First to Market is not Everything: an Analysis of Preferential Attachment with Fitness

The design of algorithms on complex networks, such as routing, ranking or recommendation algorithms, requires a detailed understanding of the growth characteristics of the networks of interest, such as the Internet, the web graph, social networks or online communities. To this end, preferential attachment, in which the popularity (or relevance) of a node is determined by its degree, is a wellknown and appealing random graph model, whose predictions are in accordance with experiments on the web graph and several social networks. However, its central assumption, that the popularity of the nodes depends only on their degree, is not a realistic one, since every node has potentially some intrinsic quality which can differentiate its attractiveness from other nodes with similar degrees. In this paper, we provide a rigorous analysis of preferential attachment with fitness, suggested by Bianconi and Barab´asi and studied by Motwani and Xu, in which the degree of a vertex is scaled by its quality to determine its attractiveness. Including quality considerations in the classical preferential attachment model provides a much more realistic description of many complex networks, such as the web graph, and allows to observe a much richer behavior in the growth dynamics of these networks. Specifically, depending on the shape of the distribution from which the qualities of the vertices are drawn, we observe three distinct phases, namely a first-mover-advantage phase, a fit-get-richer phase and an innovation-pays-off phase. We precisely characterize the properties of the quality distribution that result in each of these phases and we compute the exact growth dynamics for each phase. The dynamics provide rich information about the quality of the vertices, which can be very useful in many practical contexts, including ranking algorithms for the web, recommendation algorithms, as well as the study of social networks.