A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this paper a model is developed in which each vertex is given a random weight, and vertices with larger weights are more likely to be assigned large subsets. The distribution of the degree of a given vertex is characterized and is shown to depend on the weight of the vertex. In particular, if the weight distribution is a power law, the degree distribution will be so as well. Furthermore, an asymptotic expression for the clustering in the graph is derived. By tuning the parameters of the model, it is possible to generate a graph with arbitrary clustering, expected degree and -in the power law case -tail exponent.

We investigate the limiting behavior of trader wealth and prices in a simple prediction market with a finite set of participants having heterogeneous beliefs. Traders bet repeatedly on the outcome of a binary event with fixed Bernoulli success probability. A class of strategies, including (fractional) Kelly betting and constant relative risk aversion (CRRA) are considered. We show that when traders are willing to risk only a small fraction of their wealth in any period, belief heterogeneity can persist indefinitely; if bets are large in proportion to wealth then only the most accurate belief type survives. The market price is more accurate in the long run when traders with less accurate beliefs also survive. That is, the survival of traders with heterogeneous beliefs, some less accurate than others, allows the market price to better reflect the objective probability of the event in the long run.

We used a low-temperature scanning tunneling microscope to study Zn-and Cd-doping atoms near the ͑110͒-cleavage surfaces of GaAs and InP at 4.2 K. The filled-state images showed centro-symmetric elevations while the empty-state images showed circular depressions. We attribute these features to the influence of the Coulomb potential of the ionized doping atoms on the number of states available for tunneling. In a few empty-state images of the GaAs͑110͒ surface, the depressions were surrounded by maxima, which are probably direct observations of Friedel oscillations. For the InP͑110͒ surface, all depressions were surrounded by noncentrosymmetric maxima. Upon moving the tip Fermi level to the bottom of the conduction band, we observed that the depressions turned into elevations with a triangular shape for both the GaAs͑110͒ and the InP͑110͒ surface. This shape was independent of the depth of the dopants, and the chemical nature of the dopants ͑Zn or Cd͒ did not influence the triangular shape either. The orientation of these triangular features was the same for all observed doping atoms and was geometrically determined with respect to the host lattice. Furthermore, we determined the location of a triangular feature with respect to a doping atom. The features were only visible when tunneling to the impurity band suggesting that the features are a direct image of the acceptor state although the origin of the triangular shape is not clear at present.

We explore the manner in which the structure of a social network constrains the level of inequality that can be sustained among its members, based on the following considerations: (i) any distribution of value must be stable with respect to coalitional deviations, and (ii) the network structure itself determines the coalitions that may form. We show that if players can jointly deviate only if they form a clique in the network, then the degree of inequality that can be sustained depends on the cardinality of the maximum independent set. For bipartite networks, the size of the maximum independent set fully characterizes the degree of inequality that can be sustained. This result extends partially to general networks and to the case in which a group of players can deviate jointly if they are all sufficiently close to each other in the network. * We thank