Skip to main content Accessibility help
×
×
Home

Degree-dependent threshold-based random sequential adsorption on random trees

Abstract

We consider a special version of random sequential adsorption (RSA) with nearest-neighbor interaction on infinite tree graphs. In classical RSA, starting with a graph with initially inactive nodes, each of the nodes of the graph is inspected in a random order and is irreversibly activated if none of its nearest neighbors are active yet. We generalize this nearest-neighbor blocking effect to a degree-dependent threshold-based blocking effect. That is, each node of the graph is assumed to have its own degree-dependent threshold and if, upon inspection of a node, the number of active direct neighbors is less than that node's threshold, the node will become irreversibly active. We analyze the activation probability of nodes on an infinite tree graph, given the degree distribution of the tree and the degree-dependent thresholds. We also show how to calculate the correlation between the activity of nodes as a function of their distance. Finally, we propose an algorithm which can be used to solve the inverse problem of determining how to set the degree-dependent thresholds in infinite tree graphs in order to reach some desired activation probabilities.

Copyright

Corresponding author

* Postal address: Eindhoven University of Technology, MetaForum 4.086, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Email address: tmeyfroyt@gmail.com

References

Hide All
[1] Baccelli, F. and Blaszczyszyn, B. (2009). Stochastic Geometry and Wireless Networks, Vol. 1, Theory, Foundation and Trends in Networking. Now Publishers, Breda.
[2] Bermolen, P., Jonckheere, M. and Moyal, P. (2017). The jamming constant of uniform random graphs. Stoch. Process. Appl. 127, 21382178.
[3] Bermolen, P., Jonckheere, M., Larroca, F. and Moyal, P. (2014). Estimating the spatial reuse with configuration models. Preprint. Available at https://arxiv.org/abs/1411.0143.
[4] Coladon, T., Vučinić, M. and Tourancheau, B. (2015). Improving trickle fairness: locally-calculated redundancy constants. In Proceedings of the International Conference on Protocol Engineering (ICPE) and International Conference on New Technologies of Distributed Systems (NTDS), IEEE, pp. 16.
[5] Dehling, H. G., Fleurke, S. R. and Külske, C. (2008). Parking on a random tree. J. Statist. Phys. 133, 151157.
[6] Dhara, S., van Leeuwaarden, J. S. H. and Mukherjee, D. (2016). Generalized random sequential adsorption on Erdős–Rényi random graphs. J. Statist. Phys. 164, 12171232.
[7] Evans, J. W. (1993). Random and cooperative sequential adsorption. Rev. Mod. Phys. 65, 12811329.
[8] Fleurke, S. R. and Külske, C. (2010). Multilayer parking with screening on a random tree. J. Statist. Phys. 139, 417431.
[9] Hui, J. and Kelsey, R. (2016). Multicast protocol for low power and lossy networks (MPL). Preprint 7731, Internet RFC.
[10] Kermajani, H. and Gomez, C. (2014). On the network convergence process in RPL over IEEE 802.15.4 multihop networks: improvement and trade-offs. Sensors 14, 1199312022.
[11] Kulpa, W. (1997). The Poincaré-Miranda theorem. Amer. Math. Monthly 104, 545550.
[12] Levis, P., Patel, N., Culler, D. and Shenker, S. (2004). Trickle: a self-regulating algorithm for code propagation and maintenance in wireless sensor networks. In Proceedings of the First Symposium on Networked Systems Design and Implementation, ACM, New York, pp. 1528.
[13] Meyfroyt, T. M. M., Stolikj, M. and Lukkien, J. J. (2015). Adaptive broadcast suppression for Trickle-based protocols. In Proceedings of the 16th IEEE International Symposium on a World of Wireless, Mobile and Multimedia Networks (WoWMoM), IEEE, 9 pp.
[14] Meyfroyt, T. M. M., Borst, S. C., Boxma, O. J. and Denteneer, D. (2015). On the scalability and message count of trickle-based broadcasting schemes. Queueing Systems 81, 203230.
[15] Ni, S.-Y., Tseng, Y.-C., Chen, Y.-S. and Sheu, J.-P. (1999). The broadcast storm problem in a mobile ad hoc network. In Proceedings of the 5th Annual ACM/IEEE International Conference on Mobile Computing and Networking (MobiCom), ACM, New York, pp. 151162.
[16] Ostilli, M. (2012). Cayley trees and Bethe lattices: a concise analysis for mathematicians and physicists. Physica A 391, 34173423.
[17] Penrose, M. D. and Sudbury, A. (2005). Exact and approximate results for deposition and annihilation processes on graphs. Ann. App. Prob. 15, 853889.
[18] Rényi, A. (1958). On a one-dimensional problem concerning random space filling. Magyar Tud. Akad. Mat. Kutató Int. Közl. 3, 109127.
[19] Sanders, J., Jonckheere, M. and Kokkelmans, S. (2015). Sub-Poissonian statistics of jamming limits in ultracold Rydberg gases. Phys. Rev. Lett. 115, 043002.
[20] Sudbury, A. (2009). Random sequential adsorption on random trees. J. Statist. Phys. 136, 5158.
[21] Vallati, C. and Mingozzi, E. (2013). Trickle-F: fair broadcast suppression to improve energy-efficient route formation with the RPL routing protocol. In Proceedings of Sustainable Internet and ICT for Sustainability (SustainIT), IEEE, 9 pp.
[22] Winter, T. et al. (2012). RPL: IPv6 routing protocol for low-power and lossy networks. Preprint 6550, Internet RFC.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed