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The Computer Journal, Vol. 47 No. 4, © The British Computer Society; all rights reserved

On the Connectivity ofAd HocNetworks

Christian Bettstetter

Technische Universität München (TUM), Inst. of Communication Networks, Munich, GermanyNTT DoCoMo Euro-Labs, Future Networking Lab, Munich, Germany

Email: [email protected]

This paper presents a framework for the calculation of stochastic connectivity properties of wirelessmultihop networks. Assuming that n nodes, each node with transmission ranger0, are distributedaccording to some spatial probability density function, we study the level of connectivity of theresulting network topology from three viewpoints. First, we analyze the number of neighbors of agiven node. Second, we study the probability that there is a communication path between two givennodes. Third, we investigate the probability that the entire network is connected, i.e. each node cancommunicate with every other node via a multihop path. For the last-mentioned issue, we computea tight approximation for the critical (r0, n) pairs that are required to keep the network connectedwith a probability close to one. In fact, the problem is solved for the general case of ak-connectednetwork, accounting for the robustness against node failures. These issues are studied for uniformlydistributed nodes (with and without ‘border effects’), Gaussian distributed nodes, and nodes thatmove according to the commonly used random waypoint mobility model. The results are of practical

value for the design and simulation of wireless sensor and mobilead hoc networks.

Received 7 May 2003; revised 30 October 2003

1. INTRODUCTION

In an ad hoc network, mobile devices communicate witheach other in a peer-to-peer fashion; they establish aself-organizing wireless network without the need for basestations or any other pre-existing infrastructure. An out-standing feature of this emerging technology is multihopcommunication. If two devices cannot establish a directwireless link (because they are too far away from eachother), the devices in between act as relays to forward thedata from the source to the destination. In other words, eachdevice acts as both a terminal and a node of the network.

This paper investigates a fundamental property of anad hoc network: its connectivity. Whereas a mobile devicein a cellular network is ‘connected’ if it has a wireless link toat least one base station, the situation in anad hoc networkis more complicated. Since mobile devices also act as relays(routers) for messages of other devices, each single devicecontributes to the connectivity of the entire network. If adevice fails, the connectivity between two other devicesmight be destroyed. If the spatial density of the devices is toolow, the multihop principle for communication does not workat all. In other words, the communication among devices isnot guaranteed, but only probabilistic measures can be given.

We study various aspects of network connectivity usingprobabilistic methods. Given a certain random spatial dis-tribution of the devices and a simple radio link model, wedescribe the network topology as a random geometric graph.A device is represented by a node and a wireless link byan edge between two nodes. Based on this network model,

we answer questions such as: To how many nodes does anode typically have links? What is the probability that thecomplete network is connected, i.e. a wireless multihop pathexists from each node to each other node. How does nodemobility affect the results?

This article is organized as follows: Section 2 defines thenetwork model used and recalls some definitions from graphtheory. Section 3 studies the connectivity from the ‘localviewpoint’ of a node. We analyze the number of neighbors ofa node in the network graph and explain how ‘border effects’influence the results. Section 4 studies the connectivity ofall nodes from a ‘global’ point of view. We derive a tightapproximation for the minimum number of nodes with giventransmission range that must be distributed on a certain areato achieve an almost surely connected network. After givinggeneral expressions for an arbitrary spatial node distribution,we derive explicit results for uniformly distributed nodes withand without border effects, Gaussian distributed nodes andnodes moving according to a certain mobility model. Next,Section 5 considers a network design that is robust againstnode failures. That is: How can we achieve, with minimizedresources, anad hoc network that is guaranteed to remainconnected even if a certain number of nodes fail? Section 6addresses the probability that two nodes can communicatevia a multihop path. Section 7 outlines related work. Finally,Section 8 concludes and gives ideas for further research. Partsof this article are based on the author’s conference papers[1, 2, 3, 4].

Our results hold for any kind of wireless multihopnetwork, but they are especially interesting for wireless

The Computer Journal, Vol. 47, No. 4, 2004

On the Connectivity of AD HOC Networks 433

sensor networks. These networks may consist of many tinysensors that are distributed over a particular area, which needsto be observed. For example, a sensor network is deployed tomonitor the behavior of animals in a certain habitat [5]. Eachsensor fulfills a certain sensing task. For example, it measuresthe temperature, humidity or pressure, or it takes photos andrecords sounds. It then exchanges this data with other sensorsin a multihop fashion, such that an overall description of theenvironment is obtained.

2. NETWORK MODEL AND DEFINITIONS

This section explains the network model used and gives somedefinitions from graph theory. Throughout the paper, randomvariables are written in upper case letters; specific outcomesof these variables are written in lower case. For example, therandom variable representing the one-dimensional locationof a node is denoted byX; a specific location is written asx.The probability density function (pdf ) ofX is denoted byfX(x) and its expected value isE{X}. In two dimensions,we usex = (x, y) for Cartesian coordinates andx = (r, φ)

for polar coordinates.

2.1. Modeling assumptions

From a set ofn nodes, each node is independently randomlyplaced according to some two-dimensional pdffX(x) on asystem areaA. Several parts of this article consider a circulararea (disk) of radiusa (see Figure 1).

To describe the transmission between the nodes, we usea geometric link model. It defines the path loss betweentwo nodes as follows. Suppose a node transmits a signalwith powerpt , and another node receives this signal withpowerpr . The two nodes establish a wireless link ifpr islarger than or equal to a certain threshold powerpr,th (receiversensitivity). Assuming that all nodes have the samept andpr

and use omnidirectional antennas, two nodes can establish alink if the distance between them is less than or equal to

r0 =(

pt

pr,th

)1/α

m, (1)

which is called the transmission range (see Figure 1). Thetermα is the path loss exponent of the environment; typicalvalues are 2≤ α ≤ 5.

In a mobile scenario, each node moves independently ofother nodes according to some mobility model. The mostcommonly used mobility model in simulation-based researchonad hoc networking is the random waypoint (RWP) model(see, e.g. [6, 7]). A node randomly chooses a destinationpoint (‘waypoint’) from a uniform distribution overA. Itmoves at constant speed in a straight line to this point, restsfor a certain pause timeTp, chooses a new destination pointand speed, and so on.

A different mobility model, denoted as random direction(RD) model in this article, chooses a movement angle insteadof a destination point (see, e.g. [8]). This angleϕ ischosen from a uniform distribution on the interval[0, 2π [.A new direction is chosen after a random time that is taken,

r0

Isolated node a

A

r

A0

(x,y)

FIGURE 1. Illustration of network topology.

(a) (b) (c) (d)

FIGURE 2. Definition of network connectivity. (a) (1-)connected,(b) disconnected, (c) 2-connected and (d) MinSpan Tree.

for example, from an exponential distribution. Whenever anode reaches the border ofA, it is either ‘wrapped around’ tothe opposite side ofA, or it is ‘bounced back’ with an angle−ϕ or π − ϕ, respectively.

2.2. Definitions from graph theory

Given these modeling assumptions, the topology of anad hocnetwork can be represented at each time instant as a graphG = G(r0, n), which consists of a set ofn nodes (vertexes)and a set ofm links (edges). Since our link model considersonly bidirectional links, we always obtain an undirectedgraph. The weight of a link is the Euclidean distance betweenthe nodes. The following paragraphs recall some basicdefinitions from graph theory [9].

Two nodes are neighbors if they have a link betweeneach other. The number of neighbors of a nodeu ∈ G

is called its degreed(u). A node withd(u) = 0 is saidto be isolated (see Figure 1). The mean node degree ofG is dmean = (1/n)

∑u∈G d(u), wheredmean = 2m/n

in undirected graphs. The minimum node degree ofG isdmin = min∀u∈G{d(u)}. When we refer to a single node, weskipu and just writed.

A path between two nodes is a subgraph ofG definedby a list of consecutive links connecting the two nodes. Theshortest path between two nodes is the path with the minimumsum of the link weights between the two nodes.

A graph is said to be connected if for every pair of nodesthere exists a path between them (Figure 2a). If a graphis not connected, it is called disconnected (Figure 2b). Adisconnected graph may consist of isolated nodes and disjointgraph partitions (connected components) with more thanone node. A graph is said to bek-connected (k ∈ N) if


434 C. Bettstetter

for each node pair there exist at leastk node-disjoint pathsconnecting them (see Figure 2c,k = 2). Equivalently, agraph isk-connected if and only if no set of(k − 1) nodesexists whose removal would disconnect the graph. In otherwords, if any(k − 1) nodes fail, the graph is guaranteed to bestill connected. Clearly, ak-connected graph is also(k − i)-connected withi = 1, 2, . . . , k − 1.

The minimum spanning tree of a connected graphG isthe cycle-free subgraphG′ ⊂ G that keeps all nodes ofGconnected with minimum sum of the link weights (Figure 2d).

An important observation is that a graph created by ourmodeling assumptions is called a geometric random graphG(r0, n) in mathematics. Later in this article, we will employa recently published theorem from this nascent research field,which lies at the intersection of stochastic point processesand graph theory. The random variables corresponding tothe properties of a random graph are again written in uppercase. For example, the minimum degree of a random graphis Dmin, and the number of links isM.

3. NUMBER OF NEIGHBORS

The number of neighbors is an essential characteristic of anode in a network. If a node has no neighbors, it cannotexchange any information with other nodes and is thus uselessfor the entire network. In a mobile scenario, an isolatednode that wants to send or receive information must waituntil it moves into the range of another node or until anothernode passes by. This might cause an unacceptable messagedelivery delay. On the one hand, a high degree makes anode resistant against movement and failures of neighborsor links. On the other hand, it creates higher interference forneighboring nodes. This tradeoff has been the motivationin several publications to discuss the ‘optimal number ofneighbors’ in anad hoc network [6, 10].

Let us, therefore, show how to compute the stochasticproperties of a node’s degree, represented by the randomvariableD ∈ N0. We investigate the discrete pdfP(D = d)

and the expected valueE{D}. After giving a generalframework on how to compute these values for an arbitraryspatial node distribution, we study a uniform distributionon a bounded area, a Poisson distribution on an infinitearea and, finally, nodes moving according to the RD andRWP models.

3.1. Arbitrary node distribution

Let us consider a node at a given locationx. A second nodeis randomly placed according to some arbitrary pdffX(x′).The two nodes establish a link if the second node is placedwithin a circle of radiusr0 aroundx. As shown in Figure 1,this circular area is denoted asA0(x). The probability forthis link event is given by

P0(x) =∫∫

A0(x)

fX(x′) dx′

=∫ y+r0

y−r0

∫ x+√

r20−(y′−y)2

x−√

r20−(y′−y)2

fXY (x′, y′) dx′ dy′. (2)

In a network withn nodes, the probability that the givennode has degreed is given by the binomial distribution

P(D = d | x) =(

n − 1

d

)P0(x)d(1 − P0(x))n−d−1, (3)

with an expected valueE{D | x} = µ0(x) = (n − 1)P0(x).For smallP0(x) and largen, the binomial distribution can bewell approximated by the Poisson distribution

P(D = d | x) �µ0(x)d

d! e−µ0(x), (4)

with µ0(x) � nP0(x). The Poisson distribution yields anaccurate approximation if bothP0(x) ≤ 0.08 andn ≥1500P0(x) are fulfilled [11]. The probability that the givennode is isolated is thus

P (node iso|x) = P(D = 0 | x) � e−µ0(x). (5)

In the following, we assume that the above conditions forPoisson approximations are fulfilled. The probability that agiven node has at mostd neighbors is then

P(D ≤ d | x) = e−µ0(x)d∑

D=0

µ0(x)D

D! = �(d + 1, µ0(x))

d! ,

(6)

where �(α, β) denotes the incomplete Gamma function,defined by�(α, β) = (α−1)!e−β

∑α−1i=0 (βi/i!), α ∈ N [12].

We now regard a randomly chosen node with unknownlocation. The stochastic properties ofD are given bythe weighted sum of the above conditional values over allpossible locations:

E{D} = µ0 =∫∫

µ0(x)fX(x) dx, (7)

P(node iso) =∫∫

P (node iso|x)fX(x) dx, (8)

and

P(D ≤ d) =∫∫

P(D ≤ d | x)fX(x) dx. (9)

These properties can be interpreted in various ways: Ina static network,E{D} denotes the expected degree ofa randomly chosen node in a random network. Forn

sufficiently large, it also represents the mean degreeDmeanof a random network (ensemble average). Thus, for givenE{D}, the expected number of links of a network can becomputed byE{M} = (n/2)E{D}. In a mobile scenario,the above properties can be interpreted as time values:E{D}represents the time average of a node’s degree during itsentire movement. Equivalently, the probabilities denote thepercentage of time that a node has a certain degree.

3.2. Uniform node distribution

The majority of studies on staticad hoc networks employs auniform node distribution over a finite areaA, i.e.

fX(x) = fXY (x, y)

1

A(x, y) ∈ A,

0 else,(10)



(a) (b)

FIGURE 3. Illustration of random network topologies. (a)n = 500 nodes withr0 = 0.10a and (b)n = 500 nodes with (b)r0 = 0.14a.

with A = ‖A‖ denoting the size ofA. In this case, theprobabilityP0(x) only depends on the size of the intersectionarea betweenA and a node’s coverage areaA0(x), denotedasA′

0(x) = ‖A0(x) ∩ A‖. Nodes located at leastr0 awayfrom the border ofA are called center nodes. They haveA′

0(x) = r20π and thus an expected degreenr2

0π/A. Nodeslocated closer thanr0 to the border ofA are called bordernodes. They experience a smaller intersection area, i.e.A′

0(x) < r20π , leading to a smaller expected degree. Due

to this fact, denoted as border effect in the following, arandomly chosen node in a network has an expected degreeµ0 = nE{A′

0(x)}/A < nr20π/A.

We now regard a circular system areaA of radiusa. Twoexample topologies are shown in Figure 3. Without lossof generality, we define the origin in the middle ofA anduse polar coordinatesr andφ for the locationx. Plugging(10) into (2) with A = a2π , we observe thatP0(x) isdetermined by the normalized radial coordinater = r/a andthe normalized ranger0 = r0/a. We obtain

µ0(x) =

nr20 for 0 ≤ r ≤ 1 − r0

n

π

(r20 arccos

r2 + r20 − 1

2r r0+ arccos

r2 − r20 + 1

2r

−1

2

√ζ

)for 1 − r0 < r ≤ 1

(11)

with ζ = (r + r0 + 1)(−r + r0 + 1)(r − r0 + 1)(r + r0 − 1).This expression will be important in the computation of theoverall network connectivity later in this paper.

Let us now calculate the unconditional valuesµ0 andP(node iso). Plugging (10) and (11) into (7) withdx =rdrdφ yields

µ0 = n

2π

[4(1− r2

0) arcsinr0

2+2r2

0π−(2r0 + r3

0

)√1 − r2

0

4

].

(12)

10-6

10-5

10-4

10-3

10-2

10-1

P(n

ode

iso)

r0/a

n = 1000

n = 500

Uniformly distributed nodes on diskPoisson distributed nodes (no border effects)

0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

FIGURE 4. Probability of a node to be isolated.

To simplify this expression, we expand it into a Taylor serieswith respect tor0. Taking into account thatr0 1, sayr0 < 0.3, all terms of this series with power 5 and higher canbe ignored. We obtain

E{D} = µ0 � nr20

(1 − 4

3πr0

). (13)

This approximation yields a relative error of only−0.03% forr0 = 0.3 and lower errors for lower ranges. Note that the term−(4/3π)r0 accounts for the smaller degree of border nodes.

The isolation probability of a node at locationr can becomputed by plugging (11) into (5). Afterward, Equation (8)yields the overall isolation probabilityP(node iso). The twobold curves in Figure 4 depict this probability overr0 forn = 500 andn = 1000 nodes. For example, to achieveP(node iso) ≤10−5 with n = 500 nodes, a range of at least


436 C. Bettstetter

r0 = 0.184 is needed. Havingn = 1000 nodes allows us toreduce the range tor0 = 0.127.

3.3. Uniform distribution without border effects

Let us assume for a moment that border effects are notpresent, i.e. each node in the network has an expected degreeµ0(x) = µ0 = nr2

0π/A ∀x. Such a situation occurs if bothn andA tend toward infinity but their ratioρ = n/A remainsconstant. This limiting case of a uniform distribution iscalled a homogeneous Poisson point process of densityρ.It is completely defined by two properties [13]: (a) thenumber of nodes in each finite subarea follows a Poissondistribution, (b) the number of nodes in non-overlappingsubareas are independent random variables. This process is‘homogeneous’ becauseρ is constant over the entire infinitelylarge area.

The isolation probability of each node is then given by

P(node iso) = e−µ0 = e−ρr20π = e−nr2

0π/A. (14)

This function is plotted in Figure 4 forρ = 500/a2π and1000/a2π . Due to the missing border effect, the isolationprobability is drastically reduced compared to uniform nodes,in particular for higher ranges. The probability that a nodehas at leastk neighbors is

P (D ≥ k) = 1 − P (D ≤ k − 1) = 1 − � (k, µ0)

(k − 1)! . (15)

To generate such a homogeneous scenario in a simulation,it is not feasible to deploy an infinitely large system area. Apossible approach to simulate Poisson nodes in a boundedarea, however, follows directly from the above definition.We regard an observation areaA of finite size (see disc inFigure 5) and define a guard areaAg aroundA. The entirearea is thenA′ = A ∪ Ag. We choose the number of nodesN ′ from a Poisson distribution and uniformly place them overthe entire areaA′. The expected number of nodes inA′ isE{N ′} = ρA′. Now, all nodes located in the observationareaA—their expected number isE{N} = ρA—do notsuffer from any border effect and have an expected degreeµ0 = ρr2

0π . Only these nodes are considered for thesimulation statistics. To compare this scenario with that of agiven number ofn uniformly distributed nodes inA, we setρ = n/A. A disadvantage of this approach is the increasednumber of nodes that must be simulated.

A second simulation technique in which nodes do notsuffer from border effects is to consider a uniform nodedistribution on a bounded areaA and use a wrap-arounddistance model (also called cyclic distance model) insteadof the usual Euclidean distance model [14, 15]. Here, a nodeat the border ofA is considered as being ‘close’ to nodeslocated close to the opposite border ofA, and, thus, it canestablish links via the borderline to these nodes. In otherwords, a flat circular area becomes a sphere.

These borderless homogeneous scenarios serve as abenchmark for the connectivity properties of inhomogeneousnode distributions.

FIGURE 5. Poisson distributed nodes (ρ = 700/16a2 andr0 = 0.2a).

3.4. Nodes with random waypoint mobility

Let us now consider a scenario in which all nodes moveaccording to a given mobility model. If the RD model is usedwith one of the border strategies mentioned in Section 2.1,the resulting spatial node distribution will be uniform [16].Thus, all results derived so far are also valid for RD nodes.

Using the RWP model, however, new calculations must bemade. Although typically a uniform spatial node distribu-tion fX(x, t = 0) over A is used at the initialization of asimulation, the RWP movement changes this distribution intoa non-uniform distribution as time goes on [6, 16]. For along-running simulation, an asymptotically stationary pdf isachieved, which has its maximum in the middle ofA and iszero at the border. It can be described by [7]

fX(x, t > 0) = ppfX,p(x)︸︷︷︸pause component

+ (1 − pp)fX,m(x)︸︷︷︸mobility component

, (16)

wherepp denotes the probability that a given node pausesat a given time instant. The pause component is given by auniform pdf overA weighted bypp. The mobility componentis in general non-uniform; in a circular areaA, fX,m(x) canbe approximated by [17]

fX,m(x, y) ≈

1

a2π

(− 2

a2(x2 + y2) + 2

)0≤ x2 + y2 ≤ a

0 else.(17)

The pause probability ispp = E{Tp}/(E{Tp} + E{T }),whereE{T } denotes the expected movement time betweentwo waypoints. It is given by

E{T } = 0.9054a

νmax − νminln

(νmax

νmin

)



if the node moves on a disk and chooses a new speed in eachwaypoint from a uniform pdf on the interval[vmin, vmax] [18].

Plugging (16) into (2), the expected degree of a node at alocationx in an RWP scenario can be expressed by the sum

µ0(x) = ppµ0,p(x) + (1 − pp)µ0,m(x). (18)

The term µ0,p(x) = n∫∫

A0(x)fX,p(x) dx denotes the

expected degree in an RWP scenario in which all nodes pausefor the entire running time in their waypoints (pp = 1).Since the waypoints are uniformly distributed,µ0,p(x) isgiven by the results of Section 3.2. The termµ0,m(x) =n

∫∫A0(x)

fX,m(x) dx denotes the expected degree of a nodelocated atx in a scenario in which all nodes move ceaselesslywithout any pause periods (pp = 0). Let us compute thisterm for a circular system area. EmployingfX,m(x) given by(17), using the integration bounds in (2), and converting topolar coordinates yields the piecewise function

µ0,m(x)

=

n(−2r20 r2 + 2r2

0 − r40) for 0 ≤ r ≤ 1 − r0

n

4π

[4r2

0(2r2 + r20 − 2) arcsin

(r2 + r2

0 − 1

2r r0

)

−4 arcsin

(r2 − r2

0 + 1

2r

)+ √

ζ (r2 + 5r20 − 3)

+2π(−2r2r20 + 2r2

0 − r40 + 1)

]for 1 − r0 < r ≤ 1.

(19)

An RWP node with a ranger0 1 located in the centerof the system area (r = 0) has about twice as manyneighbors as a center node in a uniformly distributed network.For increasingr, the expected degree decreases due to thedecreasing spatial node density. Nodes at the very border ofthe system area have a very low degree.

The unconditional average degreeµ0 experienced byan RWP node during its movement can be calculated bycombining (7), (16) and (18). We have

µ0 = p2pµ0,p + (1 − pp)2µ0,m + pp(1 − pp)

×(∫∫

Aµ0,p(x)fX,m(x)dx + 1

A

∫∫A

µ0,m(x)dx)

.

This expression enables us to computeµ0 on a disc.Piecewise integration and expansion to a Taylor series withrespect tor0 yields

µ0 �nr2

0

3

((4 − 2pp + p2

p) − 4

πp2

pr0 − 3(1 − pp)r20

)(20)

for r0 ≤ 0.3. The plot ofµ0 is shown in Figure 6 fordifferent pause probabilities. We observe that the RWPmobility significantly increases the average degree comparedto uniformly distributed nodes. The difference is the highestif the nodes move without any pauses. The introduction ofpause times decreasesµ0. In the extreme casepp = 1, theexpected degree of uniformly distributed nodes is achieved.

0 0.1 0.2 0.3

E{D

}/n

r0/a

CeaselessRWP mobility

uniformlydistributednodes

increasingpause time

RWP pp=0.0pp=0.2pp=0.5pp=0.9pp=1.0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

FIGURE 6. Normalized expected degree on a disk.

Since the average degree of nodes has a significant impacton various network performance properties, these resultsshow that we must be careful when comparing propertiesof static (uniformly distributed) and mobile RWP scenarios.The fact that RWP nodes have a higher average degree means,for example, that they suffer from higher interference fromother nodes. This property also has significant consequencesfor power control and random access on the shared channel.

4. CONNECTIVITY

Whereas the previous section studied the level of connectiv-ity from the viewpoint of a single node and its neighborhood,this section investigates the connectivity of all nodes froma ‘global’ network point of view. The property of ‘beingconnected’ is one of the essential characteristics of any net-work, essentially because in a connected network each nodecan reach any other node via a multihop path. A generalquestion of interest in this context is: What is the probabilityP(con) that a wireless multihop network withn randomlydistributed nodes on an areaA, each node with transmissionranger0, is connected? Clearly, very small values ofr0 andn create networks that are disconnected, i.e.P(con) = 0. Ifwe increase one or both parameters, the probabilityP(con)increases, until finallyP(con) = 1. In this article, we are es-pecially interested in finding the(r0, n) pairs that result withhigh probability in a connected network, sayP(con) ≥ 99%.This problem arises, for example, in the design of wirelesssensor networks, as described in the following example.

Example (Planning of sensor networks): A large-scalewireless sensor network should be deployed on a circular areaof radiusa = 1000 m to perform environmental monitoring.The sensors are thrown from an aircraft in such a way thatthey are randomly distributed over the area. The sensor typeused has a transmission powerpt = 20 dBm = 100 mWand a receiver sensitivitypr,th = −70 dBm = 0.1 µW,representing two typical power values given in [19]. The pathloss exponent in the deployment area isα = 3. How manysensors must be distributed at least, such that the resultingnetwork is connected with high probability?


438 C. Bettstetter

In a mobile scenario,P(con) can be interpreted as thefraction of time that the network is connected for givenr0andn. We can then ask: Which critical(r0, n) pairs result ina connected network during at least 99% of the time? Thisquestion is relevant if we want to set the system parametersin simulations of mobilead hoc networks.

This section derives solutions to these problems.Section 4.1 shows a simulation-based study of theconnectivity with uniformly distributed nodes. This isfollowed by Section 4.2, deriving analytical bounds forP(con) for an arbitrary spatial node distribution. Next,Section 4.3 calculates a tight analytical approximation forP(con) in the case of uniformly distributed nodes takinginto account border effects. Section 4.4 illustrates practicalapplications of this theoretical result. Section 4.5 addressesthe uniform case without border effects. Sections 4.6and 4.7 study non-uniform distributions, namely nodes thatare distributed according to a Gaussian distribution and nodesthat move according to the RWP model. Throughout thesection, we say that a network is almost surely connected ifP(con) ≥ 99%.

Let us explain why we are interested in the minimum(r0, n) pairs resulting in an almost surely connected network.If the ranger0 of all nodes is set to a very high value,the network is surely connected. But this guaranteedconnectivity has its price: A higher range requires a highertransmission power, which in turn consumes more batterypower. Furthermore, a higher range causes more interferencebetween the nodes, which in turn reduces the networkcapacity. On the other hand, if the transmission ranges arelow, spatial reuse of radio resources is improved, but thenetwork might become disconnected into network partitionsand isolated nodes. Obviously, a good compromise betweenhigh and low ranges should be found (‘range assignmentproblem’ [20]). The range should be large enough to keepthe network connected, but it should still be small enough toyield a reasonable power consumption and low interferencebetween nodes.

4.1. A simulation-based study

To get an idea of the connectivity behavior of a randomad hocnetwork, let us first perform a simulation-based study. Weplacennodes using a uniform random distribution on a disk ofradiusa, add links according to the geometric link model, andcheck whether this random topologyG(r0, n) is connectedor not. For this purpose, we programmed a tool in C++that employs the random number generator Mersenne Twister[21] for node placement and the Library of Efficient Datatypes and Algorithms (LEDA) for definition of graph objects.Figure 3 illustrates two examples withn = 500 nodes. Inthe left topology each node hasr0 = 0.10a, which resultsin a disconnected network. The right topology, where thesame nodes haver0 = 0.14a, is connected. For a givenr0,the same experiment is repeated times, and, finally, thepercentage of connected topologies is computed. Clearly,

# connected random topologies

→∞−→ P(con).

P(c

on)

r0/a

n = 1000 n = 500

P(con), simulated

0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 7. ProbabilityP(con) that the network is connected.

For sufficiently large, this experiment gives us a goodestimate ofP(con) of G(r0, n).

Figure 7 shows the simulation results forP(con) overthe normalized ranger0 = r0/a for n = 500 and 1000nodes (based on = 10,000 random topologies). Startingat r0 = 0 and increasingr0, P(con) remains zero untila certain threshold range is achieved. For example, 1000nodes yield an almost surely disconnected network until therange isr0 = 0.08. Oncer0 is larger than this threshold,P(con) increases until the resulting random network isalmost surely connected. For example, 1000 nodes requirer0 ≥ 0.117 to make the network connected with probabilityP(con) ≥ 95%. Forn = 500 we must setr0 ≥ 0.162. Adesired probabilityP(con) ≥ 99% yieldsr0 ≥ 0.130 andr0 ≥ 0.180, respectively. Another observation is that thecurve for the larger number of nodes has a steeper slope.

4.2. Arbitrary node distribution

After this simulation, let us consider the question as to how wecan compute the critical(r0, n) pairs in an analytical manner.Recall that the non-existence of isolated nodes in a givennetwork is a necessary but not sufficient condition for theconnectivity of the network. Thus, for givenn andr0, theprobability that the network has no isolated node, denotedby P(no iso node), is an upper bound for the probability thatthe network is connected, i.e.P(con) ≤ P(no iso node). Theminimum transmission range that is necessary and sufficientto obtain, with a certain probabilityp, a network withno isolated node is therefore a lower bound for the rangethat is required to achieve, with the same probabilityp, aconnected network:

r0(P (con) = p, n) ≥ r0(P (no iso node) = p, n). (21)

Let these threshold ranges be abbreviated byrni0 (p, n) =

r0(P (no iso node) = p, n) andrcon0 (p, n) = r0(P (con) =

p, n), respectively. As shown in the following,rni0 (p, n) can

be computed in a straightforward manner.From Section 3.1 we can calculate the probability for the

event that a node is isolated, i.e.P(node iso) = P (D = 0).These isolation events are ‘almost independent’ from node tonode with the assumptionsn � 1 andr0/a 1. Thus, the



probability that none of then nodes is isolated is

P(no iso node) = (1 − P(node iso))n. (22)

Assuming thatP0(x) ≤ 0.08 andn ≥ 120, as given inSection 3.1, we obtain a very lowP(node iso). Thus, we canagain apply a Poisson approximation, leading to the doubleexponential expression

P(no iso node) = exp

(−n

∫∫e−µ0(x)fX(x) dx

). (23)

To computeP(no iso node), all we need to know is theexpected degree at any possible locationx ∈ A. Varying r0we can then determine the threshold rangerni

0 (p, n), whichserves as a lower bound forrcon

0 (p, n).

4.3. Uniform node distribution

As before, we first study a uniform node distribution on adisk with consideration of border effects. Equation (23) withµ0(x) from (11) simplifies to

P(no iso node)

= exp( − 2n

∫ 1

0e−µ0(r)rdr

)= exp

(−n(1 − r0)

2e−nr20

)︸︷︷︸P(no iso node in central area)

exp(−2n

∫ 1

1−r0

e−µ0(r)rdr)

︸︷︷︸P(no iso node in border area)

.

(24)

The numerical solution of this expression is plotted inFigure 8 forn = 500 andn = 1000 overr0. Simulationresults validate our analytical solution. Moreover, we notethatP(no iso node) always lies above the simulation resultson P(con), which have been copied from Figure 7. Letus consider the behavior ofP(no iso node) for n = 500nodes. The qualitative behavior is the same as that ofP(con).Up to r0 ≤ 0.1 the network contains, almost surely, oneor more isolated nodes. For ranges above this threshold,P(no iso node) increases until each node in the network hasalmost surely at least one neighbor. A probability of 99% isachieved forr0 = 0.176.

The important question is now whetherP(no iso node) isa rather loose or tight bound forP(con). From Figure 8, weobserve the following: There is a non-negligible differencebetweenP(no iso node) and P(con) at low probabilityvalues, but the two curves merge at high probabilities. IfP(no iso node) is larger than 0.95, it represents a goodapproximation forP(con). Let us express this behavior as

P(con) = P(no iso node) − ε,

with ε ≥ 0 and ε → 0 as P(no iso node) → 1.

(25)

Certainly, this phenomenon must be investigated in moredetail: Is the convergence ofP(no iso node) towardP(con)of general validity? Is there a mathematical proof for this?

P(n

o is

o no

de)

or P

(con

)

r0/a

P(con), simulated

P(no iso node), sim.

P(no iso node), analy.

n = 1000 n = 500

0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 8. ProbabilityP(no iso node) that the network has noisolated node and comparison withP(con).

The key to answering these questions is to employ a recentlypublished mathematical theorem about the ‘longest link of therandom minimal spanning tree’ among uniformly distributednodes (Penrose [22]). In terms of graph theory, the essentialstatement of this theorem is as follows: We randomly placen nodes on a given area and measure (a) the longest ofthe nearest neighbor distances between the nodes and (b)the length of the longest link in the minimum spanningtree between the same nodes. Penrose proved that the twolengths are the same in almost all random uniform nodeplacements, ifn/A is sufficiently large. In other words,limn/A→∞ P(rni

0 = rcon0 ) = 1. The bridge to our problem is

as follows: For a given node placement, the longest nearestneighbor distance is the minimum rangerni

0 that is neededto have no isolated node in the network; the longest link inthe minimum spanning tree is the minimum rangercon

0 . Thenetwork has no isolated node ifrni

0 ≤ r0 and it is connectedif rcon

0 ≤ r0.With respect to our range assignment problem, this

theorem can also be interpreted as follows (see illustrationin Figure 9). We consider a given placement ofn nodesthat has been generated by a uniform random distributionon a bounded area. First, all nodes have zero transmissionrange (r0 = 0, Figure 9a). We then slowly increaser0 inall nodes synchronously, such that links are added to thenetwork in the order of increasing link length (Figure 9b–e).The graph created is called the geometric random graphG(r0, n). Clearly, there will be some moment whenr0 issufficiently large to makeG(r0, n) connected (Figure 9e).Penrose proved the following statement: If the node densityis large enough, then with probability close to 1,G(r0, n)

becomes connectedat the same moment wherer0 is largeenough to achieve a graph with no isolated nodes.

An analogous phenomenon is known for pure (non-geometric) random graph processes, in which links betweennodes are added uniformly at random from the set of allpossible node pairs [9, 23]. Also in this case, the networkbecomes connected, with high probability, at the momentwhen we add the link connecting the last isolated node.

Note that this result was proved to be true for uniformlydistributed nodes with and without border effects in a square


440 C. Bettstetter

(a) (b) (c) (d) (e)

FIGURE 9. Illustration of Penrose’s theorem on the connectivity of geometric random graphs.

area [22]. Moreover, in [24] it was shown that the same resultholds for anylp metric in any dimension higher than one (itis not valid in one dimension). It is straightforward to applythis result also to circular areas with or without border effects,since the qualitative connectivity behavior is the same in bothtypes of areas (see [25]).

Together with our observations from simulation-basedstudies, we can conclude with the following result. Thecritical rangerni

0 (p, n) required to achieve, with probabilityp, a network with no isolated node is a tight lower boundfor the rangercon

0 (p, n) required to achieve, with the sameprobabilityp, a connected network, ifp is close to one:

rcon0 (p, n) = rni

0 (p, n) + ε,

with ε ≥ 0 and ε → 0 as p → 1. (26)

Since we are indeed interested in connectivity probabilitiesp close to 1, this result is of great value for us. Letus investigate by means of simulation how largeε is forp = 99%. For givenn, we first computerni

0 (0.99, n) usingnumerical integration. Using this(r0, n) pair, we generate = 20,000 random topologies and check the percentageof topologies with no isolated node and the percentage ofconnected topologies. We then slightly increaser0 andperform the same experiment again, and so on, until thepercentage of connected topologies is 99% within a toleranceof ±0.3%. This simulation is repeated for various values ofn. The overall result is shown in Figure 10 (upper curve). Itshows the analytical curve forrni

0 (0.99) overn, along withsome simulation results forrni

0 (0.99) and rcon0 (0.99). The

simulation results forrni0 (0.99) are directly located on the

analytical curve. Thercon0 (0.99, n) values are only slightly

aboverni0 (0.99, n), i.e. we haveε � 0 for p = 0.99. In

other words, it is sufficient to calculate the critical rangesrni0 (0.99, n) and use them as very good approximations

for rcon0 (0.99, n). Note that the critical range decreases very

slowly for largen as n increases. In summary, we havederived an analytical solution to the connectivity problemin ad hoc networks on a bounded area. We can thus give thesolution to the sensor network planning problem describedin the introduction to this section.

Example (Planning of sensor networks, solution): Using(1), the maximum range of a sensor isr0 = 100 m= 0.1a.From Figure 10 we find that at leastn = 1650 sensors mustbe distributed to obtain an almost surely connected network.In other words, the node density should ben/A = 525 km−2.

In the remaining paragraphs of this subsection, we dis-cuss two further issues: First, how good is our analytical

2500

r 0/a

n

P(no isolated node) = 0.99 = P(con) + ε

P(no isolated node) = 0.95uniformdistributionon disk

uniform distributionneglecting / avoiding border effects

P(no isolated node) = 99% ± 0.005%, simulatedP(con) = 99% ± 0.3%, simulatedP(con) = 95% ± 0.3%, simulated

0 500 1000 1500 2000 0.05

0.1

0.15

0.2

0.25

0.3

FIGURE 10. Critical (r0, n) pairs for uniform nodes on disk.

approximation if we do not requireP(con) = 99% but aresatisfied with a lower connectivity probability, sayP(con) =95%? Second, how good is the approximation for smalln?

To study the first issue, we repeat the above computationsand simulations forp = 95%. The result is shown inFigure 10 (second curve from top). We observe that, in thiscase also,rni

0 (p, n) yields a fairly good approximation forrcon0 (p, n). Nevertheless, the difference between the two

ranges is larger than in the 99%-case. For probabilitiesp ≤ 90% (not shown) the difference is not negligibleanymore.

For smalln, the derived analytical expressions cannot beapplied in a straightforward manner because both the Poissonapproximations and Penrose’s theorem are only valid forlargen. Figure 11a shows additional simulation results onP(con) for low n. Comparing the analytical results withsimulations, we observe thatn can actually be quite small. Infact, rni

0 (0.99, n) gives good bounds down ton = 30 nodes.Our result can therefore be applied for sparsely populatednetworks as well. Finally, we show the simulation results forP(con) = 5%. Choosing(r0, n) below this curve gives analmost surely disconnected network.



10 100 1000

r 0/a

n

almost surelydisconnected

almost surelyconnected

P(no iso node) = 99%, analytical

P(con) = 99% ± 0.2%, simulatedP(con) = 95% ± 0.4%, simulatedP(con) = 5% ± 0.4%, simulated

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

FIGURE 11. Critical (r0, n) pairs for uniform nodes on disk.

4.4. Application of results

The above presentations of critical(r0, n) pairs are usefulin practice for topology design and simulation ofad hoc andsensor networks. As shown in our example, for given para-meters (pt , pr,th andα), we can compute the minimum nodedensity required to obtain a connected network. The resultscan be employed in simulations with mobile nodes, if we usea mobility model that retains the uniform spatial distributionof the nodes. The probabilityP(con) can then be interpretedas the percentage of time that the network is connected. Infact, several simulation-based performance evaluations ofrouting protocols forad hoc networks [26, 27, 28] assume thatthe topology is connected during most of the simulation time.

One might argue that the(r0, n) pairs are valueless inpractice if power/topology control algorithms (e.g. [29])are used in the nodes ‘to raise power until connectivity isachieved.’ But, certainly, the power control also has its limits,namely the maximum transmission power of the nodes. Ifwe interpretr0 as the maximum possible transmission rangeresulting from the maximum transmission power, our resultscan be applied as usual. If we choose a sufficiently highnode densityn/A according to Figures 10 and 11, we canguarantee that the nodes are able to form a connected networkafter being distributed. A topology control mechanism couldproceed as follows: After initial deployment, each nodefirst transmits with its maximum power. In a later phase,distributed power control will be used in a way that nodeswith high degree reduce their power to limit interference andincrease the network capacity, while still keeping the networkconnected.

4.5. Uniform distribution without border effects

As in the section on the expected node degree, we nowregard networks in which none of the nodes suffers fromborder effects, i.e. each node has the same expected degree.We study two scenarios: First,n nodes are uniformlydistributed on a bounded areaA of size A = a2π , anda wrap-around distance model is employed to form links.We are interested in the connectivity of alln nodes. This

0.05 0.1 0.15 0.2

P(n

o is

o no

de)

or P

(con

)

r0/a

P(no iso node):- analytical curve- simulation results

P(con), simulation results:- toroidal distance- subarea Poisson process

n = 1000 n = 500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 12. Connectivity without border effects.

scenario can be applied in simulations to avoid border effects.Second, infinitely many nodes are distributed according toa homogeneous Poisson point process of densityρ on aninfinite area. We are interested in the connectivity propertiesof all nodes located in a circular subareaAa of radiusa (seeFigure 5). Nodes outside ofAa are not considered for thestatistics, but these outside nodes may act as relays to connecttwo inside nodes. In Figure 5, all nodes inside the circle ofradiusa are connected. This scenario can occur if we areinterested in the performance of a wireless network in a givenregion, say, on a university campus.

In the first scenario, the derivation ofrni0 (p, n) is quite

straightforward. Applying the Poisson approximation in (22)twice yields the double exponential expression

P(no iso node) = exp(−ne−µ0) (27)

with µ0 = nr20π/A. Figure 12 shows the plot of (27) for

n = 500 and 1000 nodes, along with some simulation resultsfor P(no iso node) andP(con). Again,rni

0 (p, n) serves as alower bound forrcon

0 (p, n), which is very tight forp = 0.99.A rearrangement of (27) yields

rni0

(p, n

) =√

A

nπ

(ln n − ln ln

1

p

). (28)

Figure 10 shows the plots forp = 0.99 and 0.95 withsimulation results. Clearly, fewer nodes or a lower rangeare sufficient in this borderless scenario to achieve thesamep as in the border scenario. Note thatp = 1 yields− ln ln (1/p) = +∞. In summary, (28) serves as a lowerbound forrcon

0 (p, n) on a bounded area of arbitrary shape.In the second scenario, we do not know how many nodes

are located inAa . We thus define the random variableNa

denoting the number of nodes in this area. The probabilitythat none of the nodes inAa is isolated, under the conditionthat na nodes are located inAa , is P(no iso node| Na =na) � exp(−naP (node iso)). The overall non-isolationprobability is thenP(no iso node) = ∑∞

na=1 P(Na =na)P (no iso node| Na = na). The probability thatna nodesare placed inAa is given by the Poisson distribution

P(Na = na) = µnaa

na ! e−µa , na ∈ N0, (29)


442 C. Bettstetter

whereµa = E{Na} = ρAa denotes the expected number ofnodes inAa . We obtain

P(no iso node)

= exp( − µa

(1 − e−P(node iso)))

� exp(−µaP (node iso))

= exp(−µae−µ0), (30)

with µ0 = ρr20π . Clearly, for a fair comparison with

the cyclic case, we setAa = A and ρ = n/A, whichgives µa = n. Comparing (30) with (27) shows us thatP(no iso node) is equal in both scenarios.

To study the probabilityP(con) that all nodes insideAa

are connected, we simulate a Poisson point process. Afteradding links, we check whether a randomly chosen nodeinside Aa has a path to all other nodes insideAa . If so,the network insideAa is connected. The results for a nodedensityρ = 500/a2π are shown as ‘+’-points in Figure 12.We clearly observe thatP(con) again yields almost the sameresults as the cyclic model on a bounded area. In summary,the connectivity properties of both scenarios can be treatedas being equivalent ifµa = n holds.

The following two sections study inhomogeneous non-uniform spatial node distributions. Our major motivation is tofind out how sensitive the connectivity is to inhomogeneitiesin the node distribution.

4.6. Gaussian node distribution

Supposen nodes are randomly distributed on an infinitearea as a two-dimensional Gaussian distribution with mean(µx, µy) = (0, 0) and varianceσ 2

x = σ 2y = σ 2. We have

fXY (x, y) = 1

2πσ 2exp

(−x2 + y2

2σ 2

). (31)

We consider the following problem: What is the prob-ability that all nodes within a circle of radiusa around themaximum of fXY (x, y) are connected? As before, thiscircular area is denoted byAa , and the unknown num-ber of nodes inAa is Na . SincefXY (x, y) is circularlysymmetric, it makes sense to introduce polar coordinates.The transformation of (31) to polar random variablesR and yields fR (r, φ) = rfXY (r cosφ, r sinφ) =(r/(2πσ 2))exp(−(r2/2σ 2)). The pdf ofR is given by theRayleigh distribution

fR(r) = r

σ 2e−r2/(2σ2). (32)

The probability that a given node is placed withinAa is

Pa = P(R ≤ a) =∫ a

0fR(r) dr = 1 − e−a2/(2σ2), (33)

and, thus,

P(Na = na) =(

n

na

)P na

a (1 − Pa)n−na . (34)

The expected number of nodes inAa is E{Na} = nPa .

P(c

on)

or P

(no

iso

node

)

r0/σ

n = 1270E{Na} = 500

n = 1000E{Na} = 394

n = 500E{Na} = 197

P(no iso node)P(con)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.15 0.2 0.25

FIGURE 13. Connectivity for Gaussian distributed nodes within acircle of radiusa = σ .

Using (2), we computeP0(x); it only depends on the node’snormalized radial coordinater = r/σ and its normalizedranger0 = r0/σ . Next, we obtainµ0(x) = nP0(x) and

P(node iso) =∫ a

0e−µ0(x)fR(r) dr. (35)

We can now calculateP(no iso node| Na = na) as inthe previous section and defineP(no iso node) =∑n

na=1 P(Na = na)P (no iso node| Na = na). Combiningall equations yields

P(no iso node) = (1 − PaP (node iso))n − (1 − Pa)n.

(36)

In summary, we computeP(no iso node) as follows: First,we calculate for givena/σ the value ofPa . Second, forgiven n and r0/σ , we computeP(node iso) by numericalintegration. Finally, both results are applied in (36).

It is not straightforward to state whetherrcon0 (0.99, n) =

rni0 (0.99, n)+ε, with ε very small, still holds in this scenario.

We, therefore, perform some simulations. From the resultsin Figure 13, we observe that the difference between boththreshold ranges is larger than in the uniform case, but it isstill acceptable for highE{Na}.

Figure 14 gives some(r0, n) pairs forP(no iso node) =99% andP(con) = 99%, respectively. Each simulation pointrepresents the average of = 20,000 random topologies.A highera/σ clearly increases the requiredn or r0.

We also observe thatP(no iso node) again represents agood approximation forP(con) in the casesa = σ/2 anda = σ . For increasinga/σ , however, the difference betweenthe two probabilities becomes larger. This behavior can beexplained as follows. If we consider a rather small circulararea (lowa/σ ), nodes outside this area help to connect insidenodes, similar to the Poisson point scenario of Section 4.5.The existence of outside nodes, therefore, increases bothP(no iso node) andP(con). Asa/σ increases, outside nodesstill help to decreaseP(no iso node) by establishing a link,but their density is not sufficient anymore to improve theoverall connectivity.



500 1000 1500 2000

r 0/σ

n

a = σ

a = σ/2

a = 2σ

P(con) = 99% ± 0.3%P(no isolated node) = 99% ± 0.3%

Analytical approx. P(no isolated node) = 99% 0.05

0.1

0.15

0.2

0.25

0.3

0.35

FIGURE 14. Critical (r0, n) pairs for Gaussian distributed nodes.

0.05 0.1 0.15 0.2 0.25

P(n

o is

o no

de)

r0/a

Uniformly distr.nodes; analyticalsolution

RWP nodes, no pauses;analytical approximation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 15. Probability that none of then = 500 nodes is isolated.

4.7. Nodes with random waypoint mobility

Finally, we investigate the connectivity of RWP nodes on adisc. To do so, we plug our results onµ0(x) = ppµ0,p(x) +(1− pp)µ0,m(x) into (23), transform to polar coordinates,and perform numerical integration. Care must be takenon the two different branches of the integral, namely thebranches for the central arear = 0 · · · a − r0 and borderarear = a − r0 · · · a. The curve on the right in Figure 15shows the numerical result forP(no iso node) for n =500 ceaselessly moving nodes. As shown, RWP mobilitydrastically reducesP(no iso node) compared to uniformlydistributed nodes. In particular, the curve experiences asmaller gradient asr0 increases. A threshold range of aboutrni0 (0.99, 500) = 0.24a is needed, compared to 0.176a for

uniformly distributed nodes.The upper curve in Figure 16 shows the critical(r0, n) pairs

required to achieveP(no iso node) = 99% with ceaselessRWP mobility. This figure illustrates very clearly that muchhigher ranges or more nodes are required than in a uniformnetwork. If we increase the expected pause time of the nodes,the spatial node distribution becomes more uniform, and thusthe behavior is improved (seepp = 50%).

The question arising for calculation of the connectivitythreshold rcon

0 (0.99, n) with RWP mobility is whetherrcon0 (0.99, n) = rni

0 (0.99, n) + ε can also be employed

0.05

0.1

0.15

0.2

0.25

0.3

0 200 400 600 800 1000 1200 1400 1600 1800 2000

r 0 /

an

P(no isolated node) = 0.99 = P(con) + ε

RWP movement

RWP movementwith pause prob. 50%

uniform distribution

uniform distributionneglecting / avoiding border effects

FIGURE 16. The critical(r0, n) pairs forP(con) � 99%.

for RWP distributions. To answer this question, let usre-interpret Penrose’s theorem. It actually says: If weincreaser0 of all nodes simultaneously (starting atr0 = 0),the final transition from disconnected to connected is mostlikely caused by a previously isolated node that obtains aneighbor, while network partitions with more than one nodetypically have already become connected at lowerr0. Thisstatement was proved to be true for uniformly distributednetworks with and without border effects. In a uniformnetwork without border effects, each node has the sameisolation probability. If border effects are present, theisolation probability for nodes close to the border is higher,and it becomes even more likely that the transition fromdisconnected to connected is due to an isolated node. Usinga spatial distribution that shows a monotonically decreasingnode density from the middle toward the border of the area,e.g. an RWP distribution, intensifies this effect.

As a result of this discussion, we can again say that thecritical (r0, n) pairs required to keep the network connectedduring at leastP(con) = 99% of the total running timecan be well approximated by the critical pairs required toavoid isolated nodes duringP(no iso node) = 99% of thetime. Figure 16 is thus useful for researchers performingsimulations of ad hoc networks with random waypointmobility. They can now set the simulation parameterssuch that the network is connected during most of thesimulation time. Note that, if the initial spatial distributionfX(x, t = 0) of the RWP nodes is uniform, the network isalso connected withP(con) ≥ 99% during the startup phaseof the mobility model, i.e. when the asymptotically stationarynode distribution is not yet achieved.

In summary, RWP mobility decreases the connectivity ofad hoc networks, although it increases the expected node


444 C. Bettstetter

degree. The shorter the expected pause time of the nodes attheir destination points,E{Tp}, the lower the connectivity.

5. k-CONNECTIVITY

The robustness of a communication network against nodeoutages is an important design aspect. This is especially truefor ad hoc networks, since, here, nodes are very likely tofail, be switched off or suffer from an empty battery. Let us,therefore, study how we obtain a network that is not merelyone-connected but has a higher level of connectivity; let usconsider two- and three-connected networks and the generalcase ofk-connected networks (k ∈ N). In the following, wederive a probabilistic expression that enables us to calculatethe critical(r0, n)pairs that are necessary to achieve an almostsurelyk-connected network.

In Section 4 we required that each node has at least oneneighbor (d(u) ≥ 1, ∀u ∈ G). A generalization of thisproblem is to require that each node has at least a certainnumber of (sayk) neighbors (d(u) ≥ k, ∀u ∈ G). Inother words, the resulting network should have a certainminimum node degreedmin = k. Assuming almost statisticalindependence of the node degrees, we can write

P(Dmin≥k) � P(D ≥ k)n � exp(−nP (D ≤ k − 1)).

(37)

For an arbitrary node distribution, taking into accountborder effects, we obtain with (6) and (9) the expression

P(Dmin≥k) � exp

(−n

∫∫�(k, µ0(x))

(k − 1)! fX(x) dx)

.

(38)

This approximation is tight ifP(D ≥ k) > 0.92 andn ≥ 1500(1 − P(D ≥ k)).

In the following, we again focus on a uniform distributionon a disc. Numerical results ofP (Dmin ≥ k) for n = 500and 1000 nodes andk = 1, 2, 3 are depicted in Figure 17(lines). Moreover, we perform a number of simulations forP (Dmin ≥ k) andP(k-con). We observe the following: fork > 1 also, the minimum range required to achieve, withprobability p, a network withDmin ≥ k serves as a tightlower bound forr0(P (k-con) = p, n), if p is close to 1. Inmathematical form,

r0(P (k-con) = p, n) = r0(P (Dmin≥k) = p, n) + ε,

with ε ≥ 0 and ε → 0 as p → 1. (39)

It is interesting to note that this bound becomes tighter forhigherk. Additionally, it seems thatr0(P (Dmin≥k) = p, n)

is a good approximation forr0(P (k-con) = p, n) even ifp islow. Moreover, as in the simple connectivity case, the boundis tighter for highern.

The convergence ofP(Dmin ≥ k) towardP(k-con) alsohas a graph-theoretical background. In fact, Penrose’stheorem can be extended to thek-connectivity case [24]:If n is high enough, then with high probability, if one startswith an empty graph (i.e. only isolated nodes) and addsthe corresponding links asr0 increases, the resulting graph

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.05 0.1 0.15 0.2 0.25

P(D

min

≥ k)

or

P(k

-con

)

r0/a

r0/an = 1000 nodes

n = 500 nodes

k = 1 2 3

P(Dmin≥ k), simulatedP(k-con), simulated

P(Dmin≥ k), Poisson approx.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.05 0.1 0.15 0.2 0.25

P(D

min

≥ k)

or

P(k

-con

)

k = 1 2 3

P(Dmin≥ k), simulatedP(k-con), simulated

P(Dmin≥ k), Poisson approx.

(a)

(b)

FIGURE 17. ProbabilitiesP(k-con) andP(Dmin≥k) on disc.

becomesk-connected at the moment it achieves a minimumdegree ofk.

Figure 18 shows our overall result: the critical(r0, n) pairsrequired to achieve, with probabilityP(k-con) = 99%, ak-connected network on a disc. These curves have beencomputed by numerical integration of (38) under variationof r0. Simulation results validate the analytical curves.

If border effects are not present, we obtain

P(Dmin ≥ k) � exp

(−n

�(k, nr20π/A)

(k − 1)!

). (40)

For k = 2 this expression can be re-arranged to give us anexplicit equation for the threshold range that is required toobtain, with a probabilityp, a network withdmin = 2:

r0(P (Dmin≥k) = p, n) �

√A

nπ

(1 − W−1

(−e · ln p

n

)).

(41)

The functionW−1(·) denotes the real-valued, non-principalbranch of the LambertW function,1 as defined in [30]. Thisequation is valid forP(2-con) close to one.

1The definition of the LambertW function is that it satisfiesW(x)eW(x) = x. If x is a real number, two real values forW(x) arepossible for−e−1 ≤ x ≤ 0: the principal branchW0(x) with W0(x) ≥ −1,and a second branchW−1(x) with W−1(x) ≤ −1. In our problem,W−1(x)

yields the desiredr0, whileW0(x) would result in a complex value. We usedLambertW(·) in Maple.



0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

100 1000

r 0 /

a

n

P(Dmin≥ k) = 0.99 = P(k-con) + εk = 1 2 3 4 5

P(k-con) = 99% ± 0.3%, simulatedP(Dmin≥ k) = 99% ± 0.005%, simulated

P(Dmin≥ k) = 99%, Poisson approximation

FIGURE 18. The (r0, n) pairs for P(k-con) ≈ P(Dmin ≥k) = 99%.

6. PATH PROBABILITY

Until now, we have studied the level of connectedness fromthe viewpoint of a single node (number of neighbors) andthe viewpoint of the entire network (connectivity). Let usnow address a third viewpoint: the connectedness oftwonodes that want to communicate with each other. We areinterested in the probability that two randomly chosen nodesin a randomad hoc network are connected via a path. Wedenote this value as path probabilityP(path).

For a given networkG with n nodes, the percentage ofconnected node pairs isPpath(G) = # connected node pairs/

# node pairs, wherePpath(G) = 1 if and only if the networkis connected. Otherwise, we have 0≤ Ppath(G) < 1, wherePpath(G) = 0 corresponds to a network withn isolated nodes.The path probabilityP(path) is defined by taking the averageof Ppath(G) over an infinitely large number of random topolo-gies G, i.e. P(path) = lim→∞(1/)

∑i=1 Ppath(Gi).

Clearly, for a random placement ofn nodes and given trans-mission capabilities, it is more likely that two given nodesare connected than that all nodes are connected. Thus, wecan already state thatP(path) ≥ P(con).

Reviewing the literature onad hoc networks, it seemsthat an analytical approach for calculatingP(path) does onlyexist for one dimension and an infinitely large system space(see Section 7). Since we are mainly interested in two-dimensional bounded areas, we approach the problem bymeans of simulation. Once more, we distributen nodes, eachwith ranger0, in a random uniform manner on a disc of radiusa. ThePpath-value of this network is computed, and the sameexperiment is repeated over random topologies. Finally,the average over allPpath-values of the samples is calculated.Varyingr0, we determine the critical(r0, n) pairs required to

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10 100 1000

r 0 /

a

n

P(no iso node) = 99%P(path) = 99% ± 0.1%

95% ± 0.1%5% ± 0.1%

FIGURE 19. Critical (r0, n) pairs forP(path).

achieveP(path) = 5%, 95% andP(path) = 99%. The finalresult is shown in Figure 19.

Example (Planning of sensor networks, continued): Usingthe given sensor type, a network should be deployed thatguarantees that a sensor can reach a randomly chosendestination sensor with a probability ofP(path) = 99%.It is necessary and sufficient to distributen = 800 sensorsusing a uniform random distribution.

7. RELATED WORK

One of the first papers on connectivity issues in wirelessmultihop networks was published by Cheng and Robertazziin 1989 [31]. It investigates how far a node’s messagepercolates for Poisson distributed nodes on an infinitelylarge area. The paper [32] studies the disconnectednessof Poisson distributed nodes. Another early paper [33]addresses uniformly distributed nodes on a one-dimensionalline segment and questions some of the results in [32].

The study of the connectivity problem in two dimensionswas resumed by Gupta and Kumar in 1998 [34]. Theyperformed a fundamental study on the connectivity ofuniform nodes in the asymptotic casen → ∞. The mainresult is as follows: Ifr0 of all nodes in a disk of sizeA = 1 is set tor0 = √

(ln n + const(n))/nπ , the resultingwireless multihop network is asymptotically connected withprobability P(con) = 1 if and only if const(n) → ∞(compare with (28)). At the same time, Chlamtac andFaragó [35] established the link betweenad hoc networks andconventional random graphs and proposed a random networkmodel based on the superposition of Kolmogorov complexityand random graph theory.

In the following years, a number of authors studiedconnectivity issues from various viewpoints using differentmethods. Santiet al. [20] computed bounds forP(node iso)and P(con) with uniform nodes on a one-dimensionalline segment[0, a]. Two-dimensional system areas wereconsidered using simulations. Krishnamachariet al. [36]described by means of simulations that the connectivityexperiences a ‘threshold effect’ for very largen: Aswe increaser0, the connectivity increases suddenly from


446 C. Bettstetter

P(con) = 0 toP(con) → 1 asr0 exceeds a certain thresholdrange. This effect is well known in the literature on randomgraphs. Bettstetter gave the relation betweenP(con) andP(no iso node) [1] and showed how to compute the tightapproximation for the critical(r0, n)-values for uniformlydistributed nodes in two dimensions if border effects areavoided [1, 3] and if they are taken into account [2, 4].During the writing of this article, an alternative solutionmethod for uniform nodes on square areas was presentedin [37]. Moreover, an upper bound forP(con) for uniformnodes on a square area was given in [38]. At the same time,Dousseet al. [39] addressed connectivity-related aspectsemploying methods from percolation theory. This papergives an expression for the probability that two Poissondistributed nodes on an infinite line with a given distancecan establish a multihop path between them. It also studieshow the placement of fixed base stations helps to increase thelevel of connectivity. Most recently, Nakanoet al. [40] studyhow long a node is able to keep a multihop path to a fixedbase station if the node moves in a straight line away fromthe base station.

All the above papers use either a Poisson or uniform nodedistribution. The analytical computation of connectivityfor inhomogeneous node distributions with mobile nodes,namely RWP nodes, is presented in the author’s paper [4].Two further recent papers [41, 42] perform a simulation-based study of topology properties of RWP nodes. Theissue ofk-connectivity and the study of different ranges arediscussed in [1, 3] respectively.

8. CONCLUSIONS

This paper was motivated by the fact that the connectivityproperties ofad hoc networks have a significant impacton their performance. We presented an analyticalframework for the calculation of different stochastic topologyproperties. In doing so, we took three different viewpoints:the viewpoint of a single node, two nodes and thecomplete network. We studied the expected node degreeE{D}, the node isolation probabilityP(node iso), thepath probability P(path), the non-isolation probabilityP(no iso node) and the connectivity probabilityP(con) ina variety of scenarios. The major results can be summarizedas follows.

Considering uniformly distributed nodes, we gave a verytight bound to the well-known problem of finding the critical(r0, n) pairs achieving an almost surely connected network(we setP(con) = 0.99). These results were derived forscenarios with and without border effects. As opposed toprevious research, we also considered inhomogeneous spa-tial node distributions. For example, we derived closed-formexpressions for the average node degree and the critical(r0, n) pairs for almost sure connectivity of RWP nodes.These results show that RWP mobility increases the expectednode degree while it drastically decreases the connectivity.We also studied in detail thek-connectivity, which accountsfor the robustness of the wireless network. Last but notleast, the critical(r0, n) pairs for a high path probability

were given. While the analytical derivations in this articlefocused on a circular area, an adaptation to rectangular areasis straightforward (see [43]).

All results are not only applicable to radio networks butto any kind of multihop system (e.g. multihop underwateracoustic networks [44]). Further work could investigateconnectivity issues with a more realistic channel model,taking into account shadowing [45] and interference [46].

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