4 research outputs found
HodgeRank is the limit of Perron Rank
We study the map which takes an elementwise positive matrix to the k-th root
of the principal eigenvector of its k-th Hadamard power. We show that as
tends to 0 one recovers the row geometric mean vector and discuss the geometric
significance of this convergence. In the context of pairwise comparison
ranking, our result states that HodgeRank is the limit of Perron Rank, thereby
providing a novel mathematical link between two important pairwise ranking
methods
Enumerating Polytropes
Polytropes are both ordinary and tropical polytopes. We show that tropical
types of polytropes in are in bijection with cones of a
certain Gr\"{o}bner fan in restricted
to a small cone called the polytrope region. These in turn are indexed by
compatible sets of bipartite and triangle binomials. Geometrically, on the
polytrope region, is the refinement of two fans: the fan of
linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite
binomial fan. This gives two algorithms for enumerating tropical types of
polytropes: one via a general Gr\"obner fan software such as \textsf{gfan}, and
another via checking compatibility of systems of bipartite and triangle
binomials. We use these algorithms to compute types of full-dimensional
polytropes for , and maximal polytropes for .Comment: Improved exposition, fixed error in reporting the number maximal
polytropes for , fixed error in definition of bipartite binomial