Topological Properties of an Exponential Random Geometric Graph Process

T opological Properties of an Exponen tial Random Geometric Graph Process 1 Y ilun Shang Department of Mathe matics Shang hai Jiao T o ng University Shang hai 200240 , China Email: shyl@sjtu.edu.cn Abstract —In this paper , we consider a one-dimensional random geometric graph proc ess with the inter -nodal gaps ev olving according to an exponent ial AR(1) process, which may ser ve as a mobile wir eless netwo rk model. The transition probability matrix and stationary d istribution ar e d eriv ed for the Markov chains in terms of network connectivity and th e number of components. W e ch aracterize an algorithm for the hitting time regar ding d isconnectivity . In addit ion, we also study topological properties f or static snapshots. W e obtain the degree distributions as well as asymptotic precise bound s and strong law of large n umbers for connectivity t hreshold distance and the largest nearest neighbor distance amongst others. Both closed form results and l imit theorems are provided. Key words -ra ndom geometric graph; autoregressi ve process; component; connectivity; mobil e network. I . I N T R O D U C T I O N Many randomly deployed networks, such as wireless sensor networks, are p roperly character ized by ra ndom geo - metric g raphs (RGGs). Giv en a specified no rm on the space under c onsideration , an RGG is u sually obta ined by placing a set of n vertices independen tly at random accordin g to some spatial p robab ility distribution and connectin g two vertices by an edge if an d only if the ir distance is less than a c ritical cutoff r . T o polog ical prop erties of RGGs are com prehen si vely summarized in [1]; also see [2] for a recent survey in the context o f wireless networks. Although extensi ve simulations and empirical studies are perfo rmed in dyn amical RGGs, a nalytical trea tments of to pologica l proper ties are merely d one in static RGGs in the previous work. A recent pap er [3] is a remarkab le exception, in which the autho rs cond uct the first analytical research on the co nnectivity of mo bile RGG in the torus [0 , 1) 2 . In this paper, we will also p resent analytical results and consider a one-dim ensional expo nential RGG pro cess G ( t, r , Λ) e volv- ing w ith time, where vertices are rand omly placed along a semi-infinite line. One-dimension al expon ential RGGs have been r ecently in vestigated b y some author s [4]– [6], which offer a significant variant from the familiar uniformly U [0 , 1] distributed nodes, see e.g. [2], [7]–[9] and references therein. 1 This paper is an extended version of a conference paper “Exponenti al Random Geometr ic Graph Proce ss Models for Mobile Wire less Netw orks” pp. 56–61, present ed in International Conference on Cyber-Enab led Dis- trib uted Computing and Knowle dge Discov ery , 2009. In [ 10], the distributions of distances b etween successi ve vertices rather than those of vertices th emselves are exam- ined, an d as it is stated in the same paper, this assumption is more natura l since “ sen sors a r e usually thr own o ne by one along a trajectory of a vehicle . ” W e will then follo w suit, and assume expo nential distributions for in ter-nodal distances of the graph p rocess G ( t, r, Λ) . Every segment between two successi ve vertices is supp osed to evolve f ollowing a stationary TEAR(1) p rocess [11] with exp onential m arginal. This lin ear process has no zero-de fect and thu s sur passes the elementary AR(1) p rocess in volved in [6]. W e b eliev e such a mo bile scheme has br oad potential app lications due to the flexible do uble random ness mechan ism (see Section II). Sin ce th e evolution of con nectivity an d the nu mber of compon ents in G ( t, r, Λ) are both Markovian, we will address the tran sition pro babilities a nd limiting distributions of these two processes G t and G ′ t by employing Markov chain theory [12], [13]. It is worth n oting that there are se veral Markov chains coupled in our model stemming from the first order auto regressiv e properties endowed in the e volution of inter-nodal distances. In addition to d ynamical pr operties, we also establish static proper ties for fixed t . V ertices in G ( t, r, Λ) , for any given t , form nearly a Poisson point p rocess (more precisely , a con tinuous time pu re birth Markov process). The connectivity of a Poisson RGG is well-studied in the literature ( see e.g. [ 14]–[17]), especially in the context of ad hoc networks. W e will invest igate som e top ological proper ties basically along th e lines o f [ 4]. In our o pinion , the aforem entioned simp le idea in [10] reflects a conceptio n of one step “memo ry” essentially . W e show (in Theore m 4) that “1 -step mem ory” + “gr owth” are no t en ough to pro duce power la w distribution reminiscen t of the architecture of Polya urn process, where ty pically infinite memory generates the power law [18]. Both finite and asym ptotic analysis are given in this paper . W e rem ark here that exact solutio ns are importan t since th e asymptotic re sults can no t be a pplied to real networks whe n not knowing the rate of conv ergence. The rest of this paper is organized as follows. Section II provides the definition of the expo nential RGG proce ss and some p reliminaries. Section II I d eals with the ev o lutionar y proper ties o f G ( t, r, Λ) , includ ing the transition probab ility , the station ary distribution and the hittin g time fo r dis- connectivity . In Section IV , we present static to pologica l proper ties of G ( t, r , Λ) for fixed t . The d egree distribution and strong laws o f co nnectivity an d the largest ne arest neighbo r distances are given amo ng other things. In Section V , some co ncludin g remarks and f uture research topics ar e discussed. I I . M O D E L A N D P R E L I M I N A R I E S The RGG process G ( t, r, Λ) is constructed as a discrete time proce ss with n vertices deployed in on e dimension on [0 , ∞ ) . Let X t 1 , · · · , X t n denote the vertices o f the network at time t , for t ≥ 0 . Set Y t l := X t l +1 − X t l , for l = 1 , 2 , · · · , n − 1 and Y t 0 := X t 1 ; see Fig.1 for an illustration. W e may en vision time evolving upward along the t -axis and n vertices possibly growing a long the x -axis. For 0 ≤ p < 1 , we assume that { Y t l } e volves following: Y t +1 l = ( Y t l + ε t l w.p. p ε t l w.p. 1 − p (1) where the in novation sequences { ε t l } t ≥ 0 consist of i.i.d . nonnegative ran dom variables. The behavior of this autore- gressiv e pr ocess { Y t l } t ≥ 0 is ch aracterized by ru ns o f r ising values (with g eometrically distributed run length) wh en choosing Y t l + ε t l , fo llowed by a sharp fall when ch oosing ε t l without inclu sion of the p revious v alues. Fur thermor e, we assume tha t Y t l , l = 0 , 1 , · · · , n − 1 are independ ent f or any t. In particular, we set ε t l := (1 − p ) Z t l , where Z t l ∼ E xp ( λ l ) is an exponential rando m variable with mean λ − 1 l > 0 . Let Λ := { λ 0 , λ 1 , · · · , λ n − 1 } . I n th is case, as is shown in [11], the above TEAR(1) process { Y t l } t> 0 would be a stationary sequence of marginally expo nentially distributed r andom variables with p arameter λ l , assuming that the initial in ter- nodal g aps Y 0 l are expo nentially distributed with p arameter λ l . That me ans Y t l ∼ E xp ( λ l ) . In this case, the auto correlation fu nction of { Y t l } is Corr( Y t l , Y t + j l ) = p j , bein g nonnegative. Refer ence [19] showed that (1) is stationary fo r each 0 ≤ p < 1 iff Y t l is g eometrically infin itely divisible. For further extension and discussion of (1) we refer the reader to [2 0]. Remark 1 : V ertices in snap shot o f G ( t, r , Λ) yield a counting process with inter -nodal distances h aving d istri- bution E xp ( λ l ) , while in standa rd expo nential RGG, the correspo nding distributions are r elev an t to n the total number of vertices (see Le mma 1 in [4]) rely ing on the g lobal informa tion. Remark 2: Notice that the cuto ff r = r ( n, t ) may depend on n an d t . However , we restrict o urselves to fixed r in order to keep calculations clear though some results may be generalized without much ef f ort. The popular assump tion lim n →∞ r ( n ) = 0 is no t necessary h ere in virtue of unbou nded suppor t resulting from in ter-nodal spacing. ✲ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✻ t ✲ ✛ ✲ ✛ ✲ ✛ Y t 0 Y t 1 Y t n − 1 0 X t 1 X t 2 X t n − 1 X t n · · · ∞ Figure 1. One-dimensional exponent ial RGG process model I I I . E VO L U T I O N A RY P RO P E RT I E S O F G ( t, r , Λ) A. Stationary Distribution of G t Let us den ote by C t and D t the events that G ( t, r , Λ) is connected a nd disconnected at time t , r espectively . Define G t as a discre te time stochastic proc ess describin g con nec- ti vity o f the gr aph p rocess G ( t, r , Λ) . T herefor e C t = { G t = “conneted ” } and D t = { G t = “disconneted ” } . It is easy to see that G t is a hom ogeneo us Markov chain, assumin g the cutoff r is independen t of t . W e abbreviate as usual the states as 1=“co nnected” ( C ) an d 2=“ disconnected ” ( D ) . Ou r main results in this section then re ad as follows: Theor em 1: G t is a time-reversible, homog eneous fin ite Markov chain, with on e step tran sition p robab ility matrix P ( n ) =  p 11 p 12 p 21 p 22  , where p 11 = n − 1 Y l =1  1 − (1 − p ) e − λ l r  1 − e − λ l r 1 − p  1 − e − λ l r  , (2) p 21 = P ∅6 = A ⊆ [ n − 1] (1 − p ) Q l ∈ A e − λ l r  1 − e − λ l r 1 − p  1 − n − 1 Q l =1 (1 − e − λ l r ) · Y l ∈ [ n − 1] \ A  1 − e − λ l r − (1 − p ) e − λ l r  1 − e − λ l r 1 − p  , (3) p 12 = 1 − p 11 and p 22 = 1 − p 21 . Pr opo sition 1: G t has a un ique stationa ry distribution π ( n ) = ( π 1 ( n ) , π 2 ( n )) , where π 1 ( n ) = (1 − p 22 ) 2 p 11 (1 − p 22 ) 2 + p 21 p 12 (2 − p 22 ) and π 2 ( n ) = (1 − p 11 ) 2 p 22 (1 − p 11 ) 2 + p 12 p 21 (2 − p 11 ) . (4) Pr opo sition 2: Suppo se λ l ≡ λ , for l = 0 , 1 , · · · , n − 1 . Let P ( ∞ ) be the transition probability ma trix of G t as n tends to infin ity , and π ( ∞ ) the (u nique) stationary distribu- tion c orrespo nding to P ( ∞ ) . Then π ( ∞ ) = (0 , 1) and lim n →∞ π ( n ) P ( n ) = π ( ∞ ) P ( ∞ ) . Proposition 2 implies that we can swap th e order of obtaining stationary distribution an d tak ing limit w .r .t. n . Pr oof of Theorem 1 : The probability density function of ε t l can be shown to be given by f l ( s ) = λ l 1 − p e − λ l s/ (1 − p ) 1 [ s> 0] . Also, the co nditional density fun ction fo r Y t l in the con- nected network is g Y l |C ( y ) = λ l e − λ l y 1 − e − λ l r 1 [0 0] + 2 λ l e − λ l (2 − p ) y 2(1 − p ) 1 − e − λ l r · sinh  λ l py 2(1 − p )  1 [ y > 0] − 2 λ l e − λ l  r + (2 − p )( y − r ) 2(1 − p )  1 − e − λ l r · sinh  λ l p ( y − r ) 2(1 − p )  1 [ y >r ] . Hence P ( Y t +1 l < r | Y t l < r ) = Z r 0 L − 1 ( L ( e Y t +1 l ))( y )d y = 1 − (1 − p ) e − λ l r  1 − e − λ l r 1 − p  1 − e − λ l r (6) which gi ves (2). Let ∅ 6 = A ⊆ [ n − 1] := { 1 , 2 , · · · , n − 1 } . Deno te th e ev ent E A := { Y t l > r , ∀ l ∈ A ; Y t l < r, ∀ l ∈ [ n − 1] \ A } , then we h av e P ( C t +1 | E A ) = Y l ∈ A P ( Y t +1 l < r | Y t l > r ) · Y l ∈ [ n − 1] \ A P ( Y t +1 l < r | Y t l < r ) = Y l ∈ A (1 − p )  1 − e − λ l r 1 − p  · Y l ∈ [ n − 1] \ A  1 − (1 − p ) e − λ l r  1 − e − λ l r 1 − p  1 − e − λ l r  . Here we used the expression P ( Y t +1 l < r | Y t l > r ) = (1 − p )  1 − e − λ l r 1 − p  . Since P ( E A ) = Q l ∈ A e − λ l r Q l ∈ [ n − 1] \ A (1 − e − λ l r ) and P ( D t ) = 1 − Q n − 1 l =1 (1 − e − λ l r ) , (3) fo llows by n oting th at p 21 = P ( C t +1 |D t ) = X ∅6 = A ⊆ [ n − 1] P ( C t +1 | E A ) · P ( E A ) /P ( D t ) . G t is time -reversible by standar d results of Mar kov ch ains [12]. Pr oof of Pr oposition 1: Sin ce G t is an ir reducib le finite Markov chain, C and D are both positive recurren t. Also since they are both n on-p eriodical, C and D are ergodic states. Set T ij := min { k : k ≥ 1 , G k = j, G 0 = i } , for i, j ∈ { 1 , 2 } . If the righ thand side of the above d efinition is ∅ , set T ij = ∞ . The first hitting pr obability is then given by f ( k ) ij = P ( T ij = k | G 0 = i ) . By a standard result fr om [13], an irreducib le ergodic Markov cha in has uniq ue stationary d istribution π ( n ) , and π i ( n ) is given by π i ( n ) = 1 / P ∞ k =1 k f ( k ) ii , for i = 1 , 2 in the present case. Thereb y , (4) follows easily fro m the facts f (1) 11 = p 11 , f ( k ) 11 = p 21 p k − 2 22 p 12 , for k ≥ 2 ; and f (1) 22 = p 22 , f ( k ) 22 = p 12 p k − 2 11 p 21 , for k ≥ 2 . Pr oof of Pr oposition 2 : When λ l ≡ λ , the rig hthand side of expression (6) b elongs to interval (0 , 1) . Henc e p 11 tends to 0 as n → ∞ in vie w of (2). Since (1 − p ) e − λr  1 − e − λr 1 − p  +  1 − e − λr − (1 − p ) e − λr  1 − e − λr 1 − p  = 1 − e − λr < 1 , p 21 tends to 0 as n → ∞ by the b inomial the orem and (3). Th en we have P ( ∞ ) =  0 1 0 1  . In th is case, C is a transien t state and D is an absorbin g and positive recurren t state. By a standar d result (see e.g. [1 3]), the stationary distribution correspo nding to P ( ∞ ) exists and is u nique. Direct calculation giv es π ( ∞ ) = (0 , 1) . It is straightfor ward to verify that π ( n ) → π ( ∞ ) as n ten ds to infinity . The th eorem is th us conclu ded by exploiting the relation π P = π . B. T ransition Pr ob ability Ma trix of G ′ t In th is section, we show a refineme nt stochastic pro cess G ′ t from G t . T o be precise, let { G ′ t = i } d enote the event that G ( t, r , Λ) ha s i c ompon ents at time t , for 1 ≤ i ≤ n . Therefo re, G ′ t is a homo geneo us Markov ch ain with state space [ n ] . It’ s clear that { G ′ t = 1 } = C t . Let th e tran sition p robab ilities of G ′ t be p ′ ij := P ( G ′ t +1 = j | G ′ t = i ) . Set A, B ⊆ [ n − 1 ] with | A | = i − 1 an d | B | = j − 1 , 1 ≤ i , j ≤ n . Denote the e vent E A := { Y t l > r, ∀ l ∈ A ; Y t l < r, ∀ l ∈ [ n − 1] \ A } a nd similarly fo r E B . W e o btain by the to tal pr obability formula, p ′ ij = X A,B ⊆ [ n − 1] | A | = i =1 , | B | = j − 1 P ( E B | E A ) · P ( E A ) /P ( G ′ t = i ) , (7) for 1 ≤ i, j ≤ n . W e hav e d erived P ( E A ) in the proo f o f Theorem 1 , and P ( G ′ t = i ) = P A ⊆ [ n − 1] , | A | = i − 1 P ( E A ) . T o ev alu ate (7), we still need the p robab ility P ( E B | E A ) , but it is also at hand already: P ( E B | E A ) = Y l ∈ A ∩ B P ( Y t +1 l > r | Y t l > r ) · Y l ∈ A \ B P ( Y t +1 l < r | Y t l > r ) · Y l ∈ B \ A P ( Y t +1 l > r | Y t l < r ) · Y l ∈ [ n − 1] \ A ∪ B P ( Y t +1 l < r | Y t l < r ) . The seco nd and fourth te rms in the above expression h av e been obtained in the pr oof o f Th eorem 1 , an d clear ly P ( Y t +1 l > r | Y t l > r ) = 1 − P ( Y t +1 l < r | Y t l > r ) , P ( Y t +1 l > r | Y t l < r ) = 1 − P ( Y t +1 l < r | Y t l < r ) . Now we arriv e at the m ain r esult. Theor em 2: The transition pro bability matrix of G ′ t is P ′ = ( p ′ ij ) n × n , which is given by (7). Of cour se, we have p ′ 11 = p 11 and P n j =2 p ′ 1 j = p 12 . Since G ′ t is a n irredu cible ergod ic chain, it has a un ique stationary distribution which may be d educed analogously as in Section III.A. C. Hitting T ime for Disconn ectivity Suppose C t holds at time t , and we will consider the Markov ch ain G t . Denote T := min { k : k ≥ 1 , D t + k holds } , then T is the hitting time for disconnectivity . W e m ay o btain th e expectatio n o f T using the transition probab ilities derived in Section III. A by a rou tine appr oach [13]. I n th is section , we will in stead d epict an algorith m for getting the distribution o f T d irectly . The ev ent { T > k } is equiv alent to { Y t +1 l < r , Y t +2 l < r , · · · , Y t + k l < r , ∀ 1 ≤ l ≤ n − 1 } . In view of (5), we can interpret th e above as fo llows Y t +1 l = ε t l + V t l Y t l < r, Y t +2 l = ε t +1 l + V t +1 l ε t l + V t +1 l V t l Y t l < r, · · · Y t + k l = ε t + k − 1 l + V t + k − 1 l ε t + k − 2 l + · · · + V t + k − 1 l · · · V t +1 l ε t l + V t + k − 1 l · · · V t l Y t l < r. Set U t + j l := V t + j l ε t + j − 1 l + · · · + V t + j l · · · V t +1 l ε t l + V t + j l · · · V t l Y t l , for 1 ≤ j ≤ k − 1 and U t l := V t l Y t l . Therefo re, cond itioned on Y t l , V t l , · · · , V t + k − 1 l , the p rob- ability that the above k inequalities h olds simultane ously is shown to b e giv en by P k l ( Y t l , { V t l , · · · , V t + k − 1 l } ) = R r − U t l 0 f l ( ε t l )d ε t l · · · · R r − U t + k − 1 l 0 f l ( ε t + k − 1 l )d ε t + k − 1 l , (8) where f l ( · ) is g iv en in the proof of Theorem 1 . Den ote th e last i + 1 integrals of (8 ) b y I l,k − i , 0 ≤ i ≤ k − 1 . For i = 0 , I l,k = Z r − U t + k − 1 l 0 λ l 1 − p e − λ l s 1 − p d s = 1 − e − λ l ( r − U t + k − 1 l ) 1 − p . For i = 1 , I l,k − 1 = Z r − U t + k − 2 l 0 λ l 1 − p e − λ l ε t + k − 2 l 1 − p I l,k d ε t + k − 2 l = 1 − e − λ l ( r − U t + k − 2 l ) 1 − p − λ l ( r − U t + k − 2 l ) 1 − p e − λ l ( r − U t + k − 2 l ) 1 − p 1 [ V t + k − 1 l =1] −  1 − e − λ l ( r − U t + k − 2 l ) 1 − p  e − λ l r 1 − p 1 [ V t + k − 1 l =0] . In g eneral, for 0 ≤ i ≤ k − 1 , I l,k − i = Z r − U t + k − i − 1 l 0 λ l 1 − p e − λ l ε t + k − i − 1 l 1 − p I l,k − i +1 d ε t + k − i − 1 l . W e can proceed using this recursi ve fo rmula by induction and integration by parts. Notice that P k l ( Y t l , { V t l , · · · , V t + k − 1 l } ) = I l, 1 from ( 8). Consequently , giv en Y t l < r , th e prob ability th at Y t +1 l < r , Y t +2 l < r , · · · , Y t + k l < r all are simu ltaneously true is seen to be given by e P k l := λ l 1 − e − λ l r · k X i =0 p i (1 − p ) k − i · X k − vector ξ consisting of k 1 ′ s,k − i 0 ′ s Z r 0 P k l ( y , ξ ) e − λ l y d y . Now we state our result as fo llows, whose p roof is straight- forward at this stage. Theor em 3: Supp ose the h itting time T of G t is d efined as a bove, then the distribution P ( T ≤ k ) = 1 − Q n − 1 l =1 e P k l and it’ s expec tation E T = P ∞ k =0 Q n − 1 l =1 e P k l . In principle, by the truncation o f k , we may approxim ate E T arbitrarily close. I V . S N A P S H O T S O F G ( t, r, Λ) For fixed t , we denote b y G ( r , Λ) th e static case which can be regarded as a snapsh ot of the dyna mical process G ( t, r , Λ) . Also, we omit the superscr ipt t typically , e.g. Y l , etc. 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Probability Number of nodes k=4 k=3 k=2 k=11 Figure 2. Probability that G ( r, Λ) contains k compone nts for dif ferent v alues of k A. Cluster Structur e Let P n ( C ) denote the proba bility that G ( r , Λ) is con - nected. W e have the following r esult regard ing con nectivity . The proof is easy and hence o mitted. Pr opo sition 3: W e have P n ( C ) = n − 1 Y l =1 (1 − e − λ l r ) . Moreover , supp ose th ere exists M > 0 such that λ l < M , for all l , then P n ( C ) → 0 as n → ∞ . Let ψ n ( k ) de note the probab ility tha t G ( r , Λ) co nsists of k compon ents and P m n ( k ) the probab ility that there are k compon ents in G ( r, Λ) , eac h of wh ich having size m (i.e. m vertices). Pr opo sition 4 : Suppose there exists M > 0 such that λ l < M , for all l . Then , for any fixed k , ψ n ( k ) → 0 as n → ∞ ; and f or any fixed k , m , P m n ( k ) → 0 as n → ∞ . Pr oof: Mimickin g the proof o f The orem 3 and 4 in [4] yields th e result. In Figur e 2, we p lot ψ n ( k ) as a fun ction of n n umber of vertices fo r different k . W e take λ i = 1 f or 1 ≤ i ≤ 10 , and λ i = 2 for i > 10 . Observe that the conver gence to the asymptotic value 0 is very fast. W e may th us co nclude that this static n etwork is almost surely divided into an in finite n umber of finite clusters. This observation was fir st m ade in [15] by a dif ferent ap proach . B. De g r ee Distribution Let G ( r, λ ) den ote the gr aph G ( r , Λ) w hen Λ = { λ, · · · , λ } . Theor em 4: In the gr aph G ( r , λ ) , the degree distribution can be d ivided into three classes: the degree distribution of X 1 and X n is P oi ( λr ) ; an d for k + 1 ≤ i ≤ n − k , that of X i is  e − 2 λr (2 λr ) k k !  k ∈ N . For 2 ≤ i ≤ k , the d egree distribution of X i and X n +1 − i is  e − 2 λr ( λr ) k k ! P i − 1 j =0  k j  k ∈ N . Pr oof: Let { Y i } , { Y ′ i } be indep enden t E xp ( λ ) . Den ote the degree of vertex X i as d i . W e get P ( d n ≥ k ) = P ( d 1 ≥ k ) = P ( Y 1 + · · · + Y k ≤ r ) = e − λr  ( λr ) k k ! + ( λr ) k +1 ( k + 1)! + · · ·  , where we used an eq uiv ale nt d efinition of gamma distribu- tion. Hence, P ( d n = k ) = P ( d 1 = k ) = e − λr ( λr ) k k ! . Next, for 2 ≤ i ≤ k , P ( d n +1 − i = k ) = P ( d i = k ) = i − 1 X j =0 P ( Y 1 + · · · + Y j ≤ r, Y 1 + · · · + Y j +1 > r ) · P ( Y ′ 1 + · · · + Y ′ k − j ≤ r, Y ′ 1 + · · · + Y ′ k − j +1 > r ) = i − 1 X j =0 Z r 0 λe − λx ( λx ) j − 1 ( j − 1)! Z ∞ r − x λe − λy d y d x · Z r 0 λe − λx ( λx ) k − j − 1 ( k − j − 1)! Z ∞ r − x λe − λy d y d x = e − 2 λr ( λr ) k k ! i − 1 X j =0  k j  . Finally , for k + 1 ≤ i ≤ n − k , P ( d i = k ) = k X j =0 P ( Y 1 + · · · + Y j ≤ r, Y 1 + · · · + Y j +1 > r ) · P ( Y ′ 1 + · · · + Y ′ k − j ≤ r, Y ′ 1 + · · · + Y ′ k − j +1 > r ) = e − 2 λr (2 λr ) k k ! which concludes the pro of. C. Str on g Law Results Define the con nectivity d istance c n := inf { r > 0 : G ( r , λ ) is co nnected } ; and the largest n earest neigh bor distance b n := max 1 ≤ i ≤ n min 1 ≤ j ≤ n,j 6 = i {| X i − X j |} . W e derive a symptotic tight bound s for c n and strong law of large nu mbers fo r b n , as n tends to infinity . Theor em 5: In the g raph G ( r , λ ) , we have (i) lim sup n →∞ λc n 2 ln n ≤ 1 and lim inf n →∞ λc n ln n ≥ 1 a.s. (ii) lim n →∞ λb n ln n = 1 a.s. Pr oof: (i) Observe that P ( c n ≥ x ) ≤ P n − 1 l =1 e − λ l x = ( n − 1 ) e − λx in voking the Boole inequality . Let ε > 0 . T ake x = x n = (2 + ε ) ln n /λ in the above expression and sum in n , then we get ∞ X n =1 P ( c n ≥ x n ) ≤ ∞ X n =1 n − (1+ ε ) < ∞ . By the Borel-Cantelli lemma, P ( c n ≥ x i . o . ) = 0 . Hence, lim sup n →∞ λc n 2 ln n ≤ 1 almost surely . On the other hand, P ( c n ≤ y ) = Q n − 1 l =1 (1 − e − λ l y ) = (1 − e − λy ) n − 1 . T a ke y = y n = (1 − ε ) ln n/λ , then ∞ X n =1 P ( c n ≤ y n ) ≤ ∞ X n =1  1 − n − (1 − ε )  n − 1 ∼ ∞ X n =1 e − n ε < ∞ . W e co nclude that lim inf n →∞ λc n ln n ≥ 1 a.s. b y using the Borel-Cantelli lemma aga in. (ii) By the indep endenc e of { Y l } , we ob tain P ( b n ≥ x ) = P  ∪ n − 1 i =2 {{ Y i − 1 ≥ x } ∩ { Y i ≥ x }} ∪{ Y 1 ≥ x } ∪ { Y n − 1 ≥ x }  ≤ n − 1 X i =2 P ( Y i − 1 ≥ x ) · P ( Y i ≥ x ) + P ( Y 1 ≥ x ) + P ( Y n − 1 ≥ x ) = ( n − 2) e − 2 λx + 2 e − λx . T ake x = x n = (2 + ε ) ln n/ (2 λ ) , th en we ge t ∞ X n =1 P ( b n ≥ x n ) ≤ ∞ X n =1  n − (1+ ε ) + 2 n − (1+ ε 2 )  < ∞ . By the Borel-Cante lli lemma, lim sup n →∞ λb n ln n ≤ 1 almost surely . On the oth er h and, P ( b n ≤ y ) = P  ∩ n − 1 i =2 {{ Y i − 1 ≤ y } ∪ { Y i ≤ y }} ∩{ Y 1 ≤ y } ∩ { Y n − 1 ≤ y }  ≤ ⌊ n 2 ⌋ Y i =1 P ( Y 2 i − 1 ≤ y ) · P ( Y 2 i ≤ y ) ∼ (1 − e − λy ) n . Arguing similarly as in (i), we ca n g et lim inf n →∞ λb n ln n ≥ 1 a.s.. Th is completes th e pr oof. V . C O N C L U D I N G R E M A R K S This paper dealt with random g eometric graph s in o ne dimension in wh ich the vertex p ositions we re evolving time. The critical assumption th at this e volution was modeled by describing an ev olu tion equation fo r the c hange in the inter- nodal spacing. W e studied some dynam ical as well as static proper ties and results were g iv en fo r fixed n total number of vertice s as well as n ten ding to infinity . It is worth poin ting out that this paper is only a prelimi- nary step on the invest igation of exponential RGG process models. 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