We investigate the cluster behavior of financial markets within the framework of a model based on a scale-free network. In this model, a cluster is formed by connected agents that are in the same state. The cumulative distribution of clusters is found to be a power-law. We find that the probability distribution of the liquidity parameter, which measures the financial markets’ energy, is rather robust. Furthermore, the time series of the liquidity parameter have the characteristics of 1/f1/f noise, which may indicate the fractal geometry of financial markets.
Co-interaction and evolution between different agents are known to be one of the ingredients of complex systems, such as—social, biological, economical and technological systems. Following the trend of research on complex systems, to find the universal rules and principles of these systems become more and more attractive [1], [2], [3], [4], [5] and [6]. In particular, the studies of financial markets prices have been found to suggest several generalized properties similar to those observed in physical systems with a large number of interacting ingredients. More and more models have been introduced to attempt to capture the universalities behind the financial markets which are the so-called stylized facts [7], [8] and [9], such as sharp peaks and fat-tail distributions for the financial prices, absence of autocorrelation in return, and long-time correlations in absolute return, etc. These models include the herding multi-agent model [10], [11], [12], [13] and [14], the related percolation model [15] and [16] and the dynamic games model [17], [18], [19], [20], [21] and [22], etc.
Among the more sophisticated approaches are the multi-agent models, based on the interactions of two different agent groups (“noise” and “fundamental” traders), which reproduce some of the stylized facts of real markets but do not account for the origin of the universal characteristics. An alternative approach, the herd behavior [23] may be capable to induce the power-law asymptotic behavior in the tail of return distribution as found in the real data. But an assumption made in its model that should not be ignored is that the probability of each cluster to sell or buy is set to be the same and remains constant throughout the whole process, which may be a good strategy for simplifying a physical model but may not be a good regulation for establishing a model which we expect to reflect the various phenomena found in the real financial markets as genuine as possible. Here we introduce a model where the probability of each agent to sell or buy varies along with the difference between the demand and supply at each time step, which may be more helpful for us to learn the nature of the markets.
It is interesting to find that the various networks of the real-world, from social networks to biological networks, display scale-free degree distributions and small-world characteristics. So recently more and more models of financial markets have been proposed based on different types of networks [24], [25] and [26]. It could characterize quantitatively the interaction between agents by means of a series of topological quantities, which could better capture the complex properties of the real-world. In Refs. [23], [24] and [27], the models are in the view of the regular lattices and the famous Cont and Bouchaud model’s network structure is that of the random graph. However, in our model, we consider a different topology on the scale-free network. More importantly, we obtain some interesting results about cluster behavior which is a very common phenomena in the real-world financial market.
This paper is organized as follows. In Section 2, we introduce our model. In Section 3, there are numerical simulations and some results. In Section 4, a discussion and main conclusions are given.
We have presented a self-organized model for the formation of clusters based on the scale-free network and applied it to the description of cluster behavior in financial markets. We make a simple assumption for the participants in financial markets that the probability of buying financial products is the same as the probability of selling them and the expression of the evolution for liquidity parameter is in the same pattern as of financial index price’s evolution.
Such a model has the only one adjustable parameter defining the liquidity parameter. Through the simulations, we have obtained some critical stylized facts which are rather similar to real financial markets. We also draw a conclusion that the time series of liquidity parameter evolves steadily with several large volatilities during the whole process, which shows the behavior of 1/f1/f noise and means the whole financial market has the fractal properties.
According to the distribution of the clusters’ sizes, We find that it follows the power-law scaling in the range of small cluster size independent of time. It suggests that in financial markets, the number of clusters of large size is small and most of clusters have small sizes. These results were not found in Refs. [23], [24] and [27] which are on the basis of the regular lattices or random graph. The behaviors in this regime need more study in a quantitative manner in the further investigation.
