74 research outputs found
Perron-based algorithms for the multilinear pagerank
We consider the multilinear pagerank problem studied in [Gleich, Lim and Yu,
Multilinear Pagerank, 2015], which is a system of quadratic equations with
stochasticity and nonnegativity constraints. We use the theory of quadratic
vector equations to prove several properties of its solutions and suggest new
numerical algorithms. In particular, we prove the existence of a certain
minimal solution, which does not always coincide with the stochastic one that
is required by the problem. We use an interpretation of the solution as a
Perron eigenvector to devise new fixed-point algorithms for its computation,
and pair them with a homotopy continuation strategy. The resulting numerical
method is more reliable than the existing alternatives, being able to solve a
larger number of problems
Convergence of Tomlin's HOTS algorithm
The HOTS algorithm uses the hyperlink structure of the web to compute a
vector of scores with which one can rank web pages. The HOTS vector is the
vector of the exponentials of the dual variables of an optimal flow problem
(the "temperature" of each page). The flow represents an optimal distribution
of web surfers on the web graph in the sense of entropy maximization.
In this paper, we prove the convergence of Tomlin's HOTS algorithm. We first
study a simplified version of the algorithm, which is a fixed point scaling
algorithm designed to solve the matrix balancing problem for nonnegative
irreducible matrices. The proof of convergence is general (nonlinear
Perron-Frobenius theory) and applies to a family of deformations of HOTS. Then,
we address the effective HOTS algorithm, designed by Tomlin for the ranking of
web pages. The model is a network entropy maximization problem generalizing
matrix balancing. We show that, under mild assumptions, the HOTS algorithm
converges with a linear convergence rate. The proof relies on a uniqueness
property of the fixed point and on the existence of a Lyapunov function.
We also show that the coordinate descent algorithm can be used to find the
ideal and effective HOTS vectors and we compare HOTS and coordinate descent on
fragments of the web graph. Our numerical experiments suggest that the
convergence rate of the HOTS algorithm may deteriorate when the size of the
input increases. We thus give a normalized version of HOTS with an
experimentally better convergence rate.Comment: 21 page
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