Persistence of the Jordan center in Random Growing Trees

TL;DR

This paper investigates the behavior of the Jordan center in random growing trees under various infection spread models, showing its persistence on a single vertex over time and analyzing its distance from the root.

Contribution

It demonstrates the persistence of the Jordan center in random growing trees under discrete and continuous infection models, providing theoretical bounds and insights.

Findings

Jordan center stabilizes on a single vertex after finite steps in discrete models

Maximum distance between Jordan center and root is bounded in continuous models

Results apply to regular trees with infection spread dynamics

Abstract

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.