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.Google Scholar

Addario-Berry, L., Broutin, N., and Goldschmidt, C. 2010. Critical random graphs: limiting constructions and distributional properties. Electron. J. Probab., 15(25), 741–775.Google Scholar

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.Google Scholar

Albert, R., Jeong, H., and Barabási, A. -L. 1999. Internet: diameter of the World-Wide Web. Nature, 401, 130–131.Google Scholar

Albert, R., Jeong, H., and Barabási, A. -L. 2001. Error and attack tolerance of complex networks. Nature, 406, 378–382.Google Scholar

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.Google Scholar

Aldous, D. 1993. Tree-based models for random distribution of mass. J. Stat. Phys., 73, 625–641.Google Scholar

Aldous, D. 1997. Brownian excursions, critical random graphs and the multiplicative coalescent. Ann.Probab., 25(2), 812–854.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Azuma, K. 1967. Weighted sums of certain dependent random variables. Tohoku Math. J., 3, 357–367.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Bavelas, A. 1950. Communication patterns in task-oriented groups. Journ. Acoust. Soc. Amer., 22(64), 725–730.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Bennet, G. 1962. Probability inequaltities for the sum of independent random variables. J. Amer. Statist.Assoc., 57, 33–45.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Bianconi, G., and Barabási, A. -L. 2001b. Competition and multiscaling in evolving networks. Europhys.Lett., 54, 436–442.Google Scholar

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.Google Scholar

Bollobás, B. 1980. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin., 1(4), 311–316.Google Scholar

Bollobás, B. 1981. Degree sequences of random graphs. Discrete Math., 33(1), 1–19.Google Scholar

Bollobás, B. 1984a. The evolution of random graphs. Trans. Amer. Math. Soc., 286(1), 257–274.Google Scholar

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.Google Scholar

Bollobás, B., and Riordan, O. 2004. The diameter of a scale-free random graph. Combinatorica, 24(1), 5–34.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Britton, T., Deijfen, M., andMartin-Löf, A. 2006. Generating simple random graphs with prescribed degree distribution. J. Stat. Phys., 124(6), 1377–1397.Google Scholar

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.Google Scholar

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.Google Scholar

Cayley, A. 1889. A theorem on trees. Q. J. Pure Appl. Math., 23, 376–378.Google Scholar

Champernowne, D. G. 1953. A model of income distribution. Econ. J., 63, 318.Google Scholar

Chen, N., and Olvera-Cravioto, M. 2013. Directed random graphs with given degree distributions. Stoch.Syst., 3(1), 147–186.Google Scholar

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.Google Scholar

Choudum, S. A. 1986. A simple proof of the Erdős-Gallai theorem on graph sequences. Bull. Austral. Math.Soc., 33(1), 67–70.Google Scholar

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).Google Scholar

Chung, F., and Lu, L. 2002b. Connected components in random graphs with given expected degree sequences. Ann. Comb., 6(2), 125–145.Google Scholar

Chung, F., and Lu, L. 2003. The average distance in a random graph with given expected degrees. InternetMath., 1(1), 91–113.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Clauset, A., Shalizi, C., and Newman, M. E. J. 2009. Power-law distributions in empirical data. SIAM. review, 51(4), 661–703.Google Scholar

Cohen, R., Erez, K., ben Avraham, D., and Havlin, S. 2000. Resilience of the Internet to random breakdowns. Phys. Rev. Letters, 85, 4626.Google Scholar

Cohen, R., Erez, K., ben Avraham, D., and Havlin, S. 2001. Breakdown of the Internet under intentional attack. Phys. Rev. Letters, 86, 3682.Google Scholar

Cooper, C., and Frieze, A. 2003. A general model of web graphs. Random Structures Algorithms, 22(3), 311–335.Google Scholar

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.Google Scholar

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.Google Scholar

Coupechoux, E., and Lelarge, M. 2014. How clustering affects epidemics in random networks. Adv. in Appl.Probab., 46(4), 985–1008.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Dereich, S., and Mörters, P. 2011. Random networks with concave preferential attachment rule. Jahresber.Dtsch. Math.-Ver., 113(1), 21–40.Google Scholar

Dereich, S., and Mörters, P. 2013. Random networks with sublinear preferential attachment: the giant component. Ann. Probab., 41(1), 329–384.Google Scholar

Dodds, P., Muhamad, R., and Watts, D. 2003. An experimental study of search in global social networks. Science, 301(5634), 827–829.Google Scholar

Dommers, S., Hofstad, R. van der, and Hooghiemstra, G. 2010. Diameters in preferential attachment graphs. Journ. Stat. Phys., 139, 72–107.Google Scholar

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.Google Scholar

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.Google Scholar

Dwass, M. 1968. A theorem about infinitely divisible distributions. Z. Wahrscheinleikheitsth., 9, 206–224.Google Scholar

Dwass, M. 1969. The total progeny in a branching process and a related random walk. J. Appl. Prob., 6, 682–686.Google Scholar

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.Google Scholar

Eckhoff, M., and Mörters, P. 2014. Vulnerability of robust preferential attachment networks. Electron. J.Probab., 19, no. 57, 47.Google Scholar

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.Google Scholar

Erdős, P., and Gallai, T. 1960. Graphs with points of prescribed degrees. (Hungarian). Mat. Lapok, 11, 264–274.Google Scholar

Erdős, P., and Rényi, A. 1959. On random graphs. I. Publ. Math. Debrecen, 6, 290–297.Google Scholar

Erdős, P., and Rényi, A. 1960. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int.Közl., 5, 17–61.Google Scholar

Erdős, P., and Rényi, A. 1961a. On the evolution of random graphs. Bull. Inst. Internat. Statist., 38, 343–347.Google Scholar

Erdős, P., and Rényi, A. 1961b. On the strength of connectedness of a random graph. Acta Math. Acad. Sci.Hungar., 12, 261–267.Google Scholar

Erdős, P., and Wilson, R. J. 1977. On the chromatic index of almost all graphs. J. Combinatorial TheorySer. B, 23(2–3), 255–257.Google Scholar

Ergün, G., and Rodgers, G. J. 2002. Growing random networks with fitness. Phys. A, 303, 261–272.Google Scholar

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.Google Scholar

Faloutsos, C., Faloutsos, P., and Faloutsos, M. 1999. On power-law relationships of the internet topology. Computer Communications Rev., 29, 251–262.Google Scholar

Feld, S. L. 1991. Why your friends have more friends than you do. American Journal of Sociology, 96(6), 1464–1477.Google Scholar

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.Google Scholar

Freeman, L. 1977. A set of measures in centrality based on betweenness. Sociometry, 40(1), 35–41.Google Scholar

Gao, P., and Wormald, N. 2016. Enumeration of graphs with a heavy-tailed degree sequence. Adv. Math., 287, 412–450.Google Scholar

Gilbert, E. N. 1959. Random graphs. Ann. Math. Statist., 30, 1141–1144.Google Scholar

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.Google Scholar

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.Google Scholar

Hagberg, O., andWiuf, C. 2006. Convergence properties of the degree distribution of some growing network models. Bull. Math. Biol., 68, 1275–1291.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Hofstad, R. van der, and Keane, M. 2008. An elementary proof of the hitting time theorem. Amer. Math.Monthly, 115(8), 753–756.Google Scholar

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.Google Scholar

Hofstad, R. van der, and Spencer, J. 2006. Counting connected graphs asymptotically. European J. Combin., 27(8), 1294–1320.Google Scholar

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.Google Scholar

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.Google Scholar

Janson, S. 2007. Monotonicity, asymptotic normality and vertex degrees in random graphs. Bernoulli, 13(4), 952–965.Google Scholar

Janson, S. 2009. The probability that a random multigraph is simple. Combin. Probab. Comput., 18(1-2), 205–225.Google Scholar

Janson, S. 2010. Asymptotic equivalence and contiguity of some random graphs. Random StructuresAlgorithms, 36(1), 26–45.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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).Google Scholar

Jin, S., and Bestavros, A. 2006. Small-world characteristics of Internet topologies and implications on multicast scaling. Computer Networks, 50, 648–666.Google Scholar

Jordan, J. 2006. The degree sequences and spectra of scale-free random graphs. Random StructuresAlgorithms, 29(2), 226–242.Google Scholar

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.Google Scholar

Karp, R. M. 1990. The transitive closure of a random digraph. Random Structures Algorithms, 1(1), 73–93.Google Scholar

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.Google Scholar

Kesten, H., and Stigum, B. P. 1966b. A limit theorem for multidimensional Galton–Watson processes. Ann.Math. Statist., 37, 1211–1223.Google Scholar

Kesten, H., and Stigum, B. P. 1967. Limit theorems for decomposable multi-dimensional Galton–Watson processes. J. Math. Anal. Appl., 17, 309–338.Google Scholar

Kleinberg, J. M. 1999. Authoritative sources in a hyperlinked environment. J. ACM, 46(5), 604–632.Google Scholar

Kleinberg, J. M. 2000a. Navigation in a small world. Nature, 406, 845.Google Scholar

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.Google Scholar

Krapivsky, P. L., and Redner, S. 2001. Organization of growing random networks. Phys. Rev. E, 63, 066123.Google Scholar

Krapivsky, P. L., and Redner, S. 2005. Network growth by copying. Phys. Rev. E, 71(3), 036118.Google Scholar

Krapivsky, P. L., Redner, S., and Leyvraz, F. 2000. Connectivity of growing random networks. Phys. Rev.Lett., 85, 4629.Google Scholar

Krioukov, D., Kitsak, M., Sinkovits, R., Rideout, D., Meyer, D., and Boguñá, M. 2012. Network cosmology. Scientific reports, 2.Google Scholar

Kumar, R., Raghavan, P., Rajagopalan, S, and Tomkins, A. 1999. Trawling the Web for emerging cyber communities. Computer Networks, 31, 1481–1493.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Lotka, A. J. 1926. The frequency distribution of scientific productivity. Journal of the Washington Academyof Sciences, 16(12), 317–323.Google Scholar

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.Google Scholar

Łuczak, T. 1990b. On the number of sparse connected graphs. Random Structures Algorithms, 1(2), 171–173.Google Scholar

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.Google Scholar

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.Google Scholar

Martin-Löf, A. 1986. Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Probab., 23(2), 265–282.Google Scholar

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.Google Scholar

Milgram, S. 1967. The small world problem. Psychology Today, May, 60–67.Google Scholar

Miller, J. C. 2009. Percolation and epidemics in random clustered networks. Phys. Rev. E, 80(Aug), 020901.Google Scholar

Mitzenmacher, M. 2004. A brief history of generative models for power law and lognormal distributions. Internet Math., 1(2), 226–251.Google Scholar

Molloy, M., and Reed, B. 1995. A critical point for random graphs with a given degree sequence. RandomStructures Algorithms, 6(2-3), 161–179.Google Scholar

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.Google Scholar

Móri, T. F. 2002. On random trees. Studia Sci. Math. Hungar., 39(1-2), 143–155.Google Scholar

Móri, T. F. 2005. The maximum degree of the Barabási–Albert random tree. Combin. Probab. Comput., 14(3), 339–348.Google Scholar

Nachmias, A., and Peres, Y. 2010. The critical random graph, with martingales. Israel J. Math., 176, 29–41.Google Scholar

Nerman, O., and Jagers, P. 1984. The stable double infinite pedigree process of supercritical branching populations. Z. Wahrsch. Verw. Gebiete, 65(3), 445–460.Google Scholar

Newman, M. E. J. 2000. Models of the small world. J. Stat. Phys., 101, 819–841.Google Scholar

Newman, M. E. J. 2001. The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA, 98, 404.Google Scholar

Newman, M. E. J. 2002. Assortative mixing in networks. Phys. Rev. Lett., 89(20), 208701.Google Scholar

Newman, M. E. J. 2003. The structure and function of complex networks. SIAM Rev., 45(2), 167–256 (electronic).Google Scholar

Newman, M. E. J. 2005. Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46(5), 323–351.Google Scholar

Newman, M. E. J. 2009. Random graphs with clustering. Phys. Rev. Lett., 103(Jul), 058701.Google Scholar

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.Google Scholar

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.Google Scholar

O'Connell, N. 1998. Some large deviation results for sparse random graphs. Probab. Theory Related Fields, 110(3), 277–285.Google Scholar

Okamoto, M. 1958. Some inequalities relating to the partial sum of binomial probabilities. Ann. Inst. Statist.Math., 10, 29–35.Google Scholar

Oliveira, R., and Spencer, J. 2005. Connectivity transitions in networks with super-linear preferential attachment. Internet Math., 2(2), 121–163.Google Scholar

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.Google Scholar

Otter, R. 1949. The multiplicative process. Ann. Math. Statist., 20, 206–224.Google Scholar

Pansiot, J.-J., and Grad, D. 1998. On routes and multicast trees in the Internet. ACM SIGCOMM ComputerCommunication Review, 28(1), 41–50.Google Scholar

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.Google Scholar

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.Google Scholar

Pittel, B. 1990. On tree census and the giant component in sparse random graphs. Random StructuresAlgorithms, 1(3), 311–342.Google Scholar

Pittel, B. 2001. On the largest component of the random graph at a nearcritical stage. J. Combin. TheorySer. B, 82(2), 237–269.Google Scholar

Pool, I., and Kochen, M. 1978. Contacts and influence. Social Networks, 1, 5–51.Google Scholar

Rényi, A. 1959. On connected graphs. I. Magyar Tud. Akad. Mat. Kutató Int. Közl., 4, 385–388.Google Scholar

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.Google Scholar

Rudas, A., Tóth, B., and Valkó, B. 2007. Random trees and general branching processes. Random StructuresAlgorithms, 31(2), 186–202.Google Scholar

Seneta, E. 1969. Functional equations and the Galton-Watson process. Advances in Appl. Probability, 1, 1–42.Google Scholar

Sierksma, G., and Hoogeveen, H. 1991. Seven criteria for integer sequences being graphic. J. Graph Theory, 15(2), 223–231.Google Scholar

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.Google Scholar

Simon, H. A. 1955. On a class of skew distribution functions. Biometrika, 42, 425–440.Google Scholar

Solomonoff, R., and Rapoport, A. 1951. Connectivity of random nets. Bull. Math. Biophys., 13, 107–117.Google Scholar

Spencer, J. 1997. Enumerating graphs and Brownian motion. Comm. Pure Appl. Math., 50(3), 291–294.Google Scholar

Spitzer, F. 1956. A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc., 82, 323–339.Google Scholar

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.Google Scholar

Strogatz, S. 2001. Exploring complex networks. Nature, 410(8), 268–276.Google Scholar

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.Google Scholar

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.Google Scholar

Travers, J., and Milgram, S. 1969. An experimental study of the small world problem. Sociometry, 32, 425–443.Google Scholar

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.Google Scholar

Wendel, J. G. 1975. Left-continuous random walk and the Lagrange expansion. Amer. Math. Monthly, 82, 494–499.Google Scholar

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.Google Scholar

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.Google Scholar

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.Google Scholar

Wright, E. M. 1977. The number of connected sparsely edged graphs. J. Graph Theory, 1(4), 317–330.Google Scholar

Wright, E. M. 1980. The number of connected sparsely edged graphs. III. Asymptotic results. J. GraphTheory, 4(4), 393–407.Google Scholar

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.Google Scholar

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.Google Scholar

Zipf, G. K. 1929. Relative frequency as a determinant of phonetic change. Harvard Studies in ClassicalPhilology, 15, 1–95.Google Scholar