Scott Elliott Fahlman (born March 21, 1948) is an American computer scientist and Professor Emeritus at Carnegie Mellon University's Language Technologies Institute and Computer Science Department. He is notable for early work on automated planning and scheduling in a blocks world, on semantic networks, on neural networks (especially the cascade correlation algorithm), on the programming languages Dylan, and Common Lisp (especially CMU Common Lisp), and he was one of the founders of Lucid Inc. During the period when it was standardized, he was recognized as "the leader of Common Lisp." From 2006 to 2015, Fahlman was engaged in developing a knowledge base named Scone, based in part on his thesis work on the NETL Semantic Network. He also is credited with coining the use of the emoticon. == Life and career == Fahlman was born in Medina, Ohio, the son of Lorna May (Dean) and John Emil Fahlman. He attended the Massachusetts Institute of Technology (MIT), where he received a Bachelor of Science (B.S.) and Master of Science (M.S.) degree in electrical engineering and computer science in 1973, and a Doctor of Philosophy (Ph.D.) in artificial intelligence in 1977. He has noted that his doctoral diploma says the degree was awarded for "original research as demonstrated by a thesis in the field of Artificial Intelligence" and suggested that it may be the first doctorate to use that term. He is a fellow of the American Association for Artificial Intelligence. Fahlman acted as thesis advisor for Donald Cohen, David B. McDonald, David S. Touretzky, Skef Wholey, Justin Boyan, Michael Witbrock, and Alicia Tribble Sagae. From May 1996 to July 2001, Fahlman directed the Justsystem Pittsburgh Research Center. === Boltzmann Machine (1983) === In 1983, Fahlman, Geoffrey Hinton, and Terry Sejnowski published a paper in Proceedings of the AAAI-83 Conference, Washington DC, August 1983. The paper was titled as "Massively Parallel Architectures for AI: NETL, Thistle and Boltzmann Machines". === Emoticons === Fahlman was not the first to suggest the concept of the emoticon – a similar concept for a marker appeared in an article of Reader's Digest in May 1967, although that idea was never put into practice. In an interview printed in The New York Times in 1969, Vladimir Nabokov noted: "I often think there should exist a special typographical sign for a smile – some sort of concave mark, a supine round bracket." Fahlman is credited with originating the first smiley emoticon, which he thought would help people on a message board at Carnegie Mellon to distinguish serious posts from jokes. He proposed the use of :-) and :-( for this purpose, and the symbols caught on. The original message from which these symbols originated was posted on 19 September 1982. The message was recovered by Jeff Baird on 10 September 2002 and read: 19-Sep-82 11:44 Scott E Fahlman :-) From: Scott E Fahlman
XLNet
The XLNet was an autoregressive Transformer designed as an improvement over BERT, with 340M parameters and trained on 33 billion words. It was released on 19 June 2019, under the Apache 2.0 license. It achieved state-of-the-art results on a variety of natural language processing tasks, including language modeling, question answering, and natural language inference. == Architecture == The main idea of XLNet is to model language autoregressively like the GPT models, but allow for all possible permutations of a sentence. Concretely, consider the following sentence:My dog is cute.In standard autoregressive language modeling, the model would be tasked with predicting the probability of each word, conditioned on the previous words as its context: We factorize the joint probability of a sequence of words x 1 , … , x T {\displaystyle x_{1},\ldots ,x_{T}} using the chain rule: Pr ( x 1 , … , x T ) = Pr ( x 1 ) Pr ( x 2 | x 1 ) Pr ( x 3 | x 1 , x 2 ) … Pr ( x T | x 1 , … , x T − 1 ) . {\displaystyle \Pr(x_{1},\ldots ,x_{T})=\Pr(x_{1})\Pr(x_{2}|x_{1})\Pr(x_{3}|x_{1},x_{2})\ldots \Pr(x_{T}|x_{1},\ldots ,x_{T-1}).} For example, the sentence "My dog is cute" is factorized as: Pr ( My , dog , is , cute ) = Pr ( My ) Pr ( dog | My ) Pr ( is | My , dog ) Pr ( cute | My , dog , is ) . {\displaystyle \Pr({\text{My}},{\text{dog}},{\text{is}},{\text{cute}})=\Pr({\text{My}})\Pr({\text{dog}}|{\text{My}})\Pr({\text{is}}|{\text{My}},{\text{dog}})\Pr({\text{cute}}|{\text{My}},{\text{dog}},{\text{is}}).} Schematically, we can write it as
AI Analytics Tools: Free vs Paid (2026)
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Rada Mihalcea
Rada Mihalcea is the Janice M. Jenkins Collegiate Professor of Computer Science and Engineering at the University of Michigan. She has made significant contributions to natural language processing, multimodal processing, computational social science, and AI for Social Good. With Paul Tarau, she invented the TextRank Algorithm, which is a classic algorithm widely used for text summarization. == Career == Mihalcea has a Ph.D. in Computer Science and Engineering from Southern Methodist University (2001) and a Ph.D. in Linguistics, Oxford University (2010). In 2017 she was named Director of the Artificial Intelligence Laboratory at University of Michigan, Computer Science and Engineering. In 2018, Mihalcea was elected as vice president for the Association for Computational Linguistics (ACL). In 2021, she was elected the president for ACL. She is a professor of Computer Science and Engineering at the University of Michigan, where she also leads the Language and Information Technologies (LIT) Lab. Before joining UofM, she was a professor at North Texas University between 2002-2013. A prolific researcher, Mihalcea has authored or coauthored over 500 articles since 1998 on topics ranging from semantic analysis of text to lie detection. Her work has been cited over 50,000 times on Google Scholar, which made her one of the most cited scholars in Multimodal Interaction and Computational Social Science. In 2008, Mihalcea received the Presidential Early Career Award for Scientists and Engineers (PECASE) She is an ACM Fellow (since 2019), AAAI Fellow (since 2021), and ACL Fellow (since 2025). Mihalcea is an outspoken promoter of diversity in computer science. She also supports an expansion of the traditional analysis of educational success, which tends to focus on academic behaviour, to include student life, personality and background outside of the classroom. Mihalcea leads Girls Encoded, a program designed to develop the pipeline of women in computer science as well as to retain the women who have entered into the program. == Awards == Elected to American Academy of Arts & Sciences, 2026 ACL Fellow, 2025 "for significant contributions to graph-based language processing, computational social science, and the advancement of NLP for social good." AAAI Fellow, 2021 "for significant contributions to natural language processing and computational social science". ACM Fellow, 2019 "for contributions to natural language processing, with innovations in data-driven and graph-based language processing". Sarah Goddard Power Award, 2019. Carol Hollenshead Award, 2018. Presidential Early Career Award for Scientists and Engineers (PECASE), 2009. Awarded by President Barack Obama. == Research == Mihalcea is known for her research in natural language processing, multimodal processing, computational social sciences. In a collaboration she leads at the University of Michigan, Mihalcea has created software that can detect human lying. In a study of video clips of high profile court cases, a computer was more accurate at detecting deception than human judges. Mihalcea's lie-detection software uses machine learning techniques to analyze video clips of actual trials. In her 2015 study, the team used clips from The Innocence Project, a national organization that works to reexamine cases where individuals were tried without the benefit of DNA testing with the aim of exonerating wrongfully convicted individuals. After identifying common human gestures, they transcribed the audio from the video clips of trials and analyzed how often subjects labeled deceptive used various words and phrases. The system was 75% accurate in identifying which subjects were deceptive among 120 videos. That puts Mihalcea's algorithm on par with the most commonly accepted form of lie detection, polygraph tests, which are roughly 85 percent accurate when testing guilty people and 56 percent accurate when testing the innocent. She notes there are still improvements to be made — in particular to account for cultural and demographic differences. A possibly unique advantage of Mihalcea's study was the real world, high stakes nature of the footage analyzed in the study. In laboratory experiments, it is difficult to create a setting that motivates people to truly lie. In 2018, Mihalcea and her collaborators worked on an algorithm-based system that identifies linguistic cues in fake news stories. It successfully found fakes up to 76% of the time, compared to a human success rate of 70%. == Publications == === Books === Rada Mihalcea and Dragomir Radev, Graph-based Natural Language Processing and Information Retrieval, Cambridge U. Press, 2011. Gabe Ignatow and Rada Mihalcea, Text Mining: A Guidebook for the Social Sciences, SAGE, 2016. Gabe Ignatow and Rada Mihalcea, An Introduction to Text Mining: Research Design, Data Collection, and Analysis, SAGE, 2017. === Journals and conferences === Textrank: Bringing order into text. R. Mihalcea, P. Tarau. Proceedings of the 2004 conference on empirical methods in natural language processing. 2004 Corpus-based and knowledge-based measures of text semantic similarity. R. Mihalcea, C. Corley, C. Strapparava. AAAI 6, 775-780. 2006 Wikify!: linking documents to encyclopedic knowledge. R. Mihalcea, A. Csomai. Proceedings of the sixteenth ACM conference on Conference on information and information management. 2007 Learning to identify emotions in text. C. Strapparava, R. Mihalcea. Proceedings of the 2008 ACM symposium on Applied computing, 1556-1560. 2008 Semeval-2007 task 14: Affective text. C. Strapparava, R. Mihalcea. Proceedings of the Fourth International Workshop on Semantic Evaluations. 2007 Learning multilingual subjective language via cross-lingual projections. R. Mihalcea, C. Banea, J. Wiebe. Proceedings of the 45th annual meeting of the association of computational linguistics. 2007 Graph-based ranking algorithms for sentence extraction, applied to text summarization. R. Mihalcea. Proceedings of the ACL Interactive Poster and Demonstration Sessions. 2004 Falcon: Boosting knowledge for answer engines. S. Harabagiu, D. Moldovan, M. Pasca, R. Mihalcea, M. Surdeanu, Razvan Bunescu, Roxana Girju, Vasile Rus, Paul Morarescu. TREC 9, 479-488. 2000 Measuring the semantic similarity of texts. C. Corley, R. Mihalcea. Proceedings of the ACL workshop on empirical modeling of semantic equivalence and entailment. 2005 R Mihalcea (2007). "Using wikipedia for automatic word-sense disambiguation". Human Language Technologies 2007: The Conference of the North American Chapter of the Association for Computational Linguistics; Proceedings of the Main Conference. CiteSeerX 10.1.1.74.3561. - see also Word-sense disambiguation Unsupervised graph-based word sense disambiguation using measures of word semantic similarity. R. Sinha, R. Mihalcea. International Conference on Semantic Computing (ICSC 2007), 363-369. 2007 == Personal life == Mihalcea was born in Cluj-Napoca, Romania, where she attended the Technical University of Cluj-Napoca. She can speak Romanian, English, Italian, and French. Mihalcea has two children - Zara (b. 2009) and Caius (b. 2013). They were both born in Dallas, Texas. She is married to an associate professor of engineering at the University of Michigan–Flint - Mihai Burzo. They met while they were both completing Ph.D.s at Southern Methodist University in 2001 and have often collaborated on research, such as the 2015 study on lie detection.
Thomas G. Dietterich
Thomas G. Dietterich is emeritus professor of computer science at Oregon State University. He is one of the pioneers of the field of machine learning. He served as executive editor of Machine Learning (journal) (1992–98) and helped co-found the Journal of Machine Learning Research. In response to the media's attention on the dangers of artificial intelligence, Dietterich has been quoted for an academic perspective to a broad range of media outlets including National Public Radio, Business Insider, Microsoft Research, CNET, and The Wall Street Journal. Among his research contributions were the invention of error-correcting output coding to multi-class classification, the formalization of the multiple-instance problem, the MAXQ framework for hierarchical reinforcement learning, and the development of methods for integrating non-parametric regression trees into probabilistic graphical models. == Biography and education == Thomas Dietterich was born in South Weymouth, Massachusetts, in 1954. His family later moved to New Jersey and then again to Illinois, where Tom graduated from Naperville Central High School. Dietterich then entered Oberlin College and began his undergraduate studies. In 1977, Dietterich graduated from Oberlin with a degree in mathematics, focusing on probability and statistics. Dietterich spent the following two years at the University of Illinois, Urbana-Champaign. After those two years, he began his doctoral studies in the Department of Computer Science at Stanford University. Dietterich received his Ph.D. in 1984 and moved to Corvallis, Oregon, where he was hired as an assistant professor in computer science. in 2013, he was named "Distinguished Professor". In 2016, Dietterich retired from his position at Oregon State University. Throughout his career, Dietterich has worked to promote scientific publication and conference presentations. For many years, he was the editor of the MIT Press series on Adaptive Computation and Machine Learning. He also held the position of co-editor of the Morgan Claypool Synthesis Series on Artificial Intelligence and Machine Learning. He has organized several conferences and workshops including serving as Technical Program Co-Chair of the National Conference on Artificial Intelligence (AAAI-90), Technical Program Chair of the Neural Information Processing Systems (NIPS-2000) and General Chair of NIPS-2001. He served as founding President of the International Machine Learning Society and he has been a member of the IMLS Board since its founding. He is currently also a member of the Steering Committee of the Asian Conference on Machine Learning. == Research interests == Professor Dietterich is interested in all aspects of machine learning. There are three major strands of his research. First, he is interested in the fundamental questions of artificial intelligence and how machine learning can provide the basis for building integrated intelligent systems. Second, he is interested in ways that people and computers can collaborate to solve challenging problems. And third, he is interested in applying machine learning to problems in the ecological sciences and ecosystem management as part of the emerging field of computational sustainability. Over his career, he has worked on a wide variety of problems ranging from drug design to user interfaces to computer security. His current focus is on ways that computer science methods can help advance ecological science and improve our management of the Earth's ecosystems. This passion has led to several projects including research in wildfire management, invasive vegetation and understanding the distribution and migration of birds. For example, Dietterich's research is helping scientists at the Cornell Lab of Ornithology answer questions like: How do birds decide to migrate north? How do they know when to land and stopover for a few days? How do they choose where to make a nest? Tens of thousands of volunteer birdwatchers (citizen scientists) all over the world contribute data to the study by submitting their bird sightings to the eBird website. The amount of data is overwhelming – in March 2012 they had over 3.1 million bird observations. Machine learning can uncover patterns in data to model the migration of species. But there are many other applications for the same techniques which will allow organizations to better manage our forests, oceans, and endangered species, as well as improve traffic flow, water systems, the electrical power grid, and more. I realized I wanted to have an impact on something that really mattered – and certainly the whole Earth's ecosystem, of which we are a part, is under threat in so many ways. And so if there's some way that I can use my technical skills to improve both the science base and the tools needed for policy and management decisions, then I would like to do that. I am passionate about that. == Dangers of AI: an academic perspective == Dietterich has argued that the most realistic risks about the dangers of artificial intelligence are basic mistakes, breakdowns and cyberattacks, and the fact that it simply may not always work, rather than machines that become super powerful or destroy the human race. Dietterich considers machines becoming self-aware and trying to exterminate humans to be more science fiction than scientific fact. But to the extent that computer systems are given increasingly dangerous tasks, and asked to learn from and interpret their experiences, he said they may simply make mistakes. Instead, much of the work done in the AI safety community does indeed focus around accidents and design flaws. == Positions held == 2014–2016: President, Association for the Advancement of Artificial Intelligence (AAAI). 2013–present: Distinguished Professor of computer science, Oregon State University. 2011–present: Chief Scientist, BigML, Corvallis, OR. 2005–present: Director of Intelligent Systems Research, School of Electrical Engineering and Computer Science, Oregon State University. 2006–2008: Chief Scientist, Smart Desktop, Inc., Seattle, WA. 2004–2005: Chief Scientist, MyStrands, Inc., Corvallis, OR. 1995-2013: Professor of computer science, Oregon State University. 1998–1999: Visiting Senior Scientist, Institute for the Investigation of Artificial Intelligence, Barcelona, Spain. (Sabbatical leave position) 1988–1995: Associate Professor of computer science, Oregon State University. 1991–1993: Senior Scientist, Arris Pharmaceutical Corporation, S. San Francisco, CA. 1985–1988: Assistant Professor of computer science, Oregon State University. 1979–1984: Research Assistant, Heuristic Programming Project, Department of Computer Science, Stanford University. 1979 (Summer): Member of Technical Staff, Bell Telephone Laboratories, Naperville, Illinois. Computer-to-computer file transfer and micro-code distribution to remote switching systems. 1977 (Summer): Assistant to the Director of Planning and Research, Oberlin College, Oberlin, Ohio. Developed institutional planning database. == Awards and honors == Thomas Dietterich was honored by Oregon State University in the spring of 2013 as a "Distinguished Professor" for his work as a pioneer in the field of machine learning and being one of the mostly highly cited scientists in his field. He has also earned exclusive "Fellow" status in the Association for the Advancement of Artificial Intelligence, the American Association for the Advancement of Science and the Association for Computing Machinery. Over his career, he obtained more than $30 million in research grants, helped build a world-class research group at Oregon State, and created three software companies. He also co-founded two of the field's leading journals and was elected first president of the International Machine Learning Society. His other awards and honors include: ACM Distinguished Lecturer, 2012-2013 Fellow, American Association for the Advancement of Science, 2007 Oregon State University, College of Engineering Collaboration Award, 2004 Winner, JAIR Award for Best Paper in Previous Five Years, 2003 Fellow, Association for Computing Machinery, elected 2003 Oregon State University, College of Engineering Research Award, 1998 Fellow, Association for the Advancement of Artificial Intelligence, elected 1994 NSF Presidential Young Investigator, 1987-92 Nominated for Carter Award for Graduate Teaching, 1987, 1988 IBM Graduate Fellow, 1982, 1983 Upsilon Pi Epsilon, 1996 Sigma Xi, 1979–present State Farm Companies Foundation Fellowship, 1978 Member, Board of Trustees, Oberlin College, 1977-1980 Graduation with Honors in Mathematics, Oberlin College, 1977 Phi Beta Kappa, 1977 National Merit Scholar, 1973 == Selected publications == Liping Liu, Thomas G. Dietterich, Nan Li, Zhi-Hua Zhou (2016). Transductive Optimization of Top k Precision. International Joint Conference on Artificial Intelligence (IJCAI-2016). pp. 1781–1787. New York, NY Md. Amran Siddiqui, Alan Fern, Thomas G. Dietterich, Shubhomoy Da
Ernie Bot
Ernie Bot (Chinese: 文心一言, Pinyin: wénxīn yīyán), full name Enhanced Representation through Knowledge Integration, is an artificial intelligence chatbot developed by the Chinese technology company Baidu. Ernie Bot rivals GPT models in Chinese NLP tasks. It is built on the company's ERNIE series of large language models, which have been in development since 2019. The service was first launched for invited testing on March 16, 2023, and was released to the general public on August 31, 2023, after receiving approval from Chinese regulators. Since its public launch, Ernie Bot has undergone several updates, with newer versions like ERNIE 4.0 and 4.5 released to improve its capabilities. The service has seen rapid user adoption, reportedly reaching over 200 million users by April 2024. It has been integrated into various products, notably powering AI features for the Chinese release of Samsung's Galaxy S24 smartphones. As a product operating in China, Ernie Bot is subject to the country's censorship regulations. It has been observed to refuse answers to politically sensitive questions, such as those regarding CCP general secretary Xi Jinping, the 1989 Tiananmen Square protests and massacre, and other topics deemed taboo by the government. == History == Ernie Bot was initially released for invited testing on March 16, 2023. The live release demo was reported to have been prerecorded, which caused Baidu's stock to drop 10 percent on the day of the launch. The company's stock gained 14 percent the following day after analysts from Citigroup and Bank of America tested Ernie Bot and gave it positive preliminary reviews. On August 31, 2023, Ernie Bot was released to the public after receiving approval from Chinese regulatory authorities. By December 2023, Baidu announced the service had surpassed 100 million users. In January 2024, Hong Kong newspaper South China Morning Post reported that a university research lab linked to the People's Liberation Army (PLA) had tested Ernie Bot for military response scenarios. Baidu denied the allegations, stating it had no connection with the academic paper. That same month, Ernie was integrated into Samsung's Galaxy S24 lineup for its launch in China. The user base reportedly grew to 200 million by April 2024 and 300 million by June 2024. In September 2024, Baidu changed the chatbot's Chinese name from "Wenxin Yiyan" (文心一言) to "Wenxiaoyan" (文小言) to position it as a search assistant. On March 16, 2025, Baidu announced version 4.5 and the reasoning model ERNIE X1. The following month, at the Create2025 Baidu AI Developer Conference, the company released the Wenxin 4.5 Turbo and Wenxin X1 Turbo models, designed to be faster and less expensive to operate. == Development == Ernie Bot is based on Baidu's ERNIE (Enhanced Representation through Knowledge Integration) series of foundation models. The general training process begins with pre-training on large datasets, followed by refinement using techniques like supervised fine-tuning, reinforcement learning with human feedback, and prompt engineering. === Foundation models === ==== Ernie 3.0 ==== The model powering the initial launch of Ernie Bot. It was trained with 10 billion parameters on a 4-terabyte corpus consisting of plain text and a large-scale knowledge graph. ==== Ernie 3.5 ==== Released in June 2023. At the time of release, its performance was reported as "slightly inferior" to OpenAI's GPT-4. ==== Ernie 4.0 ==== Unveiled in October 2023 and released to paying subscribers in November. According to Baidu, this version featured improved performance over its predecessor, with information updated to April 2023. ==== Ernie X1 ==== Announced in March 2025, with Ernie X1 positioned as a specialized reasoning model. Baidu stated that performance improvements were achieved through new technologies such as "FlashMask" dynamic attention masking and a heterogeneous multimodal mixture-of-experts architecture. === Turbo Models === In June 2024, Baidu announced Ernie 4.0 Turbo. In April 2025, Ernie 4.5 Turbo and X1 Turbo were released. These models are optimized for faster response times and lower operational costs. == Service == In its subscription options, the professional plan gives users access to Ernie 4.0 with a payment either for a month or with reduced payment for auto-renewal per month. Meanwhile, Ernie 3.5 is free of charge. Ernie 4.0, the language model for Ernie bot, has information updated to April 2023. == Censorship == Ernie Bot is subject to the Chinese government's censorship regime. In public tests with journalists, Ernie Bot refused to answer questions about CCP general secretary Xi Jinping, the 1989 Tiananmen Square protests and massacre, the persecution of Uyghurs in China in Xinjiang, and the 2019–2020 Hong Kong protests. When queried about the origin of SARS-CoV-2, Ernie Bot stated that it originated among American vape users.
Google matrix
A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links between pages. The PageRank of each page can then be generated iteratively from the Google matrix using the power method. However, in order for the power method to converge, the matrix must be stochastic, irreducible and aperiodic. == Adjacency matrix A and Markov matrix S == In order to generate the Google matrix G, we must first generate an adjacency matrix A which represents the relations between pages or nodes. Assuming there are N pages, we can fill out A by doing the following: A matrix element A i , j {\displaystyle A_{i,j}} is filled with 1 if node j {\displaystyle j} has a link to node i {\displaystyle i} , and 0 otherwise; this is the adjacency matrix of links. A related matrix S corresponding to the transitions in a Markov chain of given network is constructed from A by dividing the elements of column "j" by a number of k j = Σ i = 1 N A i , j {\displaystyle k_{j}=\Sigma _{i=1}^{N}A_{i,j}} where k j {\displaystyle k_{j}} is the total number of outgoing links from node j to all other nodes. The columns having zero matrix elements, corresponding to dangling nodes, are replaced by a constant value 1/N. Such a procedure adds a link from every sink, dangling state a {\displaystyle a} to every other node. Now by the construction the sum of all elements in any column of matrix S is equal to unity. In this way the matrix S is mathematically well defined and it belongs to the class of Markov chains and the class of Perron-Frobenius operators. That makes S suitable for the PageRank algorithm. == Construction of Google matrix G == Then the final Google matrix G can be expressed via S as: G i j = α S i j + ( 1 − α ) 1 N ( 1 ) {\displaystyle G_{ij}=\alpha S_{ij}+(1-\alpha ){\frac {1}{N}}\;\;\;\;\;\;\;\;\;\;\;(1)} By the construction the sum of all non-negative elements inside each matrix column is equal to unity. The numerical coefficient α {\displaystyle \alpha } is known as a damping factor. Usually S is a sparse matrix and for modern directed networks it has only about ten nonzero elements in a line or column, thus only about 10N multiplications are needed to multiply a vector by matrix G. == Examples of Google matrix == An example of the matrix S {\displaystyle S} construction via Eq.(1) within a simple network is given in the article CheiRank. For the actual matrix, Google uses a damping factor α {\displaystyle \alpha } around 0.85. The term ( 1 − α ) {\displaystyle (1-\alpha )} gives a surfer probability to jump randomly on any page. The matrix G {\displaystyle G} belongs to the class of Perron-Frobenius operators of Markov chains. The examples of Google matrix structure are shown in Fig.1 for Wikipedia articles hyperlink network in 2009 at small scale and in Fig.2 for University of Cambridge network in 2006 at large scale. == Spectrum and eigenstates of G matrix == For 0 < α < 1 {\displaystyle 0<\alpha <1} there is only one maximal eigenvalue λ = 1 {\displaystyle \lambda =1} with the corresponding right eigenvector which has non-negative elements P i {\displaystyle P_{i}} which can be viewed as stationary probability distribution. These probabilities ordered by their decreasing values give the PageRank vector P i {\displaystyle P_{i}} with the PageRank K i {\displaystyle K_{i}} used by Google search to rank webpages. Usually one has for the World Wide Web that P ∝ 1 / K β {\displaystyle P\propto 1/K^{\beta }} with β ≈ 0.9 {\displaystyle \beta \approx 0.9} . The number of nodes with a given PageRank value scales as N P ∝ 1 / P ν {\displaystyle N_{P}\propto 1/P^{\nu }} with the exponent ν = 1 + 1 / β ≈ 2.1 {\displaystyle \nu =1+1/\beta \approx 2.1} . The left eigenvector at λ = 1 {\displaystyle \lambda =1} has constant matrix elements. With 0 < α {\displaystyle 0<\alpha } all eigenvalues move as λ i → α λ i {\displaystyle \lambda _{i}\rightarrow \alpha \lambda _{i}} except the maximal eigenvalue λ = 1 {\displaystyle \lambda =1} , which remains unchanged. The PageRank vector varies with α {\displaystyle \alpha } but other eigenvectors with λ i < 1 {\displaystyle \lambda _{i}<1} remain unchanged due to their orthogonality to the constant left vector at λ = 1 {\displaystyle \lambda =1} . The gap between λ = 1 {\displaystyle \lambda =1} and other eigenvalue being 1 − α ≈ 0.15 {\displaystyle 1-\alpha \approx 0.15} gives a rapid convergence of a random initial vector to the PageRank approximately after 50 multiplications on G {\displaystyle G} matrix. At α = 1 {\displaystyle \alpha =1} the matrix G {\displaystyle G} has generally many degenerate eigenvalues λ = 1 {\displaystyle \lambda =1} (see e.g. [6]). Examples of the eigenvalue spectrum of the Google matrix of various directed networks is shown in Fig.3 from and Fig.4 from. The Google matrix can be also constructed for the Ulam networks generated by the Ulam method [8] for dynamical maps. The spectral properties of such matrices are discussed in [9,10,11,12,13,15]. In a number of cases the spectrum is described by the fractal Weyl law [10,12]. The Google matrix can be constructed also for other directed networks, e.g. for the procedure call network of the Linux Kernel software introduced in [15]. In this case the spectrum of λ {\displaystyle \lambda } is described by the fractal Weyl law with the fractal dimension d ≈ 1.3 {\displaystyle d\approx 1.3} (see Fig.5 from ). Numerical analysis shows that the eigenstates of matrix G {\displaystyle G} are localized (see Fig.6 from ). Arnoldi iteration method allows to compute many eigenvalues and eigenvectors for matrices of rather large size [13]. Other examples of G {\displaystyle G} matrix include the Google matrix of brain [17] and business process management [18], see also. Applications of Google matrix analysis to DNA sequences is described in [20]. Such a Google matrix approach allows also to analyze entanglement of cultures via ranking of multilingual Wikipedia articles abouts persons [21] == Historical notes == The Google matrix with damping factor was described by Sergey Brin and Larry Page in 1998 [22], see also articles on PageRank history [23], [24].