A note on general epidemic region for infinite regular graphs

Abstract

We study the contagion game with the bilingual option on infinite regular graphs introduced and modeled mathematically in [8]. In the reference, lmmorlica 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.