Group Size Distribution Research Articles

Growth and addition in a herding model with fractional orders of derivatives

This work involves an investigation of the mechanics of the herding behavior using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behavior in the real world, such as the financial markets. A herding model with fractional orders of derivatives was developed. This model involves the use of derivatives of order α where 0<α⩽1 . We have found the generalized result which indicates that number of groups of agents with size k increases linearly with time as nk=p(2p−1)(2−α)p(1−α)+1Γ(α+2−α1−p)Γ(k)Γ(k−1+α+2−α1−p)t for α∈(0,1] , where p is a growth parameter. The result reduces to that in a previous herding model with a derivative order of 1 for α = 1. The results corresponding to various values of α and p are presented. The group-size distribution at long time is found to decay as a generalized power law, with an exponent depending on both α and p, thereby demonstrating that the scale invariance property of a complex system holds regardless of the order of the derivatives. The physical interpretation of fractional calculus is also explored based on the results of this work.

Homophily-Based Social Group Formation in a Spin Glass Self-Assembly Framework.

Homophily, the tendency of humans to attract each other when sharing similar features, traits, or opinions, has been identified as one of the main driving forces behind the formation of structured societies. Here we ask to what extent homophily can explain the formation of social groups, particularly their size distribution. We propose a spin-glass-inspired framework of self-assembly, where opinions are represented as multidimensional spins that dynamically self-assemble into groups; individuals within a group tend to share similar opinions (intragroup homophily), and opinions between individuals belonging to different groups tend to be different (intergroup heterophily). We compute the associated nontrivial phase diagram by solving a self-consistency equation for "magnetization" (combined average opinion). Below a critical temperature, there exist two stable phases: one ordered with nonzero magnetization and large clusters, the other disordered with zero magnetization and no clusters. The system exhibits a first-order transition to the disordered phase. We analytically derive the group-size distribution that successfully matches empirical group-size distributions from online communities.

Enhanced cooperation in multiplayer snowdrift games with random and dynamic groupings.

An analytically tractable generalization of the N-person snowdrift (NSG) game that illustrates how cooperation can be enhanced is proposed and studied. The number of players competing within a NSG varies from one time step to another. Exact equations governing the frequency of cooperation f_{c}(r) as a function of the cost-to-benefit ratio r within an imitation strategy updating scheme are presented. For group sizes g uniformly distributed within the range g∈[1,g_{m}], an analytic formula for the critical value r_{c}(g_{m}), below which the system evolves into a totally cooperative (AllC) state, is derived. In contrast, a fixed group size NSG does not support an AllC state. The result r_{c}(g_{m}) requires the presence of sole-player groups and involves the inverse of the harmonic numbers and, more generally, the inverse first moment of the group size distribution. For r>r_{c}(g_{m}), the equationthat determines the dynamical mixed states f_{c}(r) is given, with exact solutions existing for g_{m}≤5. The exact treatment allows the study of the phase boundary between the AllC state and the mixed states. The analytic results are checked against simulation results and exact agreements are demonstrated. The analytic form of the critical r_{c}(g_{m}) illustrates the necessity of having groups of a sole player in the evolutionary process. This result is supported by simulations with group sizes excluding the sole groups for which no AllC state emerges. A physically transparent picture of the importance of the sole players in inducing an AllC state is further presented based on the last surviving pattern before the AllC state is attained. The exact expression r_{c}(g_{m}) turns out to remain valid for nonuniform group-size distributions. Our analytical tractable generalization, therefore, sheds light on how a competing environment with variable group sizes could enhance cooperation and induce an AllC state.

Isolation and exploitation of minority: Game theoretical analysis.

We investigate various group-size distributions occurring in a situation where each group’s resource is exposed to appropriation by other groups. The amount of appropriation depends on the size difference between groups. Our work focuses on the cases where the entire community isolates a small group or even an individual to maximize its gain. While people’s basic motivation to form a group can be understood based on the group-size effect on multiplying a collective asset, sensitive factors that induce a asymmetric group distribution are the group efficiency and the ratio of secured assets to assets pending in a competition. We show that social rejection to a minor group may occur when the group efficiency is relatively low and their asset is severely exposed to possible appropriation.

A generalized public goods game with coupling of individual ability and project benefit

Facing a heavy task, any single person can only make a limited contribution and team cooperation is needed. As one enjoys the benefit of the public goods, the potential benefits of the project are not always maximized and may be partly wasted. By incorporating individual ability and project benefit into the original public goods game, we study the coupling effect of the four parameters, the upper limit of individual contribution, the upper limit of individual benefit, the needed project cost and the upper limit of project benefit on the evolution of cooperation. Coevolving with the individual-level group size preferences, an increase in the upper limit of individual benefit promotes cooperation while an increase in the upper limit of individual contribution inhibits cooperation. The coupling of the upper limit of individual contribution and the needed project cost determines the critical point of the upper limit of project benefit, where the equilibrium frequency of cooperators reaches its highest level. Above the critical point, an increase in the upper limit of project benefit inhibits cooperation. The evolution of cooperation is closely related to the preferred group-size distribution. A functional relation between the frequency of cooperators and the dominant group size is found.

Variability in group size and the evolution of collective action

Models of the evolution of collective action typically assume that interactions occur in groups of identical size. In contrast, social interactions between animals occur in groups of widely dispersed size. This paper models collective action problems as two-strategy multiplayer games and studies the effect of variability in group size on the evolution of cooperative behavior under the replicator dynamics. The analysis identifies elementary conditions on the payoff structure of the game implying that the evolution of cooperative behavior is promoted or inhibited when the group size experienced by a focal player is more or less variable. Similar but more stringent conditions are applicable when the confounding effect of size-biased sampling, which causes the group-size distribution experienced by a focal player to differ from the statistical distribution of group sizes, is taken into account.

Group-size distribution of skeins of wild geese

In appropriate situations, large populations of geese exhibit dynamical rearrangements by repeated mergers and splits among the groups. We describe the grouping process in terms of a mean-field model based on the Smoluchowski equation of coagulation with fragmentation and observationally plausible kernels. To verify our model, we conducted field observations on skeins of airborne geese, noting both the group-size distribution and the group-forming processes. We found that the group-size distribution we obtained in our field measurements could be represented by a fractional power function with an exponential cutoff. This function matches the asymptotic form of the steady-state solution of our model. Furthermore, we estimated the effective number of individuals involved in interactions by comparison of the model to our field data.

GROUP-SIZE DIVERSITY IN PUBLIC GOODS GAMES

Public goods games are models of social dilemmas where cooperators pay a cost for the production of a public good while defectors free ride on the contributions of cooperators. In the traditional framework of evolutionary game theory, the payoffs of cooperators and defectors result from interactions in groups formed by binomial sampling from an infinite population. Despite empirical evidence showing that group-size distributions in nature are highly heterogeneous, most models of social evolution assume that the group size is constant. In this article, I remove this assumption and explore the effects of having random group sizes on the evolutionary dynamics of public goods games. By a straightforward application of Jensen's inequality, I show that the outcome of general nonlinear public goods games depends not only on the average group size but also on the variance of the group-size distribution. This general result is illustrated with two nonlinear public goods games (the public goods game with discounting or synergy and the N-person volunteer's dilemma) and three different group-size distributions (Poisson, geometric, and Waring). The results suggest that failing to acknowledge the natural variation of group sizes can lead to an underestimation of the actual level of cooperation exhibited in evolving populations.

Understanding Animal Group-Size Distributions

One of the most striking aspects of animal groups is their remarkable variation in size, both within and between species. While a number of mechanistic models have been proposed to explain this variation, there are few comprehensive datasets against which these models have been tested. In particular, we only vaguely understand how environmental factors and behavioral activities affect group-size distributions. Here we use observations of House sparrows (Passer domesticus) to investigate the factors determining group-size distribution. Over a wide range of conditions, we observed that animal group sizes followed a single parameter distribution known as the logarithmic distribution. This single parameter is the mean group size experienced by a randomly chosen individual (including the individual itself). For sparrows, the experienced mean group size, and hence the distribution, was affected by four factors: morning temperature, place, behavior and the degree of food spillage. Our results further indicate that the sparrows regulate the mean group size they experience, either by groups splitting more or merging less when local densities are high. We suggest that the mean experienced group size provides a simple but general tool for assessing the ecology and evolution of grouping.

SCALING PHENOMENA IN THE SLOPE SYSTEM

An agent-based model is applied for studying the group-forming behavior numerically. A fixed number of homogeneous agents are distributed on a two-dimensional square lattice system with L × L unit cells. The dynamics of the agents are described in terms of a potential and a friction factor, just like that of a weight on a slope. The potential is defined in an attempt to model the concept of the "biosocial attraction." Depending on the density of the population and the friction factor, four regimes of group-size distributions are identified. They are the fixed, the exponential, the power-law and the black-hole regimes. Especially, beyond the well-known exponential and power-law distributions, a black-hole regime is identified for the lowest values of the friction factor. In the black-hole regime, all the agents in the system are attracted to the only group remaining in the system. An example membership values for each of the four classes are suggested in terms of the two key parameters of the system, the population density and the friction factor. And a phase transition is observed between the exponential and the black-hole regimes. Also the results are discussed in relation to the city-size distributions in a country.

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.

Scaling in steady-state aggregation with injection

A mean-field approach for steady-state aggregation with injection is presented. It is shown that for a wide variety of aggregation processes the resulting steady-size distribution obeys a power law N(m) approximately m(-alpha) with alpha=(3+beta)/2 and beta the degree of homogeneity of the coagulation kernel. The general conditions for this to happen are obtained. Some applications are studied. In particular, it predicts a potential behavior for coagulation in atmospheric aerosols with exponent alpha approximately 2, in agreement with observations. The theoretical results also agree with some animal group-size distributions and with numerical simulations in fractal aggregates.

Scaling in animal group-size distributions.

An elementary model of animal aggregation is presented. The group-size distributions resulting from this model are truncated power laws. The predictions of the model are found to be consistent with data that describe the group-size distributions of tuna fish, sardinellas, and African buffaloes.

From individuals to aggregations: the interplay between behavior and physics.

This paper analyses the processes by which organisms form groups and how social forces interact with environmental variability and transport. For aquatic organisms, the latter is especially important-will sheared or turbulent flows disrupt organism groups? To analyse such problems, we use individual-based models to study the environmental and social forces leading to grouping. The models are then embedded in turbulent flow fields to gain an understanding of the interplay between the forces acting on the individuals and the transport induced by the fluid motion. Instead of disruption of groups, we find that flows often enhance grouping by increasing the encounter rate among groups and thereby promoting merger into larger groups; the effect breaks down for strong flows. We discuss the transformation of individual-based models into continuum models for the density of organisms. A number of subtle difficulties arise in this process; however, we find that a direct comparison between the individual model and the continuum model is quite favorable. Finally, we examine the dynamics of group statistics and give an example of building an equation for the spatial and temporal variations of the group-size distribution from the individual-based simulations. These studies lay the groundwork for incorporating the effects of grouping into models of the large scale distributions of organisms as well as for examining the evolutionary consequences of group formation.

Optimal language regimes for the European Union

Like other linguistically diverse institutions, the European Union promotes conflicting values as it chooses official languages. Increased activism and the admission of new states threaten a language-policy crisis. One approach to a solution is to analyze more carefully than before the universe of possible language regimes for institutions like the EU and the justifications for deeming any alternative We can define a language regime as a set of official languages and a sei ofrules permitting complete mutual comprehension in a deliberation among representatives of language groups. If only the groups' languages and one synthetic language are relevant, ifthe representatives are monolingual, if translators are bilingual, and if other simplifications are assumed, there are almost five times as many potentially optimal language regimes as language groups (e.g. with nine groups, 40). We can partition the language regimes into ten classes, distinguished by their official languages and whether they translate directly, via intermediate group languages, or via the synthetic language. The duration and urgency ofthe deliberation, the capacity of translation facilities, the importance of language equality, the distribution of group sizes, and the relative cost of learning the synthetic language are some conditions that determine which language regime is optimal. The prevailing conditions in the EU create a clear choice between twofamilies of language regimes. One family satisfies the professed norm of equal language treatment by making either none or all of the groups' languages official. The other family, by making only the largest languages official, systematizes the common EUpractice ofsacrificing language equality for cost reduction. A victoryfor one normative position wouldfavor a corresponding language regime, but a continued daily normative struggle would tend to produce complex Variation in language regimes across EU agencies.