Seminars

We use a toy model of the financial market to test the efficiency and dangers of credit regulation schemes. We find that Basle-type regulation works fine in situations of low leverage levels in the financial system, however they become destabilizing in scenarios with realistic leverage level. We further design an ideal world, where all leverage introduced risk is hedged with options. Even by assuming that option writers never default, we see that introducing the heavy requirement of complete hedging does not make the system systemically more secure.

In the last decade increasing attention has been devoted to the study of random walks on complex topologies. Various features of random walks on networks, such as passage times and spectral properties have been investigated, and random walks have also been used to to detect communities, to evaluate centrality of nodes and to coarse-grain graphs. Another quantity recently considered for the study of random walks is the entropy rate, a measure used to characterize the mixing properties of a stochastic process. In this talk we will first discuss some of the properties of random walks which might have many relevant applications. In particular, we will consider biased random walks, i.e. random walks with a jumping probability which depends on some properties of target node. We will then focus on designing biased random walks with maximal entropy rate on a given graph, i.e. choosing the transition probabilities of the random walk in such a way that the random walkers are maximally dispersing in the graph, exploring every possible walk with equal probability. Although in principle the optimization of the entropy rate requires that a walker has, at each time step, a global information on the structure of the entire graph, information which is often unavailable, we will demonstrate that it is always possible to construct maximal-entropy random walks by defining a set of transition probabilities that are markovian and that rely only local information on the graph structure.

Zentrale Teile der Stochastik kreisen um das Gesetz der großen Zahlen, wonach (bereits unter schwachen Voraussetzungen) die Konvergenz von Mittelwerten gegen den Erwartungswert fast sicher ist. Aus topologischer Sicht, d.h. im Sinne Bairescher Kategorien, ist jedoch ein gänzlich konträres Verhalten typisch. Der allgemeine mathematische Hintergrund dieses Phänomens soll ebenso beleuchtet werden wie einige charakteristische Beispiele.

A short introduction to the concept of generalized entropies: The necessity of generalized entropies is not primarily grounded in alternative means to reproduce particular types of observed distribution functions from some modified maximum entropy principle - but by the concept of Extensivity. By relaxing the axioms of Information Theory an axiomatic approach towards generalized entropies can be taken. The generic functional form of generalized entropies can be deduced as well as conditions which allow to determine which entropy will be the extensive entropy of some system - given the knowledge of how phase-space expands as a system is enlarged.

The regulation of gene expression in live-forms is subject to high > dynamics and nonlinearities. > To create a mathematical model of these systems biological processes > must be reflected adequately. > This talk will handle the biological foundations of gene expression from > DNA to RNA including eukaryotic characteristics like RNA splicing with > respect to the influence on the mathematical model.

We study long-time mobility of human-controlled characters on a network-shaped universe of a massive multiplayer online game. We take a number of mobility measurements and compare them with measures of simulated random walkers on the same topology. Mobility of players is sub-diffusive - the mean squared displacement follows a power law with exponent 0.4 - and significantly deviates from mobility patterns of random walkers. Mean first passage times and transition counts relate via a power-law with slope -1/3. We compare our results with studies where human mobility was measured via mobile phone data and find striking similarities.

Explaining the breadth of human cooperation is a prime challenge in both the social sciences and biology. One possible mechanism focuses on the role of reputation in cooperation. Usually it is assumed that such a reputation is objective--that is, the same for all agents. We develop a model where reputations are private and synchronized through the sharing of information, i.e. gossiping. Interpreting cooperation between two agents as a positive link and defection as a negative link, this model shows an interesting connection to social balance theory. Furthermore, we study its evolutionary stability.

Strongly interacting statistical systems – complex systems in particular – can change their macro- scopic properties merely as a function of the number of their constituents. Examples include neurons, state-forming insects, financial markets, etc. where systemic properties of small systems can differ drastically from those of a large system built from the same components. The origin of this property is not understood on fundamental grounds. Here we explore this phenomenon from first principles within a thermodynamical framework, by asking about the consequences of bringing interacting sub- systems in thermal contact, where the first three Kinchin axioms hold but the 4th is violated. We show that all sufficiently interacting statistical systems fall into two categories: systems which are asymptotically stable, and those which are asymptotically unstable, meaning that small changes in system size can lead to a drastic increase in entropy. We argue that complex systems belong to this unstable class which make drastic qualitative changes possible as a function of system size. Under the same conditions we then derive the unique asymptotic entropy, Scd = Gamma (d + 1,1 - c ln pi) (c, d constants) which covers all equivalence classes of asymptotically stable and unstable, i.e. all interacting and non-interacting systems. The corresponding distribution functions are special forms of Lambert-W exponentials. As special cases they contain Boltzmann, stretched exponential and Tsallis distributions (power-laws) – all widely abundant in nature. This is, to our knowledge, the first ab initio justification for the existence of generalized entropies.

The celebrated statistical mechanics introduced by Boltzmann and Gibbs more than one century ago lie (for classical systems, for instance) on hypotheses such as ergodicity and mixing. Strongly chaotic systems, with positive maximum Lyapunov exponent, satisfy requirements of this sort. Within this realm, relevant random variables are probabilistically independent or nearly so. It is for such situations, and related quantum ones, that the central limit theorem and the standard entropy (Boltzmann, Gibbs, von Neumann, Shannon) exhibit their well known utility and connections with classical thermodynamics. What can be done outside this world? Can we approach such anomalous, and nevertheless ubiquitous, cases on thermostatistical grounds similar to the usual ones? For wide classes of such systems the answer appears to be positive, by appropriately generalizing the entropy and, consistently, the central limit theorem. Some central concepts as well as typical verifications and applications for natural, artificial and social systems will be briefly presented. BIBLIOGRAPHY: (i) C. Tsallis, Entropy, in Encyclopedia of Complexity and Systems Science (Springer, Berlin, 2009); (ii) C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer, New York, 2009); (iii) S. Umarov, C. Tsallis, M. Gell-Mann and S. Steinberg, J. Math. Phys. 51, 033502 (2010); (iv) CMS Collaboration, J. High Energy Phys. 02, 041 (2010); (v) http://tsallis.cat.cbpf.br/biblio.htm

Human societies can be regarded as large numbers of locally interacting agents, connected by a broad range of social and economic relationships. These relational ties represent e.g. the feeling a person has for another (friendship, enmity, love), communication, exchange of goods, or behavioral interactions. Each type of relation spans a social network on its own. A whole society can be understood systemically only by uncovering interactions between different networks. Here we analyze a complete, multi-relational social network of a society consiting of over 300,000 players of a massive multiplayer online game. We extract networks of six different types of one-to-one interactions between the players. Three of them carry a positive connotation (friendship, communication, trade), three a negative (enmity, armed aggression, punishment). We first analyze these types of networks as separate entities and find that negative interactions differ from positive interactions by their lower reciprocity, weaker clustering and fatter-tail degree distribution. We then proceed to explore how the inter-dependence of different network types determines the organization of the social system. In particular we study correlations and overlap between different types of links and demonstrate the tendency of individuals to play different roles in different networks. As a demonstration of the power of the approach we present the first empirical large-scale verification of the long-standing structural balance theory, by focusing on the specific multiplex network of friendship and enmity relations.

Quantification of human group-behavior has so far defied an empirical, falsifiable approach. This is due to tremendous difficulties in data acquisition of social systems. Massive multiplayer online games (MMOG) provide a fascinating new way of observing hundreds of thousands of simultaneously socially interacting individuals engaged in virtual economic activities. We have compiled a data set consisting of practically all actions of all players over a period of three years from a MMOG played by 300,000 people. This large-scale data set of a socio-economic unit contains all social and economic data from a single and coherent source. Players have to generate a virtual income through economic activities to 'survive' and are typically engaged in a multitude of social activities offered within the game. Our analysis of high-frequency log files focuses on three types of social networks, and tests a series of social-dynamics hypotheses. In particular we study the structure and dynamics of friend-, enemy- and communication networks. We find striking differences in topological structure between positive (friend) and negative (enemy) tie networks. All networks confirm the recently observed phenomenon of network densification. We propose two approximate social laws in communication networks, the first expressing betweenness centrality as the inverse square of the overlap, the second relating communication strength to the cube of the overlap. These empirical laws provide strong quantitative evidence for the Weak ties hypothesis of Granovetter. Further, the analysis of triad significance profiles validates well-established assertions from social balance theory. We find overrepresentation (underrepresentation) of complete (incomplete) triads in networks of positive ties, and vice versa for networks of negative ties. Empirical transition probabilities between triad classes provide evidence for triadic closure with extraordinarily high precision. For the first time we provide empirical results for large-scale networks of negative social ties. Whenever possible we compare our findings with data from non-virtual human groups and provide further evidence that online game communities serve as a valid model for a wide class of human societies. With this setup we demonstrate the feasibility for establishing a 'socio-economic laboratory' which allows to operate at levels of precision approaching those of the natural sciences.

We present a general method to convert an ensemble of symbolic sequences into a weighted directed network. The nodes of the network are short motifs of the ensemble, while the directed links and their weights are defined from statistically significant co-occurrences of two motifs in the same sequence. The method is shown to be able in correlating sequences with functions in protein data, and might find useful applications for structure discovery as well as in computational linguistic and in the theory of dynamical systems.