  How and to what extent will new activities spread through social ties? Here,
we develop a more sophisticated framework than the standard mean-field approach
to describe the diffusion dynamics of multiple activities on complex networks.
We show that the diffusion of multiple activities follows a saddle path and can
be highly unstable. In particular, when the two activities are sufficiently
substitutable, either of them would dominate the other by chance even if they
are equally attractive ex ante. When such symmetry-breaking occurs, any
average-based approach cannot correctly calculate the Nash equilibrium - the
steady state of an actual diffusion process.
