  Diffusion processes are central to human interactions. One common prediction
of the current modeling frameworks is that initial spreading dynamics follow
exponential growth. Here, we find that, ranging from mobile handsets to
automobiles, from smart-phone apps to scientific fields, early growth patterns
follow a power law with non-integer exponents. We test the hypothesis that
mechanisms specific to substitution dynamics may play a role, by analyzing a
unique data tracing 3.6M individuals substituting for different mobile
handsets. We uncover three generic ingredients governing substitutions,
allowing us to develop a minimal substitution model, which not only explains
the power-law growth, but also collapses diverse growth trajectories of
individual constituents into a single curve. These results offer a mechanistic
understanding of power-law early growth patterns emerging from various domains
and demonstrate that substitution dynamics are governed by robust
self-organizing principles that go beyond the particulars of individual
systems.
