# A note on general epidemic region for infinite regular graphs

## TL;DR

This paper analyzes the spread of innovations in infinite regular graphs, showing which structures facilitate or hinder epidemic contagion, extending previous results on infinite trees and grids.

## Contribution

It improves prior results by identifying infinite regular trees as least conducive to epidemic spread and constructs families of graphs that maximize contagion potential.

## Key findings

- Infinite regular trees are least advantageous for epidemic spread.
- Any infinite Δ-regular graph with an infinite Δ-tree structure is also least advantageous.
- Constructed infinite Δ-regular graphs that are most conducive to epidemic spread.

## Abstract

We study the contagion game with the bilingual option on infinite regular graphs introduced and modeled mathematically in [N. Immorlica et al. (2007)]. In the reference, Immorlica et al. studied conditions for an innovation to become epidemic over infinite regular trees, the grid, and the infinite thick-lines in terms of payoff enhancement and cost of the bilingual option. We improved their results by showing that the class of infinite regular trees make an innovation least advantageous to become epidemic considering the whole class of infinite regular graphs. Moreover, we show that any infinite $\Delta$-regular graph containing the infinite $\Delta$-tree structure is also least advantageous to be epidemic. Also, we construct an infinite family of infinite $\Delta$-regular graphs (including the thick $\Delta$-line) that is the most advantageous to be epidemic as known so far.