On the duration and intensity of cumulative advantage competitions

Network growth can be framed as a competition for edges among nodes in the network. As with various other social and physical systems, skill (fitness) and luck (random chance) act as fundamental forces driving competition dynamics. In the context of networks, cumulative advantage (CA)-the rich-get-richer effect-is seen as a driving principle governing the edge accumulation process. However, competitions coupled with CA exhibit non-trivial behavior and little is formally known about duration and intensity of CA competitions. By isolating two nodes in an ideal CA competition, we provide a mathematical understanding of how CA exacerbates the role of luck in detriment of skill. We show, for instance, that when nodes start with few edges, an early stroke of luck can place the less skilled in the lead for an extremely long period of time, a phenomenon we call 'struggle of the fittest'. We prove that duration of a simple skill and luck competition model exhibit power-law tails when CA is present, regardless of skill difference, which is in sharp contrast to the exponential tails when fitness is distinct but CA is absent. We also prove that competition intensity is always upper bounded by an exponential tail, irrespective of CA and skills. Thus, CA competitions can be extremely long (infinite mean, depending on fitness ratio) but almost never very intense. The theoretical results are corroborated by extensive numerical simulations. Our findings have important implications to competitions not only among nodes in networks but also in contexts that leverage socio-physical models embodying CA competitions.