Abstract. In this work we analyze fundamental properties of Random Apollonian Networks [34,35], a popular random graph model which generates planar graphs with power law properties. Specifically, we analyze (a) the degree distribution, (b) the k largest degrees, (c) the k largest eigenvalues and (d) the diameter, where k is a constant. 1

ABSTRACT. We study distance properties of a general class of random directed acyclic graphs (DAGs). In a DAG, many natural notions of distance are possible, for there exists multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a random DAG. This completes the study of natural distances in random DAGs initiated (in the uniform case) by Devroye and Janson (2009+). We also obtain large deviation bounds for the minimum of a branching random walk with constant branching, which can be seen as a simplified version of our main result. 1.