  We consider two competing first passage percolation processes started from
uniformly chosen subsets of a random regular graph on $N$ vertices. The
processes are allowed to spread with different rates, start from vertex subsets
of different sizes or at different times. We obtain tight results regarding the
sizes of the vertex sets occupied by each process, showing that in the generic
situation one process will occupy $\Theta(1)N^\alpha$ vertices, for some $0 <
\alpha < 1$. The value of $\alpha$ is calculated in terms of the relative rates
of the processes, as well as the sizes of the initial vertex sets and the
possible time advantage of one process.
  The motivation for this work comes from the study of viral marketing on
social networks. The described processes can be viewed as two competing
products spreading through a social network (random regular graph). Considering
the processes which grow at different rates (corresponding to different
attraction levels of the two products) or starting at different times (the
first to market advantage) allows to model aspects of real competition. The
results obtained can be interpreted as one of the two products taking the lion
share of the market. We compare these results to the same process run on $d$
dimensional grids where we show that in the generic situation the two products
will have a linear fraction of the market each.
