This article has been accepted for inclusion in a future issue of this journal.Content is final as presented,with the exception of pagination HUANG AND WANG:CAPACITY SCALING OF GENERAL COGNITIVE NETWORKS 5 O Proof:Let E and F be two arbitrary points on line segment ZiZi and ZiZi,by triangle inequality Zi-E+E-F+F-Z+Zk-F+F-E+E-Zil ≥|Z-Zl+lZ-Z (1+△)Rr ● (1+△)R Since Zi,E,Zi are collinear,substituting lemma condition IZ-Z+lZ-Z+2E-F≥lZ-Z团+Z-Zl Fig.1.Examples of the hybrid protocol model:Given an active primary link ≥(1+△21Z-Z+(1+△11Z-Z the left plot shows the guard zone regarding primary interferers,and the right plot for secondary interferers. therefore E-Fl≥（△1/2川Z:-Z+(△2/2川Zk-Z: Now we prove the lemma by contradiction.Suppose the two neighborhoods overlap,then there exist points P onZiZi and Q on Zi.ZI and Z such that |Z-P<(A1/2Zi-Zil and Z-Ql<(A2/2)Zk-Zil.Then,P-Q<IZ-P+ 1 Z-Ql<(△1/2lZ:-Zl+(△2/2)lZk-Z,which is a R2 contradiction. Corollary /Under the hybrid protocol model,we have the △2R2/2I following. 7 If (Xi,Xi)and (Xi Xi)are active primary links,the AplXi-Xil/2 neighborhood of line segment XiXi and △R1/21 ApXk:-Xi/2 neighborhood of XiXi are disjoint. If (Yi,Yi)and (Yi,Yi)are active secondary links,the Fig.2.Disjoint regions of two active transmissions. AsYi-Yil/2 neighborhood of line segment YiYi and As Y-Yil/2 neighborhood of Yiyi are disjoint. If (Xi Xi)is an active primary link and (Y,Yi)is an and∀Ya,y)∈Ss) active secondary link,the AspXi-Xil/2 neighborhood of line segment XiXi and Aps Y-Yi/2 neighborhood of YY are disjoint. Xk-Y引≥(1+△s)Y-Y k∈TP) (3) For active link (Xi,XRx()and (Yj,YRx()),where function Rx indicates the index of receiver,let Ri=X:-XRx(and where Asp,Aps define internetwork guard zones(Fig.1). rYj-Yx()Notice that Ri in general is a function of n The hybrid protocol model only depends on pairwise distance and ri is a function of m.We say the secondary network adopts between transmitters and receivers.Such simplicity will facili- power assignment scheme A(C)if for iE T(s),Pi=Cr?P. tate our analysis in the next section.Moreover,it is compatible The quadratic power assignment facilitates our effort to upper- with the classic protocol interference model.Thus,rich com- bound aggregate interference by converting it to an integral over munication schemes and results based on the protocol model the network area. can be easily extended to cognitive networks,as will be shown Theorem 1:Under power assignment A(C1)and the hy- in Section V. brid protocol model,if Aps As,then for any active primary In the following,we should prove that if is used as a deci- link(Xi,XRx()),the interference suffered by XRx()from the sion model for secondary nodes,it will comply with Operation secondary network is upper-bounded by CRP,for some Rule 2.This involves correctly tuning the parameters Ap,Ape, C2=Θ(C1) △p,△，and{P}jera Proof:Let B(X,r)be the disk centered at X with radius r.Then,all B(Yi,Asri/2),jET(s)should be mutually dis- joint according to Corollary 1.As well,B(Yi,Apsri/2),j B.Interference at Primary Nodes T(s)are disjoint with B(XRx(),ApRi/).Since ApA.. then all B(Y,△s/2),j∈Ta,B(XRx),△spR/2)are We first address the challenge that primary transmissions pairwise disjoint.Denote Dij=B(XRx(),XRx()-Yi)n should not be interrupted by secondary nodes.The main task is B(Yi;Asri/2).it is clear that all Dij are disjoint (see Fig.3). to bound the interference from the secondary network.We start Denote by E.F the two points where B(XRx(i),XRx(- with a useful property of the hybrid protocol model. Yil)intersects B(Yi,Asri/2).It is clear that FYjXRx(= Lemma 2:Given arbitrary Zi,Zi:Zk,ZIEO,if (Zi,Zi). ZEYjXR)≥T/3 because IXR⊙-YA>△s/2.Thus, (Zi,Z)are active links (primary or secondary),and Zk- the areaof Dij is at least one third of B(Yj,Asrj/2).Let Isp() Z1≥(1+△1Z-Z|Z-Z1≥(1+△2Zk-2 then the A1Zi-Zil/2 neighborhood of line segment ZiZ; 3More precisely,A is the asymptotic lens generated by the and the A2Z-Z/2 neighborhood of line segment ZiZi are h4lo码4"A intersection of two disks.Let disjoint (See Fig.2 for an example). 1)022)n(102)/（2=n1)(2=+1)0x.This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. HUANG AND WANG: CAPACITY SCALING OF GENERAL COGNITIVE NETWORKS 5 Fig. 1. Examples of the hybrid protocol model: Given an active primary link, the left plot shows the guard zone regarding primary interferers, and the right plot for secondary interferers. Fig. 2. Disjoint regions of two active transmissions. and (3) where define internetwork guard zones (Fig. 1). The hybrid protocol model only depends on pairwise distance between transmitters and receivers. Such simplicity will facili￾tate our analysis in the next section. Moreover, it is compatible with the classic protocol interference model. Thus, rich com￾munication schemes and results based on the protocol model can be easily extended to cognitive networks, as will be shown in Section V. In the following, we should prove that if is used as a deci￾sion model for secondary nodes, it will comply with Operation Rule 2. This involves correctly tuning the parameters , , , , and . B. Interference at Primary Nodes We first address the challenge that primary transmissions should not be interrupted by secondary nodes. The main task is to bound the interference from the secondary network. We start with a useful property of the hybrid protocol model. Lemma 2: Given arbitrary , if , are active links (primary or secondary), and , , then the neighborhood of line segment and the neighborhood of line segment are disjoint (See Fig. 2 for an example). Proof: Let and be two arbitrary points on line segment and , by triangle inequality Since are collinear, substituting lemma condition therefore . Now we prove the lemma by contradiction. Suppose the two neighborhoods overlap, then there exist points on and on and such that and . Then, , which is a contradiction. Corollary 1: Under the hybrid protocol model, we have the following. • If and are active primary links, the neighborhood of line segment and neighborhood of are disjoint. • If and are active secondary links, the neighborhood of line segment and neighborhood of are disjoint. • If is an active primary link and is an active secondary link, the neighborhood of line segment and neighborhood of are disjoint. For active link and , where function indicates the index of receiver, let and . Notice that in general is a function of and is a function of . We say the secondary network adopts power assignment scheme if for . The quadratic power assignment facilitates our effort to upper￾bound aggregate interference by converting it to an integral over the network area. Theorem 1: Under power assignment and the hy￾brid protocol model, if , then for any active primary link , the interference suffered by from the secondary network is upper-bounded by , for some . Proof: Let be the disk centered at with radius . Then, all should be mutually dis￾joint according to Corollary 1. As well, are disjoint with . Since , then all , are pairwise disjoint. Denote , it is clear that all are disjoint (see Fig. 3). Denote by , the two points where intersects . It is clear that because . Thus, the area of is at least one third3 of . Let 3More precisely, ￾ is the asymptotic lens generated by the intersection of two disks. Let ￾ ￾  ￾  , then ￾￾ ￾￾   ￾ ￾          .