Jigsaw percolation on random hypergraphs

The jigsaw percolation process on graphs was introduced by Brummitt, Chatterjee, Dey, and Sivakoff as a model of collaborative solutions of puzzles in social networks. Percolation in this process may be viewed as the joint connectedness of two graphs on a common vertex set. Our aim is to extend a result of Bollob\'as, Riordan, Slivken, and Smith concerning this process to hypergraphs for a variety of possible definitions of connectedness. In particular, we determine the asymptotic order of the critical threshold probability for percolation when both hypergraphs are chosen binomially at random.Comment: 17 page

Uniform generation of random graphs with power-law degree sequences

We give a linear-time algorithm that approximately uniformly generates a random simple graph with a power-law degree sequence whose exponent is at least 2.8811. While sampling graphs with power-law degree sequence of exponent at least 3 is fairly easy, and many samplers work efficiently in this case, the problem becomes dramatically more difficult when the exponent drops below 3; ours is the first provably practicable sampler for this case. We also show that with an appropriate rejection scheme, our algorithm can be tuned into an exact uniform sampler. The running time of the exact sampler is O(n^{2.107}) with high probability, and O(n^{4.081}) in expectation.Comment: 50 page