A discontinuous phase transition in the threshold-θ ≥ 2 contact process on random graphs

We study the discrete-time threshold-θ ≥ 2 contact process on random graphs of general degrees. For random graphs with a given degree distribution μ, we show that if μ is lower bounded by θ + 2 and has finite kth moments for all k > 0, then the discrete-time threshold-θ contact process on the random graph exhibits a discontinuous phase transition in the emergence of metastability, thus answering a question of Chatterjee and Durrett [5]. To be specific, we establish that (i) for any large enough infection probability p > p1, the process started from the all-infected state whp survives for e-time, maintaining a large density of infection; (ii) for any p < 1, if the initial density is smaller than ε(p) > 0, then it dies out in O(log n)-time whp. We also explain some extensions to more general random graphs, including the Erdős-Rényi graphs. Moreover, we prove that the threshold-θ contact process on a random (θ + 1)-regular graph dies out in time n whp.