Asymptotic Distribution of The Number of Isolated Nodes in Wireless Ad Hoc Networks with Bernoulli Nodes

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1 Asymptotic Distribution of The Number of Isolated Nodes in Wireless Ad Hoc Networs with Bernoulli Nodes Chih-Wei Yi Peng-Jun Wan Xiang-Yang Li Ophir Frieder Department of Computer cience, Illinois Institute of Technology, Chicago, IL s: Abstract Nodes in wireless ad hoc networs may become inactive or unavailable due to, for example, internal breadown or being in the sleeping state. The inactive nodes cannot tae part in routing/relaying and thus may affect the connectivity. A wireless ad hoc networ containing inactive nodes is then said to be connected if each inactive node is adjacent to at least one active node and all active nodes form a connected networ. This paper is the first installment of our probabilistic study of the connectivity of wireless ad hoc networs containing inactive nodes. We assume that the wireless ad hoc networ consists of n nodes which are distributed independently and uniformly in a unit-area dis and are active (or available independently with probability p for some constant 0 <p. We show that if ln n+c all nodes have a maximum transmission radius r n pn for some constant c, then the total number of isolated nodes is asymptotically Poisson with mean e c and the total number of isolated active nodes is also asymptotically Poisson with mean pe c. I. INTRODUCTION A wireless ad hoc networ is a collection of radio devices (transceivers located in a geographic region. Each node is equipped with an omni-directional antenna and has limited transmission power. A communication session is established either through a single-hop radio transmission if the communication parties are close enough, or through relaying by intermediate devices otherwise. Because of the no need for a fixed infrastructure, wireless ad hoc networs can be flexibly deployed at low cost for varying missions such as decision maing in the battlefield, emergency disaster relief and environmental monitoring. In most applications, the ad hoc wireless devices are deployed in a large volume. The shear large number of devices deployed coupled with the potential harsh environment often hinders or completely eliminates the possibility of strategic device placement, and consequently, random deployment is often the only viable option. In some other applications, the ad hoc wireless devices may be continuously in motion or be dynamically switched to on or off. For all these applications, it is natural to represent the ad hoc devices by a finite random point process over the (finite deployment region. Correspondingly, the wireless ad hoc networ is represented by a random graph. The classic random graph model due to Erdős and Rényi (960 [4], in which each pair of vertices are joined by an edge independently and uniformly at some probability, is not suited to accurately represent networs of short-range radio nodes due to the presence of local correlation among radio lins. This motivated Gilbert (96 [5] to propose an alternative random graph model for radio networs. Gilbert s model assumes that all devices, represented by an infinite random point process over the entire plane, have the same maximum transmission radius r and two devices are joined by an edge if and only if their distance is at most r. For the modelling of wireless ad hoc networs which consist of finite radio nodes in a bounded geographic region, a bounded (or finite variant of the standard Gilbert s model has been used by Gupta and Kumar (998 [6] and others. In this variant, the random point precess representing the ad hoc devices is typically assumed to be a uniform n-point process X n over a dis or a square of unit area by proper scaling, and the wireless ad hoc networ, denoted by G (n, r, is exactly the r-dis graph over X n. To distinguish the random graph G (n, r from the classic random graph due to Erdős and Rényi, it is referred to as a random geometric graph. The connectivity of the random geometric graph G (n, r has been studied by Dette and Henze (989 [3] and Penrose (997 [7]. For any constant c, Dette and Henze (989 [3] showed that the graph G ( n, ln n+c n has no isolated nodes with probability exp ( e c asymptotically. Eight years later, Penrose (997 [7] established that if a random geometric graph G (n, r has no isolated nodes, then it is almost surely connected. These results are the exact analogue of the counterpart in classic random graphs. However, as pointed out by Bollobás (00 [], we should not be misled by the remembrance: the proof for the random geometric graph is much harder. In this paper, we consider an extension to the random geometric graph G (n, r by introducing an additional assumption that all nodes are active (or available independently with probability p for some constant 0 <p. uch extension is motivated by the fault-tolerance of wireless ad hoc networs. In a practical wireless ad hoc networ, a node may be inactive (or unavailable due to either internal breadown, or being in the sleeping state. In either case, the inactive nodes will not tae part in routing/relaying and thus may affect the /03/$7.00 (C 003 IEEE 585 Authorized licensed use limited to: CityU. Downloaded on May,00 at 07:59:53 UTC from IEEE Xplore. Restrictions apply.

2 connectivity. It is natural to model the availability of the nodes by a Bernoulli model, and hence we call the nodes as Bernoulli nodes. A wireless ad hoc networ of Bernoulli nodes is then said to be connected if each inactive node is adjacent to at least one active node and all active nodes form a connected networ. Our probabilistic study of the connectivity of the random geometric graph with Bernoulli nodes consists of two installments due to the lengthy analysis. The first intallment, which is the focus of this paper, addresses the distribution of the number of nodes without active neighbors. For convenience, a node is said to be isolated from active nodes, or simply isolated, it has no active neighbors. We shall prove that both the number of isolated nodes and the number of isolated active nodes have asymptotic Poisson distributions. The second installment, which will be reported in a separate paper, proves that if a random geometric graph with Bernoulli nodes has no isolated nodes, it is also connected almost surely. In what follows, x is the Euclidean norm of a point x R, and A is shorthand for -dimensional Lebesgue measure (or area of a measurable set A R. All integrals considered will be Lebesgue integrals. The topological boundary of a set A R is denoted by A. The dis of radius r centered at x is denoted by D (x, r. The special unit-area dis centered at the origin is denoted by. For any set and positive integer, the-fold Cartesian product of is denoted by. The symbols O, o, always refer to the limit n.to avoid trivialities, we tacitly assume n to be sufficiently large if necessary. For simplicity of notation, the dependence of sets and random variables on n will be frequently suppressed. The remaining of this paper is organized as follows. In section II, we present several useful geometric results and integrals. In ection III, we derive both the distribution of the number of isolated nodes and the distribution of the number of isolated active nodes. II. GEOMETRY OF DIK The results in this section are purely geometric, with no probabilistic content. Let r be the transmission radius of the nodes. For any finite set of nodes {x,,x } in, we use G r (x,,x to denote the graph over {x,,x } in which there is an edge between two nodes if and only if their Euclidean distance is at most r. For any positive integers and m with m, letc m denote the set of (x,,x satisfying that G r (x,,x has exactly m connected components. We partition the unit-area dis into three regions, (0, ( and ( as shown in Fig. : (0 is the dis of radius / r centered at the origin; ( is the annulus of radii / r and / r centered at the origin; and ( is the annulus of radii / r and / centered at the origin. Then, (0 ( r, ( r, ( r ( / r r r ( ( 0 ( Fig.. The partition of the unit-area dis. For any set and r>0,ther-neighborhood of is the set x D (x, r. Weuseν r ( to denote the area of the r-neighborhood of, and sometimes by slightly abusing the notation, to denote the r-neighborhood of itself. Obviously, for any x, ν r (x r /3. Ifx (0, ν r (x r. If x (, we have the following tighter lower bound on ν r (x. Lemma : For any x (, ν r (x r + Fig.. a x b ( x y r. The half-dis and the triangle. Proof: Let y be the point in such that y x x, and ab be the diameter of D (x, r perpendicular to xy (see Fig.. Then ν r (x contains a half-dis of D (x, r to the side of ab opposite to y, and( the triangle aby. ince the area of the triangle aby is exactly x r, the lemma follows. The next lemma gives a lower bound on the area of the r-neighborhood of more than one nodes. Lemma : Assume that r / / + / 0.45/. Let x,,x be a sequence of nodes in such that x has the largest norm, and x i x j r if and only if i j. Then ν r (x,,x ν r (x + r x i+ x i. Proof: We prove the lemma by induction on. We begin with. Let t x x and f (t D (x,r \ D (x,r. We first show that f (t (/ rt. Let y y be the common chord of D (x,r and D (x,r, and let z z be another chord of D (x,r that is parallel to y y and has the same length as y y (see Fig. 3(a. Then 586 Authorized licensed use limited to: CityU. Downloaded on May,00 at 07:59:53 UTC from IEEE Xplore. Restrictions apply.

3 y 0 t x t00 x y (a z 0 z y z t x x t y z (b l A 0 A A 3 If 3, then ν r (x,x,x 3 ν r (x +ν r (x 3 ν r (x + r 3 ν r (x + r 4r ν r (x + r x i+ x i. Fig. 3. The area of two intersecting diss. If >3, then by the induction hypothesis f (t is also equal to the area of the portion of D (x,r between the two chords y y and z z. f (t y y, which is decreasing over [0, r]. Therefore, f (t is concave over [0, r]. ince f (0 0 and f (r r,wehave f (t (/ rt. Now we are ready to prove the lemma for. If x (0, then ν r (x,x ν r (x is exactly f (t, and thus the lemma follows immediately from f (t (/ rt. o we assume that x / (0. Note that for the same distance t, ν r (x,x ν r (x achieves its minimum when both x and x are in. It is sufficient to prove the lemma for x,x. Let y y and z z be the two chords of D (x,r as above with y, and l be the line through the two intersection points between and (D (x,r D (x,r (see Fig. 3(b. We use A to denote the portion of D (x,r\ D (x,r which lie in the same side of l as y ;usea to denote the portion of D (x,r which is surrounded by y y, z z, l and the short arc between y and z ; and use A 3 to denote the rectangle surrounded by y y, z z, l and the line through x and x. Then ν r (x,x ν r (x A A f (t / A 3. Note that one side of A 3 is exactly t and the other side is at most 4r (r ν r (x,x ν r (x 4 + It is straightforward to verify that if then r + (r. 4r (r / / + / 0.45/, 4 4r + (r, rt. and thereby the lemma follows. In the next, we assume the lemma is true for at most nodes and we shall show that the lemma is true for nodes. ν r (x,,x ν r (x,,x +ν r (x ν r (x + 3 r x i+ x i + r 3 ν r (x + r x i+ x i. Therefore, the lemma is true by induction. Corollary 3: Assume that r / / + / 0.45/. Then for any (x,,x C with x being the one of the largest norm among x,,x, ν r (x,,x ν r (x + r max x i x. i Proof: Without loss of generality, we assume that x x achieves max x i x. Let P be a min-hop i path between x and x in G r (x,x,,x and t be the total length of P. Then every pair of nodes in P that are not adjacent nodes in P are separated by a distance of more than r. Thus by applying Lemma to the nodes in P, we obtain ν r ({x i x i P } ν r (x + rt. ince ν r (x,,x ν r ({x i x i P }, and t x x, the corollary follows. In the remaining of this section, we give the limits of several integrals. Lemma 4: For any z [ 0, ], e z z z e z. Proof: For any z 0, z e z z + z. If z [ 0, ], then e z z ( z + z ( z + z4 z z ( z 4 z5 ( z z. Lemma 5: Let r n ln n+ξ pn for some constant ξ. Then n e npνr(x dx e ξ, ( pν r (x n dx e ξ. Proof: We only give the proof of the first asymptotic equality. The second one can be proved in the similar manner 587 Authorized licensed use limited to: CityU. Downloaded on May,00 at 07:59:53 UTC from IEEE Xplore. Restrictions apply.

4 together with the inequalities in Lemma 4. First, we calculate the integration over (0. n e npνr(x dx ne npr (0 ne npr e ξ. (0 Now, we calculate the integration over (. n e npνr(x dx ne 3 npr ( ( nr e 3 npr o (. Next, we calculate the integration over (. By Lemma, n e npνr(x dx ( ( ne npr e npr x dx Therefore, ne npr ne npr ( ne npr ne npr p Lemma 6: Let r any fixed integer, n n r r r ( ρe npr ρ ( ρe npr ρ ( e npr ρ r r 0 e nprt dt dρ dρ dρ npr r e O ( (log n / o (. n e npνr(x dx e ξ. ln n+ξ pn C e npνr(x,x,,x ( pν r (x,x,,x n C Proof: ince for some constant ξ. Then for o (, o (. ( pν r (x,x,,x n e npνr(x,x,,x ( pr, the second equality would follow from the first one. Hence, we only have to prove the first one. Let denote the set of (x,x,,x C satisfying that x is the one with largest norm among x,,x and x is the one with longest distance from x among x,,x. Then n C e npνr(x,x,,x ( n e npνr(x,x,,x. o it suffices to prove n e npνr(x,x,,x Note that for any (x,x,,x, o (. ν r (x +cr x x ν r (x,x,,x r for some constant c by Corollary 3, and n n x i B (x, x x, 3 i ; x B (x, ( r. e npνr(x,x,,x e np(νr(x+cr x x n i3 e npνr(x dx B(x, x x B(x,( r e npcr x x dx n e npνr(x dx ( e npcr x x x x dx B(x,( r (n e npνr(x dx ( ( r n e npcrρ ρ 3 dρ 0 < (n e npνr(x dx (n e npcrρ ρ 3 dρ 0 ( 3! n ( (npcr n e npνr(x dx O ( n e npνr(x dx (ln n o (, where the last equality follows from Lemma 5. Lemma 7: Let r ln n+ξ pn for some constant ξ. Then for any fixed integers m<. n o (, n C m e npνr(x,x,,x ( pν r (x,x,,x n C m Proof: ince o (. ( pν r (x,x,,x n e npνr(x,x,,x ( pr, 588 Authorized licensed use limited to: CityU. Downloaded on May,00 at 07:59:53 UTC from IEEE Xplore. Restrictions apply.

5 the second equality would follow from the first one, and thus we only have to prove the first one. For any m-partition Π {K,K,,K m } of {,,,}, let (Π denote the set of (x,x,,x such that for any j m, the nodes {x i : i K j } form a connected component of G r (x,x,,x. Then C m is the union of (Π over all m-partitions Π of {,,,}. o it is sufficient to show that for any m-partition Π of {,,,}, n e npνr(x,x,,x o ( (Π Now fix a m-partition Π {K,K,,K m } of {,,,}, and let l j K j for j m. Then, m (Π C lj, j and for any (x,x,,x (Π, m ν r (x,x,,x ν r ({x i i K j }. n n j e npνr(x,x,,x (Π e np m j νr({xi i Kj} (Π m e npνr({xi i Kj} (Π j m n n j C lj m n lj j C lj e npνr({xi i Kj} i K j e npνr({xi i Kj} o (, i K j where the last equality follows from Lemma 6 and the fact that at least one l j. Lemma 8: Let r ln n+ξ pn for some constant ξ. Then for any fixed integer, n e ξ, C e npνr(x,x,,x n ( pν r (x,x,,x n e ξ. C Proof: We again only give the proof of the first asymptotic equality and remar that he second one can be proved in the similar manner together with the inequalities in Lemma 4. For any (x,x,,x C, ν r (x,x,,x ν r (x i. n e npνr(x,x,,x C e np νr(xi C n n n e np νr(xi e np νr(xi \C. We show the first term is asymptotically equal to e ξ, and the second term is asymptotically negligible. Indeed, n e np νr(xi e npνr(xi n e npνr(xi e ξ, where the last equality follows from Lemma 5. Note that for any (x,x,,x \ C, n n ν r (x,x,,x e np νr(xi \C ν r (x i. e npνr(x,x,,x \C e npνr(x,x,,x m C m where the last equality follows from Lemma 6. o (, III. AYMPTOTIC DITRIBUTION OF THE NUMBER OF IOLATED NODE The main result of this paper is the following theorem. Theorem 9: uppose that all nodes have a maximum transmission radius r ln n+ξ pn for some constant ξ. Then the total number of isolated nodes is asymptotically Poisson with mean e ξ, and the total number of isolated active nodes is also asymptotically Poisson with mean pe ξ. The above theorem will be proved by using Brun s sieve in the form described, for example, in [], Chapter 8, which is an implication of the Bonferroni inequalities. 589 Authorized licensed use limited to: CityU. Downloaded on May,00 at 07:59:53 UTC from IEEE Xplore. Restrictions apply.

6 Theorem 0: Let B,,B n be events and Y be the number of B i that hold. uppose that for any set {i,,i } {,,n} Pr (B i B i Pr(B B, and there is a constant µ so that for any fixed n Pr (B B µ. Then Y is also asymptotically Poisson with mean µ. For applying Theorem 0, let B i be the event that X i is isolated for i n and Y be the number of B i that hold. Then Y is exactly the number of isolated nodes. imilarly, let B i be the event that X i is isolated and active for i n and Y be the number of B i that hold. Then Y is exactly the number of isolated active nodes. Obviously, for any set {i,,i } {,,n}, In addition, Pr (B i B i Pr(B B, Pr ( B i B i Pr(B B. Pr (B B p Pr (B B. in order to prove Theorem 9, it suffices to show that if r for some constant ξ, then for any fixed, ln n+ξ pn n Pr (B B e ξ. ( The proof of this asymptotic equality will use the following two lemmas. Lemma : For any x, Pr (B X x ( pv r (x n. Proof: For any x, Pr (B X x Pr( i n, X i is either outside v r (x or inactive n ( n q i ( v r (x n i v r (x i dx i i0 ( v r (x+qv r (x n ( pv r (x n. Lemma : For any and (x,,x, Pr (B B X i x i, i ( pν r (x,,x n. In addition, the equality is achieved for (x,,x C. Proof: For any (x,,x, Pr (B B X i x i, i ( νr (x Pr,,x contains no active node in X +,,X n n ( n q j ( ν r (x,,x n j j j0 ν r (x,,x j ( pν r (x,,x n. For any (x,,x C, Pr (B B X i x i, i ( i, νr (x Pr i contains no active node in X +,,X n n Pr i, ν r (x i contains m i inactive nodes and no active nodes m + +m 0 in X +,,X n n ( ( n (q mi ν r (x i mi m,,m m + +m 0 ( ν r (x,,x n ( pν r (x,,x n. m i Now we are ready to prove the asymptotic equality (. From Lemma and Lemma 5, n Pr (B n ( pv r (x n dx e ξ. o the asymptotic equality ( is true for.nowwefix. From Lemma, Lemma 6 and Lemma 7, n Pr ( B B, and (X,,X \ C n ( pν r (x,,x n o (. \C From Lemma and Lemma 8, n Pr (B B, and (X,,X C n ( pν r (x,,x n e ξ. C the asymptotic equality ( is also true for any fixed. This completes the proof of Theorem 9. REFERENCE [] N. Alon, and J.H. pencer, The Probabilistic Method, econd Edition, Wiley, New Yor, (000. [] B. Bollobás, Random Graphs, econd Edition, Cambridge University Press, 00. [3] H. Dette, and N. Henze, The limit distribution of the largest nearest neighbor lin in the unit d-cube, J. Appl. Probab., vol. 6, pp , 989. [4] P. Erdős, and A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. ci. 5, 7-6, 960. [5] E.N. Gilbert, Random plane networs, Journal of the ociety for Industrial and Applied Mathematics 9:4 (96, [6] P. Gupta, and P.R. Kumar, Critical power for asymptotic connectivity in wireless networs, tochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W. M. McEneaney, G. Yin, and Q. Zhang (Eds. (998. [7] M.D. Penrose, The longest edge of the random minimal spanning tree, Ann. Appl. Probab., vol. 7, pp , Authorized licensed use limited to: CityU. Downloaded on May,00 at 07:59:53 UTC from IEEE Xplore. Restrictions apply.