Jordan centre in random trees: persistence and distance to root

Abstract

Abstract The Jordan centre of a graph is defined as a vertex whose maximum distance to other nodes in the graph is minimal, and it finds applications in facility location and source detection problems. We study properties of the Jordan centre in the case of random growing trees. In particular, we consider a regular tree graph on which an infection starts from a root node and then spreads along the edges of the graph according to various random spread models. For the Independent Cascade (IC) model and the discrete Susceptible Infected (SI) model, both of which are discrete-time models, we show that as the infected subgraph grows with time, the Jordan centre persists on a single vertex after a finite number of timesteps. As a corollary of our results, we also establish that the distance between the Jordan centre and the infection source (root node) is finite. Finally, we also study the continuous-time version of the SI model and bound the maximum distance between the Jordan centre and the root node at any time.