Random Bayesian networks with bounded indegree

Bayesian networks (BN) are an extensively used graphical model for representing a probability distribution in artificial intelligence, data mining, and machine learning. In this paper, we propose a simple model for large random BNs with bounded indegree, that is, large directed acyclic graphs (DAG) where the edges appear at random and each node has at most a given number of parents. Using this model, we can study useful asymptotic properties of large BNs and BN algorithms with basic combinatorics tools. We estimate the expected size of a BN, the expected size increase of moralization, the expected size of the Markov blanket, and the maximum size of a minimal d-separator. We also provide an upper bound on the average time complexity of an algorithm for finding a minimal d-separator. In addition, the estimates are evaluated against BNs learned from real world data.