17 research outputs found
Fast matrix computations for pair-wise and column-wise commute times and Katz scores
We first explore methods for approximating the commute time and Katz score
between a pair of nodes. These methods are based on the approach of matrices,
moments, and quadrature developed in the numerical linear algebra community.
They rely on the Lanczos process and provide upper and lower bounds on an
estimate of the pair-wise scores. We also explore methods to approximate the
commute times and Katz scores from a node to all other nodes in the graph.
Here, our approach for the commute times is based on a variation of the
conjugate gradient algorithm, and it provides an estimate of all the diagonals
of the inverse of a matrix. Our technique for the Katz scores is based on
exploiting an empirical localization property of the Katz matrix. We adopt
algorithms used for personalized PageRank computing to these Katz scores and
theoretically show that this approach is convergent. We evaluate these methods
on 17 real world graphs ranging in size from 1000 to 1,000,000 nodes. Our
results show that our pair-wise commute time method and column-wise Katz
algorithm both have attractive theoretical properties and empirical
performance.Comment: 35 pages, journal version of
http://dx.doi.org/10.1007/978-3-642-18009-5_13 which has been submitted for
publication. Please see
http://www.cs.purdue.edu/homes/dgleich/publications/2011/codes/fast-katz/ for
supplemental code
Random Walks: A Review of Algorithms and Applications
A random walk is known as a random process which describes a path including a
succession of random steps in the mathematical space. It has increasingly been
popular in various disciplines such as mathematics and computer science.
Furthermore, in quantum mechanics, quantum walks can be regarded as quantum
analogues of classical random walks. Classical random walks and quantum walks
can be used to calculate the proximity between nodes and extract the topology
in the network. Various random walk related models can be applied in different
fields, which is of great significance to downstream tasks such as link
prediction, recommendation, computer vision, semi-supervised learning, and
network embedding. In this paper, we aim to provide a comprehensive review of
classical random walks and quantum walks. We first review the knowledge of
classical random walks and quantum walks, including basic concepts and some
typical algorithms. We also compare the algorithms based on quantum walks and
classical random walks from the perspective of time complexity. Then we
introduce their applications in the field of computer science. Finally we
discuss the open issues from the perspectives of efficiency, main-memory
volume, and computing time of existing algorithms. This study aims to
contribute to this growing area of research by exploring random walks and
quantum walks together.Comment: 13 pages, 4 figure
From random walks to distances on unweighted graphs
Large unweighted directed graphs are commonly used to capture relations
between entities. A fundamental problem in the analysis of such networks is to
properly define the similarity or dissimilarity between any two vertices.
Despite the significance of this problem, statistical characterization of the
proposed metrics has been limited. We introduce and develop a class of
techniques for analyzing random walks on graphs using stochastic calculus.
Using these techniques we generalize results on the degeneracy of hitting times
and analyze a metric based on the Laplace transformed hitting time (LTHT). The
metric serves as a natural, provably well-behaved alternative to the expected
hitting time. We establish a general correspondence between hitting times of
the Brownian motion and analogous hitting times on the graph. We show that the
LTHT is consistent with respect to the underlying metric of a geometric graph,
preserves clustering tendency, and remains robust against random addition of
non-geometric edges. Tests on simulated and real-world data show that the LTHT
matches theoretical predictions and outperforms alternatives.Comment: To appear in NIPS 201