Peng Lu

Emergencies such as terrorist attacks, with a large number of casualties, have spread worldwide and become the global issue for a long time. Previous researchers employed traditional two-dimensional (2D) models to simulate the crowd dynamics between terrorists and civilians. However, these 2D models simplify real situations and have yet to consider individual heights and visions. Therefore, more accurate models are needed. In this work, we extend the 2D model and propose the three-dimensional (3D) model, and the core is to bring the mechanism of individuals heights into decision-making process of these 3D agents for both terrorists and civilians. We first build the 3D environment. For the mechanism of crowd dynamics, under the framework of perception-decision-behavior, our 3D model has included individualized heights for all agents. Comparing 2D and 3D models, we find that individual heights and visions have greatly shaped the outcomes. The height heterogeneity has significant effects on attack deaths and slight effects on stampede deaths because smaller heights slow the moving speed for the crowd, and the higher heterogeneity (of heights) impairs the visibility of civilians. The effects of height heterogeneity on deaths will be more obvious, as the group size of civilians is beyond 1900. We have the phase transition threshold of 3030, beyond which stampede deaths exceed attack deaths. Moreover, the size effect of heroes follows the law of diminishing marginal returns. We also find that the number of heroes should twice that of terrorists, guiding better allocations of the police force and other public resources for emergencies responses. To strengthen the counter-force, heterogeneity effects of civilian heights should be controlled, and self-motivated heroes should be encouraged, which is critical for public safety worldwide.

The laws and regulations in human history can be revealed by computational models. From 221 before Christ (BC) to 1912 Anno Domini (AD), the unification pattern has dominated the main part of Chinese history for 2132 years. Before the emergence of the first unified empire, the Qin Empire in 221 BC, there existed the Eastern Zhou dynasty (770 BC to 221 BC). This long dynasty has two stages, and here we focus on the first stage. This Spring-Autumn stage was from 770 BC (with 148 states) to 476 BC (with 32 states). The whole country (China) is modelled as a multi-agent system, which contains multiple local states. They behave autonomously under certain action rules (wars and conflicts), which forms the main reason for the annexations and disappearance of most states. Key factors (power, loyalty, bellicosity and alliance) have been considered in our model settings, and simulation outcomes will be monitored and collected. Eventually, an optimal solution is obtained, which well unveils the internal mechanism and statistical features of real big history. Furthermore, counterfactuals are used to explore the nonlinear effects of the key factors, which deepens the authors' understanding of civilisation evolutions in human history. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

It is hard for individuals to handle social common risk, such as terrorism attacks (mass shooting). However, it can be contained, if they can be organized and behave intelligently. To obtain this swarm intelligence pattern, social knowledge should be captured to guide behaviors of isolated individuals. Here, we explore how swarm intelligence can be achieved by individual behaviors during social learning process. We have solved two issues. The first is to obtain social knowledge. Based on agent-based model of real target case, we calculate the optimal solution based on which we infer outcomes of all possible situations. Then, the matrix of social knowledge can be formed; The second is social learning process of individuals. Guided by the social knowledge, they all know that others will be mobilized as well. The key information is the minimal valid size of heroes, and all social members know that this condition is not difficult to satisfy. Thus, more individuals will be mobilized and become the Heroes to fight bravely against the shooter(s). Comparing two patterns, we obtain precise outcomes of how social losses (civilian deaths and injuries) can be reduced. Therefore, guided by clear social knowledge, social welfare can be enhanced substantially.

Terrorist attacks lead to huge social cost, and, especially, mass shooting cases result in massive deaths and injuries worldwide. Hence, it is necessary to explore the accurate and dynamic processes of mass shooting cases. Taking the Cinema Shooting case in 2012 as the real target case, the agent-based model is applied to explore the dynamics of human behaviors under common risk of shooting. Parameter traverse and repeated simulations can be used to obtain robust outcomes. According to real target case, the optimal solution with the highest matchiness can be obtained, which supports the validity and robustness of our agent-based model. Under this optimal solution, the same numbers of deaths and injuries can be achieved, and the dynamic process can be also back-calculated. Besides, we can infer counterfactual outcomes, which strengthens explanatory capability of our model. Hence, the multiple outcomes under different situations (settings or strategies) can be obtained, and optimal perception range R * can be solved as well. Combining agent-based model and counterfactual simulations, emergency responses, public governance, and facility designs will be improved.

The life cycle pattern is pervasive for both natural and social sciences, from human behaviors to social systems. Based on the life cycle model of collective actions, the man-land relationship governs the rise and fall cycles, namely dynastic cycles. We combine agent-based modeling, systemic dynamics, and numerical simulations, to build the life cycle model of empires.

Under the mobile internet and big data era, more and more people are discussing and interacting online with each other. The forming process and evolutionary dynamics of public opinions online have been heavily investigated. Using agent-based modeling, we expand the Ising model to explore how individuals behave and the evolutionary mechanism of the life cycles. The big data platform of Douban.com is selected as the data source, and the online case "NeiYuanWaiFang" is applied as the real target, for our modeling and simulations to match. We run 10,000 simulations to find possible optimal solutions, and we run 10,000 times again to check the robustness and adaptability. The optimal solution simulations can reflect the whole life cycle process. In terms of different levels and indicators, the fitting or matching degrees achieve the highest levels. At the micro-level, the distributions of individual behaviors under real case and simulations are similar to each other, and they all follow normal distributions; at the middle-level, both discrete and continuous distributions of supportive and oppositive online comments are matched between real case and simulations; at the macro-level, the life cycle process (outbreak, rising, peak, and vanish) and durations can be well matched. Therefore, our model has properly seized the core mechanism of individual behaviors, and precisely simulated the evolutionary dynamics of online cases in reality.

Terrorists usually choose to attack soft targets, with low self-protection abilities and resistance strengths, such as schools, campus, public squares, railway stations, etc. Under great panic, civilians attacked tend to escape aimlessly and disorderly. Hence, colliding, pushing, and trampling will take place and lead to indirect deaths and injuries. If some heroes with the spirit of altruism, coming from civilians, stand up and fight terrorists bravely, the injuries and deaths will be greatly reduced. Agentbased modeling is built to explore the role of heroes during terrorist attacks. The particle system of three categories of agents, civilians, terrorists & heroes, is built to simulate the Peshawar School Case in 2014. Multiple action rules and mechanisms are introduced, such as swarm intelligence, information communication, self-organized behaviors, and heroic behaviors. We run each simulation repeatedly for 100 times to obtain averaged (robust) outcomes. The optimal combination of parameters, which best matches real outcomes, will be solved accordingly. It indicates that: (a) the self-organizing of crowd behaviors greatly improves the survival rate of civilians. Hence, well-planned trainings of counterterrorists emergence responses can enhance capabilities of civilians; (b) people should learn from birds' swarm behavior in crowd evacuations. As a swarm intelligence pathway, small groups and information sharing during crowd escape simulations can greatly improve survival rates; and (c) the society should encourage more people to be altruistic heroes who are acting prosocially. Besides, it suggests that more heroes bring safer overall outcomes for civilians and even for heroes themselves.

As a global problem, the terrorism leads to high death tolls each year. During terrorist attacks, the direct death is caused by terrorists attacking civilians. However, indirect death caused by stampedes should not be underestimated. Under great panic, most civilians were running disorderly, rushing into limited number of Exits, which causes stampede injuries and deaths. To explore this dual-mechanism dynamics, we build the agent-based modeling of particle system. Civilians lose blood when attacked, which is the attack mechanism, or crashed and trampled by others civilians, which is the stampede mechanism. For all civilians, the blood variable determines the physical status of being strong, healthy, weak, injured, and dead. Five key factors, such as the perception range, the number of Exits, the density of civilians, the number of terrorists, and attack strategies, are introduced into the model. We run each simulation repeatedly for multiple times and take the averaged survival rate, attack death, and stampede death as robust outcomes. The collision damage has the phase transition effects between stampede and attack deaths. The perception range R have the peak effect on the survival rate and the trough effect on both stampede and attack deaths. It expands understandings of human behavior dynamics, and helps to predict the dynamics and outcomes in advance. The optimal perception range can be solved accordingly, to practically guide the public facility planning and regular emergency training in real-life.

Capturing the whole process of collective actions, the peak model contains four stages, including Prepare, Outbreak, Peak, and Vanish. Based on the peak model, one of the key variables, factors and parameters are further investigated in this paper, which is the rate between peaks and spans. Although the durations or spans and peaks' heights are highly diversified, it seems that the ratio between them is quite stable. If the rate's regularity is discovered, we can predict how long the collective action lasts and when it ends based on the peak's height. In this work, we combined mathematical simulations and empirical big data of 148 cases to explore the regularity of ratio's distribution. It is indicated by results of simulations that the rate has some regularities of distribution, which is not normal distribution. The big data has been collected from the 148 online collective actions and the whole processes of participation are recorded. The outcomes of empirical big data indicate that the rate seems to be closer to being log-normally distributed. This rule holds true for both the total cases and subgroups of 148 online collective actions. The Q–Q plot is applied to check the normal distribution of the rate's logarithm, and the rate's logarithm does follow the normal distribution.

• Investigates the effect of sympathy on cooperation, which is not adequately revealed before. • Sympathy has a quadratic effect on cooperation, and it promotes cooperation beyond its threshold. • Temptation has a quadratic effect as well, and it even promotes cooperation beyond a threshold. • Although cooperation falls at earlier stages, the resilience is strong enough to promote cooperation later on. a b s t r a c t Cooperation is vital in human societies and therefore is widely investigated in the evolutionary game theory. Varieties of mechanisms have been proposed to overcome temptation and promote cooperation. Existing studies usually believe that agents are rational, but ir-rationalism such as emotions and feelings matters as well. Winner and loser are defined by their payoffs. In addition to admiring and imitating winners, the mechanism of sympathizing and imitating losers is introduced into the model as an alternative action rule, and each one plays the prisoners' dilemma game with eight neighbors under the influence of both irrationalism and rationalism. Rationalism refers to imitating winner to get highest payoff, and irrationalism means that people sympathize and adopt the actions of losers. As it is widely recognized that temptation reduces cooperation, this study focuses on the effect of sympathy on cooperation within a certain group or society. If it overcomes temptation that leads to defection, sympathy will be a powerful mechanism to promote cooperative behavior. Simulation results indicate that sympathy and temptation shares similar quadratic relationships with cooperation. Both sympathy and temptation undermine cooperation below their thresholds, and they both promote cooperation above their thresholds. Temptation not only reduces cooperation but also promote it as temptation goes beyond the threshold. Although sympathy is a good merit or human nature that is beneficial to society, a crisis or collapse of cooperation is inevitable when the sympathy propensity is relatively smaller. After cooperation reaches a minimal bottom, it then rises increasingly and dramatically, which brings a much brighter future of the society.

People make decisions on whether they should participate as participants or not as free riders in collective actions with heterogeneous visions. Besides of the utility heterogene-ity and cost heterogeneity, this work includes and investigates the effect of vision hetero-geneity by constructing a decision model, i.e. the revised peak model of participants. In this model, potential participants make decisions under the joint influence of utility, cost, and vision heterogeneities. The outcomes of simulations indicate that vision heterogeneity reduces the values of peaks, and the relative variance of peaks is stable. Under normal distributions of vision heterogeneity and other factors, the peaks of participants are normally distributed as well. Therefore, it is necessary to predict distribution traits of peaks based on distribution traits of related factors such as vision heterogeneity and so on. We predict the distribution of peaks with parameters of both mean and standard deviation, which provides the confident intervals and robust predictions of peaks. Besides, we validate the peak model of via the Yuyuan Incident, a real case in China (2014), and the model works well in explaining the dynamics and predicting the peak of real case.

In terms of the number of participants, almost each collective action has a life cycle where the number grows from zero to its peak where its maximum potential power or influence is acquired, then it decreases to zero eventually. Therefore, we concentrate on modeling, simulating , and predicting the peaks. The model is constructed based on previous models, and the data is collected from simulations. Preliminarily, it suggests that there exists a peak for collective action when its " jointness of supply " is less than one. Under complete homogeneity, the ideal peak is calculated and the ideal peaks function (IPF) is obtained. Then, heterogeneity is introduced into to the model, and the form of real peaks function (RPF) can be obtained based on simulations and statistical methods. For those who intend to organize a collective action and increase the peak of participants should take measures, such as ideology, leadership, and propagation, to enhance homogeneity or try to reduce heterogeneity.

In evolutionary games, the temptation mechanism reduces cooperation percentage while the reputation mechanism promotes it. Inferring reputation theory proposes that agent's imitating neighbors with the highest reputation takes place with a probability. Although reputation promotes cooperation, when and how it enhances cooperation is still a question. This paper investigates the condition where the inferring reputation probability promotes cooperation. Hence, the effects of reputation and temptation on cooperation are explored under the spatial prisoners' dilemma game, utilizing the methods of simulation and statistical analysis. Results show that temptation reduces cooperation unconditionally while reputation promotes it conditionally, i.e. reputation countervails temptation conditionally. When the inferring reputation probability is less than 0.5, reputation promotes cooperation substantially and thus countervails temptation. However, when the inferring reputation probability is larger than 0.5, its contribution to cooperation is relatively weak and cannot prevent temptation from undermining cooperation. Reputation even decreases cooperation together with temptation when the probability is higher than 0.8. It should be noticed that inferring reputation does not always succeed to countervail temptation and there is a specific interval for it to promote cooperation.

• This paper investigates the heterogeneity of inferring reputation, which is not adequately revealed before. • The effect of inferring reputation probability is decomposed into two parts, the mean effect and the heterogeneity effect. • The mean merely enhances cooperation as it is smaller, and undermines cooperation when it is larger. • The heterogeneity does not influence cooperation on the whole range of mean, but reduces cooperation with a smaller mean and propels cooperation with a larger mean. a b s t r a c t As an important mechanism designed to counteract temptation and promote cooperation, reputation is widely investigated in the spatial Prisoners' dilemma game. Existing research assumes that each agent imitates the neighbor that has the highest reputation with an inferring reputation probability p i , which is heterogeneous and enhances cooperation to some extent. So far the effect of heterogeneity has not been adequately revealed. Therefore , we will inspect the heterogeneity effect on a square lattice where agents play the prisoners' dilemma game. It is assumed that the inferring reputation probability is normally distributed, and its mean p and standard deviation sd represent its mean effect and heterogeneity effect on cooperation. Simulation results demonstrate that the mean or overall effect on cooperation fits a nonlinear relationship. It promotes cooperation substantially as the mean is smaller (p < 0.5), it stabilizes cooperation at a stable state as the mean is in the middle range, and it undermines cooperation while p is larger (p > 0.8). The hetero-geneity effect varies with p as well: In the whole range of p, sd neither promotes nor reduces cooperation. However, heterogeneity reduces cooperation when p is smaller (p < 0.5), but turns to increasing cooperation when it grows larger (p > 0.5).

• Investigates the effect of sympathy on cooperation, which is not adequately revealed before. • Sympathy has a quadratic effect on cooperation, and it promotes cooperation beyond its threshold. • Temptation has a quadratic effect as well, and it even promotes cooperation beyond a threshold. • Although cooperation falls at earlier stages, the resilience is strong enough to promote cooperation later on. a b s t r a c t Cooperation is vital in human societies and therefore is widely investigated in the evolutionary game theory. Varieties of mechanisms have been proposed to overcome temptation and promote cooperation. Existing studies usually believe that agents are rational, but ir-rationalism such as emotions and feelings matters as well. Winner and loser are defined by their payoffs. In addition to admiring and imitating winners, the mechanism of sympathizing and imitating losers is introduced into the model as an alternative action rule, and each one plays the prisoners' dilemma game with eight neighbors under the influence of both irrationalism and rationalism. Rationalism refers to imitating winner to get highest payoff, and irrationalism means that people sympathize and adopt the actions of losers. As it is widely recognized that temptation reduces cooperation, this study focuses on the effect of sympathy on cooperation within a certain group or society. If it overcomes temptation that leads to defection, sympathy will be a powerful mechanism to promote cooperative behavior. Simulation results indicate that sympathy and temptation shares similar quadratic relationships with cooperation. Both sympathy and temptation undermine cooperation below their thresholds, and they both promote cooperation above their thresholds. Temptation not only reduces cooperation but also promote it as temptation goes beyond the threshold. Although sympathy is a good merit or human nature that is beneficial to society, a crisis or collapse of cooperation is inevitable when the sympathy propensity is relatively smaller. After cooperation reaches a minimal bottom, it then rises increasingly and dramatically, which brings a much brighter future of the society.

It has been a long-lasting pursuit to promote cooperation, and this study aims to promote cooperation via the combination of social stratification and the spatial prisoners' dilemma game. It is previously assumed that agents share the identical payoff matrix, but the stratifi-cation or diversity exists and exerts influences in real societies. Thus, two additional classes , elites and scoundrels, derive from and coexist with the existing class, commons. Three classes have different payoff matrices. We construct a model where agents play the prison-ers' dilemma game with neighbors. It indicates that stratification and temptation jointly influence cooperation. Temptation permanently reduces cooperation; elites play a positive role in promoting cooperation while scoundrels undermine it. As the temptation getting larger and larger, elites play a more and more positive and critical role while scoundrels' negative effect becomes weaker and weaker, and it is more obvious when temptation goes beyond its threshold.

Cooperation is vital for our society, but the temptation of cheating on cooperative partners undermines cooperation. The mechanism of reputation is raised to countervail this temptation and therefore promote cooperation. Reputation microcosmically records individual choices, while cooperation macrocosmically refers to the group or averaged cooperation level. Reputation should be preferred in order to investigate how individual choices evolve. In this work, we study the distribution of reputation to figure out how individuals make choices within cooperation and defection. We decompose reputation into its mean and standard deviation and inspect effects of their factors respectively. To achieve this goal, we construct a model where agents of three groups or classes play the prisoners' dilemma game with neighbors on a square lattice. It indicates in outcomes that the distribution of reputation is distinct from that of cooperation and both the mean and standard deviation of reputation follow clear patterns. Some factors have negative quadratic effects on reputation's mean or standard deviation, and some have merely linear effects.

• People learn to be good with bad neighbors while bad with good ones. • Harsh environment encourages people to seek a better life via self-organizing. • Selfish neighbors or even scoundrels are preferred to enhance cooperation. • The key of enhancing cooperation is a fair environment for interactions. a b s t r a c t Cooperation is vital for the human society and this study focuses on how to promote cooperation. In our stratification model, there exist three classes: two minorities are elites who are prone to cooperate and scoundrels who are born to defect; one majority is the class of common people. Agents of these three classes interact with each other on a square lattice. Commons' cooperation and its factors are investigated. Contradicting our common sense, it indicates that elites play a negative role while scoundrels play a positive one in promoting commons' cooperation. Besides, effects of good and bad neighbors vary with temptation. When the temptation is smaller the positive effect is able to overcome the negative effect, but the later prevails when the temptation is larger. It concludes that common people are more prone to cooperate in harsh environment with bad neighbors, and a better environment with good neighbors merely leads to laziness and free riding of commons.