10 ically,m =m in cluster-sparse state,m =mnk(+28)in 02k-1 π2<4 cluster-inferior dense state and m=n in cluster-dense state Therefore,m can be regarded as the number of independent node groups in the network,and this is also the main finding and contribution of this work.In the following,we will show that this finding still holds in cluster-mixed networks,where distinct states of clusters co-exist in the network. A.Necessary Condition of Theorem I Lemma9:fr-√os+a,where0<a≤1and lim(n)=ξ<+o,we have 1→● Fig.6.2:Radius proof lim inf Pems(n,a,B,Y,rp.)20.9e(1-e).(7-4) Let r=cr-p.(c>1),we use the union bound and obtain Proof:With a same definition on K.i,and K,we 2k(a+2) have P(U(U∩)=P(U(US) Pems(n,a,(Bi},(wil,,rep) 1=1=1入=1 j=1 m nk(o+28) m2 nk(o+28,) ≤∑∑P(S) + PCx)+∑∑PK=) j=1R=1 j=1k=1 m nk(o+28) ≥∑P(one component failed) 合三--) P(two components in a same state failed) ≤n(a+28)+ye-krn°(r-4标)2 P(two components in different states failed). (7-5) =ne2-(a+2+）+o(点) The first and second terms can be characterized similar to (6-11) previous analysis.The difference lies in the third term,which For c>1,taking limits of both sides,we finish the proof however can still be bounded by the property that clusters in of the sufficient condition of rP.in Proposition 3. different states are independent. VII.CRITICAL TRANSMISSION RANGE FOR For simplicity of presentation,we let x= F.y= =1 CLUSTER-MIXED STATE (W.P.) In this section,we first present an interesting observation =1 P(Sj),2 是rU.Then we have j=1k=1 from the obtained result in previous 3 distinct states and then prove this observation will hold in a more general scenario. mi First trace back to the result in Section IV. 2P(C≥x-x2-y-r网 (7-6) ylogn log ny log m (7-1) where z-x2 is the first and second terms in Eqn.(7-5)whose Vkπna Vkπna proof is similar to (9-4),and xy+zz is the probability in the Then trace back to the proof in Section VI,where each third term of Eqn.(7-5). cluster is split into sub-clusters.The total number of sub- Similarly to Eqn.(6-5), clusters is mnk(+28).And the critical transmission range is as follows. 2 nk(a+28j) log mnk(a+28) PKx)≥y-2-∑∑P(SjnNSj) /k(a+28)+]1ogn Tc=V (7-2) j=1K≠K 1 kπna kxna yx-yz. Finally,trace back to the proof in Section V.In the proof (7-7) we treat each node as an independent cluster and the result is Similar to Eqn.(5-7), as follows. m3 m3 logn Te=kana (7-3) ∑∑PK)≥a-2-∑∑∑PUsn5N) =1=1 j=1K=1K'≠K From Egn.(7-1),Egn.(7-2)and Eqn.(7-3),we define -2-2y relative cluster number m to unify these three states.Specif- (7-8)10 ( 1) k r − r θ 2 1 k π θ = − O x y kr 2 2 cos ( sin ) 4 2 1 2 1 kr r k r k k π π = + < − − Fig. 6.2: Radius proof Let r = crw.p. c (c > 1), we use the union bound and obtain P  [m j=1 € [ϖ κ=1 ( \ k λ=1 f λ jκ) Š =P  [m j=1 € n k(α+2β) [ κ=1 P(Sjκ) Š ≤ Xm j=1 n kX (α+2β) κ=1 P(Sjκ) ≤ Xm j=1 n kX (α+2β) κ=1 € 1 − π(r − 4r¯¯) 2 Šknα ≤n k(α+2β)+γ e −kπnα(r−4r¯¯) 2 =n −(c 2−1)€ k(α+2β)+㠊 +Θ( √ 1 log n ) . (6-11) For c > 1, taking limits of both sides, we finish the proof of the sufficient condition of r w.p. c in Proposition 3. VII. CRITICAL TRANSMISSION RANGE FOR CLUSTER-MIXED STATE (W.P.) In this section, we first present an interesting observation from the obtained result in previous 3 distinct states and then prove this observation will hold in a more general scenario. First trace back to the result in Section IV. rc = r γ log n kπnα = r log nγ kπnα = r log m kπnα . (7-1) Then trace back to the proof in Section VI, where each cluster is split into sub-clusters. The total number of sub￾clusters is mnk(α+2β) . And the critical transmission range is as follows. rc = r [k(α + 2β) + γ] log n kπnα = r log mnk(α+2β) kπnα . (7-2) Finally, trace back to the proof in Section V. In the proof we treat each node as an independent cluster and the result is as follows. rc = r log n kπnα . (7-3) From Eqn. (7-1), Eqn. (7-2) and Eqn. (7-3), we define relative cluster number m¯ to unify these three states. Specif￾ically, m¯ = m in cluster-sparse state, m¯ = mnk(α+2β) in cluster-inferior dense state and m¯ = n in cluster-dense state. Therefore, m¯ can be regarded as the number of independent node groups in the network, and this is also the main finding and contribution of this work. In the following, we will show that this finding still holds in cluster-mixed networks, where distinct states of clusters co-exist in the network. A. Necessary Condition of Theorem 1 Lemma 9: If r = Èlog ¯m+ξ(n) kπnα , where 0 < α ≤ 1 and limn→∞ ξ(n) = ξ < +∞, we have lim inf n→∞ Pcms(n, α, β, γ, rw.p. c ) ≥ 0.9e −ξ (1 − e −ξ ). (7-4) Proof: With a same definition on Kc i ,Ks jκ, and Km jκ, we have Pcms(n, α, {βi}, {ϖi}, γ, rw.p. c ) ≥ Xm1 i=1 P(K c i ) +Xm2 j=1 n k(α+2βj X ) κ=1 P(K s jκ) +Xm3 j=1 Xϖj κ=1 P(K m jκ) ≥ XP(one component failed) − XP(two components in a same state failed) − XP(two components in different states failed). (7-5) The first and second terms can be characterized similar to previous analysis. The difference lies in the third term, which however can still be bounded by the property that clusters in different states are independent. For simplicity of presentation, we let x = mP1 i=1 P(Fi), y = mP2 j=1 n k(α+2βj ) P κ=1 P(Sjκ), z = mP3 j=1 Pϖj κ=1 P(fjκ). Then we have Xm1 i=1 P(K c i ) ≥ x − x 2 − xy − xz, (7-6) where x−x 2 is the first and second terms in Eqn. (7-5) whose proof is similar to (9-4), and xy + xz is the probability in the third term of Eqn. (7-5). Similarly to Eqn. (6-5), Xm2 j=1 n k(α+2βj X ) κ=1 P(K s jκ) ≥y − y 2 − Xm2 j=1 X κ̸=κ′ P(Sjκ ∩ Sjκ′ ) − yx − yz. (7-7) Similar to Eqn. (5-7), Xm3 j=1 Xϖj κ=1 P(K m jκ) ≥z − z 2 − Xm3 j=1 Xϖj κ=1 X κ′̸=κ P(fjκ ∩ fjκ′ ) − zx − zy. (7-8)