# On the Distance Between the Rumor Source and Its Optimal Estimate in a Regular Tree

## TL;DR

This paper analyzes the accuracy of rumor source detection in regular trees, showing that the estimated source is typically within three edges of the true origin with high probability.

## Contribution

It provides a probabilistic analysis of the distance between the true rumor source and its optimal estimate in regular tree networks.

## Key findings

- The estimated rumor source is within distance 3 of the true source with high probability.
- The probability distribution of the distance between the true source and the estimate is characterized.
- The analysis applies specifically to regular, cycle-free tree networks.

## Abstract

This paper addresses the rumor source identification problem, where the goal is to find the origin node of a rumor in a network among a given set of nodes with the rumor. In this paper, we focus on a network represented by a regular tree which does not have any cycle and in which all nodes have the same number of edges connected to a node. For this network, we clarify that, with quite high probability, the origin node is within the distance 3 from the node selected by the optimal estimator, where the distance is the number of edges of the unique path connecting two nodes. This is clarified by the probability distribution of the distance between the origin and the selected node.