Connectivity of a Gaussian Network

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1 Wetern Wahington Univerity Wetern CEDAR Mathematic College of Science and Engineering 28 Connectivity of a Gauian Network Paul Baliter B. Bollobá Amite Sarkar Wetern Wahington Univerity, amite.arkar@wwu.edu M. Walter Follow thi and additional work at: Part of the Mathematic Common Recommended Citation Baliter, Paul; Bollobá, B.; Sarkar, Amite; and Walter, M., "Connectivity of a Gauian Network" (28). Mathematic Thi Article i brought to you for free and open acce by the College of Science and Engineering at Wetern CEDAR. It ha been accepted for incluion in Mathematic by an authorized adminitrator of Wetern CEDAR. For more information, pleae contact weterncedar@wwu.edu.

2 Connectivity of a Gauian Network P. Baliter Department of Mathematical Science, Univerity of Memphi, Memphi TN 3852, USA pbalitr@mci.memphi.edu B. Bollobá Department of Mathematical Science, Univerity of Memphi, Memphi TN 3852, USA bolloba@mci.memphi.edu and Trinity College, Univerity of Cambridge, Cambridge CB2 TQ, UK A. Sarkar* Department of Mathematic, Wetern Wahington Univerity, Bellingham WA 98225, USA amite.arkar@wwu.edu * Correponding author M. Walter Peterhoue, Univerity of Cambridge, Cambridge, CB2 RD, UK mjw9@cam.ac.uk Abtract: Following Etherington, Hoge and Parke, we conider a network coniting of (approximately) N tranceiver in the plane R 2 ditributed randomly with denity given by a Gauian ditribution about the origin, and aume each tranceiver can communicate with all other tranceiver within ditance. We give bound for the ditance from the origin to the furthet tranceiver connected to the origin, and that of the cloet tranceiver that i not connected to the origin. Keyword: Wirele enor network; Gauian ditribution; Tranceiver; Gilbert model; Continuum percolation. Biographical note: Béla Bollobá currently hold the Jabie Hardin Chair of Excellence in Graph Theory and Combinatoric at the Univerity of Memphi and i a Fellow of Trinity College, Cambridge, England. He received PhD from Eotvo Univerity, Budapet, Hungary and the Univerity of Cambridge, England. He work in extremal and probabilitic combinatoric, polynomial of graph, combinatorial probability and percolation theory. He ha written eight book, including Extremal Graph Theory, Random Graph and Modern Graph Theory, and over 3 reearch paper. He ha upervied over forty doctoral tudent, many of whom hold leading poition in prominent univeritie throughout the world. Paul Baliter i a Profeor at the Univerity of Memphi, where he ha been ince 998. He obtained hi PhD from the Univerity of Cambridge, England in 992 and ubequently held poition at Harvard and Cambridge. He ha worked in algebraic number theory, combinatoric, graph theory and probability theory. Amite Sarkar i a Viiting Aitant Profeor at the Univerity of Memphi. He obtained hi PhD from the Univerity of Cambridge in 998, and ha been at Memphi ince 23. He ha worked in combinatoric and probability theory. Mark Walter i a Teaching Fellow at Peterhoue College, Cambridge. He obtained hi PhD from the Univerity of Cambridge in 2, and wa a Reearch Fellow of Trinity College, Cambridge. He ha worked in combinatoric and probability theory.

3 INTRODUCTION In 96, E.N. Gilbert defined and tudied the following model of a random geometric graph, known a the dic model or Gilbert model (4). Let P be a Poion proce in the plane of intenity one, and join every point of P to every other point of P within ditance r, for ome fixed r >. For mall r, mot point are iolated, that i, not connected to any other point. However, a r increae, the point form mall connected cluter, which then connect up to each other (a r increae till further), eventually forming a (connected) giant component, which contain a poitive fraction of point in any large region. Looely peaking, Gilbert derived upper and lower bound for the mallet value of r uch that the probability of the lat eventuality (alo known a percolation) i one. In recent year there ha been renewed interet in uch graph, which are now being ued to model enor network and wirele ad-hoc network in general. However, for ome application, it i deirable that the initial ditribution of enor i non-uniform. One uch model wa recently propoed by Etherington, Hoge and Parke (3), in which the location of the enor are modeled a a Poion proce whoe intenity i given by a two dimenional Gauian ditribution. Such a network might arie, for intance, if the enor were dropped from an aircraft. Specifically, Etherington, Hoge and Parke defined the following random geometric graph G = G(N, σ, ). We tart with a Poion proce in R 2 with intenity at radiu r given by ρ(r) = N 2πσ 2 e r2 /2σ 2, for ome contant σ. In addition, place a point at the origin, o that the expected total number of point i N+. Then connect each point to every other point at ditance le than. We now ak two quetion. Firt, what i the larget value of r uch that every point in D r (), the (open) dic of radiu r centered at the origin, i joined to every other point of D r ()? Second, what i the mallet value of r uch that D r () contain all the point of C = C(), the component of G containing the origin? (C() conit of all the point of G that can be connected to the origin.) Formally, we define and r = up{r : D r () V (G) V (C)}, r + = inf{r : V (C) D r ()}. In thi paper we derive lower and upper bound for r and r +, for variou range of value of. Since r and r + are random variable, our bound will only hold with high probability (whp), that i, with probability tending to one a N. By a uniform caling of R 2, we may without lo of generality fix σ 2 = /2, o that the denity of point i given by ρ(r) = N π e r2. Copyright c 2x Indercience Enterprie Ltd. We imagine N to be very large, and conider variou function = (N). Firt we give a heuritic argument which yield the aymptotically correct value of r for a large range of value of. Auming that the denity of point in a mall dic of radiu i approximately contant, the probability that a point at ditance r from the origin i an iolated vertex in G can be approximated by e ρ(r)π2 = e N2 e r2. Therefore, provided R, the expected number of iolated vertice in D R () i approximately R 2Nre r2 e N2 e r2 dr = 2( e N2 e R2 e N2). () When R r, there will be no iolated vertice in D R (). On the other hand, it eem reaonable to uppoe, by analogy with other imilar problem, that the cloet point to the origin not belonging to C i, with probability tending to one a N, an iolated vertex in G. Therefore we may aume that if R > r, there will be at leat one iolated vertex in D R (). Conequently, if we chooe R o that the expected number of iolated vertice in D R () i one, we might expect that r R. Hence, if N /2 (log N) /2, r 2 log(n 2 / log(/ 2 )). (2) One approach to proving rigorou bound for r i to apply method and reult for the related dic model G (A), decribed above for the cae A = R 2. Here, given a region A R 2, we conider a Poion proce of intenity one in A, and join each point to all other point within a radiu to obtain G (A). We have the following reult of Gupta and Kumar (5), which wa proved for the quare S N of area N by Penroe (6). Theorem. Let A N be a dic of area N, and let = (N) atify π 2 = log N + ω(). Then whp G (A N ) i connected. However, it turn out that applying thi reult yield only the weak bound r 2 log(n2 / logn). (3) A heuritic explanation i a follow. For the reult of Gupta and Kumar, the obtruction to connectivity i the exitence of iolated vertice, and it eem reaonable to uppoe that thi i alo true for our model. If we chooe o that the probability that a vertex i iolated i o(n ), then the expected number of iolated vertice i o(), o whp there are no iolated vertice and (by the above fact) whp G (A) i connected. For our model, even if we chooe r o that the probability that a vertex at ditance r from the origin i iolated i Θ(N ), a in (3), then the only vertice in D r () which have probability Θ(N ) of being iolated lie very cloe to the boundary of D r (), and there

4 are far fewer than N uch vertice. Conequently, it i likely that r i omewhat larger than uggeted by (3), and we will argue directly in the proof of Lemma 5 to how that indeed (2) i much cloer to the truth. Neverthele, we will ue Theorem in deriving a lower bound for r when i very mall. Next we turn to r +. For the dic model G (R 2 ), with an additional point at the origin, reult from (2) (ee alo (4)) how that there i a critical denity γ 4.52 o that if π 2 < γ then C() (defined a above) i finite with probability one, while if π 2 > γ there i a non-zero probability that C() i infinite. Thi ugget that whp o that whp γ π 2 ρ(r + ) = N 2 e r2 +, r 2 + = O(log(N2 )), (4) and we will how in Section 2 that in fact r 2 + = Θ(N2 ) for a large range of value of. For R, the above argument fail a the denity of point in a dic of radiu at ditance R from the origin i far from contant. Of particular interet i the value of for which G become connected. We how that thi occur when 2 log N log log N 2 log log log N. The main obtacle to connectivity i the preence of iolated vertice that are among the furthet point from the origin. Thee vertice are at ditance R log N, o R. 2 Precie reult Let r min and r max be the ditance of the nearet (repectively furthet) point from the origin. Theorem 2. Define G = G(N, ), r and r + a above. Then the following tatement hold with high probability.. If 2 log N log log N 2 log log log N + ω() then G i connected. 2. If 2 log N = log log N 2 log log log N + O() then each of the following event ha probability bounded below by ome poitive contant: (a) G i connected, (b) G i connected except for the furthet point from which i iolated, (c) G i connected except for one iolated point that i not furthet from, (d) G ha more than three component. 3. If 2 log N log log N 2 log log log N ω() then G i diconnected and r + < r max whp. Moreover, if 2 log N C log log N for ome C < and N 2 log N, r 2 = log(n 2 / log(/ 2 )) + 2 log N 3 2 log max{, log N} + O() r 2 + = log(n2 ) + Θ(( log N log log N) /2 ) + O(). [If 2 log N = o(/ loglog N), thee implify to r 2 = log(n2 / log(/ 2 )) + O() r 2 + = log(n2 ) + O().] 4. If N 2 = C log N for ome contant C > then (a) if C >, r 2 = Θ(), (b) if C =, r 2 = ( + o())log log N/ log N, (c) if C <, r 2 = NC +o(). In all cae r 2 + = log(n2 ) + O() = log log N + O() a above. 5. If N 2 then r /, while r 2 + = log(n 2 ) + O() a above. 6. If N 2 = C >, then r / ha a limiting ditribution. (a) If C > γ, where γ i the critical denity for dic percolation, then with poitive probability r+ 2 = ( + o())log(c/γ). Conditional on thi not occurring r + / ha a limiting ditribution. (b) If C γ then r + / ha a limiting ditribution. In both cae there i a poitive probability that r + =. 7. If N 2 = o() then whp the origin i iolated, o that r = r min and r + =. Part of the theorem i proved a part of Lemma 5 in Section 2., and part 2 follow from the remark following the proof of Lemma 5. Part 3 i contained in Lemma 5, 7, 8, and 9, except for the aertion relating to r max, which follow from the remark following the proof of Lemma 5. The remainder of the theorem i proved in Section 3. 3 General bound Firt we prove two eay bound on r max giving an idea of the cale of thi ditribution. Lemma 3. P(rmax 2 log N +α) = e e α, o whp log N ω() < rmax 2 < log N + ω(). Proof. The number of point outide radiu R i Poion ditributed with mean Ne R2, and thu the probability that there are no point further than R from the origin i exp( Ne R2 ). The reult follow.

5 Next we prove ome bound on r. Before we do thi, we firt prove a imple lemma concerning the mean number of point in a dic, which take into account the variation in denity acro the dic. Lemma 4. Fix a point z of R 2 at ditance r from the origin. Then the expected number E r, of vertice of G lying in D (z) i given by where E r, = E V (G) D (z) f(r, ) = min = Ne (r )2 f(r, )θ(r, ) { 2, 2, 4π(r ), 4π(r ) 3 } and c θ(r, ) for ome c > independent of r and. Remark. Numerical calculation how that we can take c =.355. Proof. We can calculate the expected number exactly a N π + 2 x 2 2 x 2 e (r+x) 2 y 2 dy dx. We can etimate z z e y2 dy = min{ π, 2z}θ (z) where < c θ (z). Writing x = + ε the above expreion become where 2 N π e (r )2 e 2(r )ε ε2 m(ε, )θ ( 2ε ε 2 )dε, m(ε, ) = min{ π, 2 2ε ε 2 }. We can bound the above integral by 2 π 2εe 2(r )ε dε = 4(r ) 3. Since thi integral i dominated by the contribution when ε (r ), thi will give the correct order of magnitude when r,,. Similarly, we can bound the integral π e 2(r )ε dε = π 2(r ) by, and thi give the correct order of magnitude when r. We can bound the integral by e ε2 π dε = π 2 and thi give the right order when r. Finally we can bound the integral by 2 2 2ε ε 2 dε = π 2, and thi give the correct order when and r. Thu we have bounded E r, a required, and hown that the bound i of the right order except on a compact region in R 2 +, where the reult follow by continuity (ee Figure ). 3. Bound for r With the aid of Lemma 4, we can obtain fairly tight bound for r. r π(r-) 3.4 4π(r-) 2 Figure : The function θ(r, ). Formulae hown are value of the minimum in the expreion in Lemma 4. Lemma 5. Suppoe that = (N) = o() and (log N)/N. Then whp Moreover, if r 2 log(n2 / log(/ 2 )) + 2 log N log max{, 3 log N} 3. 2 log N log log N log log log N + ω() 2 then G i connected whp. Proof. For any point x at ditance r from, let the region A(x) be the interection of the ball D (x) and D r (). Note that all of A(x) i cloer to than x i. Let E R be the expected number of vertice x within R of with no point in the region A(x). If E R then whp r R ince whp we can find a equence of point joining any point x D R () to. Now A(x) D (x) /3, and ince the denity of point i higher in A(x) than in D (x) \ A(x), E V (G) A(x) 3 E r,. By Lemma 4, the probability that A(x) i empty i at mot exp ( 3 E ) ( r, exp Nµe (r )2), provided µ c 3 min { 2, 4π(r ) 3 }, the other two term in the minimum in Lemma 4 being redundant when = o(). For r < 3 we can take µ = c 3 2. Then E Nre r2 e Nµe (r )2 dr 2Nre Nµe 42 dr 8N 2 e (c/3 o())n

6 By aumption N 2, o E 3. For 3 r R et µ = c 3 2 min {, (3R) 3/2}. Writing z = Nµe (r )2 + log µ + 2r 2 (note that z i decreaing in r) we have dz dr = 2(r )Nµe (r ) Thu we have dz dr > rnµe (r )2 provided r 3 and Nµe (r )2 > 2. Then, writing we have z = Nµe (R )2 + log µ + 2R 2, (5) E R E 3 = R 3 z 2Nre r2 e Nµe (r )2 dr 2µ e (r ) 2 r 2 Nµe (r )2 dz z 2e z dz = 2e z, o it i enough that z, and Nµe (R )2 > 2. (6) Set (R ) 2 = log(nµ/ log(/ 2 )) α, α >. Then Nµe (R )2 = e α log(/ 2 ) > 2 for mall, and z = (e α )log(/ 2 ) + log(µ/ 2 ) + 2R 2. Now log(µ/ 2 ) = 3 2 log max{, 3R} + log(c/3), o log(µ/ 2 )+2R i bounded below. Since α > and, we have z a required. If R >, then R 2 = ( + o())log N, o log(µ/ 2 ) 3 2 log max{, 3 log N} 2.5. Thu, taking α =., R 2 =log(nµ/ log(/ 2 )) + 2R 2. log(n 2 / log(/ 2 )) + 2 log N 3 2 log max{, 3 log N} 3, and r R ince E R. If R then the ame bound applie ince 2 log N <. unle R 2 = ( + o())log N. For the lat part, et R 2 = log N + α where α i contant. Aume 2 log N = log log N 2 log log log N + β, where β, but β = o(log log N). Then 2R = ( + o())log log N, o µ = c (2R) /2 R 2 = Θ( log log N/ log N). Thu z log µ + 2R 2 = β + O(). Similarly Nµe (R )2 = e log µ+2r α+o() = e β α+o(). Hence r 2 log N +α whp for any fixed α. Thu r > r max, o G i connected whp. Thi etablihe part of Theorem 2. For part 2, uppoe that 2 log N = log log N 2 log log log N + β, where we will initially uppoe that β i contant. The proof of Lemma 5 how that there exit an α and C, depending only on β uch that E R C when R 2 = log N +α. Since the expected number of point at ditance greater than R from the origin i e α, E C for ome contant C C + e α depending only on β. By dividing the plane into, ay, C ector, we ee that there i a large probability that all the point in any one particular ector can be connected to a point nearer the origin, ince in any one ector the expected number of point which can t be connected to a point nearer the origin i at mot.. Since event in different ector are almot independent, it follow (by, for example, the Lováz Local Lemma) that G i connected with probability bounded away from zero. On the other hand, let u etimate the probability p(α, β) that a vertex at ditance R i iolated, where R 2 = log N + α and α i contant. We have { p(α, β) = exp Ne (R )2 } 4π(R ) θ(r, ) 3 exp { c Ne log N α+2r { exp c e α+2 log N } log log N (log N) exp { c 2 e β α} = c 3. R 3 } Hence the probability that the furthet, or econd furthet point from the origin i iolated i bounded below by a contant. Since thee point are likely to be far from one another, they are iolated almot independently. Thi, together with the firt obervation, etablihe part 2. Finally, if β with α fixed, we ee that p(α, β) and o r + < r max whp, etablihing the firt tatement of part 3. If i not o() then G i connected, o r = and r + = r max. If (log N)/N, the bound for r 2 in Lemma 5 i negative, o trivially true. We hall how later that in thi cae r = o(). We call a point x V (G) iolated if D (x) V (G) = {x}. Clearly any iolated point x cannot lie in V (C), o mut be at ditance at leat r from. To obtain an upper bound for r, we follow the proof of Lemma 5 but intead etimate E R, the expected number of iolated point within ditance R of. We require the following lemma. Lemma 6. Let E R denote the expected number of iolated point in G within ditance R of. If E R then r R whp. Proof. Firt note that any two iolated point mut be at leat ditance apart. Hence there i at mot one iolated point in any quare of ide length 2/3. On the other hand, the exitence of iolated point in two uch quare i independent if the quare are at ditance at leat 2, ince the event that a quare contain an iolated point depend only on the proce within ditance of that quare. Tile R 2 with quare S ij = [, 2/3] 2 + (2i/3, 2j/3). Partition thi collection into 6 clae according to the value of (i mod 4, j mod 4) Z 2 4. Let X kl, k, l Z 4 be the number of iolated point within R of and in one of the quare in cla (k, l). The X kl are dependent, but till k,l EX kl = E R. Hence there i at leat one X k,l, ay X k l, with EX k l E R /6. Now X k l i a um of independent bernoulli random variable with mean p i, ay.

7 Thu P(X k l = ) = ( p i ) exp( p i ) i i = exp( EX k l ) exp( E R /6), o if E R, then whp there i an iolated point within ditance R of, and r R whp. Lemma 7. Suppoe that (log N)/N and alo that 2 log N ( ε)log log N for ome ε >. Then whp Moreover, if r 2 log(n2 / log(/ 2 )) + 2 log N 3 2 log max{, 2 log N} + log(2/ε). 2 log N log log N log log log N ω() 2 then G i diconnected whp. Proof. We etimate E R and ue Lemma 6. The probability that x V (G) i iolated i at leat exp( E r, ) exp( Nµe (r )2 ), where µ = min{ 2, 4πR }, when x 3 i ditance at leat R + from the origin. Hence E R = R R + R R e 2R R 2Nre r2 e Nµe (r )2 dr 2N(z + )e z 2 2z 2 e Nµe z2 dz R 2Nze z 2 e Nµe z2 dz [ ] R = e 2R µ e Nµe z2 ( z=r ) = µ e 2R e Nµe (R )2 e Nµe R 2. Set (R ) 2 = log(nµ/ log(/ 2 )) α, where α < i contant. Conider the cae when log N = O() firt. Take R =, o µ = 2. Then E R )(e α ) e 2R log(/2 2 e N2. By aumption on, 2 e N2 / logn, and R = O(), o E R when α <. Now aume log N i large. Then R 2 = ( + o())log N. Take R = ( ε)r. Now Nµe (R )2 = e α log(/ 2 ). Alo e R 2 /e (R )2 = o(), o log E R log µ 2R e α log(/ 2 )+o() = ( e α +o())log log N 2R+o() (ε e α + o())log log N for α < log ε. Now, if (R ) 2 = log(nµ/ log(/ 2 )) α then R 2 log(n 2 / log(/ 2 )) + 2 log N 3 2 log max{, 2 log N} α + o(), and r R. For the lat part, et R 2 = log N α. Then log µ + 2R. Thu Nµe (R )2 = e log µ+2r+α+o(). Therefore, uing the approximation e θ θ for mall θ, E R ( o())µ e 2R (Nµe R 2 Nµe (R )2 ) = ( o())(e log N R 2 2R e α 2 ). If R 2 = ( ε)r 2, then log N R 2 2R = ε logn + O(log log N) o E R a required. Since thi hold for all α >, r < r max and G i diconnected whp. Thi complete the proof of the etimate for r in part 3 of Theorem Bound for r + Now we turn our attention to r +. In thi cae we are intereted in the exitence of at leat one point at ditance R which i joined to the origin. Lemma 8. Suppoe that 2 log N log log N and r / whp. Then, whp, r+ 2 log(n2 ) + 2 log N log log N + O( log N + ). Proof. Firt aume log N log log N = O(). Then the tatement reduce to r 2 + log(n2 ) + O(). Conider the dic D R () where R i given by R 2 = log(n 2 ) 3. The Poion proce retricted to D R () tochatically dominate a Poion proce in D R () with contant intenity ρ(r). Cover the dic with a quare teellation where the quare have ide length / 5. The number of point inide any of the quare which are wholly inide D R () dominate a Poion ditribution with mean λ = ρ(r) 2 /5. Subtituting for ρ and R we get λ = N 5π e R2 2 = e3 5π >.27. The probability that any uch quare contain no point i at mot e λ. We compare thi proce to a ite percolation on Z 2 by declaring a ite to be open if it correponding quare contain at leat one point. The ite percolation dominate our proce in the ene that percolation in the ite model implie percolation in our model. Since e λ >.7 > p c (ite) the probability that ome quare inide D r () i in an infinite component tend to one, and thu the probability that the origin in our proce i in an infinite component tend to one. Thu, whp, the origin i joined to ome point of the proce at ditance at leat R from the origin. Thu r 2 + (R )2 R 2 2R log(n 2 ) 3 2 log N log(n 2 ) + O()

8 R R S Conider a ector of angle / 5 logn containing uch a point. For i =,, 2,... let r i = R + (i )/ 5. Let A i be the interection of the ector with the annulu with inner radiu r i and outer radiu r i+. Suppoe that k i uch that all the region A i for i k contain at leat one point of the proce. Then, for i < k, point in adjacent region A i and A i+ are adjacent in G, o we may conclude that r + i at leat R + (k )/ 5. The denity in the region A i i at leat ρ(r i+ ). The condition on implie that R 2 = ( + o())log N. Thu, the area of the region A i i at leat 2 /5.. Hence, etting λ = ( 2 /5.)ρ(R), the expected number λ i of point in A i atifie λ i λ e αi where α = 2r k+ / 5 (the definition of R implie that λ = e 3 /5.π + o() >.25). The probability p i that A i contain at leat one point i e λi e e αi. Thu the probability that all the region A i for i k contain at leat one point i at leat S Figure 2: Region S and A i in the proof of Lemma 8. The dic hown have radiu /2, o that dic correponding to adjacent vertice overlap. With high probability, at leat one of the column of mall (approximate) quare contain a point in every quare. A i If k i= k = p i exp( α ( ) k+ 2 k)..98(loglog N)/α then thi probability i at leat (log N).49 o(). (Note that log N log log N enure that αk.) However we have at leat R = Θ((log N) /2 ) dijoint ector o, whp, at leat one of the ector atifie thi. Thu whp, r+ 2 R 2 + 2(k )R/ 5 log(n 2 ) log N log log N + O( log N + ). + 2 and the reult follow. Now aume log N log log N. We aim to how that whp C() extend ome ditance beyond D R () in at leat one narrow ector, where R 2 = log(n 2 ) 3 a before. By Lemma 5, R 2 r 2 = O(log log N), and o R r = o(). Divide D R () into ector of angle about / log N. Conider the approximately quare region formed by interecting one of thee ector with an annulu with radii R and R (ee Figure 2). Subdivide thi region S into ector of angle / 5 log N and annuli of thickne / 5. The region i thu ubdivided into approximately quare region of ide length at mot / 5. We aim to how that with a reaonable probability there i a point in S C() at ditance at leat R / 5 from. Each mall quare-like region contain at leat one point with probability at leat p = exp( 2 ρ(r)/5.) > p ite. Moreover, point in any two adjacent quare are within ditance, o are connected in G. Comparing thi proce with a ite percolation with probability p, we ee that with poitive probability there i a quare near the centre of S which i connected to the ide r = R of S, and hence there i a point within r of joined to a point at leat R / 5 of in S. Lemma 9. Suppoe that = o() and N 2. Then, whp, r+ 2 log(n 2 ) + 2 log N log log N + O( log N + ). Proof. Let the radiu R be defined by π 2 ρ(r) = /3. We define a equence of area A i. Let A be D R (), and for i let A i = D R+i () \ D R+(i ) (). Let V = V A and for i 2 let V i be the vertice in A i joined to a vertex in V wholly inide D r+i (). We want to bound the ize of V i. Let W i be the point in A i that are neighbour of vertice in V i. Now, ince the denity in A i+ i bounded above by ρ(r+i) we have E( W i+ V i ) π 2 ρ(r + i) V i. Alo, any vertex in V i i either in W i or i a decendant of a vertex in W i. Moreover any vertex in V i \ W i ha an ancetor in W i which can be reached from thi vertex by a path uing no vertex of any V j for j < i. Note that thee vertice may lie in A j for j < i but cannot lie in A or A. However the expected number of decendant of a vertex

9 in W i which can be reached wholly outide D R () i at mot j= 3 j = /2. Hence E( V i W i ) 3 2 W i. Combining thee we ee that, ubtituting for R, E( V i+ V i ) 3 2 π2 ρ(r + i) V i 2 e 2Ri V i. Alo V i at mot the number of point outide D R (), which i dominated by a Poion ditribution with mean 3. Hence 2 E( V j ) 2 (j ) exp( 2R ( ) j 2 ) 3. 2 Thu if j = (R) log(/ 2 ) + then E( V j ) 2 j e R (uing 2 ( j 2) (j ) 2 + for j 2). Now ince = o(), (R)j 2. Thu either j or R i large and o E( V j ) = o(). Therefore whp r + R + j, o that whp r 2 + log(n 2 ) + 2jR + O(). The reult follow ince we may aume log(/ 2 ) log log N and R log N. Thi complete the proof of the etimate for r + in part 3 of Theorem 2. Corollary. If = o(log log N/ log N) and N 2, then whp r 2 max r 2 + ( + o())log(/ 2 ). Proof. From Lemma 9, we have whp r+ 2 log N log(/ 2 ) + 2 log N log log N + O( log N + ) log N log(/ 2 ) + o(log log N). Thu for any ε >, whp r ( ε)log(/2 ) = log N ω(). The reult now follow from Lemma 3. 4 Small The reult we have proved o far ay very little about the cae N 2 = O(log N). In thi ection we addre thi cae. Lemma. Suppoe that R i uch that πr 2 ρ(r), N 2 = O(log N), and N 2 e R2 log N log R 2. (7) Then, whp, G DR() i connected. Proof. The proce retricted to D R () tochatically dominate a Poion proce of mean ρ(r). The expected number of point of the Poion proce inide the dic i πr 2 ρ(r). The hypothee of the theorem imply that R = O() (ince if R then eventually N 2 e R2 log N log R 2 < 2 log N logr2 ), o that ρ() Cρ(R) for ome contant C. We modify the original proce inide D R () by keeping point of the proce at ditance r R from the origin with probability ρ(r)/ρ(r). Then the modified proce ha denity exactly ρ(r) in D R (). Denote by G the graph obtained from the modified proce by joining two point at ditance le than. Now uppoe that H = G DR() i not connected. Then H will contain two vertice v and v 2 which are not joined by a path in H. Thee vertice will be in H = G DR() with probability at leat /C 2, and they will certainly not be joined by a path in H. Hence if H i not connected with probability at leat p infinitely often a N, then H i not connected with probability at leat p/c 2 infinitely often a N. Thu if H i connected whp, H i alo connected whp. Provided that πr 2 ρ(r) tend to infinity a N tend to infinity, we can apply Theorem to ee that H i connected, whp, if π 2 ρ(r) log(πr 2 ρ(r)) + ω(). Subtituting for ρ(r) and rearranging give the reult. Corollary 2. Suppoe C > and N 2 = C log N. Then whp log C o(). r 2 Proof. Fix α > with α < log C. Subtitute R 2 = log C α > in (7). Then πr 2 ρ(r) and the left hand ide of (7) become e α log N log N log R 2 = (e α )logn + O() which tend to infinity. Lemma 3. If N 2 < ( ε)log N and N 2 then whp r 2 = N exp{n 2 ( + o())}. Proof. Firt we prove r 2 N exp{n 2 ( + o())}. Let R 2 = N exp{n 2 ( + δ)}. Then if < δ < ε, R = o() and the expected number of iolated point in D R () i at leat N( e R2 )exp{ N 2 } 2 NR2 exp{ N 2 } = 2 exp(δn2 ). The reult follow from Lemma 6. To prove r 2 N exp{n 2 ( + o())}, let R 2 = N exp{n 2 ( δ)} and apply Lemma. Then πr 2 ρ(r) and we have N 2 e R2 log(nr 2 ) =N 2 ( o()) N 2 ( δ) + o(). The reult follow, ince if G DR() i connected then r R. Lemma 4. If N 2 = log N then whp r 2 = ( + o()) log log N log N.

10 Proof. For the lower bound we ue Lemma. Take R 2 = ( α)log log N/ log N for α >. Certainly πr 2 ρ(r) = NR 2 e R2. Now N 2 e R2 log N log R 2 ( ) ( α)log log N (e R2 )log N log log N R 2 log N log( α) log log log N + log log N = α log log N + O(log log log N) o that whp r R. For the upper bound, we ue the proof of Lemma 7. Take R 2 = ( + α)log log N/ logn for α >. We may aume that (R ) 2 ( + α 2 )log log N/ log N. Chooe θ uch that θ( + α 2 ) > and aume that N i large enough that e (R )2 θ(r ) 2. Then with notation a in the proof of Lemma 7, it i eay to ee that there i an abolute contant C uch that E R C N e (R )2 e log 2 CN (+ N e α log log N 2 ) log N e log. log N Therefore, } log E R { e (+ α log log N 2 ) log N log N + log C log log N { } θ( + α log N 2 )log log N log N + log C log log N = { θ( + α 2 ) } log log N + log C looely peaking, decribe the width of the central component of uch a graph. It i our hope that ome of our method will find application to other problem in thi area. REFERENCES [] B. Bollobá, Random Graph, econd edition, Cambridge Univerity Pre, 2. [2] P. Baliter, B. Bollobá and M. Walter, Continuum percolation with tep in the quare or the dic, Random Structure and Algorithm 26 (25), [3] D.W. Etherington, C.K. Hoge and A.J. Parke, Global urrogate, manucript. [4] E.N. Gilbert, Random Plane Network, Journal of the Society for Indutrial Applied Mathematic 9 (96), [5] P. Gupta and P.R. Kumar, Critical power for aymptotic connectivity in wirele network, Stochatic Analyi, Control, Optimization and Application: A Volume in Honor of W.H. Fleming (W.M. McEneaney, G. Yin and Q. Zhang (Ed.)), 998. [6] M.D. Penroe, The longet edge of the random minimal panning tree, Annal of Applied Probability 7 (997), [7] M.D. Penroe, Random Geometric Graph, Oxford Univerity Pre, 23. and o whp r R by Lemma 6. The lat three lemma together etablih part 4 of Theorem 2. Part 7 i immediate, and for part 5, if N 2 then r / by Lemma 3 and o Lemma 8 applie and it together with Lemma 9 give r 2 + = log(n 2 )+O(). Part 6 follow from tandard reult on branching procee: we note only that if N 2 = C > γ, where γ i the critical denity for dic percolation, and if R 2 = log C log γ, then the probability that the origin belong to the giant component in G DR() lie trictly between and, and, conditional on thi event occurring, r + = ( + o())r. 5 Concluion In thi paper we have analyzed a model of a random geometric graph whoe vertice are given by a Poion proce of Gauian intenity: two uch vertice are connected in the graph if they lie within ditance of each other. We have given precie bound for two parameter which,