Three faces of node importance in network epidemiology: Exact results for small graphs

TL;DR

This paper provides exact calculations of node importance in small networks for disease spread, vaccination, and surveillance, revealing insights into their differences and similarities based on network structure and infection rates.

Contribution

It introduces exact formulas for three node importance measures in all small connected graphs, highlighting differences in optimal node sets and their dependence on network and disease parameters.

Findings

Node separation is more important than centrality for multiple nodes.

Vaccination and influence maximization are fundamentally different.

The three importance aspects become more similar at low infection rates.

Abstract

We investigate three aspects of the importance of nodes with respect to Susceptible-Infectious-Removed (SIR) disease dynamics: influence maximization (the expected outbreak size given a set of seed nodes), the effect of vaccination (how much deleting nodes would reduce the expected outbreak size) and sentinel surveillance (how early an outbreak could be detected with sensors at a set of nodes). We calculate the exact expressions of these quantities, as functions of the SIR parameters, for all connected graphs of three to seven nodes. We obtain the smallest graphs where the optimal node sets are not overlapping. We find that: node separation is more important than centrality for more than one active node, that vaccination and influence maximization are the most different aspects of importance, and that the three aspects are more similar when the infection rate is low.