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The network structure of mathematical knowledge according to the Wikipedia, MathWorld, and DLMF online libraries

Published online by Cambridge University Press:  10 October 2014

FLAVIO B. GONZAGA ,
VALMIR C. BARBOSA  and
GERALDO B. XEXÉO
FLAVIO B. GONZAGA
Affiliation:
Núcleo de Ciência da Computação, Universidade Federal de Alfenas, 37130-000 Alfenas - MG, Brazil (e-mail: [email protected])
VALMIR C. BARBOSA
Affiliation:
Programa de Engenharia de Sistemas e Computação, COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, 21941-972 Rio de Janeiro - RJ, Brazil (e-mail: [email protected], [email protected])
GERALDO B. XEXÉO
Affiliation:
Programa de Engenharia de Sistemas e Computação, COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, 21941-972 Rio de Janeiro - RJ, Brazil (e-mail: [email protected], [email protected])
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Abstract

We study the network structure of Wikipedia (restricted to its mathematical portion), MathWorld, and DLMF. We approach these three online mathematical libraries from the perspective of several global and local network-theoretic features, providing for each one the appropriate value or distribution, along with comparisons that, if possible, also include the whole of the Wikipedia or the Web. We identify some distinguishing characteristics of all three libraries, most of them supposedly traceable to the libraries' shared nature of relating to a very specialized domain. Among these characteristics are the presence of a very large strongly connected component in each of the corresponding directed graphs, the complete absence of any clear power laws describing the distribution of local features, and the rise to prominence of some local features (e.g., stress centrality) that can be used to effectively search for keywords in the libraries.

Keywords

online mathematical librariesWikipediaMathWorldDLMFcomplex networkstext search
Type
Research Article
Information
Network Science , Volume 2 , Issue 3 , December 2014 , pp. 367 - 386
Copyright
Copyright © Cambridge University Press 2014 

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References

Abramowitz, M., & Stegun, I. A. (1965). Handbook of mathematical functions. New York, NY: Dover Publications.Google Scholar
Albert, R., Jeong, H., & Barabási, A.-L. (1999). Diameter of the world-wide web. Nature, 401, 130–131.Google Scholar
Albert, R., Jeong, H., & Barabási, A.-L. (2000). Error and attack tolerance of complex networks. Nature, 406, 378–382.Google Scholar
Anthonisse, J. M. (1971). The rush in a directed graph. Tech. rept. BN 9/71. Stichting Mathematisch Centrum, Amsterdam, The Netherlands.Google Scholar
Baeza-Yates, R., & Ribeiro-Neto, B. (2011). Modern information retrieval (2nd ed.). Harlow, UK: Addison Wesley.Google Scholar
Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 509–512.Google Scholar
Barabási, A.-L., Albert, R., & Jeong, H. (2000). Scale-free characteristics of random networks: The topology of the world-wide web. Physica A, 281, 69–77.Google Scholar
Boisvert, R., Clark, C. W., Lozier, D., & Olver, F. (2011). A special functions handbook for the digital age. Notes of the American Mathematical Society, 58, 905–911.Google Scholar
Bollobás, B. (2001). Random graphs (2nd ed.). Cambridge, UK: Cambridge University Press.Google Scholar
Bollobás, B., Kozma, R., & Miklós, D. (Eds.) (2009). Handbook of large-scale random networks. Berlin, Germany: Springer.Google Scholar
Bornholdt, S., & Schuster, H. G. (Eds.) (2003). Handbook of graphs and networks. Weinheim, Germany: Wiley-VCH.Google Scholar
Brandes, U. (2001). A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 25, 163–177.CrossRefGoogle Scholar
Brin, S., & Page, L. (1998). The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems, 30, 107–117.CrossRefGoogle Scholar
Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., & Wiener, J. (2000). Graph structure in the Web. Computer Networks, 33, 309–320.Google Scholar
Bronshtein, I. N., Semendyayev, K. A., Musiol, G., & Muehlig, H. (2004). Handbook of mathematics (4th ed.). Berlin, Germany: Springer.CrossRefGoogle Scholar
Capocci, A., Servedio, V. D. P., Colaiori, F., Buriol, L. S., Donato, D., Leonardi, S., & Caldarelli, G. (2006). Preferential attachment in the growth of social networks: The internet encyclopedia Wikipedia. Physical Review E, 74, 036116.Google Scholar
Carbone, A. (2002). Streams and strings in formal proofs. Theoretical Computer Science, 288, 45–83.Google Scholar
Cohen, R., Erez, K., ben Avraham, D., & Havlin, S. (2000). Resilience of the Internet to random breakdowns. Physical Review Letters, 85, 4626–4628.CrossRefGoogle ScholarPubMed
Donato, D., Laura, L., Leonardi, S., & Millozzi, S. (2004). Large scale properties of the Webgraph. European Physical Journal B, 38, 239–243.CrossRefGoogle Scholar
Dorogovtsev, S. N., Mendes, J. F. F., & Samukhin, A. N. (2001). Giant strongly connected component of directed networks. Physical Review E, 64, 025101.Google Scholar
Erdős, P., & Rényi, A. (1959). On random graphs. Publicationes Mathematicae (Debrecen), 6, 290–297.CrossRefGoogle Scholar
Erdős, P., & Rényi, A. (1960). On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences A, 5, 17–61.Google Scholar
Foster, J. G., Foster, D. V., Grassberger, P., & Paczuski, M. (2010). Edge direction and the structure of networks. Proceedings of the National Academy of Sciences USA, 107, 10815–10820.Google Scholar
Freeman, L. C. (1977). A set of measures of centrality based on betweenness. Sociometry, 40, 35–41.CrossRefGoogle Scholar
Gowers, T., Barrow-Green, J., & Leader, I. (Eds.) (2008). The Princeton companion to mathematics. Princeton, NJ: Princeton University Press.Google Scholar
Gradshteyn, I. S., & Ryzhik, I. M. (2000). Table of integrals, series, and products (6th ed.). San Diego, CA: Academic Press.Google Scholar
Hage, P., & Harary, F. (1995). Eccentricity and centrality in networks. Social Networks, 17, 57–63.CrossRefGoogle Scholar
Itô, K. (ed). (1993). Encyclopedic dictionary of mathematics (2nd ed.). Cambridge, MA: The MIT Press.Google Scholar
Jia, Q.-S., & Guo, Y. (2009). Discovering the knowledge hierarchy of MathWorld for web intelligence. Proceedings of the 6th International Conference on Fuzzy Systems and Knowledge Discovery, pp. 535–539.Google Scholar
Karp, R. M. (1990). The transitive closure of a random digraph. Random Structures and Algorithms, 1, 73–93.Google Scholar
Kleinberg, J. M. (1999). Authoritative sources in a hyperlinked environment. Journal of the ACM, 46, 604–632.Google Scholar
Lozier, D. W. (2003). NIST digital library of mathematical functions. Annals of Mathematics and Artificial Intelligence, 38, 105–119.CrossRefGoogle Scholar
Molloy, M., & Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures and Algorithms, 6, 161–180.Google Scholar
Molloy, M., & Reed, B. (1998). The size of the largest component of a random graph on a fixed degree sequence. Combinatorics, Probability and Computing, 7, 295–306.CrossRefGoogle Scholar
Newman, M., Barabási, A.-L., & Watts, D. J. (Eds.) (2006). The structure and dynamics of networks. Princeton, NJ: Princeton University Press.Google Scholar
Newman, M. E. J. (2002). Assortative mixing in networks. Physical Review Letters, 89, 208701.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2003). Mixing patterns in networks. Physical Review E, 67, 026126.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46, 323–351.Google Scholar
Newman, M. E. J. (2010). Networks. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Physical Review E, 64, 026118.Google Scholar
Olver, F. W., Lozier, D. W., Boisvert, R. F., & Clark, C. W. (Eds.) (2010). NIST handbook of mathematical functions. New York, NY: Cambridge University Press.Google Scholar
Piraveenan, M., Prokopenko, M., & Zomaya, A. (2012). Assortative mixing in directed biological networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 9, 66–78.Google Scholar
Price, D. J. de S. (1976). A general theory of bibliometric and other cumulative advantage processes. Journal of the American Society for Information Science, 27, 292–306.Google Scholar
Sabidussi, G. (1966). The centrality index of a graph. Psychometrika, 31, 581–603.Google Scholar
Shimbel, A. (1953). Structural parameters of communication networks. Bulletin of Mathematical Biophysics, 15, 501–507.Google Scholar
Silva, F. N., Travençolo, B. A. N., Viana, M. P., & Costa, L. F. (2010). Identifying the borders of mathematical knowledge. Journal of Physics A, 43, 325202.CrossRefGoogle Scholar
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442.Google Scholar
Weisstein, E. W. (2002). CRC concise encyclopedia of mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC.Google Scholar
Zlatić, V., Božičević, M., Štefančić, H., & Domazet, M. (2006). Wikipedias: Collaborative web-based encyclopedias as complex networks. Physical Review E, 74, 016115.Google Scholar