Limits of dense graph sequences

We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural ``limit object'', namely a symmetric measurable 2-variable function on [0,1]. This limit object determines all the limits of subgraph densities. We also show that the graph parameters obtained as limits of subgraph densities can be characterized by ``reflection positivity'', semidefiniteness of an associated matrix. Conversely, every such function arises as a limit object. Along the lines we introduce a rather general model of random graphs, which seems to be interesting on its own right.Comment: 27 pages; added extension of result (Sept 22, 2004

Competition and the Gender Wage Gap: New Evidence from Linked Employer-Employee Data in Hungary 1986-2003

The overall gender wage gap fell from .31 to .15 between 1986 and 2003 following the transition to a free market in Hungary. During the same time period, firms faced increased competition from both new domestic and foreign firms due to the rapid liberalization measures implemented by the government. Becker's (1957) model of employer taste discrimination implies that employers that discriminate against women may be forced out of the market by competition in the long run, leading to a fall in the gender wage gap. I test this implication using data from the Hungarian Wage and Earnings Survey covering 1986-2003. I estimate the effect of variation in various measures of product market competition, including trade variables, on the within-firm endowment-adjusted gender wage gap, making use of the fact that I am able to follow firms over time. The estimates show a significant negative relationship between product market competition and the within-firm gender wage gap.Transitional labor market, wage differentials, gender discrimination

Multifractal Network Generator

We introduce a new approach to constructing networks with realistic features. Our method, in spite of its conceptual simplicity (it has only two parameters) is capable of generating a wide variety of network types with prescribed statistical properties, e.g., with degree- or clustering coefficient distributions of various, very different forms. In turn, these graphs can be used to test hypotheses, or, as models of actual data. The method is based on a mapping between suitably chosen singular measures defined on the unit square and sparse infinite networks. Such a mapping has the great potential of allowing for graph theoretical results for a variety of network topologies. The main idea of our approach is to go to the infinite limit of the singular measure and the size of the corresponding graph simultaneously. A very unique feature of this construction is that the complexity of the generated network is increasing with the size. We present analytic expressions derived from the parameters of the -- to be iterated-- initial generating measure for such major characteristics of graphs as their degree, clustering coefficient and assortativity coefficient distributions. The optimal parameters of the generating measure are determined from a simple simulated annealing process. Thus, the present work provides a tool for researchers from a variety of fields (such as biology, computer science, biology, or complex systems) enabling them to create a versatile model of their network data.Comment: Preprint. Final version appeared in PNAS