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 $k$ 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.