As batteries fail in wireless sensor networks there is an inevitable devolution of the network characterised by a breakdown in connectivity between the surviving nodes of the network. A sharp limit theorem characterising the time at which this phenomena makes an appearance is derived

Kunniyur, Srisankar S

Venkatesh, Santosh S

English

ScholarlyCommons@Penn

University of PennsylvaniaScholarlyCommonsDepartmental Papers (ESE) Department of Electrical & Systems EngineeringJune 2004Sensor Network Devolution and Breakdown inSurvivor ConnectivitySrisankar S. KunniyurUniversity of Pennsylvania, kunniyur@seas.upenn.eduSantosh S. VenkateshUniversity of Pennsylvania, venkates@seas.upenn.eduFollow this and additional works at: http://repository.upenn.edu/ese_papersCopyright 2004 IEEE. Reprinted from Proceedings of the International Symposium on Information Theory 2004 (ISIT 2004), page 81.This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of theUniversity of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish thismaterial for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE bywriting to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper is posted at ScholarlyCommons. http://repository.upenn.edu/ese_papers/94For more information, please contact repository@pobox.upenn.edu.Recommended CitationSrisankar S. Kunniyur and Santosh S. Venkatesh, "Sensor Network Devolution and Breakdown in Survivor Connectivity", . June 2004.Sensor Network Devolution and Breakdown in Survivor ConnectivityAbstractAs batteries fail in wireless sensor networks there is an inevitable devolution of the network characterised by abreakdown in connectivity between the surviving nodes of the network. A sharp limit theorem characterisingthe time at which this phenomena makes an appearance is derived.CommentsCopyright 2004 IEEE. Reprinted from Proceedings of the International Symposium on Information Theory 2004(ISIT 2004), page 81.This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any wayimply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personaluse of this material is permitted. However, permission to reprint/republish this material for advertising orpromotional purposes or for creating new collective works for resale or redistribution must be obtained fromthe IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to allprovisions of the copyright laws protecting it.This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/94Sensor Network Devolution and Breakdown in Survivor ConnectivitySrisankar S. Kunniyur and Santosh S. VenkateshDepartment of Electrical and Systems EngineeringUniversity of PennsylvaniaPhiladelphia, PA 19104{kunniyur, venkatesh}@ee.upenn.eduAbstract — As batteries fail in wireless sensor net-works there is an inevitable devolution of the networkcharacterised by a breakdown in connectivity betweenthe surviving nodes of the network. A sharp limit the-orem characterising the time at which this phenomenamakes an appearance is derived.I. System ModelPrevious work on dense sensor networks have concen-trated on establishing initial connectivity or coverage.There is much less known, however, about how connec-tivity between the surviving nodes devolves as nodes de-grade and fail over time, primarily due to limited batterypower at the nodes.We consider a sensor ﬁeld comprised of a circle of unitradius in which n sensors are to be dispersed. We will sup-pose that each sensor can communicate with any othersensor located within a distance r from it. As designparameters we will suppose that the transmission radiusr = rn is a suitably decaying function of the number ofsensors n. A known result asserts indeed that log nn is athreshold function for the radius at which network con-nectivity appears abruptly (cf. Gupta and Kumar [1] forthe result in the current framework). We will be con-cerned mainly with the situation when the graph is ini-tially connected.Each sensor is equipped with a battery which has aﬁnite lifetime determined by the usage patterns of thesensor and the selected transmission radius. We will sup-pose that the battery lifetimes are independent randomvariables with a common distribution Gr(t) for the prob-ability that the battery lifetime exceeds t.II. Breakdown in Survivor ConnectivityThe degradation of the network due to sensor lossesin time also manifests itself ultimately in a breakdownin connectivity. At the simplest level, such a breakdownoccurs when a live node is isolated though connectivitymay have broken down before such an occurrence. Moreformally, what can be said about the connectivity of thenetwork of survivors at a given time t? In particular,how long will the network of survivors remain connectedin the face of continuing losses?It is fruitful to think of the setting as follows. Initially,one starts with a connected spatial random graph on nvertices. (Of course, we are assuming tacitly that thecommunication radius rn exceeds the critical thresholdlog(n)/n so that the network is connected.) At time t arandom fraction of the nodes has expired leaving a col-lection of S(t) survivors with the induced subgraph onthose vertices. The situation may be arrived at by anequivalent probabilistic game in which random deletionsof vertices (and associated edges) are performed on theoriginal graph with each vertex removed independentlyfrom the graph with probability 1 − G(t). The numberof survivors S(t) is hence binomially distributed with pa-rameters n and G(t). The de Moivre-Laplace theoremtells us that S(t) is concentrated around its mean valuenG(t). It can be shown that S(t) = nG(t) + O(n1/2+)with asymptotic probability close to 1.Condition on S(t) = s survivors where s = nG(t) + ζand ζ = O(n1/2+). As deletions are performed indepen-dently, the locations of the s survivors are independentof each other and uniformly distributed in the unit cir-cle. It follows that log(s)/s is a threshold function forthe transmission radius to ensure survivor connectivity.More precisely, let ω(s) be any slowly growing functionof s. Bear in mind that the transmission radius is still theoriginally set radius rn and that s ∼ nG(t). We hence ob-tain that the survivors are disconnected with asymptoticprobability approaching 1 if rn ≤(log(s)−ω(s))/s whilethe survivors are connected with asymptotic probabilityapproaching 1 if rn ≥(log(s)+ω(s))/s. Write ν = nG(t)and take expectation with respect to s to get rid of theconditioning. The concentration of the binomial allows usto focus on s ∼ nG(t). It follows that a threshold functionfor the radius is log(ν)/ν to ensure survivor connectivity.Inverting the system we obtain that the critical regionof time t = tn where the survivor network passes fromconnected to disconnected satisﬁes Grn(tn) ∼ 1nrn log 1rn .For concreteness, if the failure distribution is the mem-oryless distribution seen earlier with Grn(t) = e−αr4nt andrn is initially set at just above the critical connectiv-ity threshold, then survivor connectivity breaks sharplyaround tn = n2 log log(n)/α log2 n. More precisely, forany  > 0, the probability that the survivors are con-nected tends to 1 if tn ≤ (1−)n2 log log(n)/α log2 n whilethe probability that the survivors are disconnected tendsto 1 if tn ≥ (1 + )n2 log log(n)/α log2 n.References[1] P. Gupta and P. R. Kumar. Critical power for asymptotic con-nectivity in wireless networks. Stochastic Analysis, Control,Optimization and Applications: A Volume in Honor of W.H.Fleming, pages 547–566, 1998.ISIT 2004, Chicago, USA, June 27 – July 2, 2004?????????????????????????? ?????????

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