In a community-structured population, public goods games (PGG) occur both within and between communities. Such type of PGG is referred as multilevel public goods games (MPGG). We propose a minimalist evolutionary model of the MPGG and analytically study the evolution of cooperation. We demonstrate that in the case of sufficiently large community size and community number, if the imitation strength within community is weak, i.e., an individual imitates another one in the same community almost randomly, cooperation as well as punishment are more abundant than defection in the long run; if the imitation strength between communities is strong, i.e., the more successful strategy in two individuals from distinct communities is always imitated, cooperation and punishment are also more abundant. However, when both of the two imitation intensities are strong, defection becomes the most abundant strategy in the population. Our model provides insight into the investigation of the large-scale cooperation in public social dilemma among contemporary communities.

According to the comprehensive analysis of the traditional DV-Hop, we propose an Iterative Cooperation DV-Hop localization (ICDV-Hop) algorithm, which improves the localization performance through adopting hop count threshold, collinearity test and new beacon nomination. The proposed algorithm selects the optimal beacon nodes for high localization accuracy by employing the hop count threshold to limit the distances between nodes and using collinearity degree to restrict topology relations between nodes. When the localization error of the located node is under the error threshold, it is nominated as new beacon to help the remaining unknown nodes to re-locate themselves, namely to expand localization coverage by iterative cooperation. Simulation results demonstrate that ICDV-Hop algorithm is more effective to improve the localization accuracy and coverage. Moreover, it is more reliable and robust compared to the traditional DV-Hop, especially when the ratio of beacon nodes is low and the network topology is sparse.

The public goods game is a powerful metaphor for exploring the maintenance of social cooperative behavior in a group of interactional selfish players. Here we study the emergence of cooperation in the public goods games with diverse contributions in finite populations. The theory of stochastic process is innovatively adopted to investigate the evolutionary dynamics of the public goods games involving a diversity of contributions. In the limit of rare mutations, the general stationary distribution of this stochastic process can be analytically approximated by means of diffusion theory. Moreover, we demonstrate that increasing the diversity of contributions greatly reduces the probability of finding the population in a homogeneous state full of defectors. This increase also raises the expectation of the total contribution in the entire population and thus promotes social cooperation. Furthermore, by investigating the evolutionary dynamics of optional public goods games with diverse contributions, we find that nonparticipation can assist players who contribute more in resisting invasion and taking over individuals who contribute less. In addition, numerical simulations are performed to confirm our analytical results. Our results may provide insight into the effect of diverse contributions on cooperative behaviors in the real world.

The distribution of wealth among individuals in real society can be well described by the Pareto principle or ``80-20 rule.'' How does such heterogeneity in initial wealth distribution affect the emergence of public cooperation, when individuals, the rich and the poor, engage in a collective-risk enterprise, not to gain a profit but to avoid a potential loss? Here we address this issue by studying a simple but effective model based on threshold public goods games. We analyze the evolutionary dynamics for two distinct scenarios, respectively: one with fair sharers versus defectors and the other with altruists versus defectors. For both scenarios, particularly, we in detail study the dynamics of the population with dichotomic initial wealth---the rich versus the poor. Moreover, we demonstrate the possible steady compositions of the population and provide the conditions for stability of these steady states. We prove that in a population with heterogeneous wealth distribution, richer individuals are more likely to cooperate than poorer ones. Participants with lower initial wealth may choose to cooperate only if all players richer than them are cooperators. The emergence of pubic cooperation largely relies on rich individuals. Furthermore, whenever the wealth gap between the rich and the poor is sufficiently large, cooperation of a few rich individuals can substantially elevate the overall level of social cooperation, which is in line with the well-known Pareto principle. Our work may offer an insight into the emergence of cooperative behavior in real social situations where heterogeneous distribution of wealth among individual is omnipresent.

We investigate the impact of migration on the evolutionary game dynamics of finite populations with community structures in the framework of evolutionary game theory. Rather than deterministic dynamics, our model incorporates stochastic factors induced by the finite population size. Depending on analysis of the stationary distribution of the evolutionary process in the limit of rare mutations, we prove that migration has no effect on the imitation process of strategies but has a significant influence on the competition process of communities. The population spends most time in the homogeneous state where all individuals belong to the community with the lower migration rate. Furthermore, we find that reducing the difference between migration rates of two communities can increase the first hitting time to the homogeneous absorbing state, promoting the conservation of biodiversity indirectly.

Linguistic consensus is an important way to find the origin and the evolution of language. In this paper, we propose a model to show how a population on a circle can reach consensus. Between every two agents we establish a new way of interaction which, besides direct influence, depends on everyone's insistence. We analyze its dynamics and investigate two factors that influence the consensus: one is the weight of the circle; the other is the insistence of every agent. Moreover, we prove that adjusting insistence is an effective method of reaching consensus, that is, with appropriate insistence, common language on a circle is more likely to be achieved. Furthermore, we provide a number of simulations to illustrate our analytic results.

In this paper, we propose a simple yet effective theoretical model for the evolutionary threshold public goods game with binary contributions (each individual makes decision to contribute a fixed donation amount or nothing), incorporating the effect of the collective risk. In order to investigate the evolutionary dynamics of the collective cooperative behavior, we analyze the population dynamics represented by the replicator equation. The result shows that high risk rate can enhance the emergence of social cooperation as well as the provision of public goods. Besides, other elements can also promote the cooperation, such as large initial endowment, small threshold, large cost of cooperation below the baseline of each cooperator, and large group size. In addition, our model can lead to rich dynamics. Scenarios of defection dominance, cooperation and defection bistable, cooperation and defection coexistence, and cooperation dominance may appear successively with the change of parameters.

We propose a simple yet effective theoretical model for the evolutionary threshold public goods game with binary contributions (the fair personal share or nothing), incorporating the effect of collective risk. We distinguish two distinct public goods games according to whether to return the contributions when the target is not collected. For the two cases, in order to study the impact of collective risk on cooperation, we analyze dynamics of the population which can be represented by the replicator equations. It shows that high rate of loss can enhance the emergence of social cooperation and the provision of public goods. Furthermore, other elements also can promote the cooperation, such as large initial endowment and small threshold. Interestingly, for large group size, it has a positive impact on cooperation in the case of returning the donation amount, whereas a negative impact in the case of no return.

Linguistic consensus is an important way to find the origin and the evolution of language. In this paper, we propose a "circle" model to show how a population on a circle can reach consensus. Between every two agents we establish a new way of interaction which, besides direct influence, depends on everyone's insistence. We analyze its dynamics and investigate two factors that influence the consensus: one is the weight of the circle; the other is the insistence of every agent. Moreover, we prove that adjusting insistence is an effective method of reaching consensus, that is, with appropriate insistence, common language on a circle is more likely to be achieved. Furthermore, we provide a number of simulations to illustrate our analytic results.

In real situations, people are often faced with the option of voluntary contribution to achieve a collective goal, for example, building a dam or a fence, in order to avoid an unfavorable loss. Those who do not donate, however, can free ride on others' sacrifices. As a result, cooperation is difficult to maintain, leading to an enduring collective-risk social dilemma. To address this issue, here we propose a simple yet effective theoretical model of threshold public goods game with collective risk and focus on the effect of risk on the emergence of social cooperation. To do this, we consider the population dynamics represented by replicator equation for two simplifying scenarios, respectively: one with fair sharers, who contribute the minimum average amount versus defectors and the other with altruists contributing more than average versus defectors. For both cases, we find that the dilemma is relieved in high-risk situations where cooperation is likely to persist and dominate defection in the population. Large initial endowment to individuals also encourages the risk-averse action, which means that, as compared to poor players (with small initial endowment), wealthy individuals (with large initial endowment) are more likely to cooperate in order to protect their private accounts. In addition, we show that small donation amount and small threshold (collective target) can encourage and sustain cooperation. Furthermore, for other parameters fixed, the impacts of group size act differently on the two scenarios because of distinct mechanisms: in the former case where the cost of cooperation depends on the group size, large size of group readily results in defection, while easily maintains cooperation in the latter case where the cost of cooperation is fixed irrespective of the group size. Our theoretical results of the replicator dynamics are in excellent agreement with the individual based simulation results.

Multicommunity population systems may reach a consensus state where the fractions of each species in different communities agree on a common value. In this paper, by analyzing the evolutionary dynamics based on an extended replicator equation incorporating community effects, the consensus problem of population systems with $n$ communities is studied. In particular, the simple case of two communities is investigated in detail. In general, for $n$ communities, a sufficient and necessary condition for population systems to reach a consensus of coexistent state is provided. Regarding the population dynamics for the four different types of games, whether the population systems can achieve consensus is determined. The dynamics of community-structured populations shows richer features than nonstructured populations, and some nontrivial phenomena arising from different community-structured population systems are illustrated with concrete numerical examples.

We propose a model for evolutionary game dynamics with three strategies $A$, $B$ and $C$ in the framework of Moran process in finite populations. The model can be described as a stochastic process which can be numerically computed from a system of linear equations. Furthermore, to capture the feature of the evolutionary process, we define two essential variables, the {\em global} and the {\em local} fixation probability. If the {\em global} fixation probability of strategy $A$ exceeds the neutral fixation probability, the selection favors $A$ replacing $B$ or $C$ no matter what the initial ratio of $B$ to $C$ is. Similarly, if the {\em local} fixation probability of $A$ exceeds the neutral one, the selection favors $A$ replacing $B$ or $C$ only in some appropriate initial ratios of $B$ to $C$. Besides, using our model, the famous game with AllC, AllD and TFT is analyzed. Meanwhile, we find that a single individual TFT could invade the entire population under proper conditions.

As there are different gendered assumptions in legislation and policies,there are different gender equality expectations.This paper examines the legislation and policies concerning the division of property in divorce from a gender perspective in order to reveal their different influence on men and women.It also summarizes gender-based academic discussions on questions of the division of property in divorce so as to urge policy-makers to adopt a gender perspective and to promote gender equality.