Asymmetry and structural information in preferential attachment graphs

Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically asymmetric, real networks are highly symmetric. So a natural question is whether preferential attachment graphs, where in each step a new node with $m$ edges is added, exhibit any symmetry. In recent work it was proved that preferential attachment graphs are symmetric for $m=1$, and there is some non-negligible probability of symmetry for $m=2$. It was conjectured that these graphs are asymmetric when $m \geq 3$. We settle this conjecture in the affirmative, then use it to estimate the structural entropy of the model. To do this, we also give bounds on the number of ways that the given graph structure could have arisen by preferential attachment. These results have further implications for information theoretic problems of interest on preferential attachment graphs.