Complex Systems

Systems of nature, society or man-made are constituted of highly interconnected parts on many scales, the interactions of which result in complex or emergent behaviour. Examples of this kind of complex systems are financial markets, traffic flow, networks of biology, economics, web, society etc., and chemical or biological reaction-diffusion systems to mention a few. The processes of such a complex system occur simultaneously at different scales, and the intricate behaviour of the system depends also on its constituents, units or agents in a non-trivial way. Examples of intricate behaviour include for example self-organisation, self-adaptation, and structure or pattern formation etc., which can be studied using various statistical physics, information theory, statistics and game theory methods involving computational modelling and computer simulations.

Our research on complex systems focuses on modern Econophysics including financial analysis, risk analysis, tree and graph analysis, and adaptive multiagent games, complex pattern formation with Turing's reaction diffusion systems, small world type random networks.

Statistical Analysis of Asset Returns

Researchers: Juuso Töyli, Laszlo Kullman*, Laszlo Gillemot*, Kimmo Kaski, Janos Kertesz*
* Dept. of Theoretical Physics, Budapest University of Technology and Economics

Asset returns have traditionally been modelled with methods based on the normal distribution. However, the empirical returns are characterised by stylized facts that imply non-normality. The stylised facts include heavy tails thus the empirical distribution is leptokurtic, the variances are auto-correlated although returns are not (except for very small intraday time scales), large and small returns are clustered, and there are jumps and crashes although these are typically asymmetric so that the magnitude of crashes is larger than that of jumps. Despite of these stylized facts, the shape of the distribution approaches normal distribution when the time interval is increased and the monthly returns are generally regarded as normally distributed. These characteristics suggest that the return generating stochastic process is non-linear, time dependent, and complex.

During the past century several model has been suggested but there seems to be no unanimous view. These models can be divided in time-independent and time-dependent categories. Well-known time-independent models include normal distribution, Lévy distribution, truncated Lévy distribution, generalised Lévy distribution, Student t, mixed diffusion jump, mixture normal distribution, and mixture distributions. Time dependent models contain autoregressive heteroscedastic models, stochastic volatility models assuming the volatility as stochastic process, and models based on chaos theory resulting in complex dynamics. These current models are not nevertheless able to capture the dynamics of empirical returns and the results are contradictory. The possible time-dependency of the shape of the distribution has also mainly been ignored.

The research has so far concentrated on the understanding of the return generating process. We have developed toolboxes to fit and simulate data according to the well-known time-independent models. It seems that they are able to capture the long-term distribution but not the structure of the process. We have also studied the effect of different dependencies, linear and non-linear, on the shape of the distribution and generating process. Along with these, the possible biases resulting from different terms of measurements have been researched. Finally, we have studies the changes in the return generating process when the time interval grows. Currently, it seems that the models we have used cannot completely capture the dynamics of the market. Therefore, we have also started to build artificial market in order to understand the role of micro agents in return generation process and to understand the underlying dynamics.

Taxonomy of Financial Assets

Researchers: Jukka-Pekka Onnela, Anirban Chakraborti, Kimmo Kaski, Janos Kertesz*
* Dept. of Theoretical Physics, Budapest University of Technology and Economics

Our research concentrates on two connected fields, both of which are closely related to the portfolio optimization problem. First, we have analyzed temporal properties of asset correlations. For example, in a rising or falling market, financial assets display greater correlated movement than during "business as usual" periods. We have investigated some statistical properties of correlation distributions, and this is one of the findings our study has confirmed. Second, we have investigated asset taxonomy. Building upon the first research field, distances between stocks are defined from asset correlations, and an asset tree is constructed by determining the minimum spanning tree of the distances. Thus, all stocks are connected, but each is linked only to its nearest neighbor. This produces a unique market taxonomy, in which stocks are divided into economically meaningful clusters (tree branches). Further, minimum risk Markowitz portfolio stocks are practically always located on the outskirts of the branches. We have also studied the dynamics of this system and found it to reflect upon the state of the market. Due to these and some additional properties, we believe dynamic asset trees can provide an intuition friendly approach to and facilitate incorporation of subjective judgement to the portfolio optimization problem. Overall, they can further our understanding of the stock market as an evolving complex system.

Figure 13a Figure 13b

Figure 13: Snapshots of asset taxonomy. Left: Normal topology. Right: Crash topology due to Black Monday, October 19, 1987.

Multiagent Models for Complex Adaptive Systems

Researchers: Marko Sysi-Aho, Anirban Chakraborti, Kimmo Kaski, Janos Kertesz*
* Dept. of Theoretical Physics, Budapest University of Technology and Economics

Agent based models, which try to describe features in real-world complex systems, have become more popular during the last years. Thanks to the increasing computing power, it is possible to excecute simulations for ever complicating models, although the complexity is not a favourable quality as regards to the understanding of the hided, fundamental factors in a phenomenon under interest.

An interesting, simple and powerful model including many features present in real world complex systems is a minority game. At each time step, the players of the game have to decide between two alternatives, say whether to choose side A or B, and those who, after the decisions, happen to belong to the minority win. This simple game exposes many interesting features that have been extensively studied in recent years. Our contribution to the development of this game, is the introduction of learning, or intelligent agents, who can adapt to the changing environment and try to improve their competence, if they find it too low. We allow agents to use genetic manipulations to cross their strategies in order to find good ones. This added feature describes better many real-world situations, where one is required to fight for ones survival. It is not enough to be good or best at one time, but one has to improve and fight all the time. Examples that proof the need of this continuous developing can be found from business, academy, sports, biology, evolution, ... The intelligence leads to interesting changes compared to the basic minority game. Especially, the system as a whole strives towards a state that maximizes the utility of the whole community. This is quite an interesting feature, if taking into consideration the fact, that individual agents are only interested in their own performance.

Figure 12

Figure 12: On the left: Comparison of the total utilities of the system, when agents use different kind of rules to modify their strategies. On the right: The same plot reversed and in log-log scale.

Pattern Formation in Turing Systems

Researchers: T. Leppänen, M. Karttunen, K. Kaski, and R.A. Barrio*
*Instituto de Fisica, Universidad Nacional Autonoma de Mexico

In 1952 one of the greatest mathematicians of the 20th century, Alan Turing proposed a system of reaction-diffusion equations describing chemical reactions and diffusion to account for morphogenesis, i.e., the development of patterns, shapes and structures found in nature. The recent growth in computing resources has enabled numerical simulations of Turing systems, which has brought a great deal of knowledge concerning their properties. These complex systems have been used in explaining, e.g. patterns on animal coatings and the segmentation in embryos.

We study numerically structures generated by the Turing mechanism in two and three dimensions. We investigate the dependence of the resulting structures on the system parameters, transitions between these structures, growth from two to three dimensions and percolation of chemicals in the system. In addition, we are interested in the effect of random noise on developing structures, since it is very important from the point of view of biological applications.

The goal of these studies is to develop biological growth models based on Turing systems. We are developing a model for neural patterning, i.e., the signaling mechanism that neurons use to form connections to other neurons in a developing nervous system. Figure 15 shows how connections between certain points can be grown by using a Turing system with sources of chemicals. For applications, it is very important to have a better insight of the morphological characteristics of Turing systems in order to be able to implement the required qualitative features to the model in a biologically plausible manner.

Figure 15

Figure 15: Fully connected networks generated by Turing system in two and three dimensions. The sources are seen in red and the connections are yellow in the two-dimensional case (left).

Statistical Analysis of Small World Networks

Researchers: Jani Lahtinen, János Kertész* and Kimmo Kaski
* Budapest University of Technology and Economics

The small world networks are graphs, which albeit having a large amount of vertices still on average retain small distance between individual vertices relative to traversal of links. Such networks are for example the internet, WWW, stock market trading interlinking, biochemistry signaling and metabolism in protein systems, epidemics, formation of polymers, tranportation systems or interlinked systems of social interactions. These real world examples can - to certain extent - be modelled either as random graphs of Erdös-Rényi, Watts-Strogatz, Barabasi etc. types, or as regular graphs like lattices and cages. Regular graphs are of great interest from the point of view of manageable analytical theory approach but because they offer means to algorithmically generate structures of smallest possible diameter (i.e. the longest distance in the graph) for a given amount of vertices. An example of such a regular graph is cage and it could perhaps be applied in developing efficient telecommunication network topologies. Our research consentrates on analysis of statistical properties of random graphs, and construction of deterministic graphs as approximations of cages and investigation of their properties.

Figure 16a Figure 16b

Figure 16: An Erdös-Rényi (left) and a Watts-Strogatz (right) graphs with 100 vertices.