Critical Sensor Density for Partial Connectivityin Large Area Wireless Sensor Networks

Haiyan Cai1; Xiaohua Jia2; Mo Sha2,3

1University of Missouri-St. Louis; 2City University of Hong Kong3Washington University in St. Louis

Abstract—Assume sensor deployment follows the Poissondistribution. For a given partial connectivity requirement ρ,0.5 < ρ < 1, we prove, for a hexagon model, that there existsa critical sensor density λ0, around which the probabilitythat at least 100ρ% of sensors are connected in the networkincreases sharply from ε to 1− ε within a short interval ofsensor density λ. The location of λ0 is at the sensor densitywhere the above probability is about 1/2. We also extendthe results to the disk model. Simulations are conducted toconfirm the theoretical results.

I. INTRODUCTION

The problem of connectivity of sensors deployed ran-domly in a large area is a central issue in the studies ofsuch networks. Each sensor connects only locally to thesensors within its communication range. The connectionsamong the sensors across the field rely on intermediaterelays. Extensive studies have been done on connectivityof sensor networks. Some early works focused on howmany neighbors each node should connect to in order tomaximize the network throughput, such as [1], [2], [3],[4]. More recent works studied how large the node trans-mission range is needed to ensure the full connectivityof the network (with high probability), or equivalentlyhow many nodes are needed to ensure such connectivitygiven node transmission range is fixed [5], [6], [7], [8].Some other works investigated the relationship betweennetwork connectivity and sensor coverage and studied thedeployment of minimal number of sensors in a field suchthat the network is connected and the field is fully covered[9], [10], [11].

However, most of those studies focus only on thefull connectivity (or k-connectivity) of the sensors inwhich every single sensor is connected to the entirenetwork. This is a rather stringent requirement. To addressthe issue, we investigate the partial connectivity of thenetwork in this paper. Our simulations show that thefull coverage or connectivity would require much greaternumber of sensors in the field, just to avoid the exclusionof a small fraction of the isolated sensors. The savingsfrom reducing the number of sensors will be significant ifwe are willing to give up a small portion of the sensors.

For a given percentage requirement ρ, we are interestedin the events like "the percentage of the sensors thatare connected is at least ρ". We want to find how, in

general, the probability of this event is related to thesensor density. It is clear that a higher density will alwaysincrease the chance of the occurrence of such an event,and the question of our concern is how fast the chancewill increase when the sensor density increases. Throughour analysis, we find that there exists a critical value ofthe sensor density λ0 such that the probabilities of theabove events increase sharply from some small ε > 0 to1−ε in a short interval centered near λ0. Our analysis alsoreveals that, roughly speaking, the length of this interval isof the order O(− log ε/ log A), where A is the area of thefield. We further find that λ0 is such a value at which theprobability for the event to occur is about 1/2 when A islarge. More precise statements for these results are givenin the next section for a hexagon model. We will alsoapply them to obtain some estimates for a disk model.

The concept of partial connectivity is not new [12].Indeed, in the context of continuum percolation [13],the connectivity problem can be formulated using apercolation model in Poisson blobs. It was shown in [14],for example, that for a given sensor density above thepercolation threshold, one can achieve a percentage ofconnectivity with a probability approaching to 1 expo-nentially fast, as the sensor field goes to the infinity.The percentage of the connectivity converges to the cor-responding percolation probability, and the convergencerate depends implicitly on the given sensor density. Thisresult is, however, different from ours. Here we focuson how fast the probability of a partial connectivity witha given connection percentage can get close to 1, fromnearly 0, when the sensor area is large but fixed, andthe sensor density is allow to change. This later questioncan be more important in an actual sensor deploymentsituation where a sensor area is pre-fixed and one needs todecide how many sensors to use. In this work we providean answer to the later question by establishing a sharp-threshold property [15] around a “critical value” of thesensor density.

II. PROPERTIES OF CONNECTIVITY PROBABILITIES

This section contains all the main results. Due to thespace limit, we omit the proofs of the theorems. In thenext section we will use the simulations to illustrate theseanalytical results.

This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2010This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2010 proceedings

978-1-4244-5837-0/10/$26.00 ©2010 IEEE

Let λ be the sensor density, the number of sensors perunit area. Suppose NA sensors are deployed randomlyand independently in a square region of area A, wherewe assume NA is a Poisson random variable NA ∼Poisson(λA). Note that this is equivalent to assumingthat the sensors are located according to a Poisson pointprocess. Sensor networks are formed through communica-tions between sensors. In a disk model, two sensor nodescan directly communicate with each other if they arein each other’s communication range. We are interestedin the probability that certain percentage of sensors areconnected to each other in a network under a givendensity λ . A direct analysis of this probability for a givenM is difficult. We are, however, able to study the problemin a slightly simplified model, i.e., the hexagon model. Inwhat follows, we will first formulate the hexagon modeland study the connectivity problem of the sensor networkin this model. We will then use the results from thehexagon model to estimate the connectivity probabilityof the disk model.

A. The Hexagon Model

We partition the sensor field into hexagons using a reg-ular honeycomb hexagon lattice. Hexagons are arrangedin such a way that each hexagon has two of its sixsides placed vertically. Hexagons with adjacent verticalsides line up to form a row. Suppose that in the latticethere are M rows and that in each row, there are Mhexagons. Let Hi,j denote the jth hexagon in the ith row,i, j = 1, 2, · · · ,M . Every off-boundary hexagon Hi,j

has six neighbors, including Hi,j−1 and Hi,j+1 on eachside. Let H = {Hi,j : i, j = 1, 2, · · · ,M}. Note thatfor large M , A = O(M2).

Let AH be the area of each hexagon. From the previousassumption, there is an equal probability for a sensor tofall into each Hi,j . The probability that a hexagon willcontain at least one sensor depends on λ. We denote thisprobability as p(λ). Then

p(λ) = 1 − e−λAH . (1)

We say a hexagon is occupied, if there is at leastone sensor in it. Thus, p(λ) is the probability that ahexagon is occupied when the sensor density is λ. Wesay two neighboring hexagons are directly-connected ifthey are both occupied. Two hexagons in the latticeare said to be connected, if there is a path of directly-connected hexagons that connects them. Hexagons thatare connected with each other form a cluster (a maximalconnected component) in the corresponding graph. In thismodel, sensors are divided into a set of disconnectedclusters.

In the hexagon model, we assume that two sensorscommunicate with each other if and only if they are eitherin the same hexagon or in the neighboring hexagons.According to this definition of connectivity, two sensors

are connected if and only if they are in the same cluster ofhexagons. We are interested in the percentage of sensorswhich are connected within a given cluster. Let C be acluster in the network, and NC the total number of sensorsin C. Let us consider the ratio

rN (C) =NC

NA. (2)

For a given ρ, 1/2 < ρ < 1, we defined the event

Bρ = {There is a cluster C such that rN (C) ≥ ρ} .(3)

Bρ is the event that at least 100ρ% of the sensors inthe sensor field are connected. Note that since ρ > 1/2,there can be at most one such cluster. This is the largestcluster among all the clusters in the network. Let Pλ(Bρ)be the probability of event Bρ under sensor density λ.We are interested in how Pλ(Bρ) changes as λ changes.To analyze this probability we first discuss some generalproperties of clusters of hexagons.

B. The largest cluster in the hexagon network

Here we first focus on a more coarse problem. For eachhexagon, instead of counting how many sensors in it, weonly ask if it is occupied. We then define an event similarto Bρ as follows. Suppose C is a cluster of connectedhexagons in the network. Let HC be the total number ofhexagons in cluster C. We define the relative size of Cin terms of the relative area in the network:

rH(C) =HC

M2,

and define, for a given ρ, 1/2 < ρ < 1, the event

Dρ = {There is a cluster C such that rH(C) ≥ ρ} .(4)

Let Pλ(Dρ) be the probability of event Dρ under sensordensity λ. Pλ(Dρ) represents the probability that certainpercentage of the whole sensor area is covered by thelargest cluster of connected sensors.

To analyze this probability, we introduce a set of binaryrandom variables X = {Xi,j : i, j = 1, · · · ,M} suchthat Xi,j = 1, if Hi,j is occupied and Xi,j = 0 otherwise.Then Xi,j , i, j = 1, · · · ,M , are independent randomvariables with P (Xi,j = 1) = 1 − P (Xi,j = 0) =p(λ). Let Ω = {0, 1}{1,··· ,M}×{1,··· ,M}. A realizationx = {xi,j : i, j = 1, · · · ,M} ∈ Ω of X defines anetwork configuration. Ω is the configuration space ofthe network.The event Dρ is completely determined bythe realizations x ∈ Ω of X .

If we use the center of each hexagon to representthat hexagon, and declare that a center is “open” ifthe corresponding hexagon is occupied, and “close” ifotherwise, then it is not difficult to see that our problemhere is actually a site percolation problem in a triangularlattice. A well known result from the percolation theoryasserts that there is a critical probability p0 = 1/2, such

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that to have an infinite cluster in the lattice as M → ∞,it is necessary and sufficient to have p(λ) > p0 = 1/2.Since p(log 2/AH) = 1/2, an immediate consequence ofthis result is that if λ < log 2/AH then Pλ(Dρ) → 0, asM → ∞. It is also easy to see that for every M < ∞,Pλ(Dρ) is a differentiable and strictly increasing functionof λ such that P0(Dρ) = 0 and P∞(Dρ) = 1.

The following preliminary result is the key to the mainresult of this work.

Theorem 1. There is a constant c > 0, independent ofM , and a λ0 = λ(AH ,M, ρ) > log 2/AH such that forall positive λ ≤ λ0,

Pλ(Dρ) ≤ 12M−c[p(λ0)−p(λ)] (5)

and, for all λ ≥ λ0, any small δ > 0, and any small ε1 >0, there is an M0(δ, ε1) such that for all M > M0(δ, ε),

Pλ(Dρ−δ) ≥ 1 −(

12

+ ε1

)12M−c[p(λ)−p(λ0)]. (6)

This theorem is proved based on a general sharp-threshold inequality of increasing events in a latticeBernoulli percolation model, which is a special case ofthe problem treated in [16]. We call λ0 in Theorem 1 thecritical density. Note that λ0 depends on M , ρ and, inparticularly, AH .

C. Probability of connectivity

With a simple large deviation argument, Theorem 1implies the following theorem for the connectivity prob-ability Pλ(Bρ).

Theorem 2. Let λ0 and constant c be as given in Theorem1. For every fixed small δ and small ε1, and for allsufficiently large M (depending on δ and ε1),

Pλ(Bρ+δ) ≤(

12

+ ε1

)M−c[p(λ0)−p(λ)], (7)

whenever λ ≤ λ0, and

Pλ(Bρ−δ) ≥ 1 −(

12

+ ε1

)M−c[p(λ)−p(λ0)], (8)

whenever λ ≥ λ0.

For small δ, if we agree that Pλ(Bρ−δ) ≈ Pλ(Bρ+δ) ≈Pλ(Bρ), then Theorem 2 asserts basically the following.For sufficiently large M and some small ε > 0, if λ′ andλ′′ are such that λ′ < λ0 < λ′′ and that Pλ′(Bρ) = ε andPλ′′(Bρ) = 1 − ε, then distance between λ′′ and λ′ is

λ′′ − λ′ = O

(− log ε

log M

).

In other words, if M is sufficiently large, a relativelysmall increment in sensor density λ in a neighborhoodof λ0 can result a significant increase of the probabilityPλ(Bρ). On the other hand, any change of values of λ

outside this neighborhood will have much less significantinfluence. While the existence of λ0 is established, itsexact value is unknown. It should be determined experi-mentally.

It follows immediately from Theorem 2 that for anysmall δ > 0,

lim supM→∞

Pλ0(Bρ+δ) ≤ 12≤ lim inf

M→∞Pλ0(Bρ−δ), (9)

which indicates, approximately, where the critical sensordensity is located for large M .

D. Connectivity in the disk model

Now we discuss how to extend above results to the diskmodel. According to the definition of connectivity for adisk model, we can estimate the connectivity probabilityfor the disk model with two hexagon models, H1 and H2

say, to obtain upper and lower bounds. This can be doneas follows.

Let BDρ be the event that at least 100ρ% of the sensors

are connected in a single cluster in the disk model.Suppose the transmission range for all sensors is Rc.In the first hexagon model H1, we scale the size of thehexagons so that the farthest distance between points oftwo neighboring hexagons equals Rc. Then, a connectionbetween any two hexagons in the lattice always impliesa connection between any pair of sensors inside thesetwo hexagons in the disk model. Let BH1

ρ be the eventin H1 as defined in subsection A. Then, BH1

ρ ⊂ BDρ

and P (BH1ρ ) ≤ P (BD

ρ ). Similarly, we can scale the sizeof the hexagons so that whenever the distance betweentwo sensors is less or equal to Rc, these two sensorsare either in the same hexagon or in two neighboringhexagons. We choose the smallest possible hexagon sizefor this to happen and define the corresponding latticeas H2. Let BH2

ρ be the corresponding connectivity event.Then P (BD

ρ ) ≤ P (BH2ρ ).

Applying Theorem 2 to both BH1ρ and BH2

ρ , we con-clude that there are λH1

1 and λH22 such that (7) holds for

BH1ρ and λH1

1 , and (8) holds for BH2ρ and λH2

2 .

Theorem 3. There is a constant c > 0, independent of M ,and two sensor densities λH

1 and λH2 such that for every

fixed small δ > 0 and small ε1, and for all sufficientlyM ,

Pλ(Bρ+δ) ≤(

12

+ ε1))

M−c[p(λH11 )−p(λ)], (10)

whenever λ ≤ λH11 , and

Pλ(Bρ−δ) ≥ 1 −(

12

+ ε1

)M−c[p(λ)−p(λ

H22 )], (11)

whenever λ ≥ λH22 .

This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2010This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2010 proceedings

III. SIMULATIONS

The purpose of the current simulations is twofold.First, we compare the connectivity probabilities of thehexagon model to the disk model. Second, we study howthe connectivity probability, under various connectivitypercentage ρ, is affected by the sensor density λ.

A. Methodology and Simulation Settings

The simulation code is written in Matlab. The sensorfield is chosen to be a square region with an area of50×50 units. We remark that this is actually an ap-propriate area, and we do not intend to use any largersensor field in our simulations, because this study isnot about a limiting property of sensor connectivity inwhich the area of the sensor field goes to infinity, butrather, it is about the “sharp-threshold" phenomenon. Weare interested in sensor fields of any reasonable size(preferably, smaller sizes) at which the sharp-thresholdphenomenon emerges. The number NA of sensors followsa Poisson distribution with given density λ. Sensor nodesare randomly placed in the sensor field. All sensor nodeshave a fixed transmission range Rc, which is set to 1.The size of the hexagons is scaled according to Rc tocompare the connectivity properties between the hexagonmodel and the disk model.

The algorithm, as shown in Algorithm 1, takes theconnectivity percentage ρ as input and outputs a sequenceof pairs of λ and Pλ(Bρ) (the probability of meeting ρ atdensity λ). To compute Pλ(Bρ), we obtain a sufficientlylarge number n of network samples to get a stableestimate of Pλ(Bρ), which is the total number of networksamples that meet percentage ρ divided by n. The plotsin all simulation figures are the average of 100 runs.

Algorithm 1Input: ρ.Output: Sequence of pairs (λ, Pλ(Bρ)).

1: λ = Min V alue;2: repeat3: cnt = 0;4: for i = 1 to n do5: Generate a Poisson number NA with mean λ;6: Place NA nodes randomly in the area;7: Obtain a network by connecting nodes using

hexagon model (or disk model);8: if exist a cluster larger than 100ρ% then9: cnt++;

10: end if11: end for12: Pλ(Bρ)=cnt/n; output (λ, Pλ(Bρ));13: λ+ = h; // h is a small increment of λ14: until λ > Max V alue

Figure 1. Pλ(Bρ) versus λ with different hexagon sizes

B. Hexagon Size in the Hexagon Model

In this set of simulations, we compare the connectivityprobability of the disk model to that of the hexagonmodels. In the first hexagon model, we set the hexagons tobe as large as possible, as long as any connected nodes inthe hexagon model remain connected in the disk model. Inthe second hexagon model, we set the hexagons to be assmall as possible size as long as the opposite connectivityproperty holds. We also study a third hexagon model,called an equal-area model, in which the size of thehexagon is such that the total area of any hexagon alongwith its 6 neighbors equals the area covered by a diskwith radius Rc. Figure 1 shows the comparisons, wherethe first two hexagon models provide the bounds for theconnectivity probability for the disk model and the thirdhexagon model shows a closer match to the disk model.

We observe the following. 1) The hexagon model withvarious hexagon sizes exhibits the same pattern as the diskmodel. There is always a sharp increase of Pλ(Bρ) from0 to 1 near some critical density λ0. 2) The values of λ0

for different size of hexagons have a large variance. Asthe hexagon size increases, λ0 decreases from 3.5 (forlower bound of hexagon size) to 0.2 (for upper boundof hexagon size). This is because for a larger hexagonsize, there is a higher chance for nodes to fall into thesame hexagon or neighboring hexagons resulting a higherprobability of network connection. That is, to reach thesame Pλ(Bρ), the hexagon model with a larger hexagonsize requires less node density, leading to a smaller λ0.3) The critical density value of the equal-area hexagonmodel is very close to the disk model. For the rest ofsimulation, we use the hexagon size of equal-area for thehexagon model.

C. Sharp-threshold Property near Critical DensityIn the simulations we record the relations between

the connectivity probability and the number of sensorsdeployed in the network. For ρ = .95, we observe for thedisk model that, when the number of sensors is around4127 (λ = 1.61), the connectivity probability Pλ(Bρ) is

This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2010This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2010 proceedings

(a) ρ = 0.5 (b) ρ = 0.65

(c) ρ = 0.75 (d) ρ = 0.95

Figure 2. Pλ(Bρ) versus λ with different ρ.

Figure 3. ρ versus λ when Pλ(Bρ) ≥ 0.99

almost 0; but when the number of sensors changes to4914 (λ = 1.95), a less than 20% increase in the numberof sensors, the probability Pλ(Bρ) increases to almost1. Similar sharp increase in connectivity probability isalso seen for the hexagon model. Simulations show thatin the hexagon model when the number of sensors goesfrom 4409 (λ = 1.75) to 5408 (λ = 2.14), a 22%increase in the number of sensors, the probability Pλ(Bρ)goes up from nearly 0 to almost 1. These results suggestthat it is important to find the critical density for largescale deployment of sensors. It helps us determine theright number of sensors to deploy. With a slight increasein node density over this critical point, the connectivityprobability can be increased significantly.

D. Impact of Parameter ρIn this set of simulations, we investigate the impact

of the parameter ρ. Figure 2 shows a group of chartsof Pλ(Bρ) versus λ where ρ varies from 0.5 to 0.95.All charts for different ρ exhibit the same pattern of theincrease of Pλ(Bρ) as λ increases. It suggests that theincreasing rate of Pλ(Bρ) is independent of ρ, at leastfor ρ in a range bounded away from 1.

Figure 3 shows the correlation between ρ and λ whenPλ(Bρ) ≥ 0.99 (i.e., highly sure that 100ρ% of nodes are

connected). We see that in general the increase of λ with ρis slow for small ρ. But, the increase of λ becomes sharperfor large ρ. This is particularly true when ρ reaches 95%or beyond.

Figure 3 also translates the density λ into the numberof sensor nodes (the right hand side vertical bar). We seethat when ρ is increased from 0.95 to 0.99, the numberof sensors required to meet the connectivity percentagesjumps significantly from 4828 to 6006, which will bemuch higher if ρ is further increased. To attain the fullconnectivity, the number of sensors will be prohibitlylarge. This result justifies the need for the study of partialconnectivity. It shows that it is much more economic tosacrifice a small percentage of node connection and sat-isfy with a partial (but with high percentage) connectivity.This way we can save a large number of sensor nodes.

IV. ACKNOWLEDGMENT

This work was supported by a grant from NSF ChinaGrant No. 60633020 and 60970117.

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This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2010This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 2010 proceedings