Asymptotic Critical Transmission Radius for
k-Connectivity in Wireless Ad Hoc Networks
Peng-Jun Wan, Chih-Wei Yi, Member, IEEE, and Lixin Wang
Abstract—A range assignment to the nodes in a wireless ad hoc network induces a topology in which there is an edge between two nodes if and only if both of them are within each other’s trans-mission range. The critical transtrans-mission radius fork-connectivity is the smallestr such that if all nodes have the transmission ra-diusr, the induced topology is k-connected. In this paper, we study the asymptotic critical transmission radius fork -connectivity in a wireless ad hoc network whose nodes are uniformly and inde-pendently distributed in a unit-area square or disk. We provide a precise asymptotic distribution of the critical transmission ra-dius fork-connectivity. In addition, the critical neighbor number fork-connectivity is the smallest integer l such that if every node sets its transmission radius equal to the distance between itself and itsl-th nearest neighbor, the induced (symmetric) topology is k-connected. Applying the critical transmission radius for k-con-nectivity, we can obtain an asymptotic almost sure upper bound on the critical neighbor number fork-connectivity.
Index Terms—Asymptotic distribution, critical neighbor number, critical transmission radius, random geometric graph.
I. INTRODUCTION
L
ET be the set of radio nodes in a wireless ad hoc net-work. A range assignment to specifies a transmission radius to each node in . The network topology induced by a range assignment is a graph on with an edge connecting each pair of nodes whose distance is no more than either of their transmission radii. There are two simple range assignment schemes, uniform range assignments and -nearest-neighbor range assignments, which are both completely determined by a single parameter. In a uniform range assignment with a parameter , every node has the same transmission radius of . The network topology induced by this range assignment, denoted by , is the -graph on in which each pair of nodes separated by a distance of at most is connected by an edge. In a -nearest-neighbor range assignment with an integerManuscript received September 25, 2007; revised May 25, 2009. Current ver-sion published May 19, 2010. The work of P.-J. Wan was supported in part by the NSF by Grant CNS-0831831 of USA. The work of C.-W. Yi was sup-ported in part by the NSC by Grant NSC97-2221-E-009-052-MY3 and NSC98-2218-E-009-023, by the MoEA by Grant 98-EC-17-A-02-S2-0048, by the ITRI by Grant 99-EC-17-A-05-01-0626, and by the MoE ATU plan. The material in this paper was presented at the 5th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc 2004), Roppongi Hills, Tokyo, Japan, May 24–26, 2004.
P.-J. Wan and L. Wang are with the Department of Computer Science, Illi-nois Institute of Technology, Chicago, IL 60616 USA (e-mail: [email protected]; [email protected]).
C.-W. Yi is with the Department of Computer Science, National Chiao Tung University, Hsinchu City 30010, Taiwan, R.O.C. (e-mail: [email protected]).
Communicated by S. Ulukus, Associate Editor for Communication Networks.
Digital Object Identifier 10.1109/TIT.2010.2046254
parameter , every node sets its transmission radius equal to the distance between itself and its th nearest neighbor. The network topology induced by this range assignment, denoted by , is the symmetric -nearest-neighbor graph on in which there is an edge between each pair of nodes which are both one of each other’s nearest neighbors.
In general, a range assignment has to ensure that certain topo-logical properties are met by the induced network topology. Two topological properties of interest are -connectivity and vertex degree at least . Let and denote the connectivity and the smallest vertex degree, respectively, of a graph. Then these two properties can be simply represented by and , respectively. Both properties are monotone-increasing, which means that all supergraphs of a graph with these properties also have these properties as well. For a monotone-increasing topo-logical property , the critical (or hitting) transmission radius, denoted by , is the smallest at which has prop-erty , and the critical (or hitting) neighbor number, denoted by is the smallest at which has property . Note that is always the distance between some pair of nodes, and is always an integer no more than the size of . Thus, for those which can be tested in polynomial time (such as and ), both and can be obtained in polynomial time as well.
This paper is concerned with the asymptotic critical transmis-sion radius and critical neighbor number in a random wireless ad
hoc network. Specifically, the radio devices are represented by a uniform -point process over a unit-area region , i.e., a set of independent points each of which is uniformly distributed over . Then both and are random graphs, and both and are random variables. In this paper, the region is assumed to be either a disk or a square. For such , we provide a precise asymptotic distribu-tion of when goes to infinity. As a corollary, applying the result, we can get an asymptotic almost sure upper bound on for .
In what follows, is the Euclidean norm of a point , and is shorthand for two-dimensional Lebesgue measure (or area) of a measurable set . All integrals considered will be Lebesgue integrals. The topological boundary of a set
is denoted by . The disk of radius centered at is denoted by . An event is said to be asymptotic almost
sure (abbreviated by a.a.s.) if it occurs with a probability
con-verges to one as . The symbols always refer to the limit . To avoid trivialities, we tacitly assume to be sufficiently large if necessary. For simplicity of notation, the dependence of sets and random variables on will be frequently suppressed.
The remaining of this paper is organized as follows. In Section II, we briefly describe related works. In Section III, we give the precise asymptotic distribution of . In Section IV, based on the result of the critical transmission radius, we present an asymptotic almost sure upper bound on as a corollary. Finally, we conclude this paper in Section V.
II. RELATEDWORKS
Since implies that is always at least . A fascinating result proved by Penrose [1], [2] states that they are equal a.a.s. This means when is big enough, then with high probability, if one starts with iso-lated points and adds edges connecting the points of in order of increasing length, then the resulting graph becomes -con-nected as soon as the last vertex of degree vanishes. Thus, and have the same asymptotic dis-tribution. Although Penrose [1], [2] considered only over a unit-area square, the same result can be extended to over a unit-area disk as well with proper modification.
For and over a unit-area square, the precise asymp-totic distribution of has been derived by Dette and Henze [3] much earlier: for any constant
The same asymptotic distribution also holds for over a unit-area disk. For , Penrose [2] presented the following lim-iting property of for being a unit-area square, which also holds for being a unit-area disk.
Theorem 1: [2] Let and . Then for any sequence satisfying
the probabilities of the two events and both converge to as . A better understanding of Theorem 1 necessitates a brief ex-planation of the Poissonization technique used by Penrose [2] for the proof. Let denote a homogeneous Poisson process of intensity on . Recall that is characterized by the fol-lowing property: if are arbitrarily disjoint re-gions of , then the numbers of points in on
are mutually independent Poisson random variables with inten-sity , respectively. The relevance of to is that given that there are exactly points of in a re-gion , these points are independently and uniformly dis-tributed in . Thus, can be well approximated by . Due to the extreme independence property, is much more conve-nient to be dealt with. Penrose [2] thus first proved a Poissonized version of Theorem 1 in which is replaced by , and then de-Poissonize this Poissonized version to complete the proof of Theorem 1. The value
in Theorem 1 is exactly the expected number of points of with degree in . The value is thus the limit of the expected number of points of with degree in .
However, Penrose [2] didn’t provide the explicit form of , while stating that is not so easy to find because of the domi-nance of complicated boundary effect. To explain the boundary effect, we define the -neighborhood of a point as . The area of such -neighborhood of a point in determines the distribution of the number of neighbors in . The larger this area, the higher the expected number of neighbors. As a node close to the boundary of has small -neighborhood, in-tuitively a node around the boundary have smaller vertex degree. On the other hand, the probability for a node to be around the boundary is small when the node density is large. The overall effect produced by the boundary nodes is thus complicated and even peculiar [4]. In this paper, we will present a partition of to address the boundary effect, based on which we obtain the explicit form of .
Other earlier simulation studies and/or loose analytical results on asymptotic critical transmission radius for connectivity can be found in [5]–[13].
The problem of how many neighbors is desirable for var-ious purposes in a wireless ad hoc network whose nodes are specified by a planar Poisson point process has been studied since the 1970s. For the purpose of maximizing the one-hop progress of a packet in the desired direction under the slotted ALOHA protocol, Kleinrock and Silvester [14] proposed that if all nodes have the same transmission power then six was the “magic number,” i.e., on average every node should con-nect itself to its six nearest neighbors. Later, the magic number was revised to eight by Takagi and Kleinrock [15]. The same paper [15] also considered other transmission protocols, which resulted in some other magic numbers five and seven. Hou and Li [16] considered the situation when each node is allowed to ad-just its transmission range individually, and obtained the magic numbers six and eight. For the purpose of maximizing the trans-mission efficiency defined as the ratio between the expected progress and the area covered by the transmission, Hajek [17] suggested that each node should adjust its power to cover about three nearest neighbors on average. Mathar and Mattfeldt [18] analyzed the wireless network generated by a Poisson point process on a line, and also obtained some magic numbers.
However, none of the analyses in [17], [16], [14], [18], and [15] took connectivity into consideration. Based on simulations, Ni and Chandler [10] suggested that six to eight nearest neigh-bors can make a small size network connected with high proba-bility. But it turns out that as the number of nodes in the network increases, the network becomes disconnected with probability one whether one connects to six or eight nearest neighbors. In fact, Xue and Kumar [19] proved that even if each node con-nects bidirectionally to nearest neighbors the prob-ability of network disconnectivity is asymptotically equal to one as ; on the other hand, if each node connects bidirection-ally to more than nearest neighbors, the network is asymptotically connected. Here the bidirectional nearest neighbor graph means that two nodes have a link if and only if at least one is among the other’s nearest neighbors. In [20], the upper bound was further improved to
for any constant . Recently, Balister
et al. [21] proved that the critical number is asymptotically lower
bounded by and upper bounded by . In addition, for a directional version in which node has a di-rectional link to node if is one of ’s nearest neighbors, the two asymptotic bounds are and , respectively. In this paper, as a corollary of the critical trans-mission radius, we prove that for any integer and real number is an upper bound on , where is the natural base. Note that is defined based on the symmetric -nearest-neighbor graph, and the symmetric neighbor graph, bidirectional -nearest-neighbor graph, and directional -nearest--nearest-neighbor graph all are different from each other.
III. CRITICALTRANSMISSIONRADIUSFOR -CONNECTIVITY
The main results of this section are the following two theorems.
Theorem 2: Assume that is the unit-area square. Let
where
Then the probabilities of the two events
and both converge to as .
Theorem 3: Assume that is the unit-area disk. Let
where
Then the probabilities of the two events
and both converge to as .
We notice that in Theorem 2 and Theorem 3, depends on the shape of and parameter . This can have an intu-itive explanation. A node is isolated if and only if there are no other nodes within its transmission range. Based on the Poisson point process assumption, the probability of without neighboring nodes depends on the area of the transmission range. However, if this node is near the boundary of , then its transmission range is not fully contained in and thus with higher probability to be isolated. This is exactly the boundary effect mentioned in the previous section. Actually, comparing the proof for Theorem 2 in the Section III-A with the proof for Theorem 3 in the Section III-B, we can see that the difference
Fig. 1. Area of the shaded region isa (t).
in the formulas of is due to the boundary effect. For the case of , the boundary effect is the dominating factor. In other words, nodes with degrees less than are almost surely near . Moreover, the factor in Theorem 2 and in Theorem 3 are proportional to 4 and that are the perime-ters of a unit-area square and a unit-area disk, respectively. For the case of , the boundary effect is not the only factor. Isolated nodes also can be found in the internal area of with some probability. So, the formula of is decided by the calculation for the internal area and boundary area.
Throughout of this section, we use to denote the value given either in Theorem 2 or in Theorem 3 depending on whether is a square or a disk. For any , let
(1) the area of the shaded region illustrated in Fig. 1. It is easy to see that equals to length of the boundary chord, i.e., . Remind that we will omit all subscript for simplicity.
We first present the following technical lemma.
Lemma 4: .
Proof: It is straightforward to verify that
Let
(2) Using integration by parts on the integral yields
The first term is asymptotically equal to because
The second term is asymptotically negligible because
Thus, the lemma follows.
In the next two subsections, we give the proofs for Theorem 2 and Theorem 3, respectively.
A. Proof for Theorem 2
By Theorem 1, we only need to show that
Fig. 2. Partition of the square.
To address the boundary effect of the square region , we parti-tion into three subregions and as illustrated in Fig. 2. For any , let denote the set of satisfying that intersects exactly sides of . The areas of these three regions are
For any
When is exactly , where is the distance between and the boundary of .
First, we calculate the integration over . If . Thus
Now, we calculate the integration over . If
Thus
Finally, we calculate the integration over . We further partition into two regions: consists of all points whose distance from the boundary of is at most , and . Then for any
Recall that is defined in (1). Thus
The integration over is calculated as follows. A change of integration variable yields
The last asymptotics is given by Lemma 4.
In summary, if , the integral is asymptotically equal to
If , the integral is asymptotically equal to
In either case, Theorem 2 holds.
B. Proof for Theorem 3
Again by Theorem 1, we only need to show that
To address the boundary effect of the disk region , we partition into three subregions and as illustrated in Fig. 3. Without loss of generality, is assumed to be centered at the origin . is the disk of radius centered at
is the annulus of radii and centered at ; and is the annulus of radii and centered at . The areas of these three regions are
For any
Fig. 3. Partition of a disk region.
Using the same argument as in the proof of Theorem 2, we can show that
and
Next, we calculate the integration over .
For any , let be the distance between and the chord of the circle through the two intersecting points between and (see Fig. 4). Then
In addition
Fig. 4. Forx 2 (1); t(x) denotes the distance between x and the chord of the circle@D(x; r) through the two intersecting points between @D(x; r) and .
Thus, for any
and
We partition into two regions: consists of all points with , and . Then for any
Thus, using the same argument as in the proof of Theorem 2, we can show that
Finally, we calculate the integration over . By the two inequalities just before this theorem
Recall that was defined by (2) in the proof of Lemma 4. A change of integration variable yields
The last asymptotics follows Lemma 4.
Therefore, if , the integral is asymptotically equal to
If , the integral is asymptotically equal to
In either case, Theorem 3 holds.
IV. CRITICALNEIGHBORNUMBER FOR -CONNECTIVITY
Based on Theorem 2 and Theorem 3, we can get the following a.a.s. upper bound on the critical neighbor number
. Remind that the deployment region can be either a disk or a square.
Theorem 5: For any and , the event is a.a.s.
We shall actually prove the following stronger result.
Theorem 6: For any two constants , the event is a.a.s.
Recall that the critical transmission radius for connectivity was given by Theorem 1 in [1], and the critical transmission radius for -connectivity is given by Theorem 2 and 3 in this work (for ). More precisely, the critical transmission radius for -connectivity is in the form of
for , and for . No matter what, for given and , we have if is sufficiently large. In addition, since the probability of
-connectivity is tend to 1 as , to show
is -connected with high probability, we only need to choose large enough. Then, applying Theorem 6, it is a.a.s. that is -connected. Therefore, we can conclude that Theorem 6 together with the result by Penrose [1], Theorem 2 and Theorem 3 implies Theorem 5.
Throughout this section, we let and be fixed constants as in Theorem 6. Pick another constant and let be the smallest integer which is greater than . For any integer , let
Let be the set of all open disks of radius centered at the square grid of side with one corner point at the origin which have nonempty intersections with . Let denote the event that all disks in contains less than nodes of .
We first claim that the event implies the event that . Assume that then event occurs. For any node , there exist a disk in such that the distance between and the center of is less than
. Thus
Since contains less than nodes of , so does the disk . This implies that any neighbor of in is one of its nearest neighbors. Now consider any edge in . Then both and are one of each other’s nearest neighbors. Conse-quently, is also an edge of . So our claim is true. Therefore, Theorem 6 would follows if we can prove that is an a.a.s. event. The remaining of this section is devoted to this proof.
We partition into subsets with
where consists of all disks in centered at the square grid of side with one corner point at . Correspond-ingly, for any let denote the event that all disks in contains less than nodes of . Then
Since the intersection of a constant number of a.a.s. events is also an a.a.s. event, it is sufficient to show that each is an a.a.s. event. We prove this by using the same Poissonization technique as in [19]. Fix two integers . We denote by the event that all disks in contains less than
nodes of . Recall that denotes a Poisson point process with node density over the deployment region . Since , using the similar proof of [19, Lemma 3.2.3] we can show that if is an a.a.s. event, so must be . Thus, we only to prove that is an a.a.s. event. To prove this, we number the disks in
by
where is the index set. For any , let be the number of points of which fall in . Then can be expressed as
. In the next, we show that
Let and be a Poisson random variable with rate . The following upper bound on the tail
dis-tribution of follows from [19, Lemma 3.2.5] and Stirling’s formula:
This bound implies that . Furthermore, since
we have
For any , let . Then each is a Poisson random variable with rate . Note that for any integer , the function is strictly increasing as long as
, since
As , we have
This inequality together with the independence of implies that
Therefore, each is an a.a.s. event. This completes the proof of Theorem 6.
V. CONCLUSION
In this paper, we model the wireless ad hoc network by a uni-form -point process over a unit-area disk or square . We derived the precise asymptotic distribution of the critical trans-mission radius for -connectivity . Based on the result, we also obtained an asymptotic almost sure upper bound on the critical neighbor number for -connectivity
.
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Peng-Jun Wan received the B.S. degree from Tsinghua University, the M.S.
degree from The Chinese Academy of Science, and the Ph.D. degree from the University of Minnesota, Minneapolis.
He is currently a Full Professor of computer science at the Illinois Institute of Technology, Chicago. His research interests include wireless networks, optical networks, and algorithm design and analysis.
Chih-Wei Yi (M’99) received the B.S. and M.S. degrees from the National
Taiwan University and the Ph.D. degree from the Illinois Institute of Tech-nology, Chicago.
He is currently an Associate Professor of computer science with the National Chiao Tung University. He had been a Senior Research Fellow with the Depart-ment of Computer Science, City University of Hong Kong. His research focuses on wireless ad hoc and sensor networks, vehicular ad hoc networks, network coding, and algorithm design and analysis.
Dr. Yi is a member of the ACM. He received the Outstanding Young Engineer Award by the Chinese Institute of Engineers in 2009.
Lixin Wang received the M.S. degree in computer science from the University
of Houston at Clear Lake, the M.S. degree in applied math from the University of Houston, Houston, TX, and the M.S. degree in math from the Fudan University, Shanghai, China. He is currently pursuing the Ph.D. degree in computer science at the Illinois Institute of Technology, Chicago.
