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The small-community phenomenon in networks

  • ANGSHENG LI (a1) and PAN PENG (a2)

Abstract

We investigate several geometric models of networks that simultaneously have some nice global properties, including the small-diameter property, the small-community phenomenon, which is defined to capture the common experience that (almost) everyone in society also belongs to some meaningful small communities, and the power law degree distribution, for which our result significantly strengthens those given in van den Esker (2008) and Jordan (2010). These results, together with our previous work in Li and Peng (2011), build a mathematical foundation for the study of both communities and the small-community phenomenon in various networks.

In the proof of the power law degree distribution, we develop the method of alternating concentration analysis to build a concentration inequality by alternately and iteratively applying both the sub- and super-martingale inequalities, which seems to be a powerful technique with further potential applications.

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This research was partially supported by NSFC distinguished young investigator award number 60325206, and its matching fund from the Hundred-Talent Program of the Chinese Academy of Sciences. Both authors are partially supported by the Grand Project ‘Network Algorithms and Digital Information’ of the Institute of Software, Chinese Academy of Sciences.

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Albert, R., Jeong, H. and Barabási, A.-L. (1999) The diameter of the World Wide Web. Nature 401 130131.
Allen, C. (2004) Life with alacrity: The dunbar number as a limit to group sizes. Available at http://www.lifewithalacrity.com/2004/03/the_dunbar_numb.html.
Avin, C. (2008) Distance graphs: from random geometric graphs to Bernoulli graphs and between. In: Proceedings of the fifth international workshop on Foundations of mobile computing 71–78.
Bollobás, B. (2001) Random Graphs (second edition), Cambridge University Press.
Bollobás, B. (2003) Mathematical results on scale-free random graphs. In: In Handbook of Graphs and Networks, Wiley-VCH 134.
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 (1–6)309320.
Chung, F. and Lu, L. (2004) The small world phenomenon in hybrid power law graphs. In: Ben-Naim, E. et al. (eds.) Complex Networks, Springer-Verlag 91106.
Chung, F. and Lu, L. (2006) Complex Graphs and Networks (CBMS Regional Conference Series in Mathematics), American Mathematical Society.
Devroye, L. and Lu, J. (1995) The strong convergence of maximal degrees in uniform random recursive trees and dags. Random Structures and Algorithms 7 114.
Dubhashi, D. and Panconesi, A. (2009) Concentration of Measure for the Analysis of Randomized Algorithms (first edition), Cambridge University Press.
Flaxman, A. D., Frieze, A. M. and Vera, J. (2007a) A geometric preferential attachment model of networks. Internet Mathematics 3 (2)187205.
Flaxman, A. D., Frieze, A. M. and Vera, J. (2007b) A geometric preferential attachment model of networks II. Internet Mathematics 4 (1)87111.
Fraigniaud, P. and Giakkoupis, G. (2009) The effect of power-law degrees on the navigability of small worlds [extended abstract]. In: Proceedings of the 28th ACM symposium on Principles of distributed computing: PODC '09, ACM 240249.
Groh, G. and Rappel, V. (2009) Towards demarcation and modeling of small sub-communities/groups in p2p social networks. In: CSE '09: Proceedings of the 2009 International Conference on Computational Science and Engineering, IEEE Computer Society 304311.
Horton, D. and Wohl, R. R. (1956) Mass communication and para-social interaction: Observations on intimacy at a distance. Psychiatry 19 (3)215229.
Jordan, J. (2010) Degree sequences of geometric preferential attachment graphs. Advances in Applied Probability 42 319330.
Kannan, R., Vempala, S. and Vetta, A. (2004) On clusterings: Good, bad and spectral. Journal of the ACM 51 (3)497515.
Kleinberg, J. (2000) The small-world phenomenon: an algorithmic perspective. In: Proceedings of the 32nd ACM Symposium on the Theory of Computing.
Kleinberg, J. M., Kumar, R., Raghavan, P., Rajagopalan, S. and Tomkins, A. S. (1999) The Web as a graph: measurements, models and methods. In: Proceedings of the 5th Annual International Computing and Combinatorics Conference (COCOON). Springer-Verlag Lecture Notes in Computer Science 1627 118.
Kurucz, M. and Benczúr, A. A. (2010) Geographically organized small communities and the hardness of clustering social networks. Annals of Information Systems – Data Mining for Social Network Analyis.
Lang, K. (2005) Fixing two weaknesses of the spectral method. In: NIPS '05. Advances in Neural Information Processing Systems 18.
Leskovec, J., Lang, K. J., Dasgupta, A. and Mahoney, M. W. (2008) Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. (Informal publication: CoRR abs/0810.1355.)
Leskovec, J., Lang, K. J. and Mahoney, M. (2010) Empirical comparison of algorithms for network community detection. In: Proceedings of the 19th international conference on World wide web: WWW '10, ACM 631640.
Li, A. and Peng, P. (2011) Communities structures in classical network models. Internet Mathematics 7 (2)81106.
Penrose, M. (2003) Random Geometric Graphs, Oxford University Press.
Pittel, B. (1994) Note on the heights of random recursive trees and random m-ary search trees. Random Structures and Algorithms 5 337347.
Ravasz, E. and Barabási, A.-L. (2003) Hierarchical organization in complex networks. Physical Review E 67 026112.
Smythe, R. and Mahmoud, H. (1995) A survey of recursive trees. Theory of Probability and Mathematical Statistics 51 127.
van den Esker, H. (2008) A geometric preferential attachment model with fitness. (Available at http://arxiv.org/abs/0801.1612.)

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The small-community phenomenon in networks

  • ANGSHENG LI (a1) and PAN PENG (a2)

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