Tanya Araújo

Agent based models and opinion dynamics as markov chains

This paper introduces a Markov chain approach that allows a rigorous analysis of agent based opinion dynamics as well as other related agent based models (ABM). By viewing the ABM dynamics as a micro description of the process, we show how the corresponding macro description is obtained by a projection construction. Then, well known conditions for lumpability make it possible to establish the cases where the macro model is stillMarkov. In this case we obtain a complete picture of the dynamics including the transient stage, the most interesting phase in applications. For such a purpose a crucial role is played by the type of probability distribution used to implement the stochastic part of the model which defines the updating rule and governs the dynamics. In addition, we show how restrictions in communication leading to the co–existence of different opinions follow from the emergence of new absorbing states. We describe our analysis in detail with some specific models of opinion dynamics. Generalizations concerning different opinion representations as well as opinion models with other interaction mechanisms are also discussed. We find that our method may be an attractive alternative to mean–field approaches and that this approach provides new perspectives on the modeling of opinion exchange dynamics, and more generally of other ABM.

The Geometry of Crashes - A Measure of the Dynamics of Stock Market Crises

This paper investigates the dynamics of stocks in the S&P 500 index for the last 30 years. Using a stochastic geometry technique, we investigate the evolution of the market space and define a new measure for that purpose that is a robust index of the dynamics of the market structure and provides information on the intensity and the sectoral impact of crises. With this measure, we analyse the effects of extreme phenomena on the geometry of the market. Nine crashes between 1987 and 2001 are compared by looking at the way they modify the shape of the manifold that describes the S&P 500 market space.

A Process-Reconstruction Analysis of Market Fluctuations

The statistical properties of a stochastic process may be described (1) by the expectation values of the observables, (2) by the probability distribution functions or (3) by probability measures on path space. Here an analysis of level (3) is carried out for market fluctuation processes. Gibbs measures and chains with complete connections are considered. Some other topics are also discussed, in particular the asymptotic stationarity of the processes and the behavior of statistical indicators of level (1) and (2). We end up with some remarks concerning the nature and origin of the market fluctuation process and its relation to the efficient market hypothesis.

Opinion dynamics and communication networks

This paper examines the interplay of opinion exchange dynamics and communication network formation. An opinion formation procedure is introduced which is based on an abstract representation of opinions as k-dimensional bit-strings. Individuals interact if the difference in the opinion strings is below a defined similarity threshold dI. Depending on dI, different behavior of the population is observed: low values result in a state of highly fragmented opinions and higher values yield consensus. The first contribution of this research is to identify the values of parameters dI and k, such that the transition between fragmented opinions and homogeneity takes place. Then, we look at this transition from two perspectives: first by studying the group size distribution and second by analyzing the communication network that is formed by the interactions that take place during the simulation. The emerging networks are classified by statistical means and we find that nontrivial social structures emerge from simple rules for individual communication. Generating networks allows to compare model outcomes with real-world communication patterns.

The topology of cross-border exposures: Beyond the minimal spanning tree approach

The recent financial crisis has stressed the need to understand financial systems as networks of interdependent countries, where cross-border financial linkages play the fundamental role. It has also been emphasized that the relevance of these networks relies on the representation of changes follow on the occurrence of stress events. Here, from series of interbank liabilities and claims over different time periods, we have developed networks of positions (net claims) between countries. Besides the Minimal Spanning Tree analysis of the time-constrained networks, a coefficient of residuality is defined to capture the structural evolution of the network of cross-border financial linkages. Because some structural changes seem to be related to the role that countries play in the financial context, networks of debtor and creditor countries are also developed. Empirical results allows to relate the network structure that emerges in the last years to the globally turbulent period that has characterized financial systems since the latest nineties. The residuality coefficient highlights an important modification acting in the financial linkages across countries in the period 1997–2011, and situates the recent financial crises as replica of a larger structural change going on since 1997.

Innovation and Self-Organization in a Multi-Agent Model

A model is developed to study the effectiveness of innovation and its impact on structure creation on agent-based societies. The abstract model that is developed is easily adapted to any particular field. In an interacting environment, the agents receive something from the environment (the other agents) in exchange for their effort and pay the environment a certain amount of value for the fulfilling of their needs or for the very price of existence in that environment. This is coded by two bit strings and the dynamics of the exchange is based on the matching of these strings to those of the other agents. Innovation is related to the adaptation by the agents of their bit strings to improve some utility function.

On the empirical relevance of the transient in opinion models

While the number and variety of models to explain opinion exchange dynamics is huge, attempts to justify the model results using empirical data are relatively rare. As linking to real data is essential for establishing model credibility, this Letter develops an empirical confirmation experiment by which an opinion model is related to real election data. The model is based on a representation of opinions as a vector of k bits. Individuals interact according to the principle that similarity leads to interaction and interaction leads to still more similarity. In the comparison to real data we concentrate on the transient opinion profiles that form during the dynamic process. An artificial election procedure is introduced which allows to relate transient opinion configurations to the electoral performance of candidates for which data are available. The election procedure based on the well-established principle of proximity voting is repeatedly performed during the transient period and remarkable statistical agreement with the empirical data is observed.

Aggregation and Emergence in Agent-Based Models: A Markov Chain Approach

We analyze the dynamics of agent-based models (ABMs) from a Markovian perspective and derive explicit statements about the possibility of linking a microscopic agent model to the dynamical processes of macroscopic

Education, Neighborhood Effects and Growth: An Agent-Based Model Approach

Endogenous, ideas-led growth theory and the literature on agent-based modeling with neighborhood effects are crossed. In an economic overlapping generations framework, it is shown how social interactions and neighborhood effects are of vital importance in the endogenous determination of the long run number of skilled workers and therefore of the growth prospects of an economy. Neighborhood effects interact with the initial distribution of skilled agents across space and play a key role in the long run stabilization of the number of skilled individuals. Our model implies a tendency toward segregation, with a possibly positive influence on growth, if team effects operate. The long run growth rate is also shown to depend on the rate of time preference. Initial circumstances are of vital importance for long run outcomes. A poor initial education endowment will imply a long run reduced number of skilled workers and a mediocre growth rate, so there is no economic convergence tendency. On the contrary, poor societies will grow less, or will even fall into a poverty trap, and will diverge continuously from richer ones.

Signals on graphs: Transforms and tomograms

Development of efficient tools for the representation of large datasets is a precondition for the study of dynamics on networks. Generalizations of the Fourier transform on graphs have been constructed through projections on the eigenvectors of graph matrices. By exploring mappings of the spectrum of these matrices we show how to construct more general transforms, in particular wavelet-like transforms on graphs. For time-series, tomograms, a generalization of the Radon transforms to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signals and are robust in the presence of noise. Here the notion of tomogram is also extended to signals on arbitrary graphs.