72 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
A multi-class approach for ranking graph nodes: models and experiments with incomplete data
After the phenomenal success of the PageRank algorithm, many researchers have
extended the PageRank approach to ranking graphs with richer structures beside
the simple linkage structure. In some scenarios we have to deal with
multi-parameters data where each node has additional features and there are
relationships between such features.
This paper stems from the need of a systematic approach when dealing with
multi-parameter data. We propose models and ranking algorithms which can be
used with little adjustments for a large variety of networks (bibliographic
data, patent data, twitter and social data, healthcare data). In this paper we
focus on several aspects which have not been addressed in the literature: (1)
we propose different models for ranking multi-parameters data and a class of
numerical algorithms for efficiently computing the ranking score of such
models, (2) by analyzing the stability and convergence properties of the
numerical schemes we tune a fast and stable technique for the ranking problem,
(3) we consider the issue of the robustness of our models when data are
incomplete. The comparison of the rank on the incomplete data with the rank on
the full structure shows that our models compute consistent rankings whose
correlation is up to 60% when just 10% of the links of the attributes are
maintained suggesting the suitability of our model also when the data are
incomplete
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
A note on certain ergodicity coefficients
We investigate two ergodicity coefficients and ,
originally introduced to bound the subdominant eigenvalues of nonnegative
matrices.
The former has been generalized to complex matrices in recent years and
several properties for such generalized version have been shown so far.
We provide a further result concerning the limit of its powers. Then we
propose a generalization of the second coefficient and we show
that, under mild conditions, it can be used to recast the eigenvector problem
as a particular M-matrix linear system, whose coefficient matrix can be
defined in terms of the entries of . Such property turns out to generalize
the two known equivalent formulations of the Pagerank centrality of a graph
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