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In this paper we consider so-called Google matrices and show that all eigenvalues ($lambda$) of them have a fundamental property $|lambda|leq 1$. The stochastic eigenvector corresponding to $lambda=1$ called the PageRank vector plays a central role in the Googles software. We study it in detail and present some important problems. The purpose of the paper is to make {bf the heart of Google} clearer for undergraduates.
We construct the Google matrix of the entire Twitter network, dated by July 2009, and analyze its spectrum and eigenstate properties including the PageRank and CheiRank vectors and 2DRanking of all nodes. Our studies show much stronger inter-connecti
A distinguishing property of communities in networks is that cycles are more prevalent within communities than across communities. Thus, the detection of these communities may be aided through the incorporation of measures of the local richness of th
Segregation is the separation of social groups in the physical or in the online world. Segregation discovery consists of finding contexts of segregation. In the modern digital society, discovering segregation is challenging, due to the large amount a
Massive amounts of misinformation have been observed to spread in uncontrolled fashion across social media. Examples include rumors, hoaxes, fake news, and conspiracy theories. At the same time, several journalistic organizations devote significant e
Measuring the geographical distribution of economic activity plays a key role in scientific research and policymaking. However, previous studies and data on economic activity either have a coarse spatial resolution or cover a limited time span, and t