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

Reducing the Effects of Unequal Number of Games on Rankings

Ranking is an important mathematical process in a variety of contexts such as information retrieval, sports and business. Sports ranking methods can be applied both in and beyond the context of athletics. In both settings, once the concept of a game has been defined, teams (or individuals) accumulate wins, losses, and ties, which are then factored into the ranking computation. Many settings involve an unequal number of games between competitors. This paper demonstrates how to adapt two sports rankings methods, the Colley and Massey ranking methods, to settings where an unequal number of games are played between the teams. In such settings, the standard derivations of the methods can produce nonsensical rankings. This paper introduces the idea of including a super-user into the rankings and considers the effect of this fictitious player on the ratings. We apply such techniques to rank batters and pitchers in Major League baseball, professional tennis players, and participants in a free online social game. The ideas introduced in this paper can further the scope that such methods are applied and the depth of insight they offer

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

Dynamics-based centrality for general directed networks

Determining the relative importance of nodes in directed networks is important in, for example, ranking websites, publications, and sports teams, and for understanding signal flows in systems biology. A prevailing centrality measure in this respect is the PageRank. In this work, we focus on another class of centrality derived from the Laplacian of the network. We extend the Laplacian-based centrality, which has mainly been applied to strongly connected networks, to the case of general directed networks such that we can quantitatively compare arbitrary nodes. Toward this end, we adopt the idea used in the PageRank to introduce global connectivity between all the pairs of nodes with a certain strength. Numerical simulations are carried out on some networks. We also offer interpretations of the Laplacian-based centrality for general directed networks in terms of various dynamical and structural properties of networks. Importantly, the Laplacian-based centrality defined as the stationary density of the continuous-time random walk with random jumps is shown to be equivalent to the absorption probability of the random walk with sinks at each node but without random jumps. Similarly, the proposed centrality represents the importance of nodes in dynamics on the original network supplied with sinks but not with random jumps.Comment: 7 figure

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 $\log(n)$ 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