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in this paper we study the asymptotic properties related to connectivity of random geometric graphs where the underlying distribution of the vertices may not be uniform . a random geometric graph ( rgg ) consists of a set of vertices that are distributed in space independently , according to some common probability density function . the edge set of the graph consists of the set of all pairs of points that are within a specified cut - off distance . our point of departure from usual random geometric graphs is the specification of a cut - off function @xmath7 that determines the edge set . a directed edge exists , from a vertex located at @xmath4 to another vertex located at @xmath8 , provided the distance between @xmath4 and @xmath8 is less than @xmath9 . our motivation for the study of such graphs comes from applications in wireless networks . in models of wireless networks as rggs , the nodes are assumed to be communicating entities that are distributed randomly in space according to some underlying density . nodes are assumed to communicate effectively with other nodes that are within a cut - off distance , that is , proportional to the transmission power . hence , the transmission power has to be sufficiently large for the network to be connected . however , nodes that are within each other s transmission range interfere and thus can not transmit simultaneously . in order to maximize spatial reuse , that is , the simultaneous use of the medium by several nodes to communicate , the transmission power should be minimized . thus the asymptotic behavior of the critical radius of connectivity in a random geometric graph as the number of vertices becomes large is of considerable interest . often the nodes are assumed to be distributed in @xmath10^d$ ] according to a poisson point process of intensity @xmath1 . in this case , it is known that the critical connectivity radius scales as @xmath11 if the underlying density is nonuniform but bounded away from zero , then ( see penrose  @xcite ) the asymptotic behavior of the largest nearest neighbor distance in the graph is @xmath12 where @xmath13 is the minimum of the density over its support . note that the asymptotics of the largest nearest neighbor distance is determined by the reciprocal of the minimum of the density , since it is in the vicinity of the minimum that the nodes are sparsely distributed . in many applications such as mobile ad - hoc networks and sensor networks , the distribution of the nodes may be far from uniform ( see , e.g. , foh et al . @xcite , santi  @xcite ) . in the case of nonuniform distribution of nodes , it is not efficient from the point of view of maximizing spatial reuse , for all nodes to use the same cut - off radius . nodes near the mode of the density require a much smaller radius than those at locations where the density is small . further , the infimum of the density over its support could be zero ( @xmath14 ) . in such cases the asymptotics of the largest nearest neighbor distance or the connectivity threshold will be very different from that given above . one of the major objectives in a wireless sensor network is to maximize battery life , and hence it is important to minimize the energy expended in data transmission . these considerations leads us to rggs with location - dependent choice of radii . in many applications , it is assumed that the nodes know or can effectively estimate their location ( akkaya and younis  @xcite , langendoen and reijers  @xcite ) . we prescribe a formula for a critical location dependent cut - off radius depending on the intensity , so that almost surely the resulting graphs do not have isolated nodes eventually . a useful property of the graphs we construct is that the distribution of out - degree is independent of the location of the nodes . vertex degree distributions are important in designing algorithms for distributed computations over wireless networks where the performance worsens with increasing vertex degrees ( giridhar and kumar  @xcite ) . we derive strong law bounds for the maximum and minimum vertex degrees . by considering a finer parametrization of the cut - off function , we show that the number of vertices with zero out - degree converges to a poisson distribution , under some conditions on the underlying density . we illustrate the result with some examples . a result of this nature for usual random geometric graphs with the uniform and exponentially decaying densities of nodes can be found in penrose  @xcite , gupta and iyer  @xcite , respectively . the solution to the connectivity problem for random geometric graphs for nonvanishing densities with compact support and dimensions @xmath15 can be found , for example , in chapter 13 of penrose  @xcite . in one dimension , this problem is studied for densities in @xmath10 $ ] with polynomial rate of decay to zero in han and makowski  @xcite , while in gupta , iyer and manjunath  @xcite the density is assumed to be exponential or truncated exponential . in two dimensions , the asymptotic distribution for the critical connectivity threshold for a large class of densities , including elliptically contoured distributions , distributions with independent weibull - like marginals and distributions with parallel level curves is derived in hsing and rootzen  @xcite . in dimensions @xmath15 , penrose  @xcite obtains the asymptotic distribution for the connectivity threshold when the nodes are distributed according to a standard normal distribution . in this paper , we derive a sufficient condition for the rggs with location - dependent radius to be almost surely connected eventually .    in summary , our primary motivation in proposing the study of graphs with location dependent radii is as follows . it is to enable the design and study of wireless and sensor networks that allow nonstandard distribution of nodes obtained by fitting densities to empirical data obtained from actual deployments . given such a density , each individual node can be programmed to choose a transmission radius depending on its location so that the network is connected with high probability . further , any change in the underlying distribution over time due to failures , re - deployments , etc . , can be easily accommodated by appropriately changing the transmission radii . as far as analyzing these graphs is concerned , the key features to contend with are that the edges are directed and that the cut - off radius is specified implicitly .    in order to state our results we need some notation . let @xmath2 be a continuous probability density function with support @xmath16 . for any random variable @xmath17 with density @xmath2 , we denote it by @xmath18 . let the metric on @xmath3 be given by one of the @xmath19 norms @xmath20 , denoted by @xmath21 . let @xmath22 denote the volume of the unit ball in @xmath3 . we denote by @xmath23 the closure of the set @xmath24 . let @xmath25 be a sequence of i.i.d . points distributed according to @xmath2 . let @xmath26 be a nondecreasing sequence of poisson random variables with @xmath27 = n$ ] and define the sequence of sets @xmath28 note that @xmath29 is a poisson point process with intensity @xmath30 . for any @xmath31 and @xmath32 we denote by @xmath33 the open ball of radius @xmath34 centered at @xmath4 . for any borel set @xmath35 and any point process @xmath36 , @xmath37 $ ] represents the number of points of @xmath36 in  @xmath24 , and define @xmath38 we now define the random geometric graphs of interest with location - dependent radii . [ defrgg ] for any @xmath39 , let @xmath29 be the set given by ( [ eqnpn ] ) . for any function @xmath40 , the random geometric graph @xmath41 is defined to be the graph with vertex set @xmath29 , and the directed edge set @xmath42    we will also consider an augmented version of the above random geometric graph which is obtained by making all the edges in @xmath41 bi - directional . [ defrgge ] the enhanced random geometric graph @xmath43 associated with the graph @xmath41 is defined to be the graph with vertex set @xmath29 and ( undirected ) edge set @xmath44    in the communication application described in the , the following procedure will give a graph whose edge set will _ contain _ the edges of the enhanced graph @xmath45 . upon deployment , the nodes broadcast their radius . all nodes reset their transmission radius to be the maximum of their original radius and the ones they receive from the broadcast . note that this is done only once . clearly all the directed links in the original graph now become bi - directional together with possible creation of some directed edges . thus if the enhanced graph @xmath45 is connected , then so is the graph obtained by this procedure .      for any fixed @xmath46 , define the sequence of _ cut - off _ functions @xmath47 , via the equation @xmath48 later we will have occasion to take @xmath49 to be a function of @xmath4 and @xmath6 as well . we will denote @xmath50 by  @xmath51 and the associated enhanced graph by  @xmath45 , when @xmath49 , @xmath2 are fixed and @xmath52 is as defined in ( [ eqncutoff1 ] ) . by the palm theory for poisson point processes ( theorem 1.6 , @xcite ) , the expected out - degree of any node in @xmath51 will be @xmath53 = n \int_{b(x , r_n(c , x ) ) } f(y ) \,dy = c \log n,\vadjust{\goodbreak}\ ] ] which is the same as the vertex degree in the usual random geometric graph defined on uniform points in the connectivity regime . let @xmath54 and @xmath55 denote the palm distributions of @xmath29 conditional on a vertex located at @xmath4 . by the palm theory for poisson point processes , the expected out - degree of a node located at @xmath32 in the graph @xmath51 will be @xmath56 = c \log n.\ ] ] thus the expected vertex degree of a node in @xmath51 does not depend on the location of the node . in fact , the number of points of @xmath57 that fall in @xmath58 under @xmath54 will follow a poisson distribution with mean @xmath59 . let @xmath60 be the number of nodes in @xmath51 that have zero out - degree , that is , @xmath61 = 0\}}. \label{eqnwn}\ ] ] for each @xmath39 , define @xmath62 in other words , @xmath63 is the critical cut - off parameter , that is , the smallest @xmath49 , so that the graphs @xmath50 do not have any node with zero out - degree . our first result is a strong law for this critical cut - off parameter @xmath63 . [ thmmdnstar ] let @xmath63 be the critical cut - off parameter as defined in  ( [ eqncriticald ] ) . then almost surely , @xmath64    let @xmath65 be the random geometric graphs as defined in definition  [ defrgg ] with @xmath52 as in ( [ eqncutoff1 ] ) . consider the enhanced random geometric graph @xmath66 associated with @xmath51 ; see definition  [ defrgge ] . we now state a strong law result for the critical cut - off parameter to eleminate isolated nodes in the graph  @xmath45 . let @xmath67 be the number of isolated nodes , that is , nodes with degree zero , in the enhanced graph  @xmath45 . define @xmath68 clearly @xmath69 by construction . the following theorem shows that the threshold required to eleminate isolated nodes in the enhanced graph @xmath45 is the same as for the graph @xmath51 . [ thmmtdn ] let @xmath70 be as defined in ( [ eqntdn ] ) . then , almost surely , @xmath71    the exact asymptotics for the connectivity threshold for random geometric graphs requires a lot of elaborate computations ; see chapter 13 ,  @xcite . we provide a sufficient condition that requires only a local computation at each node and makes use of the connectivity threshold for the usual uniform random geometric graphs . let @xmath17 be a random variable with probability density function @xmath2 with support @xmath72 @xmath15 . suppose that @xmath2 admits a mapping @xmath73 such that @xmath74 is uniformly distributed on @xmath10^d$ ] . for example , if the coordinates of @xmath17 are independently distributed , then the coordinate mappings of @xmath75 will be the marginal distributions . recall that @xmath22 denotes the volume of the unit ball . for any @xmath76 , let @xmath77 be the sequence defined by @xmath78 where @xmath79 for any set @xmath24 , we denote by @xmath80 the image of the set @xmath24 under @xmath75 . suppose that the functions @xmath81 are as defined in ( [ eqncutoff1 ] ) . define the sequence of functions , @xmath82 and @xmath83 to be functions on @xmath84 that satisfy the equation @xmath85 let @xmath86 be the graphs defined as in definition  [ defrgg ] with @xmath34 replaced by @xmath87 . let @xmath88 be the enhanced graphs associated with @xmath89 , that is , the graphs obtained by making the edges in @xmath86 bi - directional .    [ thmmconnectivity ] let @xmath18 , and suppose that @xmath74 is uniform on @xmath10^d$ ] , . then for any @xmath76 , almost surely , the sequence of enhanced random geometric graphs @xmath90 is connected for all sufficiently large @xmath6 . in particular , the above result implies that @xmath91 our next result is on strong law asymptotics for the maximum and minimum vertex degrees for the sequence of graphs @xmath51 . let @xmath92 be defined by @xmath93 and @xmath94 the function @xmath95 has a unique turning point at the minima @xmath96 . let @xmath97 be the inverse of @xmath95 restricted to @xmath98 and @xmath99\rightarrow[0,1]$ ] be the inverse of the restriction of @xmath95 to @xmath10 $ ] . [ thmmmaxmindeg ] for any @xmath46 , let @xmath100 be the maximum and @xmath101 be the minimum vertex out - degree of the graph @xmath102 then with probability 1 , @xmath103 if @xmath104 , then @xmath105 almost surely . if @xmath106 , then with probability 1 , @xmath107    note that ( [ eqndeln ] ) does not shed any light on what happens to the minimum vertex degree when @xmath108 . this requires a finer parametrization of the cut - off function . for any @xmath109 , define @xmath110 to be functions satisfying @xmath111 for all n sufficiently large for which @xmath112 , and arbitrarily otherwise . since  @xmath113 is fixed , we will write @xmath114 for @xmath115 . for each @xmath39 , define the sets @xmath116    [ thmmpoicgs ] let @xmath117 be the number of nodes of out - degree zero in the graph @xmath118 where @xmath119 is as defined in ( [ eqnrpoi ] ) . suppose that @xmath2 satisfies the following two conditions for all @xmath6 sufficiently large :    there exists a constant @xmath120 , such that for all @xmath6 sufficiently large @xmath121    @xmath122    then , @xmath123 as @xmath124 , where @xmath125 denotes a poisson random variable with mean @xmath126    as noted earlier , poisson approximation results for the number of isolated nodes are available only for a small class of distributions . condition ( [ eqnc2 ] ) can be replaced by the following sufficient condition which , as will be shown below , is easier to verify for some classes of densities . @xmath127    [ thmmpoi2 ] suppose that the hypothesis ( [ eqnc1 ] ) of theorem [ thmmpoicgs ] and ( [ eqnc3 ] ) are satisfied , then ( [ eqnpoicgs ] ) holds true . our final result is an illustration of the use of the above theorem to a  class of densities with compact support that have at most polynomial rates of decay to zero . let @xmath2 be a continuous density with compact support @xmath16 , @xmath15 . let @xmath128 , @xmath129 be nonintersecting balls such that @xmath130 on the boundary of these balls . for each @xmath131 , there exists integers @xmath132 and @xmath133 , @xmath134 and constants @xmath135 such that , either @xmath136 or @xmath137 and @xmath138 elsewhere in @xmath84 . the density can vanish , for example , over water bodies which may contain islands ( which are being approximated by balls ) , and the density decays to zero polynomially near the boundary of these balls . in particular , @xmath84 itself could be taken to be a ball with the density decaying polynomially to zero at the boundary of @xmath84 in a radially symmetric fashion . we will denote by @xmath139 the set of all densities of the above form . the class @xmath139 contains many of the standard distributions such as the uniform , triangular , beta , etc . , truncated versions of standard distributions with unbounded support such as the gaussian , gamma , etc . in santi @xcite the density is assumed to be continuous and bounded away from 0 over @xmath10 ^ 2 $ ] , which is contained in @xmath139 . the class @xmath139 also contains the higher dimensional extensions of the polynomial densities considered in foh et al . @xcite and han and makowski  @xcite . as remarked earlier , since our motivation was to allow for nonstandard densities , @xmath139 contains functions that are the modulus of analytic functions in the interior of @xmath84 . [ corpoi ] the conclusion of theorem  [ thmmpoi2 ] holds for any density in class @xmath139 . proof of theorem [ thmmdnstar ] first we will show that @xmath140 fix @xmath76 and let @xmath141 . let @xmath142 , @xmath143 , where the constant @xmath144 will be chosen later . let @xmath145 be as defined in ( [ eqnwn ] ) . for each @xmath146 define the events @xmath147 recall that @xmath148 $ ] denotes the number of points of the point process @xmath29 that fall in the set @xmath24 . set @xmath149 we will show that @xmath150 < \infty . \label{eqnsumbk}\ ] ] it will then follow by the borel  cantelli lemma that almost surely , only  finitely many of the events @xmath151 ( and hence the @xmath152 ) happen . consequently , with probability  1 , @xmath153 , eventually . since @xmath76 is arbitrary , this will prove ( [ eqnlimsupdnstar ] ) . for each @xmath154 let @xmath155 be the event that there is a vertex @xmath156 that has out - degree zero in the graph @xmath157 that is , @xmath158 = 0 \bigr\}.\ ] ] since we have assumed the variables @xmath159 to be nondecreasing and the functions @xmath81 are nonincreasing in @xmath6 for each fixed @xmath49 and @xmath4 , we have @xmath160 consequently , @xmath161 by the palm theory for poisson point processes ( theorem 1.6 ,  @xcite ) , ( [ eqnf ] ) and  ( [ eqncutoff1 ] ) , we have @xmath162 & \leq & e\biggl[\sum_{x \in\cp_{n_{k+1 } } } 1_{\{\cp_{n_k}[b(x , r_{n_{k+1}}(c))\setminus\{x\ } ] = 0 \ } } \biggr ] \nonumber\\ & = & n_{k+1 } \int_{\mr^d } e^{-n_k f(b(x , r_{n_{k+1}}(c ) ) ) } f(x ) \,dx \nonumber \\[-8pt ] \\[-8pt ] \nonumber & = & n_{k+1 } \exp\biggl ( - c \frac{n_k}{n_{k+1 } } \log n_{k+1 } \biggr ) \\ & = & ( k+1)^b \exp\biggl ( - ( 1+\ep ) \biggl(\frac{k}{k+1 } \biggr)^b \log(k+1)^b \biggr).\nonumber\end{aligned}\ ] ] choose @xmath163 so that @xmath164 , and pick @xmath144 such that @xmath165 for sufficiently large @xmath166 , we have @xmath167 using this in ( [ eqnhk ] ) , we get for all @xmath168 sufficiently large @xmath169 \leq\frac{1}{(k+1)^{((1-\ga)(1+\ep ) - 1)b}},\ ] ] which is summable in @xmath166 . ( [ eqnsumbk ] ) now follows from the above inequality and  ( [ eqnbk ] ) . this proves ( [ eqnlimsupdnstar ] ) by the arguments following ( [ eqnsumbk ] ) . to complete the proof , we need to show that @xmath170 fix @xmath104 and pick @xmath171 such that @xmath172 . choose @xmath173 such that @xmath174 since @xmath2 is continuous , we can and do fix a @xmath175 satisfying @xmath176 , where @xmath177 let @xmath178 be such that @xmath179 choose @xmath180 such that @xmath181 and let @xmath182 define the sequence of functions @xmath183 by @xmath184 let @xmath185 be the maximum number such that there exists @xmath185 many disjoint balls of radius @xmath186 with centers in @xmath187 . then ( see lemma 2.1 ,  @xcite ) , we can find a constant  @xmath188 such that for all @xmath6 sufficiently large , @xmath189 let @xmath190 be the deterministic set of points in @xmath187 such that the balls @xmath191 , @xmath192 , are disjoint . let @xmath193 be the event that there is exactly one point of @xmath29 in @xmath194 with no other point in @xmath195 , that is , @xmath196 = 1 , \cp _ n[b(x,\bar{r}_n(u))\setminus b(x,\bar{r}_n(\ep ) ) ] = 0 \}. \label{eqnen}\vadjust{\goodbreak}\ ] ] note that the events @xmath197 = 1\}$ ] and @xmath198 = 0 \}$ ] are independent . hence for any @xmath199 , we have @xmath200 & = & n f(b(x,\bar{r}_n(\ep ) ) ) e^{- n f(b(x,\bar{r}_n(\ep ) ) ) } e^{- n f(b(x,\bar{r}_n(u)))\setminus b(x,\bar{r}_n(\ep ) ) } \\ & = & n f(b(x,\bar{r}_n(\ep ) ) ) e^{- n f(b(x,\bar{r}_n(u)))}.\end{aligned}\ ] ] note that for all @xmath199 and all @xmath6 sufficiently large , @xmath201 . hence for all @xmath6 sufficiently large and @xmath199 , we have @xmath202 & \geq & n f_0 \th_d \bar{r}_n(\ep)^d \exp ( - n g_0 \th _ d \bar{r}_n(u)^d ) \nonumber\\ & = & \ep\log n \exp\biggl ( - \frac{g_0 u}{f_0 } \log n \biggr ) \\ & = & \ep n^{- { g_0u}/{f_0 } } \log n.\nonumber\end{aligned}\ ] ] using the fact that the events @xmath203 , @xmath204 , are independent , the inequality @xmath205 , and ( [ eqnsg ] ) , ( [ eqnlben ] ) , we get @xmath206 \leq\exp\bigl ( - \ep\sg_n n^{- { ( g_0u)}/{f_0 } } \log n \bigr ) \leq\exp\bigl ( - c_1 \ep n^{1 - { ( g_0u)}/{f_0 } } \bigr),\ ] ] which is summable in @xmath6 since @xmath207 hence , by the borel  cantelli lemma , almost surely , for all sufficiently large @xmath6 the event @xmath203 happens for some @xmath208 . hence w.p . 1 , for all n sufficiently large , we can find a random sequence @xmath209 such that @xmath210 , and there is no other point of @xmath29 within a distance @xmath211 of @xmath212 . by ( [ eqnep ] ) , @xmath213 since @xmath214 , from ( [ eqncutoff1 ] ) and the remark above ( [ eqnlben ] ) , we get for sufficiently large @xmath6 , @xmath215 \\[-8pt ] \nonumber & = & \th_d f_0 \bar{r}_n(c)^d \leq \int_{b(x_{j(n)},\bar{r}_n(c ) ) } f(y ) \,dy.\end{aligned}\ ] ] hence @xmath216 . it follows that there is no other point of @xmath29 in @xmath217 , that is , @xmath212 has out - degree zero in @xmath218 consequently w.p . 1 , @xmath219 for all @xmath6 sufficiently large . since @xmath220 , this proves  ( [ eqnliminfdnstar ] ) . proof of theorem [ thmmtdn ] since the graph @xmath45 is obtained by making all the edges in @xmath51 bi - directional , @xmath221 . consequently @xmath69 , and hence by theorem  [ thmmdnstar ] we have @xmath222 thus it suffices to show that @xmath223 fix @xmath220 , and let @xmath224 and @xmath225 be as in the second part of the proof of theorem  [ thmmdnstar ] . recall the random sequence @xmath209 , defined on a set of probability one , such that for sufficiently large @xmath6 , the point @xmath210 has no other point of @xmath29 within a distance @xmath226 . since @xmath181 , by the same arguments as in ( [ eqnrnbound ] ) , we have @xmath227 for all @xmath228 and all  @xmath6 sufficiently large . thus almost surely , none of the points @xmath229 , that fall in the ball @xmath230 , have an out - going edge to @xmath212 for all sufficiently large @xmath6 . on the other hand , for any point @xmath231 , suppose @xmath232 . then we can find a point @xmath8 such that @xmath233 such that @xmath234 however , @xmath235 which clearly is not possible for sufficiently large @xmath6 . thus there can be no edge leading from a point of @xmath29 in @xmath236 to @xmath237 hence the points @xmath210 have zero in - degree as well for sufficiently large @xmath6 . consequently , with probability  1 , @xmath238 for all sufficiently large @xmath6 for any @xmath239 this proves ( [ eqnliminftdn ] ) and thus completes the proof of theorem  [ thmmtdn ] . proof of theorem [ thmmconnectivity ] for any @xmath76 , let @xmath240 , @xmath39 , be as defined in ( [ eqncn ] ) . consider the random geometric graph @xmath241 induced by the mapping  @xmath75 and the enhanced graph @xmath242 . the vertex set of the graph @xmath241 is the set @xmath243 with edges between any two vertices @xmath244 and @xmath245 provided there is an edge between @xmath246 and @xmath247 in the graph @xmath45 . the vertices of @xmath241 are distributed according to a homogenous poisson point process on @xmath10^d$ ] with intensity @xmath6 . let @xmath248 be as defined in ( [ eqnm ] ) . now by theorem 13.2 ,  @xcite ( with @xmath249 ) , we have @xmath250 almost surely , where @xmath251 is the threshold for simple connectivity in the usual uniform random geometric graph on @xmath10^d$ ] ( nodes being distributed according to a homogenous poisson point process with intensity @xmath6 ) .    by definition of @xmath252 , in the graph @xmath241 , each vertex is connected to all its neighbors that are within a distance @xmath253 almost surely for all sufficiently large @xmath6 . it follows that almost surely , the graphs @xmath241 , and hence the graphs  @xmath45 are connected for all sufficiently large @xmath6 . proof of theorem [ thmmmaxmindeg ] we first show ( [ eqndeln ] ) . fix @xmath46 and @xmath254 define the sequence @xmath255 fix a constant @xmath144 such that @xmath256 , and let @xmath257 @xmath258 define the events @xmath259 , @xmath39 . let @xmath260 since @xmath159 , @xmath261 are increasing , and @xmath81 is decreasing pointwise in @xmath6 , we have @xmath262 \geq c_{n_k } + 1 \}. \label{eqnmaxbk}\ ] ] by the palm theory for poisson point processes ( theorem 1.6 ,  @xcite ) , @xmath263 \geq c_{n_k } \ } } \biggr ] \nonumber \\[-8pt ] \\[-8pt ] \nonumber & = & n_{k+1 } \int_{\mr^d } p \bigl ( \operatorname{po } ( n_{k+1 } f(b(x , r_{n_k}(c , x ) ) ) ) \geq c_{n_k } \bigr ) f(x ) \,dx,\end{aligned}\ ] ] where @xmath125 denotes a poisson random variable with mean @xmath126 from ( [ eqncutoff1 ] ) , we have for sufficiently large @xmath166 , @xmath264 hence for sufficiently large @xmath166 , by definition of @xmath265 , we get @xmath266 \\[-8pt ] \nonumber & = & h_+^{-1}\biggl ( \frac{1+\ep}{c } \biggr ) > 1.\end{aligned}\ ] ] hence using the chernoff bound for the poisson distribution ( see lemma  1.2 ,  @xcite ) , we get @xmath267 since @xmath95 is increasing in @xmath268 and @xmath269 , using ( [ eqnratio ] ) we can bound the probability on the left - hand side in the above equation by @xmath270 substituting this bound in ( [ eqnmaxbbd ] ) , we get for sufficiently large @xmath166 , @xmath271 which is summable in @xmath166 , since @xmath272 hence , by the borel cantelli lemma , almost surely only finitely many of the events @xmath151 and hence @xmath152 happen . hence , almost surely , @xmath273 eventually . the result now follows since @xmath76 is arbitrary . the result for @xmath274 in case @xmath104 follows from theorem  [ thmmdnstar ] . for the case @xmath275 the proof of ( [ eqndeln ] ) is entirely analogous to that of ( [ eqndeln ] ) , and so we provide the corresponding expressions . fix @xmath276 such that @xmath277 . let @xmath278 be such that @xmath256 , and define the sequence @xmath142 , @xmath143 . define the events @xmath279 where @xmath280 let the events @xmath151 be as defined in ( [ eqndefbk ] ) . the expression analogous to ( [ eqnmaxbk ] ) will be @xmath281 \leq c_{n_{k+1 } } + 1 \}. \label{eqnminbk}\ ] ] for sufficiently large @xmath166 , @xmath282 hence for sufficiently large @xmath166 , by our choice of @xmath283 , we get @xmath284 \\[-12pt ] \nonumber & = & h_-^{-1 } \biggl ( \frac{1+\ep}{(1 - \ep)c } \biggr ) < 1.\hspace*{-35pt}\end{aligned}\ ] ] again using the chernoff bound and proceeding as in the previous proof , we will get @xmath285 which is summable in @xmath166 . since @xmath76 is arbitrary , ( [ eqndeln ] ) now follows by the borel cantelli lemma and the arguments used earlier to infer ( [ eqndeln ] ) . the proof of theorem  [ thmmpoicgs ] uses the following lemma , which is a straightforward extension of theorem 6.7 ,  @xcite . let @xmath286 denote the total variation distance between two random variables . let @xmath287 and @xmath288 be as in theorem  [ thmmpoicgs ] . we will use the notation , @xmath289 , @xmath290 and @xmath291 recall that @xmath292 denotes the number of points of @xmath29 that fall in the set @xmath24 . [ lempoiapprox ] let @xmath2 be a continuous density on @xmath3 . then , for the graph  @xmath293 , we have @xmath294 ) ) \leq \min\biggl ( 3 , \frac{1}{e[\hw_n ] } \biggr ) \bigl(i_n^{(1 ) } + i_n^{(2)}\bigr ) , \label{eqnpoiapprox}\ ] ] where @xmath295 \\[-8pt ] \nonumber & & { } \times p [ \cp_n(b_n(x ) ) = 0 ] p [ \cp_n(b_n(y ) ) = 0 ] , \\ % \label{eqni2}i_n^{(2 ) } & = & n^2 \int_{\mr^d } f(x ) \,dx \int_{\hat{b}_n(x ) } f(y ) \,dy \nonumber \\[-8pt ] \\[-8pt ] \nonumber & & { } \times p [ \cp_n^y(b_n(x ) ) = 0 , \cp_n^x(b_n(y ) ) = 0].\end{aligned}\ ] ]    the proof is identical to the proof of theorem 6.7 ,  @xcite with the following obvious change . in the definition of dependency neighborhood , the parameter  @xmath34 is replaced by the function @xmath296 . proof of theorem [ thmmpoicgs ] by the palm theory for poisson point processes , @xmath297 = n \int_{\mr^d } f(x ) \,dx\ , e^{- n \int_{b_n(x ) } f(y ) \,dy } , \ ] ] where @xmath298 is as defined above lemma  [ lempoiapprox ] . using ( [ eqnrpoi ] ) , we get @xmath297 = e^{-\be}.\ ] ] hence , by lemma  [ lempoiapprox ] , it suffices to show that @xmath299 converge to zero as . again , using the palm theory and ( [ eqnrpoi ] ) , we get @xmath300 \\[-8pt ] \nonumber & = & e^{-2 \be } \int_{\mr^d } f(x ) \,dx \int_{\bb_n(x ) } f(y ) \,dy \to 0\hspace*{-35pt}\end{aligned}\ ] ] as @xmath124 , by the dominated convergence theorem , since @xmath301 for each @xmath32 . next we will show that @xmath302 , as @xmath124 . @xmath303 \\[-8pt ] \nonumber & = & i_n^{(21 ) } + i_n^{(22)},\end{aligned}\ ] ] where @xmath304 where @xmath305 is as defined in ( [ eqnhan ] ) . consider the inner integral in @xmath306 . on the set @xmath307 , we have @xmath308 , and hence by  ( [ eqnrpoi ] ) @xmath309 thus , @xmath306 converges to zero using the same arguments as in ( [ eqnin1 ] ) . it remains to show that @xmath310 as @xmath124 . since @xmath311 , where @xmath312 , we get from ( [ eqnrpoi ] ) , ( [ eqnc1 ] ) , that @xmath313 is bounded by a  constant times @xmath314 where @xmath315 is as defined in ( [ eqnan ] ) . the expression in ( [ eqnfinalbound ] ) converges to zero as @xmath124 by ( [ eqnc2 ] ) . proof of theorem [ thmmpoi2 ] note that in the proof of theorem  [ thmmpoicgs ] , ( [ eqnc2 ] ) is used only to prove that ( [ eqnfinalbound ] ) converges to zero . hence it suffices to show that  ( [ eqnfinalbound ] ) converges to zero under ( [ eqnc3 ] ) . let @xmath315 be as defined in ( [ eqnan ] ) . then @xmath316 using this , the expression in ( [ eqnfinalbound ] ) is bounded by @xmath317 applying fubini s theorem to the second term above , we see that the above expression is bounded by @xmath318 which converges to zero by ( [ eqnc3 ] ) . proof of corollary [ corpoi ] let @xmath319 . consider the following two classes of densities : @xmath320 where @xmath321 is the support of @xmath2 . denote by @xmath322 the set of functions @xmath323 such that for some @xmath324 and @xmath325 , @xmath326 and @xmath327 @xmath328 with @xmath329 . let @xmath330 be the set of functions @xmath323 such that for some @xmath325 , @xmath331 @xmath332 with @xmath329 . we first prove the result for densities @xmath333 . to do this we need to verify conditions ( [ eqnc1 ] ) and ( [ eqnc3 ] ) . _ step _ 1 . fix @xmath334 . let @xmath335 and @xmath336 and @xmath337 . by ( [ eqnrpoi ] ) , for any @xmath338 and for all @xmath6 large enough , we get @xmath339 or @xmath340 to prove ( [ eqnc1 ] ) , note that for any @xmath341 , we can inscribe a ball of radius @xmath342 inside @xmath343 . from this and ( [ boundsrn ] ) , for all @xmath6 sufficiently large , we get @xmath344 this proves ( [ eqnc1 ] ) with @xmath345 . by ( [ boundsrn ] ) we have @xmath346 this proves ( [ eqnc3 ] ) . thus the poisson convergence result holds for any @xmath334 . _ step _ 2 . next we prove the result for @xmath347 . proof for @xmath348 is similar and so we omit it . let @xmath349 . recall that @xmath350 over @xmath351 and is of the form @xmath352 over @xmath353 . for any @xmath354 , we have by ( [ eqnrpoi ] ) , @xmath355 note that the density is radially increasing , that is , @xmath356 if @xmath357 . if @xmath341 , then @xmath358 . if @xmath357 , then using ( [ eqnequalityoverball ] ) and the monotonicity of @xmath2 we get @xmath359 on the other hand if @xmath341 and @xmath360 , then by ( [ eqnequalityoverball ] ) and the monotonicity of @xmath2 we have @xmath361 hence @xmath362 thus ( [ eqnc1 ] ) holds with @xmath363 . next we verify ( [ eqnc3 ] ) over @xmath353 . by ( [ eqnrpoi ] ) , we have @xmath364 by changing to polar coordinates , the integral on the right - hand side of the above equation has the bounds @xmath365 for some positive constants @xmath366 from ( [ eqnfr ] ) and ( [ eqnboundr ] ) , we get @xmath367 and hence for some constant @xmath49 , @xmath368 this proves ( [ eqnc3 ] ) . the result now follows for any @xmath369 by theorem  [ thmmpoi2 ] .
we propose a _ distribution - free _ approach to the study of random geometric graphs . 
 the distribution of vertices follows a poisson point process with intensity function @xmath0 , where @xmath1 , and @xmath2 is a probability density function on @xmath3 . 
 a vertex located at @xmath4 connects via directed edges to other vertices that are within a _ cut - off _ 
 distance @xmath5 . 
 we prove strong law results for ( i ) the critical cut - off function so that almost surely , the graph does not contain any node with out - degree zero for sufficiently large @xmath6 and ( ii ) the maximum and minimum vertex degrees . 
 we also provide a characterization of the cut - off function for which the number of nodes with out - degree zero converges in distribution to a poisson random variable . 
 we illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero . 
 finally , we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually .    .