Substituting (n)andr into Eqn.(4-8).and for all sufficient large n,we further have Then wen bound boud P() log(L.H.S of Eqn.(4-7)) PUF)≤∑PF) 2≥osm-kr(et+P+(xr+P）) ≥1ogm-km(ar+R2+8（a2rjP)） 26-- (414) =-5-kre(n+28)-2vk(n logn+5) ≤me-r(r-R2kna 40(y logn +)2 1 e2cvk(n log n) ekTe(na+28) e-ξ-1 -n(e-1) (4-10) where the second equality is due to Lemma 1. Since a+28<0 and a>0,u can be arbitrarily close to Since >0 and a+28 <0,there exists some constant 0 when n is large.Taking the exponent of both sides and let c>1 such that Eqn.(4-13)holds. 6=e#i<1,Lemma 2 follows. Then,we have the following lemma. V.CRITICAL TRANSMISSION RANGE FOR Lemma 3:If r =logmis(n)where a+28<0,0< CLUSTER-DENSE STATE kπ a≤1,0<y≤1 and lim(n)=ξ<+o,we have n-oo In this state,.we have a+23≥元andπR2=w(a): Different from the cluster-sparse state focusing on the cluster lim inf Peas(n,a,B,Y,r0p）≥e-(1-e-). (4-11) connectivity perspective,we investigate cluster-dense state mainly from the perspective of cluster-member connectivity, Proof:Denote K.as the event that Ci is the only since cluster members behave more like independent nodes disconnected cluster during the k time slots. now.Let Peds(n,a,B,,rp)denote the probability that g(n,,B,,rP)has some cluster members disconnected. Pess(n,a,B,Y:rp) ≥∑P(K) Proposition 2:In a correlated mobile k-hop clustered net- work g(n,B,Y,r)for the cluster-dense state,the crit- 2-Gn) ical transmission range is where ,0<a≤1,0<y≤1. ≠ ≥∑P)-(∑PF)月 A.Necessary Condition of Proposition 2 The probability that a cluster member is not connected ≥m1-r+2)n-(m(1-r-2))》 during k timeslots is (5-1) >0e--m2e-2xkn"(r-R)2 P(fis)=(1-πr2)km =0e-5-e-2e2kr8(na+2a0)+4vKme(n2V√iogn+） The following lemma will show the probability that there exists one cluster member not connected,if we regard each 0e-f-(1+2)e-2 cluster member as independent nodes. (4-12) Lemma4:fr=Va志，a+28≥，0<a≤l,for for sufficient large n and u2>0.Note that the fourth any fixed 0<0<1 and sufficiently large n,we have inequality is due to Lemma I and the fifth inequality is due to Lemma 2.Recall that 6 can be arbitrarily close to 1 and u2 e-f≤n1-r2)n ≤e-】 (5-2) is arbitrarily close to 0 for sufficient large n,we can obtain Lemma 3. ◆ Proof:For the right hand side of Eqn.(5-2). B.Sufficient Condition of Proposition I nl-r2)r≤ne-tamr-e- Recall that f is the event that cluster member is dis- Then employing similar technique in the proof of Lemma connected during.Therefore,for the sufficient condition of 2.for all sufficient large n we have Proposition 1,it suffices to show that when r =crep.(c >1), the following equality holds. sa-r2))2osm-an（(am2+8car2) ▣(g心0)=▣U)= =--50ogn+)2 6kna 会-ξ-43 (4-13) (5-3)6 Substituting δ(n) and r = Èγ log n+ξ kπnα into Eqn. (4-8), and for all sufficient large n, we further have log€ L.H.S of Eqn. (4-7) Š ≥log m − knα  π(r + R) 2 + 5 6 € π(r + R) 2 Š2  ≥ log m − knα  π(r + R) 2 + 5 6 € π(2r) 2 Š2  = − ξ − kπΘ(n α+2β ) − 2 √ kπΘ(n α+2β 2 È γ log n + ξ) − 40(γ log n + ξ) 2 3knα , − ξ − µ1. (4-10) Since α + 2β < 0 and α > 0, µ1 can be arbitrarily close to 0 when n is large. Taking the exponent of both sides and let θ = e −µ1 < 1, Lemma 2 follows. Then, we have the following lemma. Lemma 3: If r = Èγ log n+ξ(n) kπnα where α + 2β < 0, 0 < α ≤ 1, 0 < γ ≤ 1 and limn→∞ ξ(n) = ξ < +∞, we have lim inf n→∞ Pcss(n, α, β, γ, rw.p. c ) ≥ e −ξ (1 − e −ξ ). (4-11) Proof: Denote Kc i as the event that Ci is the only disconnected cluster during the k time slots. Pcss(n, α, β, γ, rw.p. c ) ≥ Xm i=1 P(K c i ) ≥ Xm i=1  P(Fi) − X j̸=i P(Fi ∩ Fj )  ≥ Xm i=1 P(Fi) − Xm i=1 P(Fi) 2 ≥m € 1 − π(r + R) 2 Šknα −  m € 1 − π(r − R) 2 Šknα 2 ≥θe−ξ − m2 e −2πknα(r−R) 2 =θe−ξ − e −2ξ e −2kπΘ(n α+2β )+4√ kπΘ € n α+2β 2 √ γ log n+ξ Š ,θe−ξ − (1 + µ2)e −2ξ (4-12) for sufficient large n and µ2 > 0. Note that the fourth inequality is due to Lemma 1 and the fifth inequality is due to Lemma 2. Recall that θ can be arbitrarily close to 1 and µ2 is arbitrarily close to 0 for sufficient large n, we can obtain Lemma 3. B. Sufficient Condition of Proposition 1 Recall that f λ jκ is the event that cluster member ψ λ jκ is dis￾connected during λ. Therefore, for the sufficient condition of Proposition 1, it suffices to show that when r = crw.p. c (c > 1), the following equality holds. limn→∞ P  [m j=1 € [ϖ κ=1 ( \ k λ=1 f λ jκ) Š = limn→∞ P( [m j=1 Fj ) = 0. (4-13) Then we use union bound to bound P( Sm j=1 Fj ): P( [m j=1 Fj ) ≤ Xm j=1 P(Fj ) ≤ Xm j=1  1 − π(r − R) 2 knα ≤me−π(r−R) 2knα = 1 n(c 2−1)γ · e 2c √ kπγΘ(n α+2β 2 log n) e kπΘ(nα+2β) , (4-14) where the second equality is due to Lemma 1. Since γ > 0 and α + 2β < 0, there exists some constant c > 1 such that Eqn. (4-13) holds. V. CRITICAL TRANSMISSION RANGE FOR CLUSTER-DENSE STATE In this state, we have α + 2β ≥ ϵ k and πR2 = ω( 1 nα ). Different from the cluster-sparse state focusing on the cluster connectivity perspective, we investigate cluster-dense state mainly from the perspective of cluster-member connectivity, since cluster members behave more like independent nodes now. Let Pcds(n, α, β, γ, rw.p. c ) denote the probability that G(n, α, β, γ, rw.p. c ) has some cluster members disconnected. Proposition 2: In a correlated mobile k-hop clustered net￾work G(n, α, β, γ, rw.p. c ) for the cluster-dense state, the crit￾ical transmission range is r w.p. c = È log n kπnα , where α + 2β ≥ ϵ k , 0 < α ≤ 1, 0 < γ ≤ 1. A. Necessary Condition of Proposition 2 The probability that a cluster member is not connected during k timeslots is P(fjκ) = (1 − πr2 ) knα . (5-1) The following lemma will show the probability that there exists one cluster member not connected, if we regard each cluster member as independent nodes. Lemma 4: If r = Èlog n+ξ kπnα , α + 2β ≥ ϵ k , 0 < α ≤ 1, for any fixed 0 < θ < 1 and sufficiently large n, we have θe−ξ ≤ n € 1 − πr2 Šknα ≤ e −ξ . (5-2) Proof: For the right hand side of Eqn. (5-2), n € 1 − πr2 Šknα ≤ ne−kπnαr 2 = e −ξ . Then employing similar technique in the proof of Lemma 2, for all sufficient large n, we have log € n € 1 − πr2 Šknα Š ≥ log n − knα € πr2 + 5 6 (πr2 ) 2 Š = − ξ − 5(log n + ξ) 2 6knα , − ξ − µ3. (5-3)