Statistics of shared components in complex component systems

Many complex systems are modular. Such systems can be represented as "component systems", i.e., sets of elementary components, such as LEGO bricks in LEGO sets. The bricks found in a LEGO set reflect a target architecture, which can be built following a set-specific list of instructions. In other component systems, instead, the underlying functional design and constraints are not obvious a priori, and their detection is often a challenge of both scientific and practical importance, requiring a clear understanding of component statistics. Importantly, some quantitative invariants appear to be common to many component systems, most notably a common broad distribution of component abundances, which often resembles the well-known Zipf's law. Such "laws" affect in a general and non-trivial way the component statistics, potentially hindering the identification of system-specific functional constraints or generative processes. Here, we specifically focus on the statistics of shared components, i.e., the distribution of the number of components shared by different system-realizations, such as the common bricks found in different LEGO sets. To account for the effects of component heterogeneity, we consider a simple null model, which builds system-realizations by random draws from a universe of possible components. Under general assumptions on abundance heterogeneity, we provide analytical estimates of component occurrence, which quantify exhaustively the statistics of shared components. Surprisingly, this simple null model can positively explain important features of empirical component-occurrence distributions obtained from data on bacterial genomes, LEGO sets, and book chapters. Specific architectural features and functional constraints can be detected from occurrence patterns as deviations from these null predictions, as we show for the illustrative case of the "core" genome in bacteria.Comment: 18 pages, 7 main figures, 7 supplementary figure

Empirical analysis of countervailing power in business-to-business bargaining

This paper provides a comprehensive econometric framework for the empirical analysis of countervailing power. It encompasses the two main features of pricing schemes in business-to-business relationships: nonlinear price schedules and bargaining over rents. Disentangling them is critical to the empirical identification of countervailing power. Testable predictions from the theoretical analysis for a pragmatic reduced form empirical pricing model are delineated. This model is readily implementable on the basis of transaction data, routinely collected by antitrust authorities and illustrated using data from the UK brick industry. The paper emphasizes the importance of controlling for endogeneity of volumes and established supply chains and for heterogeneity across buyers and sellers due to intrinsically unobservable outside options