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  1. Home
  2. Books
  3. Random Graphs and Complex Networks
  • Cited by 298
Publisher:
Cambridge University Press
Online publication date:
January 2017
Print publication year:
2016
Online ISBN:
9781316779422
DOI:
https://doi.org/10.1017/9781316779422
Subjects:
Algorithmics, Complexity, Computer Algebra, Computational Geometry, Computer Science, General Statistics and Probability, Applied Probability and Stochastic Networks, Statistics and Probability
Series:
Cambridge Series in Statistical and Probabilistic Mathematics (43)
Information

Book description

This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.

Reviews

‘… a modern and deep, yet accessible, introduction to the models that make up [the] basis for the theoretical study of random graphs and complex networks. The book strikes a balance between providing broad perspective and analytic rigor that is a pleasure for the reader.’

Adam Wierman - California Institute of Technology

‘This text builds a bridge between the mathematical world of random graphs and the real world of complex networks. It combines techniques from probability theory and combinatorics to analyze the structural properties of large random graphs. Accessible to network researchers from different disciplines, as well as masters and graduate students, the material is suitable for a one-semester course, and is laced with exercises that help the reader grasp the content. The exposition focuses on a number of core models that have driven recent progress in the field, including the Erdős–Rényi random graph, the configuration model, and preferential attachment models. A detailed description is given of all their key properties. This is supplemented with insightful remarks about properties of related models so that a full panorama unfolds. As the presentation develops, the link to complex networks provides constant motivation for the routes that are being chosen.’

Frank den Hollander - Leiden University

‘The first volume of Remco van der Hofstad's Random Graphs and Complex Networks is the definitive introduction into the mathematical world of random networks. Written for students with only a modest background in probability theory, it provides plenty of motivation for the topic and introduces the essential tools of probability at a gentle pace. It covers the modern theory of Erdős–Rényi graphs, as well as the most important models of scale-free networks that have emerged in the last fifteen years. This is a truly wonderful first volume; the second volume, leading up to current research topics, is eagerly awaited.’

Peter Mörters - University of Bath

‘This new book on random graph models for complex networks is a wonderful addition to the field. It takes the uninitiated reader from the basics of graduate probability to the classical Erdős–Rényi random graph before terminating at some of the fundamental new models in the discipline. The author does an exemplary job of both motivating the models of interest and building all the necessary mathematical tools required to give a rigorous treatment of these models. Each chapter is complemented by a comprehensive set of exercises allowing the reader ample scope to actively master the techniques covered in the chapter.’

Shankar Bhamidi - University of North Carolina, Chapel Hill

‘This book is invaluable for anybody who wants to learn or teach the modern theory of random graphs and complex networks. I have used it as a textbook for long and short courses at different levels. Students always like the book because it has all they need: exciting high-level ideas, motivating examples, very clear proofs, and an excellent set of exercises. Easy to read, extremely well structured, and self-contained, the book builds proficiency with random graph models essential for state-of-the-art research.’

Nelly Litvak - University of Twente

'What makes the book particularly interesting is that it provides all important preliminaries for readers not having the basic background knowledge of random graphs. … The book is well-suited for a graduate course on random graphs, where students may only have minimal background in probability theory, as the book provides plenty of motivation for the topic and covers all important preliminaries. All the chapters are supplemented by extensive exercises to develop better intuition and to progressively master the models covered in the book. In a nutshell, the book is easy to follow and well-organized for developing proficiency in random graph models necessary for state of the art research.'

Ghulam Abbas Source: Complex Adaptive Systems Modeling

'The writing is of a high standard. I would certainly recommend it to a starting graduate student, even if their first degree was not mathematics.'

Tobias Muller Source: Nieuw Archief voor Weskunde

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Contents

Contents
References
References
Achlioptas, D., Clauset, A., Kempe, D., and Moore, C. 2005. On the bias of traceroute sampling or, power-law degree distributions in regular graphs. In: STOC-05: Proceedings of the 37th Annual ACMSymposium on Theory of Computing.
Adamic, L. A. 1999. The small world web. Pages 443–454 of: Lecture Notes in Computer Science, vol. 1696. Springer.
Addario-Berry, L., Broutin, N., and Goldschmidt, C. 2010. Critical random graphs: limiting constructions and distributional properties. Electron. J. Probab., 15(25), 741–775.
Aiello, w., chung, f., and lu, l. 2002. random evolution in massive graphs. pages 97–122 of: handbookof massive data sets. Massive Comput., vol. 4. Dordrecht: Kluwer Acad. Publ.
Aigner, M., and Ziegler, G. 2014. Proofs from The Book. Fifth edn. Springer-Verlag, Berlin. Including illustrations by Karl H., Hofmann.
Albert, R., and Barabási, A. -L. 2002. Statistical mechanics of complex networks. Rev. Modern Phys., 74(1), 47–97.
Albert, R., Jeong, H., and Barabási, A. -L. 1999. Internet: diameter of the World-Wide Web. Nature, 401, 130–131.
Albert, R., Jeong, H., and Barabási, A. -L. 2001. Error and attack tolerance of complex networks. Nature, 406, 378–382.
Aldous, D. Random graphs and complex networks. Transparencies available from http://www.stat.berkeley.edu/∼aldous/Talks/net.ps.
Aldous, D. 1991. Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab., 1(2), 228–266.
Aldous, D. 1993. Tree-based models for random distribution of mass. J. Stat. Phys., 73, 625–641.
Aldous, D. 1997. Brownian excursions, critical random graphs and the multiplicative coalescent. Ann.Probab., 25(2), 812–854.
Alili, L., Chaumont, L., and Doney, R. A. 2005. On a fluctuation identity for random walks and Lévy processes. Bull. London Math. Soc., 37(1), 141–148.
Alon, N., and Spencer, J. 2000. The probabilistic method. Second edn.Wiley-Interscience Series in Discrete Mathematics and Optimization. New York: John Wiley & Sons.
Amaral, L. A. N., Scala, A., Barthélémy, M., and Stanley, H. E. 2000. Classes of small-world networks. Proc. Natl. Acad. Sci. USA, 97, 11149–11152.
Angel, O., Hofstad, R. van der, and Holmgren, C. Preprint (2016). Limit laws for self-loops and multipleedges in the configuration model. Available from http://arxiv.org/abs/1603.07172.
Anthonisse, J. 1971. The rush in a graph. Technical Report University of Amsterdam Mathematical Center.
Arratia, R., and Liggett, T. 2005. How likely is an i.i.d. degree sequence to be graphical. Ann. Appl. Probab., 15(1B), 652–670.
Athreya, K., and Ney, P. 1972. Branching processes. New York: Springer-Verlag. Die Grundlehren der mathematischen Wissenschaften, Band 196.
Austin, T. L., Fagen, R. E., Penney, W. F., and Riordan, J. 1959. The number of components in random linear graphs. Ann. Math. Statist, 30, 747–754.
Azuma, K. 1967. Weighted sums of certain dependent random variables. Tohoku Math. J., 3, 357–367.
Backstrom, L., Boldi, P., Rosa, M., Ugander, J., and Vigna, S. 2012. Four degrees of separation. Pages 33–42 of: Proceedings of the 3rd Annual ACM Web Science Conference. ACM.
Bak, P. 1996. How Nature Works: The Science of Self-Organized Criticality. New York: Copernicus.
Ball, F., Sirl, D., and Trapman, P. 2009. Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv. in Appl. Probab., 41(3), 765–796.
Ball, F., Sirl, D., and Trapman, P. 2010. Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math. Biosci., 224(2), 53–73.
Barabási, A. -L. 2002. Linked: the new science of networks. Cambridge, Massachusetts: Perseus Publishing.
Barabási, A. -L., and Albert, R. 1999. Emergence of scaling in random networks. Science, 286(5439), 509–512.
Barabási, A. -L., Albert, R., and Jeong, H. 2000. Scale-free characteristics of random networks: the topology of the world-wide web. Phys. A, 311, 69–77.
Barabási, A. -L., Jeong, H., Néda, Z., Ravasz, E., Schubert, A., and Vicsek, T. 2002. Evolution of the social network of scientific collaborations. Phys. A, 311(3-4), 590–614.
Barraez, D., Boucheron, S., and Fernandez de la Vega, W. 2000. On the fluctuations of the giant component. Combin. Probab. Comput., 9(4), 287–304.
Bavelas, A. 1950. Communication patterns in task-oriented groups. Journ. Acoust. Soc. Amer., 22(64), 725–730.
Behrisch, M., Coja-Oghlan, A., and Kang, M. 2014. Local limit theorems for the giant component of random hypergraphs. Combin. Probab. Comput., 23(3), 331–366.
Bender, E. A., and Canfield, E. R. 1978. The asymptotic number of labelled graphs with given degree sequences. Journal of Combinatorial Theory (A), 24, 296–307.
Bender, E. A., Canfield, E. R., and McKay, B. D. 1990. The asymptotic number of labeled connected graphs with a given number of vertices and edges. Random Structures Algorithms, 1(2), 127–169.
Bennet, G. 1962. Probability inequaltities for the sum of independent random variables. J. Amer. Statist.Assoc., 57, 33–45.
Berger, N., Borgs, C., Chayes, J. T., D'Souza, R. M., and Kleinberg, R. D. 2004. Competition-induced preferential attachment. Pages 208–221 of: Automata, languages and programming. Lecture Notes in Comput. Sci., vol. 3142. Berlin: Springer.
Berger, N., Borgs, C., Chayes, J. T., D'Souza, R. M., and Kleinberg, R. D. 2005a. Degree distribution of competition-induced preferential attachment graphs. Combin. Probab. Comput., 14(5-6), 697–721.
Berger, N., Borgs, C., Chayes, J. T., and Saberi, A. 2005b. On the spread of viruses on the Internet. Pages 301–310 of: SODA–05: Proceedings of the sixteenth annual ACM-SIAM symposium on Discretealgorithms. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics.
Berger, N., Borgs, C., Chayes, J., and Saberi, A. 2014. Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Probab., 42(1), 1–40.
Bertoin, J. 2006. Random fragmentation and coagulation processes. Cambridge Studies in Advanced Mathematics, vol. 102. Cambridge: Cambridge University Press.
Bhamidi, S. In preparation (2007). Universal techniques to analyze preferential attachment trees: globaland local analysis. Available from http://www.unc.edu/∼bhamidi/preferent.pdf.
Bhamidi, S., Hofstad, R. van der, and Hooghiemstra, G. 2010. First passage percolation on random graphs with finite mean degrees. Ann. Appl. Probab., 20(5), 1907–1965.
Bhamidi, S., Hofstad, der R. van, and Hooghiemstra, G. Universality for first passage percolation on sparserandom graphs. Preprint (2012). To appear in Ann. Probab.
Bianconi, G., and Barabási, A. -L. 2001a. Bose–Einstein condensation in complex networks. Phys. Rev.Lett., 86(24), 5632–5635.
Bianconi, G., and Barabási, A. -L. 2001b. Competition and multiscaling in evolving networks. Europhys.Lett., 54, 436–442.
Billingsley, P. 1968. Convergence of probability measures. New York: John Wiley and Sons.
Billingsley, P. 1995. Probability and measure. Third edn. Wiley Series in Probability and Mathematical Statistics. New York: John Wiley & Sons Inc. A Wiley-Interscience Publication.
Boldi, P., and Vigna, S. 2014. Axioms for centrality. Internet Math., 10(3-4), 222–262.
Bollobás, B. 1980. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin., 1(4), 311–316.
Bollobás, B. 1981. Degree sequences of random graphs. Discrete Math., 33(1), 1–19.
Bollobás, B. 1984a. The evolution of random graphs. Trans. Amer. Math. Soc., 286(1), 257–274.
Bollobás, B. 1984b. The evolution of sparse graphs. Pages 35–57 of: Graph theory and combinatorics(Cambridge, 1983). London: Academic Press.
Bollobás, B. 1998. Modern graph theory. Graduate Texts in Mathematics, vol. 184. Springer-Verlag, New York.
Bollobás, B. 2001. Random graphs. Second edn. Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge: Cambridge University Press.
Bollobás, B., and Riordan, O. 2003a. Mathematical results on scale-free random graphs. Pages 1–34 of: Handbook of graphs and networks. Wiley-VCH, Weinheim.
Bollobás, B., and Riordan, O. 2003b. Robustness and vulnerability of scale-free random graphs. InternetMath., 1(1), 1–35.
Bollobás, B., and Riordan, O. 2004. The diameter of a scale-free random graph. Combinatorica, 24(1), 5–34.
Bollobás, B., Riordan, O., Spencer, J., and Tusnády, G. 2001. The degree sequence of a scale-free random graph process. Random Structures Algorithms, 18(3), 279–290.
Bollobás, B., Borgs, C., Chayes, J., and Riordan, O. 2003. Directed scale-free graphs. Pages 132–139 of: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD,2003). New York: ACM.
Bollobás, B., Janson, S., and Riordan, O. 2007. The phase transition in inhomogeneous random graphs. Random Structures Algorithms, 31(1), 3–122.
Bonato, A. 2008. A course on the web graph. Graduate Studies in Mathematics, vol. 89. Providence, RI: American Mathematical Society.
Borgs, C., Chayes, J. T., Kesten, H., and Spencer, J. 1999. Uniform boundedness of critical crossing probabilities implies hyperscaling. Random Structures Algorithms, 15(3-4), 368–413.
Borgs, C., Chayes, J. T., Kesten, H., and Spencer, J. 2001. The birth of the infinite cluster: finite-size scaling in percolation. Comm. Math. Phys., 224(1), 153–204. Dedicated to Joel L. Lebowitz.
Borgs, C., Chayes, J., Hofstad, R. van der, Slade, G., and Spencer, J. 2005a. Random subgraphs of finite graphs. I. The scaling window under the triangle condition. Random Structures Algorithms, 27(2), 137–184.
Borgs, C., Chayes, J., Hofstad, R. van der, Slade, G., and Spencer, J. 2005b. Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab., 33(5), 1886–1944.
Borgs, C., Chayes, J., Hofstad, R. van der, Slade, G., and Spencer, J. 2006. Random subgraphs of finite graphs. III. The phase transition for th. n-cube. Combinatorica, 26(4), 395–410.
Borgs, C., Chayes, J. T., Daskalis, C., and Roch, S. 2007. First to market is not everything: an analysis of preferential attachment with fitness. Pages 135–144 of: STOC–07: Proceedings of the thirty-ninthannual ACM symposium on Theory of computing. New York, NY, USA: ACM Press.
Breiman, L. 1992. Probability. Classics in Applied Mathematics, vol. 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Corrected reprint of the 1968 original.
Bressloff, P. 2014. Waves in neural media. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, New York. From single neurons to neural fields.
Brin, S., and Page, L. 1998. The anatomy of a large-scale hypertextual Web search engine. Pages 107–117 of: Computer Networks and ISDN Systems, vol. 33.
Britton, T., Deijfen, M., andMartin-Löf, A. 2006. Generating simple random graphs with prescribed degree distribution. J. Stat. Phys., 124(6), 1377–1397.
Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., and Wiener, J. 2000. Graph structure in the Web. Computer Networks, 33, 309–320.
Buckley, P. G., and Osthus, D. 2004. Popularity based random graph models leading to a scale-free degree sequence. Discrete Math., 282(1-3), 53–68.
Cayley, A. 1889. A theorem on trees. Q. J. Pure Appl. Math., 23, 376–378.
Champernowne, D. G. 1953. A model of income distribution. Econ. J., 63, 318.
Chen, N., and Olvera-Cravioto, M. 2013. Directed random graphs with given degree distributions. Stoch.Syst., 3(1), 147–186.
Chernoff, H. 1952. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statistics, 23, 493–507.
Choudum, S. A. 1986. A simple proof of the Erdős-Gallai theorem on graph sequences. Bull. Austral. Math.Soc., 33(1), 67–70.
Chung, F., and Lu, L. 2002a. The average distances in random graphs with given expected degrees. Proc.Natl. Acad. Sci. USA, 99(25), 15879–15882 (electronic).
Chung, F., and Lu, L. 2002b. Connected components in random graphs with given expected degree sequences. Ann. Comb., 6(2), 125–145.
Chung, F., and Lu, L. 2003. The average distance in a random graph with given expected degrees. InternetMath., 1(1), 91–113.
Chung, F., and Lu, L. 2006a. Complex graphs and networks. CBMS Regional Conference Series in Mathematics, vol. 107. Published for the Conference Board of the Mathematical Sciences, Washington, DC.
Chung, F., and Lu, L. 2006b. Concentration inequalities and martingale inequalities: a survey. InternetMath., 3(1), 79–127.
Chung, F., and Lu, L. 2006c. The volume of the giant component of a random graph with given expected degrees. SIAM J. Discrete Math., 20, 395–411.
Clauset, A., and Moore, C. Preprint (2003). Traceroute sampling makes random graphs appear to havepower law degree distributions. Available from https://arxiv.org/abs/cond-mat/0312674.
Clauset, A., and Moore, C. 2005. Accuracy and scaling phenomena in Internet mapping. Phys. Rev. Lett., 94, 018701: 1–4.
Clauset, A., Shalizi, C., and Newman, M. E. J. 2009. Power-law distributions in empirical data. SIAM. review, 51(4), 661–703.
Cohen, R., Erez, K., ben Avraham, D., and Havlin, S. 2000. Resilience of the Internet to random breakdowns. Phys. Rev. Letters, 85, 4626.
Cohen, R., Erez, K., ben Avraham, D., and Havlin, S. 2001. Breakdown of the Internet under intentional attack. Phys. Rev. Letters, 86, 3682.
Cooper, C., and Frieze, A. 2003. A general model of web graphs. Random Structures Algorithms, 22(3), 311–335.
Cooper, C., and Frieze, A. 2004. The size of the largest strongly connected component of a random digraph with a given degree sequence. Combin. Probab. Comput., 13(3), 319–337.
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., and Knuth, D. E. 1996. On the Lambert W function. Adv. Comput. Math., 5, 329–359.
Coupechoux, E., and Lelarge, M. 2014. How clustering affects epidemics in random networks. Adv. in Appl.Probab., 46(4), 985–1008.
De Castro, R., and Grossman, J. W. 1999a. Famous trails to Paul Erdős. Rev. Acad. Colombiana Cienc.Exact. Fí s. Natur., 23(89), 563–582. Translated and revised from the English.
De Castro, R., and Grossman, J. W. 1999b. Famous trails to Paul Erdős. Math. Intelligencer, 21(3), 51–63. With a sidebar by Paul M. B., Vitanyi.
Deijfen, M., Esker, H. van den, Hofstad, R. van der, and Hooghiemstra, G. 2009. A preferential attachment model with random initial degrees. Ark. Mat., 47(1), 41–72.
Dembo, A., and Zeitouni, O. 1998. Large deviations techniques and applications. 2nd Applications of Mathematics (New York), vol. 38. New York: Springer-Verlag.
Dereich, S., and Mörters, P. 2009. Random networks with sublinear preferential attachment: Degree evolutions. Electronic Journal of Probability, 14, 122–1267.
Dereich, S., and Mörters, P. 2011. Random networks with concave preferential attachment rule. Jahresber.Dtsch. Math.-Ver., 113(1), 21–40.
Dereich, S., and Mörters, P. 2013. Random networks with sublinear preferential attachment: the giant component. Ann. Probab., 41(1), 329–384.
Dodds, P., Muhamad, R., and Watts, D. 2003. An experimental study of search in global social networks. Science, 301(5634), 827–829.
Dommers, S., Hofstad, R. van der, and Hooghiemstra, G. 2010. Diameters in preferential attachment graphs. Journ. Stat. Phys., 139, 72–107.
Dorogovtsev, S. N. 2010. Lectures on complex networks. Oxford Master Series in Physics, vol. 20. Oxford University Press, Oxford. Oxford Master Series in Statistical Computational, and Theoretical Physics.
Dorogovtsev, S. N., Ferreira, A. L., Goltsev, A. V., and Mendes, J. F. F. 2010. Zero Pearson coefficient for strongly correlated growing trees. Phys. Rev. E, 81(3), 031135.
Dudley, R. M. 2002. Real analysis and probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge: Cambridge University Press. Revised reprint of the 1989 original.
Durrett, R. 2007. Random graph dynamics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press.
Durrett, R. 2010. Probability: theory and examples. Fourth edn. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge.
Dwass, M. 1962. A fluctuation theorem for cyclic random variables. Ann. Math. Statist., 33, 1450–1454.
Dwass, M. 1968. A theorem about infinitely divisible distributions. Z. Wahrscheinleikheitsth., 9, 206–224.
Dwass, M. 1969. The total progeny in a branching process and a related random walk. J. Appl. Prob., 6, 682–686.
Easley, D., and Kleinberg, J. 2010. Networks, crowds, and markets: Reasoning about a highly connectedworld. Cambridge University Press.
Ebel, H., Mielsch, L. -I., and Bornholdt, S. 2002. Scale-free topology of e-mail networks. Phys. Rev. E, 66, 035103.
Eckhoff, M., and Mörters, P. 2014. Vulnerability of robust preferential attachment networks. Electron. J.Probab., 19, no. 57, 47.
Embrechts, P., Klüppelberg, C., and Mikosch, T. 1997. Modelling extremal events. Applications of mathematics (New York), vol. 33. Berlin: Springer-Verlag. For insurance and finance.
Erdős, P. 1947. Some remarks on the theory of graphs. Bull. Amer. Math. Soc., 53, 292–294.
Erdős, P., and Gallai, T. 1960. Graphs with points of prescribed degrees. (Hungarian). Mat. Lapok, 11, 264–274.
Erdős, P., and Rényi, A. 1959. On random graphs. I. Publ. Math. Debrecen, 6, 290–297.
Erdős, P., and Rényi, A. 1960. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int.Közl., 5, 17–61.
Erdős, P., and Rényi, A. 1961a. On the evolution of random graphs. Bull. Inst. Internat. Statist., 38, 343–347.
Erdős, P., and Rényi, A. 1961b. On the strength of connectedness of a random graph. Acta Math. Acad. Sci.Hungar., 12, 261–267.
Erdős, P., and Wilson, R. J. 1977. On the chromatic index of almost all graphs. J. Combinatorial TheorySer. B, 23(2–3), 255–257.
Ergün, G., and Rodgers, G. J. 2002. Growing random networks with fitness. Phys. A, 303, 261–272.
Esker, H. van den, Hofstad, R. van der, and Hooghiemstra, G. 2008. Universality for the distance in finite variance random graphs. J. Stat. Phys., 133(1), 169–202.
Faloutsos, C., Faloutsos, P., and Faloutsos, M. 1999. On power-law relationships of the internet topology. Computer Communications Rev., 29, 251–262.
Feld, S. L. 1991. Why your friends have more friends than you do. American Journal of Sociology, 96(6), 1464–1477.
Feller, W. 1968. An introduction to probability theory and its applications. Volume I. 3rd edn. New York: Wiley.
Feller, W. 1971. An introduction to probability theory and its applications. Volume II. 2nd edn. New York: Wiley.
Fortunato, S. 2010. Community detection in graphs. Physics Reports, 486(3), 75–174.
Freeman, L. 1977. A set of measures in centrality based on betweenness. Sociometry, 40(1), 35–41.
Gao, P., and Wormald, N. 2016. Enumeration of graphs with a heavy-tailed degree sequence. Adv. Math., 287, 412–450.
Gilbert, E. N. 1959. Random graphs. Ann. Math. Statist., 30, 1141–1144.
Gladwell, M. 2006. The tipping point: How little things can make a big difference. Hachette Digital, Inc.
Gnedin, A., Hansen, B., and Pitman, J. 2007. Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws. Probab. Surv., 4, 146–171.
Gradshteyn, I. S., and Ryzhik, I. M. 1965. Table of integrals, series, and products. Fourth edition prepared by Ju. V., Geronimus and M. Ju., Ceitlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey. New York: Academic Press.
Granovetter, M. S. 1973. The strength of weak ties. American Journal of Sociology, 1360–1380.
Granovetter, M. S. 1995. Getting a job: A study of contacts and careers. University of Chicago Press.
Grimmett, G. 1999. Percolation. 2nd edn. Berlin: Springer.
Grimmett, G. R., and Stirzaker, D. R. 2001. Probability and random processes. Third edn. New York: Oxford University Press.
Grossman, J. W. 2002. The evolution of the mathematical research collaboration graph. Pages 201–212 of: Proceedings of the Thirty-third Southeastern International Conference on Combinatorics, Graph Theoryand Computing (Boca Raton, FL, 2002), vol. 158.
Hagberg, O., andWiuf, C. 2006. Convergence properties of the degree distribution of some growing network models. Bull. Math. Biol., 68, 1275–1291.
Halmos, P. 1950. Measure theory. D. Van Nostrand Company, Inc., New York N. Y.
Harary, F. 1969. Graph theory. Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.- London.
Harris, T. 1963. The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119. Berlin: Springer-Verlag.
Heuvel, M. van den, and Sporns, O. 2011. Rich-club organization of the human connectome. The Journalof Neuroscience, 31(44), 15775–15786.
Hirate, Y., Kato, S., and Yamana, H. 2008. Web structure in 2005. Pages 36–46 of: Algorithms and modelsfor the web-graph. Springer.
Hoeffding, W. 1963. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 58, 13–30.
Hofstad, R. van der. 2010. Percolation and random graphs. Pages 173–247 of: New perspectives instochastic geometry. Oxford Univ. Press, Oxford.
Hofstad, R. van der. 2015. Random graphs and complex networks. Vol. II. In preparation, see http://www.win.tue.nl/∼rhofstad/NotesRGCNII.pdf.
Hofstad, R. van der, Kager, W., and Müller, T. 2009. A local limit theorem for the critical random graph. Electron. Commun. Probab., 14, 122–131.
Hofstad, R. van der, and Keane, M. 2008. An elementary proof of the hitting time theorem. Amer. Math.Monthly, 115(8), 753–756.
Hofstad, R. van der, Leeuwaarden, J. H. S. van, and Stegehuis, C. Preprint (2015). Hierarchical configurationmodel.
Hofstad, R. van der, and Litvak, N. 2014. Degree-degree dependencies in random graphs with heavy-tailed degrees. Internet Math., 10(3-4), 287–334.
Hofstad, R. van der, and Spencer, J. 2006. Counting connected graphs asymptotically. European J. Combin., 27(8), 1294–1320.
Hollander, F. den. 2000. Large deviations. Fields Institute Monographs, vol. 14. Providence, RI: American Mathematical Society.
Hoorn, P. van der, and Litvak, N. 2015. Upper bounds for number of removed edges in the Erased Configuration Model. Proceedings of the 12th International Workshop on Algorithms and Models for the Web-Graph, WAW 2015, 10–11 Dec 2015, Eindhoven, pp. 54–65. Lecture Notes in Computer Science 9479. Springer International Publishing. ISSN 0302-9743 ISBN 978-3-319-26783-8.
Huffaker, B., Fomenkov, M., and Claffy, K. 2012 (May). Internet topology data comparison. Tech. rept. Cooperative Association for Internet Data Analysis (CAIDA).
Jagers, P. 1975. Branching processes with biological applications. London: Wiley-Interscience [John Wiley & Sons]. Wiley Series in Probability and Mathematical Statistics—Applied Probability and Statistics.
Jagers, P., and Nerman, O. 1984. The growth and composition of branching populations. Adv. in Appl.Probab., 16(2), 221–259.
Jagers, P., and Nerman, O. 1996. The asymptotic composition of supercritical multi-type branching populations. Pages 40–54 of: Séminaire de Probabilités, XXX. Lecture Notes in Math., vol. 1626. Berlin: Springer.
Janson, S. 2005. Asymptotic degree distribution in random recursive trees. Random Structures Algorithms, 26(1-2), 69–83.
Janson, S. 2007. Monotonicity, asymptotic normality and vertex degrees in random graphs. Bernoulli, 13(4), 952–965.
Janson, S. 2009. The probability that a random multigraph is simple. Combin. Probab. Comput., 18(1-2), 205–225.
Janson, S. 2010. Asymptotic equivalence and contiguity of some random graphs. Random StructuresAlgorithms, 36(1), 26–45.
Janson, S. 2011. Probability asymptotics: notes on notation. Available at http://arxiv.org/pdf/1108.3924.pdf.
Janson, S. 2014. The probability that a random multigraph is simple. II. J. Appl. Probab., 51A (Celebrating 50 Years of The Applied Probability Trust), 123–137.
Janson, S., Knuth, D. E., Łuczak, T., and Pittel, B. 1993. The birth of the giant component. RandomStructures Algorithms, 4(3), 231–358. With an introduction by the editors.
Janson, S., Łuczak, T., and Rucinski, A. 2000. Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York.
Janson, S., and Spencer, J. 2007. A point process describing the component sizes in the critical window of the random graph evolution. Combin. Probab. Comput., 16(4), 631–658.
Járai, A., and Kesten, H. 2004. A bound for the distribution of the hitting time of arbitrary sets by random walk. Electron. Comm. Probab., 9, 152–161 (electronic).
Jin, S., and Bestavros, A. 2006. Small-world characteristics of Internet topologies and implications on multicast scaling. Computer Networks, 50, 648–666.
Jordan, J. 2006. The degree sequences and spectra of scale-free random graphs. Random StructuresAlgorithms, 29(2), 226–242.
Karinthy, F. 1929. Chains. In: Everything is different. Publisher unknown.
Karlin, S. 1967. Central limit theorems for certain infinite urn schemes. J. Math. Mech., 17, 373–401.
Karp, R. M. 1990. The transitive closure of a random digraph. Random Structures Algorithms, 1(1), 73–93.
Kemperman, J. H. B. 1961. The passage problem for a stationary Markov chain. Statistical Research Monographs, Vol. I. The University of Chicago Press, Chicago, Ill.
Kesten, H., and Stigum, B. P. 1966a. Additional limit theorems for indecomposable multidimensional Galton–Watson processes. Ann. Math. Statist., 37, 1463–1481.
Kesten, H., and Stigum, B. P. 1966b. A limit theorem for multidimensional Galton–Watson processes. Ann.Math. Statist., 37, 1211–1223.
Kesten, H., and Stigum, B. P. 1967. Limit theorems for decomposable multi-dimensional Galton–Watson processes. J. Math. Anal. Appl., 17, 309–338.
Kleinberg, J. M. 1999. Authoritative sources in a hyperlinked environment. J. ACM, 46(5), 604–632.
Kleinberg, J. M. 2000a. Navigation in a small world. Nature, 406, 845.
Kleinberg, J. M. 2000b (May). The small-world phenomenon: an algorithm perspective. Pages 163–170 of: Proc. of the twenty-third annual ACM symposium on Principles of distributed computing.
Kleinberg, J. M., Kumar, R., Raghavan, P., Rajagopalan, S, and Tomkins, A. 1999. The Web as a graph: measurements, models, and methods. Pages 1–17 of: Computing and Combinatorics: 5th Annual InternationalConference, COCOON-99, Tokyo, Japan, July 1999. Proceedings. Lecture Notes in Computer Science.
Konstantopoulos, T. 1995. Ballot theorems revisited. Statist. Probab. Lett., 24(4), 331–338.
Krapivsky, P. L., and Redner, S. 2001. Organization of growing random networks. Phys. Rev. E, 63, 066123.
Krapivsky, P. L., and Redner, S. 2005. Network growth by copying. Phys. Rev. E, 71(3), 036118.
Krapivsky, P. L., Redner, S., and Leyvraz, F. 2000. Connectivity of growing random networks. Phys. Rev.Lett., 85, 4629.
Krioukov, D., Kitsak, M., Sinkovits, R., Rideout, D., Meyer, D., and Boguñá, M. 2012. Network cosmology. Scientific reports, 2.
Kumar, R., Raghavan, P., Rajagopalan, S, and Tomkins, A. 1999. Trawling the Web for emerging cyber communities. Computer Networks, 31, 1481–1493.
Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., and Upfal, E. 2000. Stochastic models for the web graph. Pages 57–65 of: 42st Annual IEEE Symposium on Foundations of ComputerScience.
Lakhina, A., Byers, J. W., Crovella, M., and Xie, P. 2003. Sampling biases in IP topology measurements. Pages 332–341 of: Proceedings of IEEE INFOCOM 1.
Lengler, J., Jug, F., and Steger, A. 2013. Reliable neuronal systems: the importance of heterogeneity. PLoS ONE, 8(12), 1–10.
Leskovec, J., Kleinberg, J., and Faloutsos, C. 2005. Graphs over time: densification laws, shrinking diameters and possible explanations. Pages 177–187 of: Proceedings of the eleventh ACM SIGKDDinternational conference on Knowledge discovery in data mining. ACM.
Leskovec, J., Kleinberg, J., and Faloutsos, C. 2007. Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data (TKDD), 1(1), 2.
Leskovec, J., Lang, K., Dasgupta, A., and Mahoney, M. 2009. Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Math., 6(1), 29–123.
Leskovec, J., Lang, K., and Mahoney, M. 2010. Empirical comparison of algorithms for network community detection. Pages 631–640 of: Proceedings of the 19th International Conference on World Wide Web. WWW–10. New York, NY, USA: ACM.
Liljeros, F., Edling, C. R., Amaral, L. A. N., and Stanley, H. E. 2001. The web of human sexual contacts. Nature, 411, 907.
Lindvall, T. 2002. Lectures on the coupling method. Dover Publications, Inc., Mineola, NY. Corrected reprint of the 1992 original.
Lint, J. H. van, and Wilson, R. M. 2001. A course in combinatorics. 2nd Cambridge: Cambridge University Press.
Litvak, N., and Hofstad, R. van der. 2013. Uncovering disassortativity in large scale-free networks. Phys.Rev. E, 87(2), 022801.
Lotka, A. J. 1926. The frequency distribution of scientific productivity. Journal of the Washington Academyof Sciences, 16(12), 317–323.
Lovász, L. 2012. Large networks and graph limits. American Mathematical Society Colloquium Publications, vol. 60. American Mathematical Society, Providence, RI.
Lu, L. 2002. Probabilistic methods in massive graphs and Internet computing. Ph.D. thesis, University of California, San Diego. Available at http://math.ucsd.edu/∼llu/thesis.pdf.
Łuczak, T. 1990a. Component behavior near the critical point of the random graph process. RandomStructures Algorithms, 1(3), 287–310.
Łuczak, T. 1990b. On the number of sparse connected graphs. Random Structures Algorithms, 1(2), 171–173.
Łuczak, T., Pittel, B., and Wierman, J. 1994. The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc., 341(2), 721–748.
Lyons, R., Pemantle, R., and Peres, Y. 1995. Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab., 23(3), 1125–1138.
Martin-Löf, A. 1986. Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Probab., 23(2), 265–282.
Martin-Löf, A. 1998. The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Probab., 35(3), 671–682.
Milgram, S. 1967. The small world problem. Psychology Today, May, 60–67.
Miller, J. C. 2009. Percolation and epidemics in random clustered networks. Phys. Rev. E, 80(Aug), 020901.
Mitzenmacher, M. 2004. A brief history of generative models for power law and lognormal distributions. Internet Math., 1(2), 226–251.
Molloy, M., and Reed, B. 1995. A critical point for random graphs with a given degree sequence. RandomStructures Algorithms, 6(2-3), 161–179.
Molloy, M., and Reed, B. 1998. The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput., 7(3), 295–305.
Móri, T. F. 2002. On random trees. Studia Sci. Math. Hungar., 39(1-2), 143–155.
Móri, T. F. 2005. The maximum degree of the Barabási–Albert random tree. Combin. Probab. Comput., 14(3), 339–348.
Nachmias, A., and Peres, Y. 2010. The critical random graph, with martingales. Israel J. Math., 176, 29–41.
Nerman, O., and Jagers, P. 1984. The stable double infinite pedigree process of supercritical branching populations. Z. Wahrsch. Verw. Gebiete, 65(3), 445–460.
Newman, M. E. J. 2000. Models of the small world. J. Stat. Phys., 101, 819–841.
Newman, M. E. J. 2001. The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA, 98, 404.
Newman, M. E. J. 2002. Assortative mixing in networks. Phys. Rev. Lett., 89(20), 208701.
Newman, M. E. J. 2003. The structure and function of complex networks. SIAM Rev., 45(2), 167–256 (electronic).
Newman, M. E. J. 2005. Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46(5), 323–351.
Newman, M. E. J. 2009. Random graphs with clustering. Phys. Rev. Lett., 103(Jul), 058701.
Newman, M. E. J. 2010. Networks: an introduction. Oxford University Press.
Newman, M. E. J., Strogatz, S., and Watts, D. 2002. Random graph models of social networks. Proc. Nat.Acad. Sci., 99, 2566–2572.
Newman, M. E. J., Watts, D. J., and Barabási, A. -L. 2006. The structure and dynamics of networks. Princeton Studies in Complexity. Princeton University Press.
Norros, I., and Reittu, H. 2006. On a conditionally Poissonian graph process. Adv. in Appl. Probab., 38(1), 59–75.
O'Connell, N. 1998. Some large deviation results for sparse random graphs. Probab. Theory Related Fields, 110(3), 277–285.
Okamoto, M. 1958. Some inequalities relating to the partial sum of binomial probabilities. Ann. Inst. Statist.Math., 10, 29–35.
Oliveira, R., and Spencer, J. 2005. Connectivity transitions in networks with super-linear preferential attachment. Internet Math., 2(2), 121–163.
Olivieri, E., and Vares, M. E. 2005. Large deviations and metastability. Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press.
Ostroumova, L., and Grechnikov, E. 2012. The distribution of second degrees in the Bollobás–Riordan random graph model. Mosc. J. Comb. Number Theory, 2(2), 85–110.
Otter, R. 1949. The multiplicative process. Ann. Math. Statist., 20, 206–224.
Pansiot, J.-J., and Grad, D. 1998. On routes and multicast trees in the Internet. ACM SIGCOMM ComputerCommunication Review, 28(1), 41–50.
Pareto, V. 1896. Cours d'economie politique. Geneva, Switserland: Droz.
Pastor-Satorras, R., and Vespignani, A. 2007. Evolution and structure of the Internet: A statistical physicsapproach. Cambridge University Press.
Peköz, E., Röllin, A., and Ross, N. 2013. Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab., 23(3), 1188–1218.
Peköz, E., Röllin, A., and Ross, N. Preprint (2014). Joint degree distributions of preferential attachmentrandom graphs. Available from http://arxiv.org/pdf/1402.4686.pdf.
Pitman, J., and Yor, M. 1997. The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab., 25(2), 855–900.
Pittel, B. 1990. On tree census and the giant component in sparse random graphs. Random StructuresAlgorithms, 1(3), 311–342.
Pittel, B. 2001. On the largest component of the random graph at a nearcritical stage. J. Combin. TheorySer. B, 82(2), 237–269.
Pool, I., and Kochen, M. 1978. Contacts and influence. Social Networks, 1, 5–51.
Rényi, A. 1959. On connected graphs. I. Magyar Tud. Akad. Mat. Kutató Int. Közl., 4, 385–388.
Resnick, S. 2007. Heavy-tail phenomena. Springer Series in Operations Research and Financial Engineering. Springer, New York. Probabilistic and statistical modeling.
Ross, N. 2013. Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Adv. in Appl. Probab., 45(3), 876–893.
Rudas, A., Tóth, B., and Valkó, B. 2007. Random trees and general branching processes. Random StructuresAlgorithms, 31(2), 186–202.
Seneta, E. 1969. Functional equations and the Galton-Watson process. Advances in Appl. Probability, 1, 1–42.
Sierksma, G., and Hoogeveen, H. 1991. Seven criteria for integer sequences being graphic. J. Graph Theory, 15(2), 223–231.
Siganos, G., Faloutsos, M., Faloutsos, P., and Faloutsos, C. 2003. Power laws and the AS-level Internet topology. IEEE/ACM Trans. Netw., 11(4), 514–524.
Simon, H. A. 1955. On a class of skew distribution functions. Biometrika, 42, 425–440.
Solomonoff, R., and Rapoport, A. 1951. Connectivity of random nets. Bull. Math. Biophys., 13, 107–117.
Spencer, J. 1997. Enumerating graphs and Brownian motion. Comm. Pure Appl. Math., 50(3), 291–294.
Spitzer, F. 1956. A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc., 82, 323–339.
Spitzer, F. 1976. Principles of random walk. 2nd edn. New York: Springer.
Sporns, O. 2011. Networks of the Brain. MIT Press.
Strassen, V. 1965. The existence of probability measures with given marginals. Ann. Math. Statist., 36, 423–439.
Strogatz, S. 2001. Exploring complex networks. Nature, 410(8), 268–276.
Szemerédi, E. 1978. Regular partitions of graphs. Pages 399–401 of: Problèmes combinatoires et théoriedes graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). Colloq. Internat. CNRS, vol. 260. CNRS, Paris.
Szymanski, J. 2005. Concentration of vertex degrees in a scale-free random graph process. RandomStructures Algorithms, 26(1-2), 224–236.
Thorisson, H. 2000. Coupling, stationarity, and regeneration. Probability and its applications (New York). New York: Springer-Verlag.
Trapman, P. 2007. On analytical approaches to epidemics on networks. Theoretical Population Biology, 71(2), 160–173.
Travers, J., and Milgram, S. 1969. An experimental study of the small world problem. Sociometry, 32, 425–443.
Ugander, J., Karrer, B., Backstrom, L., and Marlow, C. Preprint (2011). The anatomy of the Facebook social graph. Available from http://arxiv.org/pdf/1111.4503.pdf.
Vega-Redondo, F. 2007. Complex social networks. Econometric Society Monographs, vol. 44. Cambridge University Press, Cambridge.
Viswanath, B., Mislove, A., Cha, M., and Gummadi, K. 2009. On the evolution of user inter-action in Facebook. Pages 37–42 of: Proceedings of the 2Nd ACM Workshop on Online Social Networks. WOSN-09. New York, NY, USA: ACM.
Watts, D. J. 1999. Small worlds. The dynamics of networks between order and randomness. Princeton Studies in Complexity. Princeton, NJ: Princeton University Press.
Watts, D. J. 2003. Six degrees. The science of a connected age. New York: W. W. Norton & Co. Inc.
Watts, D. J., and Strogatz, S. H. 1998. Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442.
Wendel, J. G. 1975. Left-continuous random walk and the Lagrange expansion. Amer. Math. Monthly, 82, 494–499.
Williams, D. 1991. Probability with martingales. Cambridge Mathematical Textbooks. Cambridge: Cambridge University Press.
Willinger, W., Alderson, D., and Doyle, J. C. 2009. Mathematics and the Internet: A source of enormous confusion and great potential. Notices of the American Mathematical Society, 56(5), 586–599.
Willinger, W., Govindan, R., Jamin, S., Paxson, V., and Shenker, S. 2002. Scaling phenomena in the Internet: Critically examining criticality. Proc. Natl. Acad. Sci., 99, 2573–2580.
Wilson, R., Gosling, S., and Graham, L. 2012. A review of Facebook research in the social sciences. Perspectives on Psychological Science, 7(3), 203–220.
Wright, E. M. 1977. The number of connected sparsely edged graphs. J. Graph Theory, 1(4), 317–330.
Wright, E. M. 1980. The number of connected sparsely edged graphs. III. Asymptotic results. J. GraphTheory, 4(4), 393–407.
Yook, S.-H., Jeong, H., and Barabási, A. -L. 2002. Modeling the Internet's large-scale topology. Proc. Natl.Acad. Sci., 99(22), 13382–13386.
Yule, G. U. 1925. A mathematical theory of evolution, based on the conclusions of Dr. J. C., Willis F. R. S. Phil. Trans. Roy. Soc. London, B, 213, 21–87.
Zipf, G. K. 1929. Relative frequency as a determinant of phonetic change. Harvard Studies in ClassicalPhilology, 15, 1–95.

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