Complex Systems and Networks

In the recent years we have seen much progress in the analysis, modelling, and theoretical studies of complex systems, with the result that seemingly very different systems share similar characteristics. Typically, these natural, social, and man-made systems comprise highly interconnected parts on many scales. Examples include financial markets, biological regulatory networks, and the Internet to mention a few. In such systems, the interaction patterns and topology can be highly intricate. Together with stochasticity, these often result in emergent system-level behavior, such as self-organisation and pattern formation. The interactions between the constituent elements can also result in highly non-trivial structural properties of the system, such as the scale-freeness of the connectivity distribution discovered in several kinds of network systems.

Complex systems are typically analyzed in an interdisciplinary way from several viewpoints, combining methods and frameworks from e.g. statistical physics, theoretical biology, information theory, game theory and social sciences. Further, computational modelling and large-scale computer simulations are often required. Our research on complex systems focuses on theoretical and numerical studies of the characteristics of complex networks, as well as dynamic phenomena taking place on such networks. We also approach complex systems from an agent-based modelling perspective, e.g. in modelling economic systems and evolutionary games.

Dynamic Phenomena on Complex Networks

Researchers: Jari Saramäki, Jukka-Pekka Onnela, Jörkki Hyvönen, Kimmo Kaski

During the recent years, the network approach to complex systems has turned out to be extremely fruitful. In this approach, diverse systems are viewed as networks, so that the interacting elements are described by network vertices and their interactions and/or relationships by edges connecting the said vertices. Systems well suited for studies in this framework are ubiquitous in Nature – neural networks, social networks, the Internet, networks of epidemic spreading, metabolic networks in cells. Perhaps the main strength of this approach is its ability to capture the salient features of the systems in question with simple building blocks - the edges and vertices - and then derive system-level properties from their relationships. The most surprising result has been that the systems often share similar properties, such as the common short average vertex-to-vertex distances (the small-world property) and the ubiquitous scalefree connectivity distributions.

In addition to the structural properties of these systems, there has been wide interest in dynamic phenomena taking place on such networks. Examples include models of opinion formation and the dynamics of spreading via contact processes, such as the propagation of rumours and information in social networks and epidemic spreading of disease. Related to the latter, one example of our research is a model of the spreading of influenza-like disease on dynamically changing small-world contact networks; this framework yields readily to an analytical treatment in addition to numerical simulations. Although the model is rather minimal and the structure of the contact network is simple, the spreading dynamics produced by the model matches real-world data with very good accuracy (see Fig. 42).

Figure 42

Figure 42: The dynamics of two influenza A epidemics, a) in the US during winter 2001-2002, and b) in the UK during winter 2003-2004. The solid circles indicate laboratory-confirmed weekly cases, and the solid and dashed lines theoretical curves produced by fitting our model to the data. For the dashed lines, only a small number of data points in the beginning of each epidemic were used.

Weighted Complex Networks

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

Complex networks provide a very general framework, based on the concepts of statistical physics, for studying systems with large numbers of interacting agents. The nodes of the network represent the agents and the links correspond to interactions between them. The network approach to complex systems, ranging from the WWW to the metabolism of cells, has turned out to be extremely fruitful during the last few years. Studies of network characteristics have produced unexpected findings, e.g. the ubiquity of scale freeness and the small-world phenomenon. The local structure of unweighted networks can be characterized by the appearance of small subgraphs, so-called motifs, and the networks’ clustering properties, which have been related to network functionality. We have generalized these approaches by providing a set of theoretical and practical tools for analyzing weighted complex networks, an effort motivated by the community’s growing need in this direction. We have studied these concepts for both empirical networks and theoretical network models. We have also studied the formation and self-organization of weighted complex networks starting from local microscopic mechanisms and transmission of information on weighted networks.

Figure 43

Figure 43: The weighted metabolic network of Escherichia coli. The nodes correspond to chemicals (metabolites) and they are linked if connected by a metabolic reaction. The weight of the link is associated with the net reaction fluxes between the connected chemicals. We have characterized this network by studying the intensity and coherence of its subgraphs, including paths and cycles. Our results show that inclusion of weights in these types of motif ensemble studies may considerably modify the conclusions drawn from them. The network was visualized using Himmeli software package by Ville Mäkinen.

Multiagent models for complex adaptive systems

Researchers: Marko Sysi-Aho, Jari Saramäki, Kimmo Kaski, Janos Kertesz*
*Budapest University of Technology and Economics

The self-organization of a population of agents with limited capabilities for scarce resources is an interesting problem and has potential applications in biology, economics and social sciences. For instance, predators that choose their turfs for hunting prays, and data routing in mobile network fit into this framework. The minority game (MG) is a simple agent-based model reminiscent of these systems. The game consists of N (odd) agents who decide between two alternatives, A and B. Those who belong to the minority, win. The agents have access to a global history, a historical record of the past M winning sides, and they are endowed with S strategies that assign a choice for each possible history. Regardless of the individual agents’ self-interested pursuit, the population of agents shows coordinated behaviour. Previous studies indicate that in case the agents are allowed to evolve, they tend to evolve such that the population as a whole performs the best. To allow evolution, we have applied genetic algorithms to the MG. Our results show that natural selection and genetic algorithms are efficient methods to boost the performance of the population as well as individual agents in this toy world.

Figure 44

Figure 44: Coordination among agents in minority games: lower values mean good coordination. Application of genetic algorithms lead to a considerable improvement in coordination compared to the basic game.

Multi-agent models of economy systems

Researchers: Marco Patriarca, Anirban Chakraborti, and Kimmo Kaski

The importance of studying economy models is related to the fact that there are well known regularities in the structure of real systems. For instance the century old Pareto law in Economics states that the higher end of the distribution of income follows a power-law, , where is the income (money) and the exponent has a value in the interval \\\\\\ We study a simple statistical model of closed economy, in which N agents can exchange money in pairs between themselves, which can be solved either numerically or analytically, in order to investigate the relation between its internal dynamics and the corresponding final equilibrium distribution. All the agents are initially assigned the same money amount. Agents are then let to interact. At every “time step”, a pair (i,j) is randomly chosen and the transaction carried out. During the transaction, a fraction of the agent money and is randomly reassigned between the two agents. The exchange law is such that the money is conserved during the transaction, i. e. ,

Formula 1 (1)

where is a random number and the saving propensity (0 < < 1). Money distributions for arbitrary ’s are well fitted by the function

Formula 2 (2)

where is the Gamma function and is a normalization factor. This particular form of was suggested by a mechanical analogy between the closed economy model with N agents and the dynamics of a gas of N interacting particles. In fact this is just the Maxwell-Boltzmann distribution for the kinetic energy of a gas in D = 2n dimensions and at a temperature . The fitting curves for the distribution (continuous curves) are compared with the numerical data in the figure [for details see M. Patriarca, A. Chakraborti, and K. Kaski, Physica A 340(2004)334 and Phys. Rev. E 70(2004) 016104].

Figure 45

Figure 45: Money distributions and interpolating functions (linear and log scale in the y axis).

Simple models for language evolution and diffusion

Researchers: Marco Patriarca, Teemu Leppänen

History and distribution of languages still represent open problems. Recently, a master equation based model was succesfully used for describing the high rate at which world’s languages are disappearing [D. M. Abrams and S. H. Strogatz, Nature 424 (2003) 900]. In this model languages are treated similarly to competing species, – i. e. as non-evolving entities – despite their evolution is probably described more realistically from an evolutionary point of view [Christian Schulze and Dietrich Stauffer, cond-mat/0411162]. The model however provides a satisfactory quantitative description of their interactions and predicts that, in a situation of homogeneity and close interaction between all the speakers, only one language survives. We suggest a generalization by introducing a space-dependence in terms of a reactiondiffusion equation. The model can take into account more special situations, in which, due to geographical constraints, a language can actually survive, despite the presence of a more influentials language, as in the example shown in the figure [for details see M. Patriarca and T. Leppänen, Physica A 338(2004)296].

Figure 46

Figure 46: Population densities of language A (lower row) and B (upper row). From left to right: initial, intermediate, and final stationary configuration. A and B zones are explicitly shown only in the initial state.

Network Approaches in Sociological Data Analysis - Mediated Relationships

Researchers: Teemu Suna, Jari Saramäki, Jukka Heikkonen, Kimmo Kaski, Michael Hardey*, Mika Ala-Korpela
*School of Geography, Politics and Sociology, University of Newcastle, UK

Internet and computerized production and use of information have by now created one of the largest societies of over 800 million associates. Within this context online relationships and dating have existed for roughly ten years. In the Western world a growing proportion of the population is composed of singles and career and time pressures are increasing. For example, 40 per cent of American adults are single, and half of them - more than 40 million Americans - are currently using online dating services, and, generating multi-million dollar market along the way. The recent rise and popularity of Internet dating services is an indication that online dating has gone mainstream, i.e., online dating attracts - regular - people, or at least regular people who use the Internet. In fact, it has recently been indicated that online daters are sociable and self-confident thereby thrusting away the late 90’s allegory of online daters being as social isolates lacking social skills.

Hardey has recently emphasized that Internet dating sites are only one example of a growing number of virtual places that have developed to bear a potential impact on users’ offline lifestyles. Indeed many of the new resources that have developed for the Internet have been designed to address offline needs. Therefore, in contrast to visions of another "life-world" occupied by users with multiple identities, the Internet, for many, is just a different space where they meet others and make use of a vast number of services and resources. The anonymity of individuals that characterizes the online dating, rarely seems to facilitate the construction of fantasy selves, but acts as a foundation for the building of trust and establishing real world relationships. Thus, rather than forming a distant cyberspace culture, the Internet is opening up new opportunities to shape the existing line and contents of social life. The vision of a logical, disembodied late modern intimacy, based on talk rather than passion, negotiation rather than commitment, and the advancement of the self rather than the development of the couple, suggest that the Internet is uniquely placed to facilitate such relationships. Indeed, recent studies have clearly shown a strong truthful connection sought by majority of online daters between the disembodied anonymous online selves and the real, offline selves. However, this does not mean that the rules governing how people construct and negotiate virtual identities and relationships online would match the rules governing formation of romantic relationships offline. In fact, little is known about the mating rituals in the digital domain.

Since the online dating seems to have taken its place in our repertoire of romantic relationship formation, it would be important to gain understanding of how people find the mutual trust online that leads to meetings offline. It would also be essential to be able to link what kind of people use the Internet dating services and how their characteristics affect their behavior both online and offline. In addition, keeping in mind the risen business in the area of online dating, the dating services can truly exemplify the issues associated with information disclosure intended for marketing use. It is well known that people are, in general, particularly reluctant to provide any personal information over the Internet. However, as the most popular Internet dating sites reveal, if information is meant for love and matchmaking aiming to find a potentially optimal mate, this reluctance partly disappears. This is also our experience for "intelligent" dating and political matchmaking services in the Internet in Finland. Although dating services are relatively unique as a service point of view, the actual use of customer data is not so unique. The user data obtained from online daters can thus be utilized for management of customer relationships in general.

In our work we are focusing on the capabilities of self-organizing maps (SOMs) in sociological data analysis and on the small-world aspects in social networks. Currently we are analyzing a data set, acquired via an online dating service, consisting of answers for an extensive Internet questionnaire from 3321 men and from 1655 women. Such an extensive data set is exceptional in the area of social networks where data is traditionally obtained through painstaking interviews. Using this data, we aim to illustrate the power of SOM in profiling and visualization of complex sociological data, to analyze and discuss the importance of various individual properties on online and offline relationships, and, to analyze and understand principles of online interaction and social networks.

Figure 47

Figure 47: Illustration of a SOM in a sociological data analysis. The SOM transforms the individual and multidimensional data into a two dimensional plane. Each node in the SOM is described by a feature vector representing the original multidimensional parameter space, i.e., the input data. The feature vectors form, via the self-organizing process, from the parameter vectors of all individuals. Thus, the feature vectors characterize average individuals using exactly the same multidimensional parameter space as originally utilized for the individuals. The SOM shown in this figure is for 3321 men and labeled for the age-parameter.