# Random Walker Coverage Analysis for Information Dissemination in Wireless Sensor Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Analysis

**Theorem**

**1.**

**Corollary**

**1.**

**Corollary**

**2.**

## 4. Simulation Results

#### Geometric Networks on a Grid

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Random Walker’s Spatial Displacement after One Step

## Appendix B. Overlapping Surface

## Appendix C. Proof of Corollary 1

## Appendix D. The Analytical Expression of C(t)

## Appendix E. Equivalence of Equations (5) and (7)

## References

- Anastasi, G.; Farruggia, O.; Re, G.L.; Ortolani, M. Monitoring High-Quality Wine Production using Wireless Sensor Networks. In Proceedings of the 2009 42nd Hawaii International Conference on System Sciences, Big Island, HI, USA, 5–8 January 2009; pp. 1–7. [Google Scholar]
- Catania, P.; Vallone, M.; Re, G.L.; Ortolani, M. A wireless sensor network for vineyard management in Sicily (Italy). Agric. Eng. Int. CIGR J.
**2013**, 15, 139–146. [Google Scholar] - Anastasi, G.; Re, G.L.; Ortolani, M. WSNs for structural health monitoring of historical buildings. In Proceedings of the 2009 IEEE 2nd Conference on Human System Interactions, (HSI’09), Catania, Italy, 21–23 May 2009; pp. 574–579. [Google Scholar]
- Peiris, V. Highly Integrated Wireless Sensing for Body Area Network Applications. 2013. Available online: http://spie.org/newsroom/5120-highly-integrated-wireless-sensing-for-body-area-network-applications (accessed on 25 August 2016).
- Hart, J.K.; Martinez, K. Environmental Sensor Networks: A revolution in the earth system science? Earth Sci. Rev.
**2006**, 78, 177–191. [Google Scholar] [CrossRef] - Huang, X.; Yi, J.; Chen, S.; Zhu, X. A Wireless Sensor Network-Based Approach with Decision Support for Monitoring Lake Water Quality. Sensors
**2015**, 15, 29273–29296. [Google Scholar] [CrossRef] [PubMed] - Gungor, V.C.; Hancke, G.P. Industrial wireless sensor networks: Challenges, design principles, and technical approaches. IEEE Trans. Ind. Electron.
**2009**, 56, 4258–4265. [Google Scholar] [CrossRef] - Akyildiz, I.; Vuran, M. Wireless Sensor Networks (Advanced Texts in Communications and Networking); Wiley: New York, NY, USA, 2010. [Google Scholar]
- Akyildiz, I.F.; Su, W.; Sankarasubramaniam, Y.; Cayirci, E. Wireless Sensor Networks: A Survey. Comput. Netw.
**2002**, 38, 393–422. [Google Scholar] [CrossRef] - Gong, P.; Xu, Q.; Chen, T.M. Energy Harvesting Aware routing protocol for wireless sensor networks. In Proceedings of the 2014 9th International Symposium on Communication Systems, Networks and Digital Signal Processing (CSNDSP), Manchester, UK, 23–25 July 2014. [Google Scholar]
- Yick, J.; Mukherjee, B.; Ghosal, D. Wireless sensor network survey. Comput. Netw.
**2008**, 52, 2292–2330. [Google Scholar] [CrossRef] - Huang, C.F.; Tseng, Y.C. The coverage problem in a wireless sensor network. Mob. Netw. Appl.
**2005**, 10, 519–528. [Google Scholar] [CrossRef] - Nakamura, E.F.; Loureiro, A.A.F.; Frery, A.C. Information Fusion for Wireless Sensor Networks: Methods, Models, and Classifications. ACM Comput. Surv.
**2007**, 39. [Google Scholar] [CrossRef] - Hoffmann, T.; Porter, M.A.; Lambiotte, R. RandomWalks on Stochastic Temporal Networks. In Temporal Networks; Holme, P., Saramäki, J., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; pp. 295–313. [Google Scholar]
- Starnini, M.; Baronchelli, A.; Barrat, A.; Pastor-Satorras, R. Random walks on temporal networks. Phys. Rev. E
**2012**, 85, 056115. [Google Scholar] [CrossRef] [PubMed] - Tzevelekas, L.; Oikonomou, K.; Stavrakakis, I. Random walk with jumps in large-scale random geometric graphs. Comput. Commun.
**2010**, 33, 1505–1514. [Google Scholar] [CrossRef] - Zuniga, M.; Avin, C.; Hauswirth, M. Querying Dynamic Wireless Sensor Networks with Non-revisiting Random Walks. In Proceedings of the 7th European Conference on Wireless Sensor Networks (EWSN 2010), Coimbra, Portugal, 17–19 February 2010; Silva, J.S., Krishnamachari, B., Boavida, F., Eds.; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Penrose, M.D. Random Geometric Graphs; Oxford Studies in Probability; Oxford University Press: Oxford, UK, 2004. [Google Scholar]
- Costa, L.D.F.; Batista, J.L.; Ascoli, G.A. Communication Structure of Cortical Networks. Front. Comput. Neurosci.
**2011**, 5, 6. [Google Scholar] [CrossRef] [PubMed] - Kaiser, M.; Martin, R.; Andras, P.; Young, M.P. Simulation of robustness against lesions of cortical networks. Eur. J. Neurosci.
**2007**, 25, 3185–3192. [Google Scholar] [CrossRef] [PubMed] - Newman, M.E. The structure and function of complex networks. SIAM Rev.
**2003**, 45, 167–256. [Google Scholar] [CrossRef] - Gkantsidis, C.; Mihail, M.; Saberi, A. Random walks in peer-to-peer networks. In Proceedings of the Twenty-Third AnnualJoint Conference of the IEEE Computer and Communications Societies, (INFOCOM 2004), Hong Kong, China, 7–11 March 2004; Volume 1. [Google Scholar]
- Boyd, S.; Ghosh, A.; Prabhakar, B.; Shah, D. Mixing times for random walks on geometric random graphs. In Proceedings of the SIAM Workshop on Analytic Algorithmics and Combinatorics (ANALCO), Vancouver, BC, Canada, 22 January 2005; pp. 240–249. [Google Scholar]
- Beraldi, R.; Baldoni, R.; Prakash, R. A biased random walk routing protocol for wireless sensor networks: The lukewarm potato protocol. IEEE Trans. Mob. Comput.
**2010**, 9, 1649–1661. [Google Scholar] [CrossRef] - Oikonomou, K.; Kogias, D.; Stavrakakis, I. A Study of Information Dissemination Under Multiple Random Walkers and Replication Mechanisms. In Proceedings of the Second International Workshop on Mobile Opportunistic Networking, (MobiOpp’10), Pisa, Italy, 22–23 February 2010; ACM: New York, NY, USA, 2010; pp. 118–125. [Google Scholar]
- Kenniche, H.; Ravelomananana, V. Random geometric graphs as model of wireless sensor networks. In Proceedings of the 2nd International Conference on Computer and Automation Engineering (ICCAE), Singapore, 26–28 February 2010. [Google Scholar]
- Ren, Y.; Qin, Y.; Wang, B.; Zhang, H.; Zhang, S. A random geometric graph coverage model of wireless sensor networks. In Proceedings of the 2006 IET International Conference on Wireless, Mobile and Multimedia Networks, Hangzhou, China, 6–9 November 2006. [Google Scholar]
- Karamchandani, N.; Manjunath, D.; Yogeshwaran, D.; Iyer, S.K. Evolving Random Geometric Graph Models for Mobile Wireless Networks. In Proceedings of the 4th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, Boston, MA, USA, 26 February–2 March 2006. [Google Scholar]
- Lima, L.; Barros, J. Random Walks on Sensor Networks. In Proceedings of the 2007 5th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks and Workshops, Limasso, Cyprus, 16–20 April 2007; pp. 1–5. [Google Scholar]
- Dall, J.; Christensen, M. Random geometric graphs. Phys. Rev. E
**2002**, 66, 016121. [Google Scholar] [CrossRef] [PubMed] - Huang, K. Throughput of wireless networks powered by energy harvesting. In Proceedings of the 2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), Pacific Grove, CA, USA, 6–9 November 2011; pp. 8–12. [Google Scholar]
- Oikonomou, K.; Kogias, D.; Stavrakakis, I. Probabilistic Flooding for Efficient Information Dissemination in Random Graph Topologies. Comput. Netw.
**2010**, 54, 1615–1629. [Google Scholar] [CrossRef] - Stauffer, A.O.; Barbosa, V.C. Probabilistic heuristics for disseminating information in networks. IEEE ACM Trans. Netw.
**2007**, 15, 425–435. [Google Scholar] [CrossRef] - Crisóstomo, S.; Schilcher, U.; Bettstetter, C.; Barros, J. Analysis of probabilistic flooding: How do we choose the right coin? In Proceedings of the 2009 IEEE International Conference on Communications (ICC’09), Dresden, Germany, 14–18 June 2009; pp. 1–6. [Google Scholar]
- Gaeta, R. Generalized Probabilistic Flooding in Unstructured Peer-to-Peer Networks. IEEE Trans. Parallel Distrib. Syst.
**2011**, 22, 2055–2062. [Google Scholar] [CrossRef] - Jamoos, A. Improved Decision Fusion Model for Wireless Sensor Networks over Rayleigh Fading Channels. Technologies
**2017**, 5, 19. [Google Scholar] [CrossRef] - Broutin, N.; Devroye, L.; Lugosi, G. Almost optimal sparsification of random geometric graphs. Ann. Appl. Probab.
**2016**, 26, 3078–3109. [Google Scholar] [CrossRef] - Mao, G. Phase Transitions in Large Networks. In Connectivity of Communication Networks; Springer: Cham, Switzerland, 2017; pp. 149–174. [Google Scholar]
- Lunagómez, S.; Mukherjee, S.; Wolpert, R.L.; Airoldi, E.M. Geometric representations of random hypergraphs. J. Am. Stat. Assoc.
**2016**, 363–383. [Google Scholar] [CrossRef] - Kahle, M. Topology of random simplicial complexes: A survey. AMS Contemp. Math.
**2014**, 620, 201–222. [Google Scholar] - Estrada, E.; Delvenne, J.C.; Hatano, N.; Mateos, J.L.; Metzler, R.; Riascos, A.P.; Schaub, M.T. Random Multi-Hopper Model. Super-Fast Random Walks on Graphs. arXiv
**2016**. [Google Scholar] - Mabrouki, I.; Lagrange, X.; Froc, G. Random Walk Based Routing Protocol for Wireless Sensor Networks. In Proceedings of the 2nd International Conference on Performance Evaluation Methodolgies and Tools, (ValueTools ’07), Nantes, France, 22–27 October 2007; ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering): Brussels, Belgium, 2007; pp. 71:1–71:10. [Google Scholar]
- Tian, H.; Shen, H.; Matsuzawa, T. Random Walk Routing in WSNs with Regular Topologies. J. Comput. Sci. Technol.
**2006**, 21, 496–502. [Google Scholar] [CrossRef] - Mian, A.N.; Beraldi, R.; Baldoni, R. On the coverage process of random walk in wireless ad hoc and sensor networks. In Proceedings of the 2010 IEEE 7th International Conference on Mobile Adhoc and Sensor Systems (MASS), San Francisco, CA, USA, 8–12 November 2010; pp. 146–155. [Google Scholar]
- Avin, C.; Ercal, G. On the cover time of random geometric graphs. In Proceedings of the International Colloquium on Automata, Languages, and Programming, Lisbon, Portugal, 11–15 July 2005; Springer: Berlin/Heidelberg, Germany, 2005; pp. 677–689. [Google Scholar]
- Weisstein, E.W. A Wolfram Web Resource: Random Walk–2-Dimensional. Available online: http://mathworld.wolfram.com/RandomWalk2-Dimensional.html (accessed on 8 April 2016).
- Kim, S.; Park, S.Y.; Kwon, D.; Ham, J.; Ko, Y.B.; Lim, H. Two-hop distance estimation in wireless sensor networks. Int. J. Distrib. Sens. Netw.
**2017**, 13. [Google Scholar] [CrossRef]

**Figure 1.**Common neighborhood surfaces of nodes sequentially visited by the random walker. Eventually, the random walker’s steps are not straight as in the figure; the Euclidean distance from the original node A is given by Equation (1). The colored gray area is still the overlapping surface for $t=8$.

**Figure 2.**This figure depicts the evolution of C(t) (i.e., network coverage) over time for a geometric random graph of 1000 nodes using the random walker mechanism. The curves produced using Equation (5) (for various values of ${r}_{c}$) are close to the one produced using Equation (7), except from the red curve that corresponds to the smallest ${r}_{c}$, which is where the network is marginally connected.

**Figure 3.**A snapshot of a geometric random graph with $N=1000$ and transmission range ${r}_{c}=0.052$. It depicts a marginally-connected WSN from which it can be visually understood why there is a significant time lag in the network coverage of a random walker, compared to denser networks. One can easily observe the existence of small bottlenecks dispersed over the network.

**Figure 4.**Mean number of neighbors depending on the radius ${r}_{c}$ for a geometric random graph of 1000 nodes.

**Figure 5.**A bar graph showing the spatial displacement of the random walker after each step on a network with ${10}^{4}$ nodes and ${r}_{c}$ = 0.05. There are ${10}^{5}$ steps, and each bar corresponds to the number of steps (×50), where the spatial displacement of the random walker is in the range of ${10}^{-3}$.

**Figure 6.**Random walker’s spatial displacement from the starting node. The blue line is from the analytical part of this study (i.e., Equation (1)), while the red one is from the simulation results. One can see that the walker is permitted to move in all directions; hence, the distance from the starting node could be either decreased or increased.

**Figure 7.**Simulation results for 100 sparse networks with number of nodes $N=1000$, radius ${r}_{c}=0.06$ and mean number of neighbors $=12$. One can see that the simulation’s outcome is in accordance with the expected results from the analysis.

**Figure 8.**Simulation results for networks of various densities of links $N=1000$, ${r}_{c}=0.100$ (red), 0.200 (green), 0.500 (magenta), 1.000 (yellow), number of neighbors $=35,158,834,997$. It depicts the coverage over the number of time steps. Full lines are for the analytic results and dashed for simulation results. Similarly to previous outcomes, the modeling is in accordance with the simulation. Moreover, it is clear that there is a difference only in the case of marginally-connected networks with the minimum value for radius ${r}_{c}$.

**Figure 9.**Simulation results for the grid geometric network (green dashed line), $N={10}^{4}$, ${r}_{c}=0.011$, mean number of neighbors $=4$. Coverage is depicted over the number of time steps. There is a strong correlation between the analytical part and the simulation one.

**Figure 10.**Simulation results for the grid geometric network, $N={10}^{4}$, ${r}_{c}=0.033$, mean number of neighbors $=34$. Similarly to the previous simulation results, there is a fit between the modeling and simulation.

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**MDPI and ACS Style**

Skiadopoulos, K.; Giannakis, K.; Oikonomou, K.
Random Walker Coverage Analysis for Information Dissemination in Wireless Sensor Networks. *Technologies* **2017**, *5*, 33.
https://doi.org/10.3390/technologies5020033

**AMA Style**

Skiadopoulos K, Giannakis K, Oikonomou K.
Random Walker Coverage Analysis for Information Dissemination in Wireless Sensor Networks. *Technologies*. 2017; 5(2):33.
https://doi.org/10.3390/technologies5020033

**Chicago/Turabian Style**

Skiadopoulos, Konstantinos, Konstantinos Giannakis, and Konstantinos Oikonomou.
2017. "Random Walker Coverage Analysis for Information Dissemination in Wireless Sensor Networks" *Technologies* 5, no. 2: 33.
https://doi.org/10.3390/technologies5020033