulation of the discrete system. Over limited ranges of group size, the frequency distributions were shown to closely exhibit a power-law dependence on group size, as found in many realizations of this type of system, ranging from colonies of mutated bacteria to the distribution of surnames in a given population. As an example, the modeled distributions were successfully fit to the distribution of surnames in several countries by adjusting the parameters specifying growth, death and creation rates.     

Migratory charr schools exhibit population and kin associations beyond juvenile stages

Few studies have critically investigated the genetic composition of wild fish schools. Yet, such investigations may have profound implications for the understanding of social organization and population differentiation in both fundamental and applied research. Using 20 microsatellite loci, we investigated the composition of 53 schools (total n = 211) of adult and subadult migratory brook charr (Salvelinus fontinalis) sampled from the known feeding areas of two populations inhabiting Mistassini Lake (Québec, Canada). We specifically tested whether (i) school members originated from the same population, (ii) individuals from the same population within schools were kin (half- or full-siblings), and (iii) kin schooling relationships differed between sexes. Randomization tests revealed a tendency for most schools to be population specific, although some schools were population mixtures. Significantly more kin were found within schools than expected at random for both populations (approximately 21-34% of the total number of school members). This result, combined with the observed size range of individuals, indicated that stable associations between kin may occur beyond juvenile stages for up to 4 years. Nevertheless, a high proportion of school members were non-kin (approximately 66-79%). No differences were detected between sexes in the propensity to school with kin. We discuss the hypothesis that the stable kin groups, rather than arising from kin selection, may instead be a by-product of familiarity based on individual selection for the maintenance of local adaptations related to migration (natal and feeding area philopatry). Our results are noteworthy because they suggest that there is some degree of permanence in the composition of wild fish schools. Additionally, they support the hypothesis that schools can be hierarchically structured (from population members down to family groups) and are thus nonrandom genetic entities.     

Recreational Fish-Finders—An Inexpensive Alternative to Scientific Echo-Sounders for Unravelling the Links between Marine Top Predators and Their Prey

Studies investigating how mobile marine predators respond to their prey are limited due to the challenging nature of the environment. While marine top predators are increasingly easy to study thanks to developments in bio-logging technology, typically there is scant information on the distribution and abundance of their prey, largely due to the specialised nature of acquiring this information. We explore the potential of using single-beam recreational fish-finders (RFF) to quantify relative forage fish abundance and draw inferences of the prey distribution at a fine spatial scale. We compared fish school characteristics as inferred from the RFF with that of a calibrated scientific split-beam echo-sounder (SES) by simultaneously operating both systems from the same vessel in Algoa Bay, South Africa. Customized open-source software was developed to extract fish school information from the echo returns of the RFF. For schools insonified by both systems, there was close correspondence between estimates of mean school depth (R² = 0.98) and school area (R² = 0.70). Estimates of relative school density (mean volume backscattering strength; S_v) measured by the RFF were negatively biased through saturation of this system given its smaller dynamic range. A correction factor applied to the RFF-derived density estimates improved the comparability between the two systems. Relative abundance estimates using all schools from both systems were congruent at scales from 0.5 km to 18 km with a strong positive linear trend in model fit estimates with increasing scale. Although absolute estimates of fish abundance cannot be derived from these systems, they are effective at describing prey school characteristics and have good potential for mapping forage fish distribution and relative abundance. Using such relatively inexpensive systems could greatly enhance our understanding of predator-prey interactions.     

Food web framework for size-structured populations

We synthesise traditional unstructured food webs, allometric body size scaling, trait-based modelling, and physiologically structured modelling to provide a novel and ecologically relevant tool for size-structured food webs. The framework allows food web models to include ontogenetic growth and life-history omnivory at the individual level by resolving the population structure of each species as a size-spectrum. Each species is characterised by the trait ‘size at maturation’, and all model parameters are made species independent through scaling with individual body size and size at maturation. Parameter values are determined from cross-species analysis of fish communities as life-history omnivory is widespread in aquatic systems, but may be reparameterised for other systems. An ensemble of food webs is generated and the resulting communities are analysed at four levels of organisation: community level, species level, trait level, and individual level. The model may be solved analytically by assuming that the community spectrum follows a power law. The analytical solution provides a baseline expectation of the results of complex food web simulations, and agrees well with the predictions of the full model on biomass distribution as a function of individual size, biomass distribution as a function of size at maturation, and relation between predator-prey mass ratio of preferred and eaten food. The full model additionally predicts the diversity distribution as a function of size at maturation.