Int. J. Ad Hoc and Ubiquitous Computing, Vol. 11, Nos. /3, 1 169 Tansition phase of connectivity fo wieless netwoks with gowing pocess Bo Gu and Xiaoyan Hong* Depatment of Compute Science, Univesity of Alabama, Tuscaloosa, AL 35487, USA E-mail: bgu@cs.ua.edu E-mail: hxy@cs.ua.edu *Coesponding autho Abstact: The connectivity of wieless netwok is citical fo effective and efficient communications. Yet, it is challenged by mobility, tansmission ange and density. In tems of inceasing node density, thee is a tansition phase of connectivity when a netwok stats tansfoming fom patition to connected state. In this pape, we study the uppe- and lowebounds of citical density fo the tansition phase using pecolation theoy. We futhe deive citical time points fo the bounds accoding to thee gowth functions of node density. The simulation demonstates the evolution of netwok connectivity and validates the connectivity tansition times. Keywods: wieless netwok; connectivity; Pecolation theoy. Refeence to this pape should be made as follows: Gu, B. and Hang, X. (1) Tansition phase of connectivity fo wieless netwoks with gowing pocess, Int. J. Ad Hoc and Ubiquitous Computing, Vol. 11, Nos. /3, pp.169 177. Biogaphical notes: Bo Gu is a PhD Student in the Depatment of Compute Science at the Univesity of Alabama. He eceived MS Degee in Compute Science fom the Univesity of Alabama at 9 and MS Degee in Compute Science fom Beihang Univesity, China at 7. His eseach inteests include wieless netwoking, mobile netwoking, delay toleant netwok and netwok modelling. Xiaoyan Hong is an Associate Pofesso in the Compute Science Depatment at the Univesity of Alabama. She completed he PhD Degee fom the Univesity of Califonia, Los Angeles in 3. He eseach inteests include wieless netwoks, mobile computing and wieless senso netwoks. 1 Intoduction In a wieless netwok, the connectivity in multi-hop way is often affected by node density, tansmission ange, esidual enegy and node mobility. A patitioned wieless ad hoc netwok can tun into a connected netwok when some of the above factos change. The numbe of nodes being deployed in a designated aea can influence the netwok connectivity geatly. Undestanding the tansition phase of the netwok between patition and connectivity can help netwok management, planning, maintenance and pefomance monitoing. The impotance of the connectivity issue has dawn geat attention ecently to obtain fundamental popeties of the poblem in many application domains. Fo example, the minimum numbe of aveage neighbous fo netwok connectivity in static topology (Xue and Kuma, 4); the last connection time and the fist patition time with egad to node failue models in wieless netwoks (Xing and Wang, 8); the impact of intefeence on the connectivity (Dousse et al., 5); the elationship between powe saving on senso netwoks and maintaining connectivity (Dousse et al., 4; Kong and Yeh, 7). In these woks, the pecolation theoy has been used leading to esults of citical densities defined based on statistical concept of giant component (Kesten, 198). Viewing the need of pasimonious use of netwoking devices in some applications, fo example, a lage aea with scace available esouces like in some suveillance scenaios, an undestanding of the tansition phase becomes extemely impotant. The tansition phase can be defined by the lowe and the uppe bounds of the citical density following the gowth of numbe of nodes. In the ealy elated woks, the poblem has been simplified fo unifomed netwoks which ae not sufficient fo some heteogeneous and multi-hop wieless netwok scenaios. In heteogeneous and multihop wieless netwoks, nodes beaing diffeences in thei capability of communications, e.g., the tansmission powe, the esidual enegy at wieless nodes, will lead to vaious tansmission anges. And futhe, nodes equipped with adjustable antennas may have Copyight 1 Indescience Entepises Ltd.
17 B. Gu and X. Hong adaptable tansmission anges. In this pape, we evise the analytic modeling pocedue to eflect the need fo a vaiable tansmission ange and the multi-hop fowading using the pecolation theoy in deiving the two bounds. The tansition phase of the wieless netwok connectivity can be illustated accoding to the gowth of node density in the following way. In the beginning, a small amount of wieless nodes ae unifomly distibuted in the aea and the low density causes the netwok to be patitioned. By inceasing the numbe of nodes, the netwok becomes dense and tuns into a connected phase when the density exceeds a citical value. The evolution of connectivity is helpful fo undestanding the states of a netwok and leads to diffeent applications coesponding to the states. In this pape, we fist deive the two bounds of the citical density of wieless ad hoc netwoks based on the pecolation theoy. Hee, the lowe bound defines the moment (called the fist connection time) befoe which the netwok is patitioned because of the spase density and afte which the nodes in netwok gadually become connected in the same component. The uppe bound defines the moment when all the nodes ae connected and become the giant component with high pobability (called total connection time). Then, we study the duation of the tansition phase, i.e., the latency to each the total connection stating fom the fist connection based on pecolation theoy. This latency highly depends on the potential gowth pocess of the node density. Hee we offe thee gowth functions to deive the fist connection time and the total connection time which denote the points when the node densities each the lowe and uppe bound of citical value espectively. The pape is oganised as follows. Section summaises the elated woks that use the pecolation theoy fo connectivity issue. In Section 3, we intoduce the netwok model and peliminay knowledge elated to the pecolation that we use in this pape. The main analytical modeling of the bounds and esults ae given in Section 4. We also pefom simulations and show case the connectivity based on the thee gowth functions and the elated bounds in Section 5. The conclusion is pesented in Section 6. Related wok Pecolation theoy is widely used to analyse the netwok connectivity unde vaious applications in ecent publications. In Dousse et al. (4), the powe saving mechanism with completely uncoodinated patten is discussed using continuum pecolation theoy. The bound of message latency, fom a sende to its eceive, is deived by using an extension of the fist passage pecolation theoy (Liggett, 1985). The authos in Kong and Yeh (7) popose a distibuted enegy management algoithm fo wieless senso netwoks. By using the node degee and the pecolation condition, the mechanism fo saving enegy can guaantee the netwok to be in connected state all the time. In addition, the wok shows that the message delay scales sub-linealy o linealy with Euclidean distance between the souce and the destination depending on netwok phase in tems of connectivity. In Xing and Wang (8), the pecolation-theoy based connectivity issue is discussed in ode to deive the last connection time and the fist patition time fo wieless netwoks, whee the failue nodes may cause the netwok to ente into a patitioned phase. In thei analytic model, the netwok is mapped to a lattice and the netwok popeties ae conveted to the coesponding featues in the lattice. Then, the condition of connectivity in wieless netwok is obtained fom pecolation condition in lattice. Ou wok is elevant to the pape (Xing and Wang, 8), but the numbe of wieless node in netwok gows as time goes, which is the opposite to the case in Xing and Wang (8). In addition, a diffeent method is employed to deive the citical density and elated time points. The citical tansmission ange fo connectivity of wieless netwok is studied in Gupta and Kuma (1998). By deiving the lowe and uppe bounds of the pobability fo the connected netwok, the conclusion is that the netwok gaph logn c( n) with ( n) is connected with pobability one n as n if and only if cn ( ). In Ramanathan and Rosales-hain (), the poblem of tansmission powe adjustment fo topology contol is poposed and the objective is to minimise the maximum tansmit powe subjecting to connectivity constaint. In addition, the distibuted algoithms ae intoduced fo mobile netwok so as to adapt to the dynamic topology change. The topology contol depending on the alteation of tansmission ange is also discussed in Sanchez et al. (1999) whee the so called Diect neighbou Gaph is geneated fist and the optimal ange is deived based on Minimum Spanning Tee. Based on the geometic andom gaph and spatial analysis of wieless node, Bettstette () povides an analytical esult to obtain the minimum node degee fo k-connectivity netwok. A continuum famewok of wieless and mobile netwok (Chen et al., 11) is poposed to identify the connectivity status of netwok. The continuum space consists of two dimensions: node density and node speed. Accoding to diffeent combination of density and speed, thee classes of netwok ae found including the total connected netwok, disupted netwok and patitioned netwok. In this pape, we follow an altenative appoach based on the technique mentioned in Meeste and Roy (1996) to study the elation of node density and tansmission ange in tems of connectivity. And we conside the incement of node density, a diffeent netwok scenaio fom the pevious woks. Futhemoe, given the cuent node density, the density model can help to find the appoximate citical tansmission ange fo netwok connection. 3 Poblem modelling 3.1 Netwok model The netwok is descibed as the Poisson Boolean Model in two dimensions (Meeste and Roy, 1996). It is intoduced as follows. (X,, λ) is a union of andomly scatteed disks: ( X,, ) B( x,,) ) i i
Tansition phase of connectivity fo wieless netwoks with gowing pocess 171 whee Bx ( i, i, ) is a disk centeed at x i having adius i. x i denotes the point of the Poisson pocess X of density λ in two dimensions. In ode to model the wieless netwok, x i can be viewed as the wieless node i and i denotes its tansmission ange. Fo simplicity, we assume the tansmission anges of node ae equal and denoted by, so the connective condition fo any two nodes is x x. i j 3. Pecolation theoy Pecolation theoy (Kesten, 198; Bollobas and Riodan, 6; Penose, 6) is initially used to model the spead of fluid o gas though a andom medium. The oiginal poblem is that: Suppose a lage poous ock is submeged unde wate fo a long time, will the wate each the cente of the stone? The theoy assumes the suface is coveed by points and points could be connected by edges. Given a lage two dimensional gid of edges, an edge can be open with pobability p and closed with pobability q 1 p. The collection W of points connected to souce by an open edge is called open cluste of souce. Most questions in pecolation theoy concen some aspects of the distibution of W. Let P p denote the pobability on the configuation of open edge with pobability p, and let θ(p) be the pecolation pobability defined as: ( p) P { W } p whee W is the numbe of points in W. In geneal, θ(p) is popotional to the pobability p, and the following popeties hold: (), (1) 1. And pc: sup{ p: ( p) } is called citical pobability The pecolation poblem in geometic stuctue has been studied extensively in last thee decades (Meeste and Roy, 1996). Howeve, when the position of node is expessed in a continuous space, the continuum model is esponsible to descibe the pocess of pecolation. A fundamental esult of continuum pecolation is that thee exists a citical density λ c, defined by c inf{ : P { W } }, so that: if c, the netwok is in the supe-citical phase; while if c, the netwok is said to be sub-citical and P { W }. When the gaph stays in supe-citical phase, W is nomally called the giant component since it would contain infinite numbe of nodes in netwok. 3.3 Connectivity latency In ode to model the tansit phase fom patition to connected state, we assume that the netwok initially contains only a small numbe of nodes expessed by the density λ within a fixed netwok aea. Afte that, we intoduce the gowth function of netwok denoted by x(t) which descibes the incement of the numbe of nodes against time. Then the density function is defined as follows accoding to the Thinning theoem (Penose, 6): x() t () t x() whee x() epesents the initial numbe of nodes in the netwok. When the density inceases, two time-based metics ae citical to undestand the citical density. One metic is the fist connection time T f and the othe is the total connection time T t. They ae defined in equation (1) and equation () espectively. T sup{ t : P { W } } (1) f t T inf{ t : P { W } }. () t t Typically, the metic T f defines the latest time point in a density gowing pocess befoe which the pobability P is zeo, i.e., t the netwok is patitioned in tems of the giant component. The metic T t depicts the ealiest time point in a density gowth afte which the pobability becomes lage than zeo, i.e., the netwok is connected in tems of the giant component. Thus, the two metics define the time instances accoding to the lowe bound and the uppe bound of the citical density. Given the time function of node density and the ange of citical density, we can deive these two time points. 4 Analysis and esult In this section, we fist deive the lowe bound and uppe bound of λ c. The vaiable of tansmission ange is consideed in the function of density bound fo studying the tend as the ange vaies. Secondly, the fist connection time and total connection time ae deived. Specifically, the spase netwok with low density of node at time t gows with cetain distibution until the density eaches the lowe bound of citical density at t 1. Then the fist connection time will be t 1. Afte that, when the density inceases to the uppe bound at t, then t becomes the total connection time. 4.1 Lowe bound of λ c We deive the lowe bound following the appoach used in Meeste and Roy (1996) by using Banching Pocess (Gimmett and Stizake, 199). The ationale behind the appoach is that we fist deive the function about the numbe of nodes in the connected component and then find the minimum density equiement fo a pecolated netwok whee the pobability of having infinite numbe of node is geate than zeo. We apply a two dimensional Poisson Boolean model (X,, λ) and a multi-type banching pocess whee the type of node denotes the distance between a node and its pedecesso. The distances ae distibuted ove all eal numbes in (, ) and the maximum value guaantees the link between the nodes in adjacent geneation Geneally, the membes of the kth geneation of the banching pocess stating fom x ae x,1, x,,, x,. the k k k n k
17 B. Gu and X. Hong childen of xki,, i [1, nk] ae those points fom anothe independent pocess X k 1 which fall in the egion S( xki,, )\ S( xk 1, j, ), whee xk 1, j is the paent of x ki, and cicle egion S( x, ) { y: yx }, also the type of node xk 1, l at the (k+1)th geneation fom the pocess xk 1 is t: xki, xk 1, l (,), in consequence, the distibution of node xk 1, l whose types lie in (, ) depends on the egion R: R ( S( x, )\ S( x, )) { y: yx (, )}. (3) ki, k1, j ki, Assume that x ki, is of type u and xk 1, l has type v, then the egion R becomes a cuve defined as ( Sx ( ki,, )\ Sx ( k 1, j, )) { y: yxki, v}. Let g( v u ) be the length of that cuve and its value is: gv ( u) u v v acos if u v; (4) uv if < v u. Figue 1 demonstates the banching pocess descibed hee. Initially, the paent node xk 1, j and the child node x ki, with type u ae shown in Figue 1(a). The gandchilden nodes having type v could occu along the cicle centeed at node x. ki, The valid ones that contibute to pecolation only appea along the cuve fom A to B with counte-clockwise diection in Figue 1(b). On the cuve AB, a gandchilden nodes 1,. xk l is placed. The equation (4) calculates the length of the cuve AB. Figue 1 Multi-type banching pocess: (a) the fist banching and (b) the second banching (see online vesion fo colous) As a esult, given that an individual is of type u, the expected total numbe of its childen whose types lie in (, ) is given by g ( v u ) dv. (5) Fom Theoem 3.1 in Meeste and Roy (1996), the expected numbe of membes of the nth geneation having types in (, ) coming fom a paticula individual of type u is given by n g ( v u) dv. (6) n Then, the expected total numbe of individuals is n gn ( v u) dv. (7) n1 By taking advantage of the linea opeato T f, the equation is futhe educed to n n T1 ( u) (8) n1 whee T ( u) f( v) g( v u) dv and 1( v) 1 fo all f v (, ). If λ is sufficiently small, the esult of banching pocess 1 will convege. Specifically, λ should be less than T whee T denotes the usual opeato nom of T. Using the theoy of Hilbet-Schmidt opeato (Moiseiwitsch, 1977), T is geate ( ) u than g v u dvdu, so the condition of convegence is. (9) g( v u) dvdu u The consequence of convegence demonstates that the expected numbe of nodes in a component is finite, so the citical density λ c should be geate than the density with which the total numbe of nodes is convegent. Then the expession of lowe bound of citical density is c. (1) g( v u) dvdu u 4. Uppe bound of λ c In ode to obtain the uppe bound of citical density, the site pecolation model on the tiangula lattice is applied and the pocess is based on the Theoem 3.1 in Meeste and Roy (1996). Figue shows a pat of the tiangula lattice. The vetex of each tiangle is called site. It epesents the aea of the yellow flowe shown in that figue. The theoy of site pecolation (Kesten, 198) says that any nodes located in the flowe can be equal to the case that the incident vetex is occupied. So the condition of pecolation on the tiangula lattice is applied to the lattice coveed by flowes centeed at the vetex of tiangle.
Tansition phase of connectivity fo wieless netwoks with gowing pocess 173 Figue The tiangula lattice with aea A (see online vesion fo colous) The ational is that if the site pecolation of the tiangula lattice occus, the pecolation occus in the netwok. Fo the multihop netwok we study in this pape, a link occus only if the distance between two nodes is within the tansmission ange. Applying the site pecolation model, we design the edge length of tiangula lattice to be. The secto foming the flowe has a adius of and centes at the midpoints of the six edges adjacent to the same site. An occupied site is defined if thee is a node with Poisson Pocess X situated inside the inteio of the associated flowe of the site. In Figue, that is the site A, epesenting the yellow flowe. If thee ae two adjacent sites A and B both being occupied, the distances between the nodes cented at these two flowes must be less than o equal to twice the edge length of the tiangula lattice. Since the edge length is /, the lagest distance between the two nodes in the adjacent flowes is. Thus, the two nodes ae linked consequently. Given that the lattice coves the entie netwok, we have poofed that the site pecolation model descibed hee allows the pecolation to occu in the tiangula lattice, hence in the netwok. We use this model to deive the uppe bound of the citical density. Assume the aea of each site that has a flowe shape be A and the node density is λ. Fom Theoem 3.1 in Meeste and Roy (1996), the pobability of A site occupation is p 1 e. Fom the theoy of site pecolation (Kesten, 198), if p 1/ then with positive pobability thee is pecolation in the site pecolation model on the tiangula lattice. Thus if A ln, then thee is pecolation with positive pobability in the Boolean model. To obtain A, we efe to Figue, in that the aea A consists of six sectos of A. Each secto has a adius of x. It is easy to get the length of x, which is ( 15 3. Then, 8 the aea A is equal to 6A which is appoximately.5 *. Thus the uppe bound of the citical density is 4.3 Discussion of λ c As fo the lowe bound of citical density, the auxiliay function a() is poposed to simplify the integal expession in equation (1). Then the non-zeo pat of g( v u ) u v becomes va() whee a ( ) acos and the uv 3 a () lowe bound of citical density is educed to. It is difficult to deive the concete fom of a(), howeve its value ange can be detemined. Lemma 4.1: Given which denotes the tansmission ange u v of node and a ( ) accos whee < u, v < uv and u +v >, then a (),. 3 u v Poof: Let f(, u,) v whee < u, v < uv u v and u + v >, then f u and f v vu v u. So the extema of function f(, u, v) can uv be found at the definition bounday. When u + v =, f(, u, v) = 1. While u, v =, f(, u, v) = 1/. Since u v a ( ) acos, then a (),. uv 3 Accoding to the lowe bound and uppe bound of node density, the citical density falls in the following ange: 3 a ()3.3779,. Figue 3 shows the cuves of the lowe bound and the uppe bound of the citical density with the tansmission ange inceasing fom 1 1. As seen fom the figue, both cuves decline when the tansmission ange inceases. This explains the fact that when tansmission ange is lage, a node can still maintain its connectivity while the nodes aound it can be less. The figue also shows that when tansmission ange inceases, the lowe and the uppe bounds become close. Figue 3 Uppe bound and lowe bound of λ c (see online vesion fo colous) ln 3.3779 c. A (11)
174 B. Gu and X. Hong 4.4 Connectivity latency The time elapsed to each the lowe o uppe bound depends on the gowth function of the netwok density. We use thee gowth functions to deive the fist connection time and total connection time accoding to the bounds we obtained peviously. The thee functions potentially depict diffeent ways that nodes can be inseted to the netwok. The fist function follows linea gowth shown in the equation x() t kt a whee x() a. The coefficient k denotes the gowth ate and a can be viewed as the initial amount of nodes. Such a function eflects a constant ate inceasing of nodes in the netwok, which can be tue in some scenaios. The second is the exponential function to eflect the case that the gowth of node happens in shot time. The expession is x(t) = ae kt whee x() = a. In this function, k and a epesent the gowth ate and initial numbe espectively. The last one is the logistic function which is often used to descibe the population gowth and vius popagation. Diffeent fom the othe functions, the logistic function incopoates an uppe bound on the gowth. We a denote the gowth function by xt () whee k( t t ) 1 e a xt ( ). The constant a can be egaded as the maximum numbe of infected nodes, k denotes the aveage infection ate and t is the time when the gowth ate eaches the maximum. Thee sample functions ae illustated in Figue 4 and the cuves show the diffeent gowth pattens. The paametes fo the thee gowth functions ae selected in such a way that all of them will each the maximum numbe of nodes 14 by the simulation time of 9s. With this special constaint, the exponential function gows slowe than linea at the beginning. It shows apid gowth ate towad the end of the simulation. The logistic function shows the tend of slowe down when appoaching the uppe limit. Figue 4 Gowth functions (see online vesion fo colous) 5 Simulation We simulate the netwok evolution in Matlab with diffeent gowth functions to veify the bounds of citical density deived fom the pevious section. The simulation aea is with unifomly distibuted nodes in two dimensional space. The netwok evolution stats fom time and time step is set to be 1 s. The esults ae obtained fom total 5 uns. The aveaged values and confidence intevals with pobability.9 ae pesented in many figues. We measue the connectivity using Relative Giant Component Size. It is the atio of the numbe of nodes in the lagest netwok component to the total nodes. The lagest netwok component can be found though Beadth Fist Seach in the geneated netwok. As the density of nodes inceases, the atio appoaches to 1. Figues 5 7 descibe the pocess of netwok evolution in tems of connection using diffeent gowth functions. The ed cicle denotes the node and blue line is the link between two nodes. As the netwok gows, moe nodes and connections appea in the simulation aea. In the linea gowth function x() t kt a used in Figue 5, both paamete k and a ae set to be 15. Fo the exponential kt function x() t ae in Figue 6, we let the a = 15 and k = 1/ then the amount of node gows quickly next to the end of simulation. In Figue 7, the specific fom of logistic 14 gowth function is xt (). The configuation.15( t 3) 1 e of function shows that the most nodes will be geneated aound 3 s. Futhe, the elation between connectivity and time is illustated. In Figue 8, the tansmission ange is set to be 1. The ed and geen line in each subfigues (a) (c) indicate the fist connection time and the total connection time along with the coesponding densities. All the cuves in Figue 8(a) (c) oughly show the upwad tend of elative size of giant netwok component. In the ealie phase, due to the smalle total numbe of nodes, the nodes in the lagest component account fo a elative high atio. Howeve, in the phase afte the fist connection time, the atio shows an inceasing mode and eaches to 1 when all the nodes ae connected. In addition, the time instants of the fist connection time and those of the total connection time in these subgaphs agee with the gowth function each epesents (see Figue 4). They also show that the coesponding tansition peiods, i.e., the length between the ed line and geen line, eflect the gowth functions. As fo the confidence inteval with.9 pobability, the intevals close to the lowe bound denoted by geen line have wide anges. The obsevations eveal that the netwok state in tems of connectivity has lage vaiance duing that peiod of time. The fist connection time appoximately captues the moment when the netwok connectivity begins to change.
Tansition phase of connectivity fo wieless netwoks with gowing pocess 175 Figue 5 Netwok evolution with linea gowth at time: (a) t = ; (b) t = 3; (c) t = 6 and (d) t = 9 (see online vesion fo colous) Figue 6 Netwok evolution with exponential gowth at time: (a) t = ; (b) t = 3; (c) t = 6 and (d) t = 9 (see online vesion fo colous) Figue 7 Netwok evolution with logistic gowth at time: (a) t = ; (b) t = 3; (c) t = 6 and (d) t = 9 (see online vesion fo colous) Figue 8 Phase tansition against time with = 1 at time: (a) linea gowth; (b) exponential gowth and (c) logistic gowth (see online vesion fo colous) Figue 9 shows the expected numbe of node neighbous with shot confidence intevals accoding to the thee node gowth functions. The cuves in each subfigue appoximate the gowth functions in Figue 4. The expected numbe of neighbous coesponding to the lowe bound of citical density is aound fou and the amount is about ten fo the uppe bound. The obsevation indicates that the netwok is connected when the amount of neighbou inceases fom fou to ten. The connectivity metic in tems of expected neighbou confoms to the esult of connection degee in Xue and Kuma (4). Compaed to the conventional knowledge of the magic numbe of node density, the lowe bound we show hee is less. This is because the conventional numbe seves moe like an aveage case, while the lowe bound hee depicts the latest time fo the netwok to have no pobability of the occuence of the giant component. It is not the time that the giant component will definitely occu. This also explains that the uppe bound is lage than the conventional numbe..
176 B. Gu and X. Hong Figue 9 Expected numbe of neighbous against time with = 1: (a) linea gowth; (b) exponential gowth and (c) logistic gowth (see online vesion fo colous) Figue 1 Phase tansition against time with = : (a) linea gowth; (b) exponential gowth and (c) logistic gowth (see online vesion fo colous) Figue 11 Expected numbe of neighbous against time with = : (a) linea gowth; (b) exponential gowth and (c) logistic gowth (see online vesion fo colous) We then change the tansmission ange to be = to study the connectivity tansition. The gowth functions ae the same as above. The esults ae given in Figue 1. Compaed with the cuves fo = 1 (Figue 8), these esults show that less time is needed to each the total connected state due to the lage. Moeove, the fist connection time occus when the size of giant component is elative small. Figue 11 futhe shows the evolution of the expected neighbous in this = case whee the values appoximately ange fom two to ten duing the citical phase. Although the connection time becomes shot due to the lage tansmission ange, the expected numbe of neighbou is elatively stable. citical densities that denote the tansfomation fom patition to connected state ae deived, which ae coesponding to the fist connection time and total connection time. We studied thee node gowth functions fo modeling the incement of node. The simulations use thee instances of these gowth functions and the esults show that the elated time metics and the tend of change in connectivity when time goes by. In futue wok, paametes can be studied fo moe ealistic adio models in which the tansmission ange could have iegula shapes athe than a cicle. Also the heteogeneity of wieless nodes can be consideed such as diffeent tansmission anges and heteogeneous distibution in deployments. 6 Conclusion In the pape, we study the connectivity tansition of wieless netwok when the density gows. The bounds of the elated Acknowledgement This wok is suppoted in pat by NSF Awads No. 8987.
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