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SMALL-WORLD EFFECT IN GEOGRAPHICAL ATTACHMENT NETWORKS

Part of: Operations research and management science Graph theory Markov processes

Published online by Cambridge University Press:  12 September 2019

Qunqiang Feng ,
Yongkang Wang  and
Zhishui Hu Open the ORCID record for Zhishui Hu [Opens in a new window]
Qunqiang Feng
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, 230026, China E-mails: [email protected]; [email protected]; [email protected]
Yongkang Wang
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, 230026, China E-mails: [email protected]; [email protected]; [email protected]
Zhishui Hu
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, 230026, China E-mails: [email protected]; [email protected]; [email protected]
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Abstract

In this work, we use rigorous probabilistic methods to study the asymptotic degree distribution, clustering coefficient, and diameter of geographical attachment networks. As a type of small-world network model, these networks were first proposed in the physical literature, where they were analyzed only with heuristic arguments and computational simulations.

Keywords

degree distributiondiametergeographical attachmentrandom networkssmall-world

MSC classification

Primary: 05C80: Random graphs 90B15: Network models, stochastic
Secondary: 60J85: Applications of branching processes
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 2 , April 2021 , pp. 276 - 296
Copyright
Copyright © Cambridge University Press 2019

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References

1.Athreya, K.B. & Ney, P.E. (1972). Branching processes. Berlin: Springer-Verlag.CrossRefGoogle Scholar
2.Azuma, K. (1967). Weighted sums of certain dependent variables. Tohoku Mathematical Journal 3: 357–367.Google Scholar
3.Bartle, R.G. & Sherbert, D.R. (2011). Introduction to Real analysis (4th ed.). USA: John Wiley & Sons.Google Scholar
4.Bellman, R. & Harris, T.E. (1948). On the theory of age-dependent stochastic branching processes. Proceedings of the National Academy of Sciences of the United States of America 34: 601–604.CrossRefGoogle ScholarPubMed
5.Bhamidi, S., Steele, J.M., & Zaman, T. (2015). Twitter event networks and the superstar model. The Annals of Applied Probability 25: 2462–2502.CrossRefGoogle Scholar
6.Bilke, S. & Peterson, C. (2001). Topological properties of citation and metabolic networks. Physical Review E 64: 036106.CrossRefGoogle ScholarPubMed
7.Bollobás, B., Riordan, O., Spencer, J., & Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures and Algorithms 18: 279–290.CrossRefGoogle Scholar
8.Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., & Wiener, J. (2000). Graph structure in the web. Computer Networks 33: 309–320.CrossRefGoogle Scholar
9.Broutin, N. & Devroye, L. (2006). Large deviations for the weighted height of an extended class of trees. Algorithmica 46: 271–297.CrossRefGoogle Scholar
10.Ebrahimzadeh, E., Farczadi, L., Gao, P., Mehrabian, A., Sato, C.M., Wormald, N., & Zung, J. (2014). On the longest paths and diameter in random Apollonian networks. Random Structures and Algorithms 45: 703–725.CrossRefGoogle Scholar
11.Eggenberger, F. & Pólya, G. (1923). Über die statistik verketteter vorgäge. Zeitschrift für Angewandte Mathematik und Mechanik 3: 279–289.CrossRefGoogle Scholar
12.Frieze, A. & Tsourakakis, C.E. (2012). On certain properties of random Apollonian networks. In Algorithms and Models for the Web Graph. Springer, Berlin, pp. 93–112.CrossRefGoogle Scholar
13.Hayashi, Y. (2006). A review of recent studies of geographical scale-free networks. IPSJ Digital Courier 2: 155–164.CrossRefGoogle Scholar
14.Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58: 13–30.CrossRefGoogle Scholar
15.Kleinberg, J.M. (2000). The small-world phenomenon: An algorithmic perspective. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing. Association of Computing Machinery, New York, pp. 163–170.Google Scholar
16.Kolossváry, I., Komjáthy, J., & Vágó, L. (2016). Degrees and distances in random and evolving apollonian networks. Advances in Applied Probability 48: 865–902.CrossRefGoogle Scholar
17.Mahmoud, H. (2008). Pólya urn models. Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
18.Milgram, S. (1967). The small world problem. Psychology Today 1: 60–67.Google Scholar
19.Moore, C. & Newman, M.E.J. (1999). Epidemics and percolation in small-world networks. Physical Review E 61: 5678–5682.CrossRefGoogle Scholar
20.Newman, M.E.J. (2001). The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences of the USA 98: 404–409.CrossRefGoogle ScholarPubMed
21.Newman, M.E.J. (2003). The structure and function of complex networks. SIAM Review 45: 167–256.CrossRefGoogle Scholar
22.Newman, M.E.J. & Watts, D.J. (1999). Renormalization group analysis of the small-world network model. Physics Letters A 263: 341–346.CrossRefGoogle Scholar
23.Ozik, J., Hunt, B.R., & Ott, E. (2004). Growing networks with geographical attachment preference: Emergence of small worlds. Physical Review E 69: 026108.CrossRefGoogle ScholarPubMed
24.Pittel, B. (1994). Note on the heights of random recursive trees and random m-ary search trees. Random Structures and Algorithms 5: 337–347.CrossRefGoogle Scholar
25.Van der Hofstad, R. (2018). Random graphs and complex networks: Volume II. Unpublished manuscript. http://www.win.tue.nl/rhofstad/NotesRGCNII.pdf.Google Scholar
26.Van Noort, V., Snel, B., & Huynen, M.A. (2004). The yeast coexpression network has a small-world, scale-free architecture and can be explained by a simple model. EMBO Reports 5: 280–284.CrossRefGoogle Scholar
27.Wasserman, S. & Faust, K. (1994). Social network analysis: Methods and applications. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
28.Watts, D.J. & Strogatz, S.H. (1998). Collective dynamics of ‘small-world’ networks. Nature 393: 440–442.CrossRefGoogle ScholarPubMed
29.Wu, J., Tse, C.K., Lau, F.C.M., & Ho, I.W.H. (2013). Analysis of communication network performance from a complex network perspective. IEEE Transactions on Circuits and Systems 60: 3303–3316.CrossRefGoogle Scholar
30.Zaidi, F. (2013). Small world networks and clustered small world networks with random connectivity. Social Network Analysis and Mining 3: 51–63.CrossRefGoogle Scholar
31.Zhang, Z., Rong, L., & Comellas, F. (2006). Evolving small-world networks with geographical attachment preference. Journal of Physics A: Mathematical and General 39: 3253–3261.CrossRefGoogle Scholar
32.Zhang, Z., Rong, L., & Guo, C. (2006). A deterministic small-world network created by edge iterations. Physica A 363: 567–572.CrossRefGoogle Scholar
33.Zhou, T., Yan, G., & Wang, B.-H. (2005). Maximal planar networks with large clustering coefficient and power-law degree distribution. Physical Review E 71: 046141.CrossRefGoogle ScholarPubMed