Self-similarity of complex networks and hidden metric spaces

We demonstrate that the self-similarity of some scale-free networks with respect to a simple degree-thresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization.

Percolation and epidemic thresholds in clustered networks

We develop a theoretical approach to percolation in random clustered networks. We find that, although clustering in scale-free networks can strongly affect some percolation properties, such as the size and the resilience of the giant connected component, it cannot restore a finite percolation threshold. In turn, this implies the absence of an epidemic threshold in this class of networks, thus extending this result to a wide variety of real scale-free networks which shows a high level of transitivity. Our findings are in good agreement with numerical simulations.

Topology of the world trade web

Economy, and consequently trade, is a fundamental part of human social organization which, until now, has not been studied within the network modeling framework. Here we present the first, to the best of our knowledge, empirical characterization of the world trade web, that is, the network built upon the trade relationships between different countries in the world. This network displays the typical properties of complex networks, namely, scale-free degree distribution, the small-world property, a high clustering coefficient, and, in addition, degree-degree correlation between different vertices. All these properties make the world trade web a complex network, which is far from being well described through a classical random network description.

Efficiency of informational transfer in regular and complex networks

We analyze the process of informational exchange through complex networks by measuring network efficiencies. Aiming to study nonclustered systems, we propose a modification of this measure on the local level. We apply this method to an extension of the class of small worlds that includes declustered networks and show that they are locally quite efficient, although their clustering coefficient is practically zero. Unweighted systems with small-world and scale-free topologies are shown to be both globally and locally efficient. Our method is also applied to characterize weighted networks. In particular we examine the properties of underground transportation systems of Madrid and Barcelona and reinterpret the results obtained for the Boston subway network.

Reconstruction of networks from their betweenness centrality

In this paper we study the reconstruction of a network topology from the values of its betweenness centrality, a measure of the influence of each of its nodes in the dissemination of information over the network. We consider a simple metaheuristic, simulated annealing, as the combinatorial optimization method to generate the network from the values of the betweenness centrality. We compare the performance of this technique when reconstructing different categories of networks –random, regular, small-world, scale-free and clustered–. We show that the method allows an exact reconstruction of small networks and leads to good topological approximations in the case of networks with larger orders. The method can be used to generate a quasi-optimal topology fora communication network from a list with the values of the maximum allowable traffic for each node.

Evolutionary Games in Networked Populations: Models and Experiments

Cooperation and coordination are desirable behaviors that are fundamental for the harmonious development of society. People need to rely on cooperation with other individuals in many aspects of everyday life, such as teamwork and economic exchange in anonymous markets. However, cooperation may easily fall prey to exploitation by selfish individuals who only care about short- term gain. For cooperation to evolve, specific conditions and mechanisms are required, such as kinship, direct and indirect reciprocity through repeated interactions, or external interventions such as punishment. In this dissertation we investigate the effect of the network structure of the population on the evolution of cooperation and coordination. We consider several kinds of static and dynamical network topologies, such as Baraba´si-Albert, social network models and spatial networks. We perform numerical simulations and laboratory experiments using the Prisoner's Dilemma and co- ordination games in order to contrast human behavior with theoretical results. We show by numerical simulations that even a moderate amount of random noise on the Baraba´si-Albert scale-free network links causes a significant loss of cooperation, to the point that cooperation almost vanishes altogether in the Prisoner's Dilemma when the noise rate is high enough. Moreover, when we consider fixed social-like networks we find that current models of social networks may allow cooperation to emerge and to be robust at least as much as in scale-free networks. In the framework of spatial networks, we investigate whether cooperation can evolve and be stable when agents move randomly or performing Le´vy flights in a continuous space. We also consider discrete space adopting purposeful mobility and binary birth-death process to dis- cover emergent cooperative patterns. The fundamental result is that cooperation may be enhanced when this migration is opportunistic or even when agents follow very simple heuristics. In the experimental laboratory, we investigate the issue of social coordination between indi- viduals located on networks of contacts. In contrast to simulations, we find that human players dynamics do not converge to the efficient outcome more often in a social-like network than in a random network. In another experiment, we study the behavior of people who play a pure co- ordination game in a spatial environment in which they can move around and when changing convention is costly. We find that each convention forms homogeneous clusters and is adopted by approximately half of the individuals. When we provide them with global information, i.e., the number of subjects currently adopting one of the conventions, global consensus is reached in most, but not all, cases. Our results allow us to extract the heuristics used by the participants and to build a numerical simulation model that agrees very well with the experiments. Our findings have important implications for policymakers intending to promote specific, desired behaviors in a mobile population. Furthermore, we carry out an experiment with human subjects playing the Prisoner's Dilemma game in a diluted grid where people are able to move around. In contrast to previous results on purposeful rewiring in relational networks, we find no noticeable effect of mobility in space on the level of cooperation. Clusters of cooperators form momentarily but in a few rounds they dissolve as cooperators at the boundaries stop tolerating being cheated upon. Our results highlight the difficulties that mobile agents have to establish a cooperative environment in a spatial setting without a device such as reputation or the possibility of retaliation. i.e. punishment. Finally, we test experimentally the evolution of cooperation in social networks taking into ac- count a setting where we allow people to make or break links at their will. In this work we give particular attention to whether information on an individual's actions is freely available to poten- tial partners or not. Studying the role of information is relevant as information on other people's actions is often not available for free: a recruiting firm may need to call a job candidate's refer- ences, a bank may need to find out about the credit history of a new client, etc. We find that people cooperate almost fully when information on their actions is freely available to their potential part- ners. Cooperation is less likely, however, if people have to pay about half of what they gain from cooperating with a cooperator. Cooperation declines even further if people have to pay a cost that is almost equivalent to the gain from cooperating with a cooperator. Thus, costly information on potential neighbors' actions can undermine the incentive to cooperate in dynamical networks.

Numerical study of Petrov-Galerkin formulations for the shallow water wave equations

The behavior of Petrov-Galerkin formulations for shallow water wave equations is evaluated numerically considering typical one-dimensional propagation problems. The formulations considered here use stabilizing operators to improve classical Galerkin approaches. Their advantages and disadvantages are pointed out according to the intrinsic time scale (free parameter) which has a particular importance in this kind of problem. The influence of the Courant number and the performance of the formulation in dealing with spurious oscillations are adressed.

The von Neumann Minimax Theorem and its relatives and a study of externality in on-line auctions

This work consists of a theoretical part and an experimental one. The first part provides a simple treatment of the celebrated von Neumann minimax theorem as formulated by Nikaid6 and Sion. It also discusses its relationships with fundamental theorems of convex analysis. The second part is about externality in sponsored search auctions. It shows that in these auctions, advertisers have externality effects on each other which influence their bidding behavior. It proposes Hal R.Varian model and shows how adding externality to this model will affect its properties. In order to have a better understanding of the interaction among advertisers in on-line auctions, it studies the structure of the Google advertisements networ.k and shows that it is a small-world scale-free network.

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

TU Graz: Course: 707.000 Web Science and Web Technology: Lecture 3: Network Theory and Terminology

In this class, we will discuss network theory fundamentals, including concepts such as diameter, distance, clustering coefficient and others. We will also discuss different types of networks, such as scale-free networks, random networks etc. Readings: Graph structure in the Web, A. Broder and R. Kumar and F. Maghoul and P. Raghavan and S. Rajagopalan and R. Stata and A. Tomkins and J. Wiener Computer Networks 33 309--320 (2000) [Web link, Alternative Link] Optional: The Structure and Function of Complex Networks, M.E.J. Newman, SIAM Review 45 167--256 (2003) [Web link] Original course at: http://kmi.tugraz.at/staff/markus/courses/SS2008/707.000_web-science/

De lo micro a lo macro prudencial: Un análisis de la supervisión bancaria desde la econometría espacial

La crisis que se desató en el mercado hipotecario en Estados Unidos en 2008 y que logró propagarse a lo largo de todo sistema financiero, dejó en evidencia el nivel de interconexión que actualmente existe entre las entidades del sector y sus relaciones con el sector productivo, dejando en evidencia la necesidad de identificar y caracterizar el riesgo sistémico inherente al sistema, para que de esta forma las entidades reguladoras busquen una estabilidad tanto individual, como del sistema en general. El presente documento muestra, a través de un modelo que combina el poder informativo de las redes y su adecuación a un modelo espacial auto regresivo (tipo panel), la importancia de incorporar al enfoque micro-prudencial (propuesto en Basilea II), una variable que capture el efecto de estar conectado con otras entidades, realizando así un análisis macro-prudencial (propuesto en Basilea III).

Network externalities across financial institutions

We propose and estimate a financial distress model that explicitly accounts for the interactions or spill-over effects between financial institutions, through the use of a spatial continuity matrix that is build from financial network data of inter bank transactions. Such setup of the financial distress model allows for the empirical validation of the importance of network externalities in determining financial distress, in addition to institution specific and macroeconomic covariates. The relevance of such specification is that it incorporates simultaneously micro-prudential factors (Basel 2) as well as macro-prudential and systemic factors (Basel 3) as determinants of financial distress. Results indicate network externalities are an important determinant of financial health of a financial institutions. The parameter that measures the effect of network externalities is both economically and statistical significant and its inclusion as a risk factor reduces the importance of the firm specific variables such as the size or degree of leverage of the financial institution. In addition we analyze the policy implications of the network factor model for capital requirements and deposit insurance pricing.

Modelling disease spread and control in networks: implications for plant sciences

Networks are ubiquitous in natural, technological and social systems. They are of increasing relevance for improved understanding and control of infectious diseases of plants, animals and humans, given the interconnectedness of today's world. Recent modelling work on disease development in complex networks shows: the relative rapidity of pathogen spread in scale-free compared with random networks, unless there is high local clustering; the theoretical absence of an epidemic threshold in scale-free networks of infinite size, which implies that diseases with low infection rates can spread in them, but the emergence of a threshold when realistic features are added to networks (e.g. finite size, household structure or deactivation of links); and the influence on epidemic dynamics of asymmetrical interactions. Models suggest that control of pathogens spreading in scale-free networks should focus on highly connected individuals rather than on mass random immunization. A growing number of empirical applications of network theory in human medicine and animal disease ecology confirm the potential of the approach, and suggest that network thinking could also benefit plant epidemiology and forest pathology, particularly in human-modified pathosystems linked by commercial transport of plant and disease propagules. Potential consequences for the study and management of plant and tree diseases are discussed.

Dynamics of brand competition: Effects of unobserved social networks

Brand competition is modelled using an agent based approach in order to examine the long run dynamics of market structure and brand characteristics. A repeated game is designed where myopic firms choose strategies based on beliefs about their rivals and consumers. Consumers are heterogeneous and can observe neighbour behaviour through social networks. Although firms do not observe them, the social networks have a significant impact on the emerging market structure. Presence of networks tends to polarize market share and leads to higher volatility in brands. Yet convergence in brand characteristics usually happens whenever the market reaches a steady state. Scale-free networks accentuate the polarization and volatility more than small world or random networks. Unilateral innovations are less frequent under social networks.

Universality of rain event size distributions

We compare rain event size distributions derived from measurements in climatically different regions, which we find to be well approximated by power laws of similar exponents over broad ranges. Differences can be seen in the large-scale cutoffs of the distributions. Event duration distributions suggest that the scale-free aspects are related to the absence of characteristic scales in the meteorological mesoscale.