Compression of Preferential Attachment Graphs

We study structural properties of preferential attachment graphs (with parameter m ≥ 1 giving the number of attachment choices that each new vertex makes) which intervene in two complementary algorithmic/statistical/information-theoretic problems involving the information shared between a random graph’s labels and its structure: in structural compression, we seek to compactly describe a graph’s structure by a bit string, throwing away its label information; in node arrival order recovery, we seek to recover node labels, given only a graph structure. In particular, we study the typical size of the automorphism group, as well as some shape parameters (such as the number of linear extensions and height) of the directed version of the graph, which in turn allows us to estimate the typical number of admissible labeled representatives of a given graph structure. Our result on the automorphism group positively settles a conjecture to the effect that, provided that m ≥ 3, preferential attachment graphs are asymmetric with high probability, and completes the characterization of the number of symmetries for a broad range of parameters of the model (i.e., for all fixed m). These results allow us to give an algorithmically efficient, asymptotically optimal algorithm for compression of unlabeled preferential attachment graphs. To show the optimality of our scheme, we also derive new, precise estimates of the Shannon entropy of both the unlabeled and labeled version of the model. Our results also imply inapproximability results for the problem of node arrival order recovery.