We consider the problem of generating uniformly random graphs from a constrained distribution. A graph is valid if it obeys certain constraints such as a given number of nodes, edges, k-stars or degree sequence, and each graph must occur with equal probability. A typical application is to confirm the correctness of a model by repeated sampling and comparing statistical properties against empirical data. Markov Chain Monte Carlo (MCMC) algorithms are often used, but have certain difficulties such as the inability to search the space of all possible valid graphs. We propose an improved algorithm which overcomes these difficulties. Although each individual iteration of the MCMC algorithm takes longer, we obtain better coverage of the search space in the same amount of time. This leads to better estimates of various quantities such as the expected number of transitive triads given the constraints. The algorithm should be of general interest with many possible applications, including the world wide web, biological, and social networks.
