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This paper introduces new efficient algorithms for two problems: samplingconditional on vertex degrees in unweighted graphs, and sampling conditional onvertex strengths in weighted graphs. The algorithms can sample conditional onthe presence or absence of an arbitrary number of edges. The resultingconditional distributions provide the basis for exact tests. Existing samplersbased on MCMC or sequential importance sampling are generally not scalable;their efficiency degrades in sparse graphs. MCMC methods usually requireexplicit computation of a Markov basis to navigate the complex state space;this is computationally intensive even for small graphs. We use state-dependentkernel selection to develop new MCMC samplers. These do not require a Markovbasis, and are efficient both in sparse and dense graphs. The key idea is tointelligently select a Markov kernel on the basis of the current state of thechain. We apply our methods to testing hypotheses on a real network andcontingency table. The algorithms appear orders of magnitude more efficientthan existing methods in the test cases considered.

State-Dependent Kernel Selection for Conditional Sampling of Graphs