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The Zipf's law establishes that if the words of a (large) text are ordered by decreasing frequency, the frequency versus the rank decreases as a power law with exponent close to −1. Previous work has stressed that this pattern arises from a conflict of interests of the participants of communication. The challenge here is to define a computational multi-agent language game, mainly based on a parameter that measures the relative participant's interests. Numerical simulations suggest that at critical values of the parameter a human-like vocabulary, exhibiting scaling properties, seems to appear. The appearance of an intermediate distribution of frequencies at some critical values of the parameter suggests that on a population of artificial agents the emergence of scaling partly arises as a self-organized process only from local interactions between agents.
An exact characterization of the different dynamical behavior that exhibit the space phase of a reversible and conservative cellular automaton, the so called Q2R model, is shown in this paper. Q2R is a cellular automaton which is a dynamical variation of the Ising model in statistical physics and whose space of configurations grows exponentially with the system size. As a consequence of the intrinsic reversibility of the model, the phase space is composed only by configurations that belong to a fixed point or a limit cycle. In this work we classify them in four types accordingly to well differentiated topological characteristics. Three of them -which we call of type S-I, S-II and S-III-share a symmetry property, while the fourth, which we call of type AS, does not. Specifically, we prove that any configuration of Q2R belongs to one of the four previous limit cycles. Moreover, at a combinatorial level, we are able to determine the number of limit cycles for some small periods which are almost always present in the Q2R. Finally, we provide a general overview of the resulting decomposition of the arbitrary size Q2R phase space, in addition, we realize an exhaustive study of a small Ising system (4 × 4) which is fully analyzed under this new framework.
A master equation approach is applied to a reversible and conservative cellular automaton model (Q2R). The Q2R model is a dynamical variation of the Ising model for ferromagnetism that possesses quite a rich and complex dynamics. The configuration space is composed of a huge number of cycles with exponentially long periods. Following Nicolis and Nicolis [G. Nicolis and C. Nicolis, Phys. Rev. A 38, 427 (1988)0556-279110.1103/PhysRevA.38.427], a coarse-graining approach is applied to the time series of the total magnetization, leading to a master equation that governs the macroscopic irreversible dynamics of the Q2R automata. The methodology is replicated for various lattice sizes. In the case of small systems, we show that the master equation leads to a tractable probability transfer matrix of moderate size, which provides a master equation for a coarse-grained probability distribution. The method is validated and some explicit examples are discussed.
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