Google matrix analysis of directed networks Ermann, Leonardo; Frahm, Klaus M.; Shepelyansky, Dima L.
Reviews of modern physics,
11/2015, Letnik:
87, Številka:
4
Journal Article
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In the past decade modern societies have developed enormous communication and social networks. Their classification and information retrieval processing has become a formidable task for the society. ...Because of the rapid growth of the World Wide Web, and social and communication networks, new mathematical methods have been invented to characterize the properties of these networks in a more detailed and precise way. Various search engines extensively use such methods. It is highly important to develop new tools to classify and rank a massive amount of network information in a way that is adapted to internal network structures and characteristics. This review describes the Google matrix analysis of directed complex networks demonstrating its efficiency using various examples including the World Wide Web, Wikipedia, software architectures, world trade, social and citation networks, brain neural networks, DNA sequences, and Ulam networks. The analytical and numerical matrix methods used in this analysis originate from the fields of Markov chains, quantum chaos, and random matrix theory.
We consider a system of linear oscillators, or quantum states, described by Random Matrix Theory and analyze how its time evolution is affected by a nonlinear perturbation. Our numerical results show ...that above a certain chaos border a weak or moderate nonlinearity leads to a dynamical thermalization of a finite number of degrees of freedom with energy equipartition over linear eigenmodes as expected from the laws of classical statistical mechanics. The system temperature is shown to change in a broad range from positive to negative values and the dependence of system characteristics on the initial injected energy is determined. Below the chaos border the dynamics is described by the Kolmogorov-Arnold-Moser integrability. Due to universal features of Random Matrix Theory we argue that the obtained results describe the generic properties of its nonlinear perturbation.
Wikipedia is a huge global repository of human knowledge that can be leveraged to investigate interwinements between cultures. With this aim, we apply methods of Markov chains and Google matrix for ...the analysis of the hyperlink networks of 24 Wikipedia language editions, and rank all their articles by PageRank, 2DRank and CheiRank algorithms. Using automatic extraction of people names, we obtain the top 100 historical figures, for each edition and for each algorithm. We investigate their spatial, temporal, and gender distributions in dependence of their cultural origins. Our study demonstrates not only the existence of skewness with local figures, mainly recognized only in their own cultures, but also the existence of global historical figures appearing in a large number of editions. By determining the birth time and place of these persons, we perform an analysis of the evolution of such figures through 35 centuries of human history for each language, thus recovering interactions and entanglement of cultures over time. We also obtain the distributions of historical figures over world countries, highlighting geographical aspects of cross-cultural links. Considering historical figures who appear in multiple editions as interactions between cultures, we construct a network of cultures and identify the most influential cultures according to this network.
Signaling pathways represent parts of the global biological molecular network which connects them into a seamless whole through complex direct and indirect (hidden) crosstalk whose structure can ...change during development or in pathological conditions. We suggest a novel methodology, called Googlomics, for the structural analysis of directed biological networks using spectral analysis of their Google matrices, using parallels with quantum scattering theory, developed for nuclear and mesoscopic physics and quantum chaos. We introduce analytical "reduced Google matrix" method for the analysis of biological network structure. The method allows inferring hidden causal relations between the members of a signaling pathway or a functionally related group of genes. We investigate how the structure of hidden causal relations can be reprogrammed as a result of changes in the transcriptional network layer during cancerogenesis. The suggested Googlomics approach rigorously characterizes complex systemic changes in the wiring of large causal biological networks in a computationally efficient way.
In a dissipative regime, we study the properties of several qubits coupled to a driven resonator in the framework of a Jaynes-Cummings model. The time evolution and the steady state of the system are ...numerically analyzed within the Lindblad master equation, with up to several million components. Two semi-analytical approaches, at weak and strong (semiclassical) dissipations, are developed to describe the steady state of this system and determine its validity by comparing it with the Lindblad equation results. We show that the synchronization of several qubits with the driving phase can be obtained due to their coupling to the resonator. We establish the existence of two different qubit synchronization regimes: In the first one, the semiclassical approach describes well the dynamics of qubits and, thus, their quantum features and entanglement are suppressed by dissipation and the synchronization is essentially classical. In the second one, the entangled steady state of a pair of qubits remains synchronized in the presence of dissipation and decoherence, corresponding to the regime non-existent in classical synchronization.
We study numerically, in the framework of the Cooper approach from 1956, mechanisms of pair formation in a model of La-based cuprate superconductors with longer-ranged hopping parameters reported in ...the literature at different values of center of mass momentum. An efficient numerical method allows to study lattices with more than a million sites. We consider the cases of attractive Hubbard and d-wave type interactions and a repulsive Coulomb interaction. The approach based on a frozen Fermi sea leads to a complex structure of accessible relative momentum states which is very sensitive to the total pair momentum of static or mobile pairs. It is found that interactions with attraction of approximately half of an electronvolt give a satisfactory agreement with experimentally reported results for the critical superconducting temperature and its dependence on hole doping. Ground states exhibit d-wave symmetries for both attractive Hubbard and d-wave interactions which is essentially due to the particular Fermi surface structure and not entirely to an eventual d-wave symmetry of the interaction. We also find pair states created by Coulomb repulsion at excited energies above the Fermi energy and determine the different mechanisms of their formation. In particular, we identify such pairs in a region of negative mass at rather modest excitation energies which is due to a particular band structure.
Graphic Abstract
In the frame of Cooper approach, the pair formation is studied numerically in La-based cuprate superconductors. The approach based on a frozen Fermi sea leads to a complex Fermi surface structure for static and mobile pairs. The attractive interactions (Hubbard or d-wave) of about 0.5 eV give a satisfactory agreement with experimental results for the critical superconducting temperature and its dependence on hole doping (shown in Figure by dashed and color curves for experiment and numerics). There are also pairs created by Coulomb repulsion at excited energies.
Protein-protein interactions is a longstanding challenge in cardiac remodeling processes and heart failure. Here, we use the MetaCore network and the Google matrix algorithms for prediction of ...protein-protein interactions dictating cardiac fibrosis, a primary cause of end-stage heart failure. The developed algorithms allow identification of interactions between key proteins and predict new actors orchestrating fibroblast activation linked to fibrosis in mouse and human tissues. These data hold great promise for uncovering new therapeutic targets to limit myocardial fibrosis.
We apply the Google matrix algorithms for analysis of interactions and influence of 37 cancer types, 203 cancer drugs and 195 world countries using the network of 5 416 537 English Wikipedia articles ...with all their directed hyperlinks. The PageRank algorithm provides a ranking of cancers which has 60% and 70% overlaps with the top 10 deadliest cancers extracted from World Health Organization GLOBOCAN 2018 and Global Burden of Diseases Study 2017, respectively. The recently developed reduced Google matrix algorithm gives networks of interactions between cancers, drugs and countries taking into account all direct and indirect links between these selected 435 entities. These reduced networks allow to obtain sensitivity of countries to specific cancers and drugs. The strongest links between cancers and drugs are in good agreement with the approved medical prescriptions of specific drugs to specific cancers. We argue that this analysis of knowledge accumulated in Wikipedia provides useful complementary global information about interdependencies between cancers, drugs and world countries.
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