9 roughly requiresr(n)Nonetheless,the order in Then we evaluate the two terms on the right hand side of (17). terms of n is the same. respectively,and we have Also note that by assuming the network area is a torus we have eliminated the possibility that nodes get reflected P{isa failed sessio)≥(-m2o）n) (18) at the boundary.If reflection is allowed,it will lead to minimal change to our conclusions.Specifically,the critical and transmission range under this case is(n)where 0<d 1.Note that it is two times larger than the critical P([si and sj are failed sessions}) transmission range in a torus.An intuitive explanation for this result is that,if nodes get reflected and follow their impinging ≤ 4rr2(n(1-r2(m)】 path,the sweep area might be kv(m)r(n)under the worst situation,m E M.Thereby r(n)should be two times larger +1-4rm0-2r2)"） than that in a torus.We omit the proof due to space constraints (19) since it is very similar to the earlier proofs in this section. The two terms in the parenthesis on the right hand side of(19) take into account the cases where the two nodes carrying the V.THE CRITICAL TRANSMISSION RANGE FOR MOBILE packet of si and sj are at a distance less than 2r(n)and greater K-HOP CLUSTERED NETWORKS,I.I.D.MOBILITY than 2r(n)when initially distributed,respectively.Using(18) and (19)in (17),we obtain The main result of this section is as follows. Theorem 5.1:Under the i.i.d.mobility assumption,the crit- Pf_iid(n,r(n)) ical transmission range is r(n)whered ≥n(1-r2(m)-n2(4r2m1-r2)” A.Necessary condition on r(n)of Theorem 5.1 +(1-2r2m) We start with the following technical lemma. (20 Lemma 5.1:If r2(n)m,for any fixed<1 and Using Lemma 5.1 and (13),for any fixed 0<1,we have μ≤l,and for all sufficiently large n n1-umr2(m)kn≥8e-, Pf_iid(n,r(n)) (16 ≥fe-- where0<d≤l. n2(4r26m)enr+e-2nr2)）】 Proof:The proof of this lemma follows the same argu- ment as that of Lemma 4.1,and we thus omit this proof. =0e-r- 4(logn+)e-tx+e 、knd-t Let gia(n,r(n))denote the network where two nodes can ≥fe-K-(1+e)e-2s communicate if their Euclidean distance is at most r(n)and Pf_iid(n,r(n))be the probability that Giid(n,r(n))has failed for any e>0 and for all n Ne.o..Note that we need d> sessions (i.e.,it has a node that is not connected).Then we in the last step. have the following proposition. Since limnK(n)=K,for any e there exits N'(e)such that for all n N'(e),K(n)<K+e.Considering that the Proposition 5.1:If(n)whered1.probability of disconnectedness is monotonously decreasing in knd then K,then we have liminf Pf_iid(n,r(n))>e-(1-e-), Ps_id(n,r(n))20e-(x+e)-(1+e)e-2(x+e), where K limnoo K(n),K>0. for n max{Ne..+N2).Taking limits Proof:Unlike the techniques employed in the proof of Proposition 4.1,we will consider all pairs of node i and j that liminf Pf_iid(n,r(n))>0e-(x+e)-(1+e)e-2(x+e). are disconnected cluster members. To evaluate Pfiid(n,r(n),we have Since this holds for all e>0 and 1,the result follows. Then we have the following corollary to prove the necessity Pi_iia(n,r(n)) part of Theorem 5.1. Corollary 5.1:Under the i.i.d.mobility assumption,the P(is a failed session in Gn)) mobile k-hop clustered network is to have failed sessions with positive probability bounded away from zero if mr2(n)= -∑∑P(fs,and号are failed session oand limn-→ek(m)<+oo,which meansπr2(n)≥ =1j≠ is necessary for the connectivity of mobile khop clus- in Giia(n,r(n))}). (17) tered networks with i.i.d.mobility model.9 roughly requires r(n) = log n 2kv⋆nd . Nonetheless, the order in terms of n is the same. Also note that by assuming the network area is a torus, we have eliminated the possibility that nodes get reflected at the boundary. If reflection is allowed, it will lead to minimal change to our conclusions. Specifically, the critical transmission range under this case is r(n) = log n kv⋆nd , where 0 < d ≤ 1. Note that it is two times larger than the critical transmission range in a torus. An intuitive explanation for this result is that, if nodes get reflected and follow their impinging path, the sweep area might be kv(m) r(n) under the worst situation, m ∈ M. Thereby r(n) should be two times larger than that in a torus. We omit the proof due to space constraints since it is very similar to the earlier proofs in this section. V. THE CRITICAL TRANSMISSION RANGE FOR MOBILE K-HOP CLUSTERED NETWORKS, I.I.D. MOBILITY The main result of this section is as follows. Theorem 5.1: Under the i.i.d. mobility assumption, the crit￾ical transmission range is r(n) = √ log n kπnd , where 1 k < d ≤ 1. A. Necessary condition on r(n) of Theorem 5.1 We start with the following technical lemma. Lemma 5.1: If πr2 (n) = log n+κ knd , for any fixed θ < 1 and µ ≤ 1, and for all sufficiently large n n ( 1 − µπr2 (n) )knd ≥ θe−κ , (16) where 0 < d ≤ 1. Proof: The proof of this lemma follows the same argu￾ment as that of Lemma 4.1, and we thus omit this proof. Let Giid(n, r(n)) denote the network where two nodes can communicate if their Euclidean distance is at most r(n) and Pf iid(n, r(n)) be the probability that Giid(n, r(n)) has failed sessions (i.e., it has a node that is not connected). Then we have the following proposition. Proposition 5.1: If πr2 (n) = log n+κ(n) knd , where 1 k < d ≤ 1, then lim inf n→∞ Pf iid(n, r(n)) ≥ e −κ (1 − e −κ ), where κ = limn→∞ κ(n), κ > 0. Proof: Unlike the techniques employed in the proof of Proposition 4.1, we will consider all pairs of node i and j that are disconnected cluster members. To evaluate Pf iid(n, r(n), we have Pf iid(n, r(n)) ≥ ∑n i=1 P({si is a failed session in Giid(n, r(n))}) − ∑n i=1 ∑ j̸=i P({si and sj are failed sessions in Giid(n, r(n))}). (17) Then we evaluate the two terms on the right hand side of (17), respectively, and we have P({si is a failed session}) ≥ (( 1 − πr2 (n) )n d )k (18) and P({si and sj are failed sessions}) ≤ ( 4πr2 (n) ( 1 − πr2 (n) )n d + ( 1 − 4πr2 (n) ) ( 1 − 2πr2 (n) )n d )k . (19) The two terms in the parenthesis on the right hand side of (19) take into account the cases where the two nodes carrying the packet of si and sj are at a distance less than 2r(n) and greater than 2r(n) when initially distributed, respectively. Using (18) and (19) in (17), we obtain Pf iid(n, r(n)) ≥ n ( 1 − πr2 (n) )knd − n 2 ( 4πr2 (n) ( 1 − πr2 (n) )n d + ( 1 − 2πr2 (n) )n d )k . (20) Using Lemma 5.1 and (13), for any fixed θ < 1, we have Pf iid(n, r(n)) ≥ θe−κ − ( n 2 k ( 4πr2 (n)e −n dπr2 (n) + e −2n dπr2 (n) ))k = θe−κ − ( 4(log n + κ) knd− 1 k e − 1 k κ + e − 2 k κ )k ≥ θe−κ − (1 + ϵ)e −2κ , for any ϵ > 0 and for all n > Nϵ,θ,κ. Note that we need d > 1 k in the last step. Since limn→∞ κ(n) = κ, for any ϵ there exits N′ (ϵ) such that for all n ≥ N′ (ϵ), κ(n) ≤ κ + ϵ. Considering that the probability of disconnectedness is monotonously decreasing in κ, then we have Pf iid(n, r(n)) ≥ θe−(κ+ϵ) − (1 + ϵ)e −2(κ+ϵ) , for n ≥ max{Nϵ,θ,κ+ϵ, N′ ϵ}. Taking limits lim inf n→∞ Pf iid(n, r(n)) ≥ θe−(κ+ϵ) − (1 + ϵ)e −2(κ+ϵ) . Since this holds for all ϵ > 0 and θ < 1, the result follows. Then we have the following corollary to prove the necessity part of Theorem 5.1. Corollary 5.1: Under the i.i.d. mobility assumption, the mobile k-hop clustered network is to have failed sessions with positive probability bounded away from zero if πr2 (n) = log n+κ(n) knd and limn→∞ κ(n) < +∞, which means πr2 (n) ≥ log n knd is necessary for the connectivity of mobile k–hop clus￾tered networks with i.i.d. mobility model