  The Jordan center 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
Center 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 center persists on a single
vertex after a finite number of timesteps. Finally, we also study the
continuous time version of the SI model and bound the maximum distance between
the Jordan center and the root node at any time.
