Asymmetry and Structural Information in Preferential Attachment Graphs∗

Symmetries of graphs intervene in diverse applications, ranging from enumeration to graph structure compression, to the discovery of graph dynamics (e.g., inference of the arrival order of nodes in a growing network). It has been known for some time that ErdősRényi 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 harder than in the Erdős-Rényi case and reveals unexpected results: in recent work it was proved that preferential attachment graphs are symmetric for m = 1 (as expected), and there is some non-negligible probability of symmetry for m = 2. The question was left open for m ≥ 3. 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. We then use it to estimate the entropy of unlabelled preferential attachment graphs (called the structural entropy of the model). To do this, we prove a new, precise asymptotic result for the labeled graph entropy, and we give bounds on another combinatorial parameter of the graph model: the number of ways in which the given graph structure could have arisen by preferential attachment (a measure of the extent to which the graph structure encodes information about the order in which nodes arrived). These results have further implications for information theoretic and statistical problems of interest on preferential attachment graphs.