De-evolution of Preferential Attachment Trees

Abstract

Given a graph $G_t$ which is a result of a t time, evolutionary process, the goal of graph de-evolution of $G_t$ is to infer what was the structure of the graph $G_t'$ for $t' < t$. This general inference problem is very important for understanding the mechanisms behind complex systems like social networks and their asymptotic behavior. In this work we take a step in this direction and consider undirected, unlabeled trees that are the result of the well known random preferential attachment process. We compute the most likely root set (possible isomorphic patient zero candidates) of the tree, as well as the most likely previous graph $G_{t-1}$ structure. While the one step forward reasoning in preferential attachment is very simple, the backward (past) reasoning is more complex and includes the use of the automorphism and isomorphism of graphs, which we elucidate here.