for any u>0 and sufficient large n.The second equality is B.Necessary Condition of Proposition 3 due to Egn.(5-8)and Egn.(5-9). ■ For convenience,we re-index the sub-clusters in each cluster from 1 to nk(+28)and define several notations. B.Sufficient Condition of Proposition 2 .Cix:the kth sub-cluster in the jth cluster. Similar to previous section,we let r =cr2-p.(c>1),and .Six:the event that the Cix is disconnected. use the union bound to obtain .Ki:the event that the Cix is the only sub-cluster in the whole system that is disconnected. P(U(U∩)≤∑∑（∩） 1 =1k=1 入= Similar to the proof in Section IV,we will first bound the a-wr) probability that a given sub-cluster is not connected during k (5-13) timeslots.Later,we will show that the probability that there exists a sub-cluster that is not connected,is bounded from ≤ne~kanar2 0.Since these sub-clusters are not really independent,the 1 disconnected probability should minus the probability that two -ne-1' sub-clusters are disconnected concurrently.The detail analysis is as follows. Taking limits of both sides,and note that c>1,we finish the proof of the sufficient condition of rp.in Proposition 2. Note that the sub-clusters have the feature of clusters in cluster-sparse state,we can use similar way to bound P(Sjs). VI.THE CRITICAL TRANSMISSION RANGE FOR （1-πr+2)≤PS)≤（1-r-3)m.61D CLUSTER-INFERIOR DENSE STATE The cluster-inferior dense state is a transition state between If we regard each cluster member as independent nodes,the the cluster-sparse and cluster-dense state and 0<a+28<. following lemma presents the probability that there exists one Therefore().Let Pcid (n,,Br)denote sub-cluster that is not connected. the probability that g(n,a,B,,r2p)is disconnected. Lemma7:fr=V/a+292osn+长，0≤a+2B< Proposition 3:In a correlated mobile k-hop clustered net- ,下= forany fixed and suficiently largen, work g(n,a,B,r)for the cluster-inferior dense state, we have the critical transmission range isr kana whee0<a≤1,0≤a+2B<元，0<Y≤1. nk()+(1()2)20e-t. (6-2) In this state,since 0<a+28<,we have rt.p.=o(R). Nodes in a same cluster can neither be treated totally inde- Proof:Employing similar technique in the proof of Lem- pendently,nor as a whole.The proof for necessary condition ma 2,for all sufficient large n,we have is conducted from the perspective of sub-cluster,which is introduced as follows. log(n(+2)+71-+))n") ≥(ka+28)+7）1ogn-kn(r++5(a2r2)） 6 A.Basis of Sub-Cluster =--2kT1/ k(a+23)+Y kx For each cluster,we cut out n+28 small circular areas(with radius of=niog元 log n log?n log?n )not overlapping with each other.Since 2 the total area of these small circular areas is na+28.2 40(k(a+28)+Y)1ogn+5) o(n28),the division is feasible. 3kna After segmentation,we getn+sub-areas in each cluster. e-ξ-6· We denote a sub-cluster as a set of k sub-areas,in which there (6-3) is one sub-area at one timeslot.Specifically,a sub-cluster is Taking the exponent of both sides and let 0=e#s<1, indexed by (a1,a2,..ax),where ai na+28-1,i= the result follows. 1,2,...,k.A node x E (a1,a2,...ak)if and only if x is in Lemma&:Ifr=√a+2tosn+d,where0≤a+ subarea ai during time slot i,where i=1,2,...,k.It is easy to see that there are totally nk(+28)sub-clusters in k time 2B<元，0<a≤1,0<y≤1and1im(m)=￡<+oo, 40 we have slots.The average number of nodes in each sub-cluster is w. (器)产=o(2-.Therefore,.whenn is sufficiently lim inf Pcids(n,a,B,y,rep.)>e(1-e-). (6-4) large,each sub-cluster will not be empty.Furthermore,these sub-clusters have the feature of clusters in cluster-sparse state because their radius r=o(rc),and we will virtually regard Proof:For a fixed value of we evaluate the nodes in each sub-cluster as a whole. lim inf Pcids(n,a,B,,rp)from the perspective of8 for any µ4 > 0 and sufficient large n. The second equality is due to Eqn. (5-8) and Eqn. (5-9). B. Sufficient Condition of Proposition 2 Similar to previous section, we let r = crw.p. c (c > 1), and use the union bound to obtain P  [m j=1 € [ϖ κ=1 ( \ k λ=1 f λ jκ) Š ≤ Xm j=1 Xϖ κ=1 P( \ k λ=1 f λ jκ) ≤ Xm j=1 Xϖ κ=1  (1 − πr2 ) n α k ≤ne−kπnαr 2 = 1 nc 2−1 . (5-13) Taking limits of both sides, and note that c > 1, we finish the proof of the sufficient condition of r w.p. c in Proposition 2. VI. THE CRITICAL TRANSMISSION RANGE FOR CLUSTER-INFERIOR DENSE STATE The cluster-inferior dense state is a transition state between the cluster-sparse and cluster-dense state and 0 ≤ α+2β < ϵ k . Therefore πR2 = ω( 1 nα ). Let Pcids(n, α, β, γ, rw.p. c ) denote the probability that G(n, α, β, γ, rw.p. c ) is disconnected. Proposition 3: In a correlated mobile k-hop clustered net￾work G(n, α, β, γ, rw.p. c ) for the cluster-inferior dense state, the critical transmission range is r w.p. c = È[k(α+2β)+γ] log n kπnα , where 0 < α ≤ 1, 0 ≤ α + 2β < ϵ k , 0 < γ ≤ 1. In this state, since 0 ≤ α + 2β < ϵ k , we have r w.p. c = o(R). Nodes in a same cluster can neither be treated totally inde￾pendently, nor as a whole. The proof for necessary condition is conducted from the perspective of sub-cluster, which is introduced as follows. A. Basis of Sub-Cluster For each cluster, we cut out n α+2β small circular areas (with radius of r¯ = 1 n α 2 log n ), not overlapping with each other. Since the total area of these small circular areas is n α+2β · πr¯ 2 = o(n 2β ), the division is feasible. After segmentation, we get n α+2β sub-areas in each cluster. We denote a sub-cluster as a set of k sub-areas, in which there is one sub-area at one timeslot. Specifically, a sub-cluster is indexed by (a1, a2, ..., ak), where 0 ≤ ai ≤ n α+2β − 1, i = 1, 2, ..., k. A node x ∈ (a1, a2, ..., ak) if and only if x is in subarea ai during time slot i, where i = 1, 2, ..., k. It is easy to see that there are totally n k(α+2β) sub-clusters in k time slots. The average number of nodes in each sub-cluster is ϖ · ( πr¯ 2 πR2 ) k = Θ( n k[ ϵ k −(α+2β)] log2k n ). Therefore, when n is sufficiently large, each sub-cluster will not be empty. Furthermore, these sub-clusters have the feature of clusters in cluster-sparse state because their radius r¯ = o(rc), and we will virtually regard the nodes in each sub-cluster as a whole. B. Necessary Condition of Proposition 3 For convenience, we re-index the sub-clusters in each cluster from 1 to n k(α+2β) and define several notations. • Cjκ: the κth sub-cluster in the jth cluster. • Sjκ: the event that the Cjκ is disconnected. • Ks jκ: the event that the Cjκ is the only sub-cluster in the whole system that is disconnected. Similar to the proof in Section IV, we will first bound the probability that a given sub-cluster is not connected during k timeslots. Later, we will show that the probability that there exists a sub-cluster that is not connected, is bounded from 0. Since these sub-clusters are not really independent, the disconnected probability should minus the probability that two sub-clusters are disconnected concurrently. The detail analysis is as follows. Note that the sub-clusters have the feature of clusters in cluster-sparse state, we can use similar way to bound P(Sjκ). € 1 − π(r + ¯r) 2 Šknα ≤ P(Sjκ) ≤ € 1 − π(r − r¯) 2 Šknα . (6-1) If we regard each cluster member as independent nodes, the following lemma presents the probability that there exists one sub-cluster that is not connected. Lemma 7: If r = È[k(α+2β)+γ] log n+ξ kπnα , 0 ≤ α + 2β < ϵ k , r¯ = 1 n α 2 log n , for any fixed θ < 1 and sufficiently large n, we have n k(α+2β)+γ € 1 − π(r + ¯r) 2 Šknα ≥ θe−ξ . (6-2) Proof: Employing similar technique in the proof of Lem￾ma 2, for all sufficient large n, we have log n k(α+2β)+γ € 1 − π(r + ¯r) 2 Šknα  ≥ € k(α + 2β) + 㠊 log n − knα  π(r + ¯r) 2 + 5 6 € π(2r) 2 Š2  = − ξ − 2kπÊ k(α + 2β) + γ log n + ξ log2 n − kπ log2 n − 40€ k(α + 2β) + 㠊 log n + ξ 2 3knα , − ξ − µ5. (6-3) Taking the exponent of both sides and let θ = e −µ5 < 1 , the result follows. Lemma 8: If r = È[k(α+2β)+γ] log n+ξ(n) kπnα , where 0 ≤ α+ 2β < ϵ k , 0 < α ≤ 1, 0 < γ ≤ 1 and limn→∞ ξ(n) = ξ < +∞, we have lim inf n→∞ Pcids(n, α, β, γ, rw.p. c ) ≥ e −ξ (1 − e −ξ ). (6-4) Proof: For a fixed value of ξ, we evaluate lim inf n→∞ Pcids(n, α, β, γ, rw.p. c ) from the perspective of