W∞W_\infty and Anomalies of Self-Dual Einstein Theories

This manuscripts corrects some minor error in the paper, Mod. Phys. Lett. A 6 1893 (1991)Comment: (revised due to TeXnical errors), 11 page

Growing random networks with fitness

Three models of growing random networks with fitness dependent growth rates are analysed using the rate equations for the distribution of their connectivities. In the first model (A), a network is built by connecting incoming nodes to nodes of connectivity $k$ and random additive fitness $\eta$, with rate $(k-1)+ \eta$. For $\eta >0$ we find the connectivity distribution is power law with exponent $\gamma=+2$. In the second model (B), the network is built by connecting nodes to nodes of connectivity $k$, random additive fitness $\eta$ and random multiplicative fitness $\zeta$ with rate $\zeta(k-1)+\eta$. This model also has a power law connectivity distribution, but with an exponent which depends on the multiplicative fitness at each node. In the third model (C), a directed graph is considered and is built by the addition of nodes and the creation of links. A node with fitness $(\alpha, \beta)$, $i$ incoming links and $j$ outgoing links gains a new incoming link with rate $\alpha(i+1)$, and a new outgoing link with rate $\beta(j+1)$. The distributions of the number of incoming and outgoing links both scale as power laws, with inverse logarithmic corrections

Models for the size distribution of businesses in a price driven market

A microscopic model of aggregation and fragmentation is introduced to investigate the size distribution of businesses. In the model, businesses are constrained to comply with the market price, as expected by the customers, while customers can only buy at the prices offered by the businesses. We show numerically and analytically that the size distribution scales like a power-law. A mean-field version of our model is also introduced and we determine for which value of the parameters the mean-field model agrees with the microscopic model. We discuss to what extent our simple model and its results compare with empirical data on company sizes in the U.S. and debt sizes in Japan. Finally, possible extensions of the mean-field model are discussed, to cope with other empirical data.Comment: 12 pages, 2 figures, submitted for publicatio

Strategy Selection in the Minority Game

We investigate the dynamics of the choice of an active strategy in the minority game. A history distribution is introduced as an analytical tool to study the asymmetry between the two choices offered to the agents. Its properties are studied numerically. It allows us to show that the departure from uniformity in the initial attribution of strategies to the agents is important even in the efficient market. Also, an approximate expression for the variance of the number of agents at one side in the efficient phase is proposed. All the analytical propositions are supported by numerical simulations of the system.Comment: Latex file, 17 page, 4 figure

Spectral Density of Complex Networks with a Finite Mean Degree

In order to clarify the statistical features of complex networks, the spectral density of adjacency matrices has often been investigated. Adopting a static model introduced by Goh, Kahng and Kim, we analyse the spectral density of complex scale free networks. For that purpose, we utilize the replica method and effective medium approximation (EMA) in statistical mechanics. As a result, we identify a new integral equation which determines the asymptotic spectral density of scale free networks with a finite mean degree $p$. In the limit $p \to \infty$, known asymptotic formulae are rederived. Moreover, the $1/p$ corrections to known results are analytically calculated by a perturbative method.Comment: 18 pages, 1 figure, minor corrections mad

Applications of W-algebras to BF theories, QCD and 4D Gravity

We are able to show that BF theories naturally emerge from the coadjoint orbits of $W_2$ and $w_\infty$ algebras which includes a Kac-Moody sector. Since QCD strings can be identified with a BF theory, we are able to show a relationship between the orbits and monopole-string solutions of QCD. Furthermore, we observe that when 4D gravitation is cast into a BF form through the use of Ashtekar variables, we are able to get order $\hbar$ contributions to gravity which can be associated with the $W_2$ anomaly. We comment on the relationship to gravitational monopoles.Comment: 14 page