98 research outputs found
Delocalization transition for the Google matrix
We study the localization properties of eigenvectors of the Google matrix,
generated both from the World Wide Web and from the Albert-Barabasi model of
networks. We establish the emergence of a delocalization phase for the PageRank
vector when network parameters are changed. In the phase of localized PageRank,
a delocalization takes place in the complex plane of eigenvalues of the matrix,
leading to delocalized relaxation modes. We argue that the efficiency of
information retrieval by Google-type search is strongly affected in the phase
of delocalized PageRank.Comment: 4 pages, 5 figures. Research done at
http://www.quantware.ups-tlse.fr
Influence, originality and similarity in directed acyclic graphs
We introduce a framework for network analysis based on random walks on
directed acyclic graphs where the probability of passing through a given node
is the key ingredient. We illustrate its use in evaluating the mutual influence
of nodes and discovering seminal papers in a citation network. We further
introduce a new similarity metric and test it in a simple personalized
recommendation process. This metric's performance is comparable to that of
classical similarity metrics, thus further supporting the validity of our
framework.Comment: 6 pages, 4 figure
Thermodynamic formalism for dissipative quantum walks
We consider the dynamical properties of dissipative continuous-time quantum
walks on directed graphs. Using a large-deviation approach we construct a
thermodynamic formalism allowing us to define a dynamical order parameter, and
to identify transitions between dynamical regimes. For a particular class of
dissipative quantum walks we propose a quantum generalization of the the
classical PageRank vector, used to rank the importance of nodes in a directed
graph. We also provide an example where one can characterize the dynamical
transition from an effective classical random walk to a dissipative quantum
walk as a thermodynamic crossover between distinct dynamical regimes.Comment: 8 page
Adiabatic quantum algorithm for search engine ranking
We propose an adiabatic quantum algorithm for generating a quantum pure state
encoding of the PageRank vector, the most widely used tool in ranking the
relative importance of internet pages. We present extensive numerical
simulations which provide evidence that this algorithm can prepare the quantum
PageRank state in a time which, on average, scales polylogarithmically in the
number of webpages. We argue that the main topological feature of the
underlying web graph allowing for such a scaling is the out-degree
distribution. The top ranked entries of the quantum PageRank state
can then be estimated with a polynomial quantum speedup. Moreover, the quantum
PageRank state can be used in "q-sampling" protocols for testing properties of
distributions, which require exponentially fewer measurements than all
classical schemes designed for the same task. This can be used to decide
whether to run a classical update of the PageRank.Comment: 7 pages, 5 figures; closer to published versio
Ranking Spaces for Predicting Human Movement in an Urban Environment
A city can be topologically represented as a connectivity graph, consisting
of nodes representing individual spaces and links if the corresponding spaces
are intersected. It turns out in the space syntax literature that some defined
topological metrics can capture human movement rates in individual spaces. In
other words, the topological metrics are significantly correlated to human
movement rates, and individual spaces can be ranked by the metrics for
predicting human movement. However, this correlation has never been well
justified. In this paper, we study the same issue by applying the weighted
PageRank algorithm to the connectivity graph or space-space topology for
ranking the individual spaces, and find surprisingly that (1) the PageRank
scores are better correlated to human movement rates than the space syntax
metrics, and (2) the underlying space-space topology demonstrates small world
and scale free properties. The findings provide a novel justification as to why
space syntax, or topological analysis in general, can be used to predict human
movement. We further conjecture that this kind of analysis is no more than
predicting a drunkard's walking on a small world and scale free network.
Keywords: Space syntax, topological analysis of networks, small world, scale
free, human movement, and PageRankComment: 11 pages, 5 figures, and 2 tables, English corrections from version 1
to version 2, major changes in the section of introduction from version 2 to
Schroedinger-like PageRank equation and localization in the WWW
The WorldWide Web is one of the most important communication systems we use
in our everyday life. Despite its central role, the growth and the development
of the WWW is not controlled by any central authority. This situation has
created a huge ensemble of connections whose complexity can be fruitfully
described and quantified by network theory. One important application that
allows to sort out the information present in these connections is given by the
PageRank alghorithm. Computation of this quantity is usually made iteratively
with a large use of computational time. In this paper we show that the PageRank
can be expressed in terms of a wave function obeying a Schroedinger-like
equation. In particular the topological disorder given by the unbalance of
outgoing and ingoing links between pages, induces wave function and potential
structuring. This allows to directly localize the pages with the largest score.
Through this new representation we can now compute the PageRank without
iterative techniques. For most of the cases of interest our method is faster
than the original one. Our results also clarify the role of topology in the
diffusion of information within complex networks. The whole approach opens the
possibility to novel techniques inspired by quantum physics for the analysis of
the WWW properties.Comment: 5 page
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs
Transport through generalized trees is considered. Trees contain the simple
nodes and supernodes, either well-structured regular subgraphs or those with
many triangles. We observe a superdiffusion for the highly connected nodes
while it is Brownian for the rest of the nodes. Transport within a supernode is
affected by the finite size effects vanishing as For the even
dimensions of space, , the finite size effects break down the
perturbation theory at small scales and can be regularized by using the
heat-kernel expansion.Comment: 21 pages, 2 figures include
Googling the brain: discovering hierarchical and asymmetric network structures, with applications in neuroscience
Hierarchical organisation is a common feature of many directed networks arising in nature and technology. For example, a well-defined message-passing framework based on managerial status typically exists in a business organisation. However, in many real-world networks such patterns of hierarchy are unlikely to be quite so transparent. Due to the nature in which empirical data is collated the nodes will often be ordered so as to obscure any underlying structure. In addition, the possibility of even a small number of links violating any overall “chain of command” makes the determination of such structures extremely challenging. Here we address the issue of how to reorder a directed network in order to reveal this type of hierarchy. In doing so we also look at the task of quantifying the level of hierarchy, given a particular node ordering. We look at a variety of approaches. Using ideas from the graph Laplacian literature, we show that a relevant discrete optimization problem leads to a natural hierarchical node ranking. We also show that this ranking arises via a maximum likelihood problem associated with a new range-dependent hierarchical random graph model. This random graph insight allows us to compute a likelihood ratio that quantifies the overall tendency for a given network to be hierarchical. We also develop a generalization of this node ordering algorithm based on the combinatorics of directed walks. In passing, we note that Google’s PageRank algorithm tackles a closely related problem, and may also be motivated from a combinatoric, walk-counting viewpoint. We illustrate the performance of the resulting algorithms on synthetic network data, and on a real-world network from neuroscience where results may be validated biologically
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