Quantum Google in a Complex Network

TL;DR

This paper explores the quantum PageRank algorithm's ability to analyze large complex networks, revealing their structure and node importance more effectively than classical methods, with implications for network analysis and robustness.

Contribution

It demonstrates the quantum PageRank's enhanced capability to identify network topology, hierarchy, and node importance, outperforming classical PageRank in various network types.

Findings

Quantum PageRank reveals scale-free topology.

It improves identification of secondary hubs.

It distinguishes different network classes effectively.

Abstract

We investigate the behavior of the recently proposed quantum Google algorithm, or quantum PageRank, in large complex networks. Applying the quantum algorithm to a part of the real World Wide Web, we find that the algorithm is able to univocally reveal the underlying scale-free topology of the network and to clearly identify and order the most relevant nodes (hubs) of the graph according to their importance in the network structure. Moreover, our results show that the quantum PageRank algorithm generically leads to changes in the hierarchy of nodes. In addition, as compared to its classical counterpart, the quantum algorithm is capable to clearly highlight the structure of secondary hubs of the network, and to partially resolve the degeneracy in importance of the low lying part of the list of rankings, which represents a typical shortcoming of the classical PageRank algorithm. Complementary to this study, our analysis shows that the algorithm is able to clearly distinguish scale-free networks from other widespread and important classes of complex networks, such as Erd\H{o}s-R\'enyi networks and hierarchical graphs. We show that the ranking capabilities of the quantum PageRank algorithm are related to an increased stability with respect to a variation of the damping parameter $\alpha$ that appears in the Google algorithm, and to a more clearly pronounced power-law behavior in the distribution of importance among the nodes, as compared to the classical algorithm. Finally, we study to which extent the increased sensitivity of the quantum algorithm persists under coordinated attacks of the most important nodes in scale-free and Erd\H{o}s-R\'enyi random graphs.