# Ranking influential spreaders is an ill-defined problem

## TL;DR

This paper challenges the traditional node-ranking approach for identifying influential spreaders in networks, showing that the most influential set varies with size and proposing a method to quantify this variability.

## Contribution

It demonstrates the limitations of ranking methods for influential spreaders and introduces a new approach to measure the variability of influential node sets.

## Key findings

- Ranking methods can be unreliable for identifying influential nodes.
- The set of top influential nodes changes with the set size.
- The variability phenomenon is common in empirical and model networks.

## Abstract

Finding influential spreaders of information and disease in networks is an important theoretical problem, and one of considerable recent interest. It has been almost exclusively formulated as a node-ranking problem -- methods for identifying influential spreaders rank nodes according to how influential they are. In this work, we show that the ranking approach does not necessarily work: the set of most influential nodes depends on the number of nodes in the set. Therefore, the set of $n$ most important nodes to vaccinate does not need to have any node in common with the set of $n+1$ most important nodes. We propose a method for quantifying the extent and impact of this phenomenon, and show that it is common in both empirical and model networks.