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Estimating and Sampling Graphs with Multidimensional Random Walks
Estimating characteristics of large graphs via sampling is a vital part of
the study of complex networks. Current sampling methods such as (independent)
random vertex and random walks are useful but have drawbacks. Random vertex
sampling may require too many resources (time, bandwidth, or money). Random
walks, which normally require fewer resources per sample, can suffer from large
estimation errors in the presence of disconnected or loosely connected graphs.
In this work we propose a new -dimensional random walk that uses
dependent random walkers. We show that the proposed sampling method, which we
call Frontier sampling, exhibits all of the nice sampling properties of a
regular random walk. At the same time, our simulations over large real world
graphs show that, in the presence of disconnected or loosely connected
components, Frontier sampling exhibits lower estimation errors than regular
random walks. We also show that Frontier sampling is more suitable than random
vertex sampling to sample the tail of the degree distribution of the graph
Localization of discrete time quantum walks on the glued trees
In this paper, we consider the time averaged distribution of discrete time
quantum walks on the glued trees. In order to analyse the walks on the glued
trees, we consider a reduction to the walks on path graphs. Using a spectral
analysis of the Jacobi matrices defined by the corresponding random walks on
the path graphs, we have spectral decomposition of the time evolution operator
of the quantum walks. We find significant contributions of the eigenvalues of the Jacobi matrices to the time averaged limit distribution of the
quantum walks. As a consequence we obtain lower bounds of the time averaged
distribution.Comment: 10 page
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