Agent-based models describing social interactions among individuals can help to better understand emerging macroscopic patterns in societies. One of the topics which is worth tackling is the formation of different kinds of hierarchies that emerge in social spaces such as cities. Here we propose a Bonabeau-like model by adding a second class of agents. The fundamental particularity of our model is that only a pairwise interaction between agents of the opposite class is allowed. Agent fitness can thus only change by competition among the two classes, while the total fitness in the society remains constant. The main result is that for a broad range of values of the model parameters, the fitness of the agents of each class show a decay in time except for one or very few agents which capture almost all the fitness in the society. Numerical simulations also reveal a singular shift from egalitarian to hierarchical society for each class. This behaviour depends on the control parameter $\eta$, playing the role of the inverse of the temperature of the system. Results are invariant with regard to the system size, contingent solely on the quantity of agents within each class. Finally, a couple of scaling laws are provided thus showing a data collapse from different model parameters and they follow a shape which can be related to the presence of a phase transition in the model.

翻译：描述个体间社会交互的基于智能体的模型有助于更好地理解社会中涌现的宏观模式。其中一个值得探讨的议题是城市等社会空间中出现的不同类型等级结构的形成。本文通过引入第二类智能体，提出了一个类Bonabeau模型。我们模型的基本特性在于，仅允许不同类别智能体之间进行成对交互。因此，智能体的适应度只能通过两个阶级之间的竞争而改变，而社会中的总适应度保持恒定。主要研究结果表明，在模型参数取值较广的范围内，每个阶级中智能体的适应度随时间呈衰减趋势，但存在一个或极少数智能体几乎攫取了社会中的全部适应度。数值模拟还揭示了每个阶级从平等社会向等级社会转变的奇异跃迁。该行为取决于控制参数$\eta$，其作用相当于系统温度的倒数。结果具有系统规模不变性，仅取决于每个阶级内的智能体数量。最后，研究给出了若干标度律，展示了不同模型参数下的数据塌缩现象，其遵循的函数形式可能与模型中存在的相变有关。