Preferential Attachment Graphs Are All But Asymmetric∗

Recent years have seen a resurgence of interest in graph asymmetry and many of its novel applications ranging from graph compression to discovery of graph dynamics (e.g., age of a node). It has been known for some time that Erdős-Rényi graphs are asymmetric with high probability, but it is also known that real world graphs (web, biological networks) have a significant amount of symmetry. So the first natural question to ask is whether preferential attachment graphs in which in each step a new node with m edges is added exhibit any symmetry. It turns out that the problem is much harder to crack than in the Erdős-Rényi case and reveals unexpected results. First of all, in recent work it was proved that attachment graphs are symmetric for m = 1 (as expected) but surprisingly there is some non-negligible probability of symmetry for m = 2. The question was open for m ≥ 3. Based on empirical results it was conjectured, however, that the preferential attachment graphs are asymmetric with high probability when more than two edges are added each time a new node arrives. In this paper we settle this conjecture in the positive. This result has implications for other questions of interest, such as structural entropy of preferential attachment graphs, their structural compression, and even recovering node arrival times given an unlabeled preferential attachment graph.