  We study the competition of two spreading entities, for example innovations,
in complex contagion processes in complex networks. We develop an analytical
framework and examine the role of dual users, i.e. agents using both
technologies. Searching for the spreading transition of the new innovation and
the extinction transition of a preexisting one, we identify different phases
depending on network mean degree, prevalence of preexisting technology, and
thresholds of the contagion process. Competition with the preexisting
technology effectively suppresses the spread of the new innovation, but it also
allows for phases of coexistence. The existence of dual users largely modifies
the transient dynamics creating new phases that promote the spread of a new
innovation and extinction of a preexisting one. It enables the global spread of
the new innovation even if the old one has the first-mover advantage.
