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arXiv:1202.3471 · Scientific Reports

Quantum Navigation and Ranking in Complex Networks

Eduardo Sánchez-Burillo, Jordi Duch, Jesús Gómez‐Gardeñes, David Zueco

Published Aug 28, 2012• 115 citations
arXivHTMLPDF

Abstract

Complex networks are formal frameworks capturing the interdependencies between the elements of large systems and databases. This formalism allows to use network navigation methods to rank the importance that each constituent has on the global organization of the system. A key example is Pagerank navigation which is at the core of the most used search engine of the World Wide Web. Inspired in this classical algorithm, we define a quantum navigation method providing a unique ranking of the elements of a network. We analyze the convergence of quantum navigation to the stationary rank of networks and show that quantumness decreases the number of navigation steps before convergence. In addition, we show that quantum navigation allows to solve degeneracies found in classical ranks. By implementing the quantum algorithm in real networks, we confirm these improvements and show that quantum coherence unveils new hierarchical features about the global organization of complex systems.

DOI
10.1038/srep00605

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Complex networks are formal frameworks capturing the interdependencies between the elements of large systems and databases. This formalism allows to use network navigation methods to rank the importance that each constituent has on the global organization of the system. A key example is Pagerank navigation which is at the core of the most used search engine of the World Wide Web. Inspired in this classical algorithm, we define a quantum navigation method providing a unique ranking of the elements of a network. We analyze the convergence of quantum navigation to the stationary rank of networks and show that quantumness decreases the number of navigation steps before convergence. In addition, we show that quantum navigation allows to solve degeneracies found in classical ranks. By implementing the quantum algorithm in real networks, we confirm these improvements and show that quantum coherence unveils new hierarchical features about the global organization of complex systems.
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We investigate the behaviour of the recently proposed Quantum PageRank algorithm, in large complex networks. We find that the algorithm is able to univocally reveal the underlying topology of the network and to identify and order the most relevant nodes. Furthermore, it is capable to clearly highlight the structure of secondary hubs and to resolve the degeneracy in importance of the low lying part of the list of rankings. The quantum algorithm displays an increased stability with respect to a variation of the damping parameter, present in the Google algorithm, and a more clearly pronounced power-law behaviour in the distribution of importance, as compared to the classical algorithm. We test the performance and confirm the listed features by applying it to real world examples from the WWW. Finally, we raise and partially address whether the increased sensitivity of the quantum algorithm persists under coordinated attacks in scale-free and random networks.
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The Google's PageRank algorithm is one of the most famous algorithms in data mining. A quantum version of this algorithm has been proposed in 2012. It is a meaningful step towards the quantum search engine but it can not search solutions. In this paper, we integrate a special operator of quantum search into the quantum PageRank algorithm to make a new quantum algorithm. We called it the Search Rank algorithm. This new algorithm is able to search solutions and rank results according to their importance at the same time. It is the first algorithm possessing this ability as far as we know. And it outperforms all classical algorithms when searching an unsorted database.
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HodgeRank generalizes ranking algorithms, e.g. Google PageRank, to rank alternatives based on real-world (often incomplete) data using graphs and discrete exterior calculus. It analyzes multipartite interactions on high-dimensional networks with a complexity that scales exponentially with dimension. We develop a quantum algorithm that approximates the HodgeRank solution with complexity independent of dimension. Our algorithm extracts relevant information from the state such as the ranking consistency, which achieves a superpolynomial speedup over similar classical methods.