We study the lattice width of lattice-free polyhedra given by ${A}{x}\leq{b}$ in terms of $\Delta({A})$, the maximal $n\times n$ minor in absolute value of ${A}\in\mathbb{Z}^{m\times n}$. Our main contribution is to link the lattice width of lattice-free polyhedra to the height of Hilbert bases and to the diameter of finite abelian groups. This leads to a bound on the lattice width of lattice-free pyramids which solely depends on $\Delta({A})$ provided a conjecture regarding the height of Hilbert bases holds. Further, we exploit a combination of techniques to obtain novel bounds on the lattice width of simplices. A second part of the paper is devoted to a study of the above-mentioned Hilbert basis conjecture. We give a complete characterization of the Hilbert basis if $\Delta({A}) = 2$ which implies the conjecture in that case and prove its validity for simplicial cones.

In this paper we make the case that software dependencies are a form of innovation adoption. We then test this on the time-evolution of the Gentoo package dependency graph. We find that the Bass model of innovation diffusion fits the growth of the number of packages depending on a given library. Interestingly, we also find that low-level packages have a primarily imitation driven adoption and multimedia libraries have primarily innovation driven growth.

Open source distributions such as Gentoo need to accurately track dependency relations between software packages in order to install working systems. To do this, Gentoo has a carefully authored database containing those relations. In this paper, we extract the Gentoo package dependency graph and its changes over time. The final dependency graph spans 15 thousand open source projects and 80 thousand dependency relations. Furthermore, the development of this graph is tracked over time from the beginning of the Gentoo project in 2000 to the first quarter of 2012, with monthly resolution. The resulting dataset provides many opportunities for research. In this paper we explore cluster analysis to reveals meaningful relations between packages and in a separate paper we analyze changes in the dependencies over time to get insights in the innovation dynamics of open source software.