Skip to main content Accessibility help
×
Home
Hostname: page-component-564cf476b6-pxp6n Total loading time: 0.143 Render date: 2021-06-21T01:06:28.900Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

On the evolution of topology in dynamic clique complexes

Published online by Cambridge University Press:  11 January 2017

Gugan C. Thoppe
Affiliation:
Technion – Israel Institute of Technology
D. Yogeshwaran
Affiliation:
Indian Statistical Institute
Robert J. Adler
Affiliation:
Technion – Israel Institute of Technology
Corresponding
E-mail address:

Abstract

We consider a time varying analogue of the Erdős–Rényi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous-time Markov chains. Our main result is that when the edge inclusion probability is of the form p=n α, where n is the number of vertices and α∈(-1/k, -1/(k + 1)), then the process of the normalised kth Betti number of these dynamic clique complexes converges weakly to the Ornstein–Uhlenbeck process as n→∞.

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.

References

[1] Babson, E., Hoffman, C. and Kahle, M. (2011). The fundamental group of random 2-complexes. J.~Amer. Math. Soc. 24, 128.CrossRefGoogle Scholar
[2] Barbour, A. D., Karoński, M.and Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. J.~Combin. Theory B 47, 125145.CrossRefGoogle Scholar
[3] Basu, P., Bar-Noy, A., Ramanathan, R. and Johnson, M. P. (2010). Modeling and analysis of time-varying graphs. Preprint. Available at http://arxiv.org/abs/1012.0260.Google Scholar
[4] Billingsley, P. (2012). Probability and Measure. John Wiley, Hoboken, NJ.Google Scholar
[5] Burke, C. J. and Rosenblatt, M. (1958). A Markovian function of a Markov chain. Ann. Math. Statist. 29, 11121122.CrossRefGoogle Scholar
[6] Clementi, A. E. F. (2010). Flooding time of edge-Markovian evolving graphs. SIAM J. Discrete Math. 24, 16941712.CrossRefGoogle Scholar
[7] Clementi, A. et al. (2015). Distributed community detection in dynamic graphs. Theoret. Comput. Sci. 584, 1941.CrossRefGoogle Scholar
[8] Edelsbrunner, H. and Harer, J. L. (2010). Computational Topology: An Introduction, American Mathematical Society, Providence, RI.Google Scholar
[9] Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6, 290297.Google Scholar
[10] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence. John Wiley, Hoboken, NJ.Google Scholar
[11] Gut, A. (YEAR). An Intermediate Course in Probability, 2nd edn. Springer, New York.Google Scholar
[12] Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.Google Scholar
[13] Kahle, M. (2007). The neighborhood complex of a random graph. J. Combin. Theory A 114, 380387.CrossRefGoogle Scholar
[14] Kahle, M. (2009). Topology of random clique complexes. Discrete Math. 309, 16581671.CrossRefGoogle Scholar
[15] Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geometry 45, 553573.CrossRefGoogle Scholar
[16] Kahle, M. (2014). Sharp vanishing thresholds for cohomology of random flag complexes. Ann. Math. (2) 179, 10851107.CrossRefGoogle Scholar
[17] Kahle, M. (2014). Topology of random simplicial complexes: a survey. In Algebraic Topology: Applications and New Directions, (Contemp. Math. 620), American Mathematical Society, Providence, RI, pp.201221.Google Scholar
[18] Kahle, M. and Meckes, E. (2013). Limit theorems for Betti numbers of random simplicial complexes. Homology Homotopy Appl. 15, 343374. (Erratum: 18 (2016), 129–142.)CrossRefGoogle Scholar
[19] Linial, N. and Meshulam, R. (2006). Homological connectivity of random 2-complexes. Combinatorica 26, 475487.CrossRefGoogle Scholar
[20] Meshulam, R. and Wallach, N. (2009). Homological connectivity of random k-dimensional complexes. Random Structures Algorithms 34, 408417.CrossRefGoogle Scholar
[21] Musolesi, M., Hailes, S. and Mascolo, C. (2005). Adaptive routing for intermittently connected mobile ad hoc networks. In Sixth IEEE International Symposium on a World of Wireless Mobile and Multimedia Networks, IEEE, New York, pp.183189.CrossRefGoogle Scholar
[22] Spyropoulos, T., Psounis, K. and Raghavendra, C. S. (2005). Spray and wait: an efficient routing scheme for intermittently connected mobile networks. In WDTN `OS Proceedings of the 2005 ACM SIGCOMM Workshop on Delay-Tolerant Networking, ACM, New York, pp.252259.CrossRefGoogle Scholar
[23] Yogeshwaran, D., Subag, E. and Adler, R. J. (2016). Random geometric complexes in the thermodynamic regime. To appear in Prob. Theory Relat. Fields Google Scholar
2
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@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 sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

On the evolution of topology in dynamic clique complexes
Available formats
×

Send article to Dropbox

To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On the evolution of topology in dynamic clique complexes
Available formats
×

Send article to Google Drive

To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On the evolution of topology in dynamic clique complexes
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? *