  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.
