Understanding power-law early growth in substitutive systems.

Diffusion processes are central to human interactions. Despite extensive studies that span multiple disciplines, our knowledge is limited to spreading dynamics in non-substitutive systems. Yet, a considerable number of ideas, products and behaviors spread by substitution---to adopt a new one, agents often need to give up an existing one. Here, we find that, ranging from mobile handsets to automobiles to smart-phone apps, early growth patterns follow a power law with non-integer exponents, in sharp contrast to the exponential growth customary in spreading phenomena. Tracing 3.6M individuals substituting for mobile handsets for over a decade, we uncover three generic ingredients governing substitutive processes, allowing us to develop a minimal substitution model, which not only predicts analytically the observed growth patterns, but also collapses growth trajectories of constituents from rather diverse systems into a single universal curve. These results not only offer a mechanistic understanding of power-law early growth patterns emerging from various domains; they also demonstrate that substitution dynamics are governed by robust self-organizing principles that go beyond the particulars of individual systems, suggesting that power laws characterizing early growth may be more common than we realize.