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, selfadaptation, 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.

Dynamic Phenomena on Complex Networks

Researchers: Jari Saramäki, Kimmo Kaski

Networks composed of interlinked elements are ubiquitous in Nature – neural networks, social networks, the Internet, networks of epidemic spreading, metabolic networks in cells. However, the mathematical methods for studying such networks have been developed only recently, and in particular dynamic phenomena on such networks still remains a subject with lots of unanswered questions. We study these phenomena from a computational and theoretical perspective. One of our focus areas is developing models of spreading on such networks. These may be used to explain various dynamic phenomena, ranging from the spreading of fads and ideas on social networks to epidemics caused by biological viruses, as well as propagation of excitatory patterns on brain-like neural networks. As an example, we modeled spreading of influenza-like, randomly infectious diseases on dynamically changing spreading networks, and obtained excellent fits to real-world epidemic data. The model also allows predicting the development of an epidemic at its beginning stages. We have also discovered stable oscillations of excitations on two-dimensional small-world networks. Here, an excited network node may excite its neighbors, if these are in a susceptible state. Excited nodes become refractory, and can be excited again only after some period of time. Figure 12 depicts propagation of excitation on such network. After the initial wave of excitations has passed, the process is “re-ignited” through a small fraction of long-range connections in the system. This process could be related to those giving rise to spontaneously emerging cortical rhythms. Furthermore, one should note that the process itself is similar to that of epidemic spreading of diseases, re- flecting the fact that many seemingly unrelated natural processes result from similar types of simple mechanisms taking place on complex networks.

Figure 12

Figure 12: Propagation of excitations on a two-dimensional small world. Panel (a) indicates amount of excited nodes as function of time; red circles show times at which the snapshots (b)-(e) where taken. In the snapshots, yellow pixels indicate excited nodes, black those susceptible to excitation and red those in a refractory phase. After some nodes become excited (b), excitation rapidly spreads across the network (c,d). Then, refractory nodes start to “cool down” (e), enabling another excitatory cycle.

Taxonomy of Financial Assets

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

Network theory provides an approach to complex systems with many interacting units, where the details of the interactions are of lesser importance. In the financial market companies interact with one another, creating an evolving complex system. These complicated interactions are reflected in temporal correlations of asset returns and flows of capital.

We study certain properties of these networks numerically, where the nodes correspond to stocks and the edges to correlation based distances between them. As studies based on random matrix theory have shown, a large majority of eigenvalues of empirical correlation matrices fall within the spectrum predicted for random matrices, i.e. they are predominantly noise. A central issue, therefore, is to prune these systems so that preferably only information is retained.

The goal of our work is to improve our understanding of interdependencies, clustering and dynamics of the financial market. One approach is to construct a minimum spanning tree of edges. We have demonstrated that this leads to a scale-free network, where the scaling exponent is fairly stable over time, except for crash periods, which are characterized by a lower exponent. During crash periods the tree shrinks both topologically and in terms of its overall length. We have also demonstrated how the stocks of the minimum risk Markowitz portfolio lie practically at all times on the outskirts of the tree. Another approach is based on agglomerative clustering, which seems to capture well the clustering present on the market. We have compared some properties of empirical graphs with those of random graphs, for which results are well known. It is postulated that deviations from theoretical predictions are indicative of genuine information. As expected, the market behaves very differently from random graph models.

Figure 13

Figure 13: Snapshots of financial networks. Left: MST approach. Right: Agglomerative clustering approach.

Gibbs versus non-Gibbs distributions in money dynamics

Researchers: Marco Patriarca, Anirban Chakraborti* and Kimmo Kaski
*Dept. of Physics, Brookhaven National Laboratory, USA.

We study simple models of money conserving economy, in which agents can exchange money in pairs. At every “time step”, a pair is randomly chosen, and the agent money amounts and undergo a variation,

Formula 1 (1)

where is is a uniform random number in the interval (0,1). This system relaxes toward an equilibrium state with a Gibbs distribution (curve () in the Figure), where represents the average money. However, if agents save a fraction represents the average money. However, if agents save a fraction

Figure 14

Figure 14: Numerical data (dots) and fitting functions (continuous lines) of equilibrium money distributions for different values of the saving propensity .

(saving propensity) Eqs. (1) become

Formula 2 (2)

Our numerical simulations lead to the final equilibrium distributions shown in Fig. 14. We also found the corresponding exact solution for an generic value of , with , by fitting the results of numerical simulations, which reads

Formula 3 (3)

This is a -distribution for the variable , where . The fitting curves for the distribution (continuous lines for ) are compared with the numerical data in Fig. 14.

Multiagent models for complex adaptive systems

Researchers: Marko Sysi-Aho, Kimmo Kaski, Janos Kertesz*
*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 genetic adaptation. 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 invisible hand effect has its analogies in real world, for example in commodities price forming process.

Figure 15

Figure 15: Normalized fluctuations describe the society utility in minority games: lower values mean higher utility. Learning mechanism leads to a considerable improvement in effieciency compared to the basic game.

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 (mammals, fish, butterflies).

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. We use linear stability analysis and nonlinear bifurcation analysis to study the pattern selection in the system analytically.

The goal of these studies is to gain insight into the properties of Turing systems and to facilitate developing biological growth models based on Turing systems. 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 16 shows two typical three-dimensional Turing structures, namely the lamellar and spherical morphologies.

Figure 16

Figure 15: The morphologies observed in three-dimensional Turing systems are the lamellar phase with twisted grain boundaries (left) and the spherical phase with organized droplets (right). The visualization has been carried out by plotting the isosurface of the chemical concentration data.

Statistical Analysis of Asset Returns

Researchers: Juuso Töyli, Laszlo Kullman*, Kimmo Kaski, Janos Kertesz*
*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 nonnormality. 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 timeindependent 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.

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. Cascading failurs occur when failure of few agents inititate a domino effect of collapsing a large number agents. These massive breakdowns can be analyzed with simple models of sandpiles of Bak, Tang & Wiesenfeld, which resemble collapsing effect on sandpiles.