Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-8hm5d Total loading time: 0.283 Render date: 2022-05-17T18:05:45.938Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Joint degree distributions of preferential attachment random graphs

Published online by Cambridge University Press:  26 June 2017

Erol Peköz*
Affiliation:
Boston University
Adrian Röllin*
Affiliation:
National University of Singapore
Nathan Ross*
Affiliation:
University of Melbourne
*
* Postal address: Questrom School of Business, Boston University, 595 Commonwealth Avenue, Room 607, Boston, MA 02215, USA. Email address: pekoz@bu.edu
** Postal address: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore. Email address: adrian.roellin@nus.edu.sg
*** Postal address: School of Mathematics and Statistics, University of Melbourne, Richard Berry Building, VIC 3010, Australia. Email address: nathan.ross@unimelb.edu.au

Abstract

We study the joint degree counts in linear preferential attachment random graphs and find a simple representation for the limit distribution in infinite sequence space. We show weak convergence with respect to the p-norm topology for appropriate p and also provide optimal rates of convergence of the finite-dimensional distributions. The results hold for models with any general initial seed graph and any fixed number of initial outgoing edges per vertex; we generate nontree graphs using both a lumping and a sequential rule. Convergence of the order statistics and optimal rates of convergence to the maximum of the degrees is also established.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aldous, D. (1991). The continuum random tree. I. Ann. Prob. 19, 128. CrossRefGoogle Scholar
[2] Aldous, D. (1993). The continuum random tree. III. Ann. Prob. 21, 248289. CrossRefGoogle Scholar
[3] Antunović, T., Mossel, E. and Rácz, M. Z. (2016). Coexistence in preferential attachment networks. Combinatorics Prob. Comput. 25, 797822. CrossRefGoogle Scholar
[4] Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509512. Google ScholarPubMed
[5] Berger, N., Borgs, C., Chayes, J. T. and Saberi, A. (2014). Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Prob. 42, 140. CrossRefGoogle Scholar
[6] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279290. CrossRefGoogle Scholar
[7] Bubeck, S., Mossel, E. and Rácz, M. Z. (2015). On the influence of the seed graph in the preferential attachment model. IEEE Trans. Network Sci. Eng. 2, 3039. CrossRefGoogle Scholar
[8] Collevecchio, A., Cotar, C. and LiCalzi, M. (2013). On a preferential attachment and generalized Polya's urn model. Ann. Appl. Prob. 23, 12191253. CrossRefGoogle Scholar
[9] Curien, N., Duquesne, T., Kortchemski, I. and Manolescu, I. (2015). Scaling limits and influence of the seed graph in preferential attachment trees. J. Éc. Polytech. Math. 2, 134. CrossRefGoogle Scholar
[10] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York. Google Scholar
[11] Goldstein, L. and Reinert, G. (2013). Stein's method for the beta distribution and the Pólya–Eggenberger urn. J. Appl. Prob. 50, 11871205. CrossRefGoogle Scholar
[12] James, L. F. (2015). Generalized Mittag Leffler distributions arising as limits in preferential attachment models. Preprint. Available at https://arxiv.org/abs/1509.07150v4. Google Scholar
[13] Janson, S. (2006). Limit theorems for triangular urn schemes. Prob. Theory Relat. Fields 134, 417452. CrossRefGoogle Scholar
[14] Krapivsky, P. L., Redner, S. and Leyvraz, F. (2000). Connectivity of growing random networks. Phys. Rev. Lett. 85, 4629. CrossRefGoogle Scholar
[15] Lieb, E. H. and Loss, M. (2001). Analysis (Graduate Stud. Math. 14), 2nd edn. American Mathematical Society, Providence, RI. CrossRefGoogle Scholar
[16] Móri, T. F. (2005). The maximum degree of the Barabási–Albert random tree. Combinatorics Prob. Comput. 14, 339348. CrossRefGoogle Scholar
[17] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167256. CrossRefGoogle Scholar
[18] Newman, M., Barabási, A.-L. and Watts, D. J. (2006). The Structure and Dynamics of Networks. Princeton University Press. Google Scholar
[19] Peköz, E. A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Prob. 23, 11881218. CrossRefGoogle Scholar
[20] Peköz, E. A., Röllin, A. and Ross, N. (2013). Total variation error bounds for geometric approximation. Bernoulli 19, 610632. CrossRefGoogle Scholar
[21] Peköz, E. A., Röllin, A. and Ross, N. (2016). Generalized gamma approximation with rates for urns, walks and trees. Ann. Prob. 44, 17761816. CrossRefGoogle Scholar
[22] Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surveys 4, 179. CrossRefGoogle Scholar
[23] Pitman, J. (1999). Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electron. J. Prob. 4, 33pp. CrossRefGoogle Scholar
[24] Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin. Google Scholar
[25] Ross, N. (2013). Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Adv. Appl. Prob. 45, 876893. CrossRefGoogle Scholar
[26] Rudas, A., Tóth, B. and Valkó, B. (2007). Random trees and general branching processes. Random Structures Algorithms 31, 186202. CrossRefGoogle Scholar
[27] Suquet, C. (1999). Tightness in Schauder decomposable Banach spaces. In Proc. St. Petersburg Mathematical Society (Amer. Math. Soc. Transl. Ser. 193), Vol. V, American Mathematical Society, Providence, RI, pp. 201224. Google Scholar
[28] Van der Hofstad, R. (2016). Random Graphs and Complex Networks, Vol. 1. Cambridge University Press. Google Scholar
12
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Joint degree distributions of preferential attachment random graphs
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Joint degree distributions of preferential attachment random graphs
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Joint degree distributions of preferential attachment random graphs
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *