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The contagion threshold for diffusion of innovations is defined andcalculated in finite graphs (two-dimensional regular lattices, regular randomnetworks (RRNs), and two kinds of scale-free networks (SFNs)) with and withoutthe bilingual option. Without the bilingual option, degree inhomogeneity andclustering enhance the contagion threshold in non-regular networks except forthose with an unrealistically small average degree. It is explained by thefriendship paradox and detour effect. We found the general boundary of the costthat makes the bilingual option effective. With a low-cost bilingual option,among regular lattices, SFNs, and RRNs, the contagion threshold is largest inregular lattices and smallest in RRNs. The contagion threshold of regularrandom networks is almost the same as that of the regular trees, which is theminimum among regular networks. We show that the contagion threshold increasesby clustering with a low-cost bilingual option.

Diffusion of innovations in finite networks: effects of heterogeneity, clustering, and bilingual option on the threshold in the contagion game model