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We study the topological and geographic structure of the national road networks of the United States, England, and Denmark. By transforming these networks into their dual representation, where roads are vertices and an edge connects two vertices if the corresponding roads ever intersect, we show that they exhibit both topological and geographic scale invariance. That is, we show that for sufficiently large geographic areas, the dual degree distribution follows a power law with exponent 2.2< or = alpha < or =2.4, and that journeys, regardless of their length, have a largely identical structure. To explain these properties, we introduce and analyze a simple fractal model of road placement that reproduces the observed structure, and suggests a testable connection between the scaling exponent and the fractal dimensions governing the placement of roads and intersections.

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Abstract. It is exponentially unlikely that a sparse random graph or hypergraph is connected, but such graphs occur commonly as the giant components of larger random graphs. This simple observation allows us to estimate the number of connected graphs, and more generally the number of connected d-uniform hypergraphs, on n vertices with m = O(n) edges. We also estimate the probability that a binomial random hypergraph H d (n, p) is connected, and determine the expected number of edges of H d (n, p) conditioned on its being connected. This generalizes prior work of Bender, Canfield, and McKay [2] on the number of connected graphs; however, our approach relies on elementary probabilistic methods, extending an approach of O'Connell, rather than using powerful tools from enumerative combinatorics. We also estimate the probability for each t that, given k = O(n) balls in n bins, every bin is occupied by at least t balls.

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