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Gene Networks@ GenealogiesBiological/Social Networks7Individuals <-> Vertices, Nodes Relationships <-> Edges`GraphdA graph G=(V,E) consists of a set V={1,2,& n} of vertices (nodes, points, individuals) and a set E={(i,j), i,j  V} of edges (joins, lines, relationships). We deal here only with simple, undirected graphs i.e. there are no self-edges (i,i), no multiple edges, and edges have no direction i.e. (i,j)=(j,i). (3n xd|8_Reproducing Graphs_We investigate a class of models in which graphs reproduce, corresponding to a growing social/relationship network. We concentrate here initially on models without mortality, then make a few comments on age-specific-mortality. We have also studied models in which vertices are eliminated according to degree and/or payoff in a game against neighbours.aGraph ProductsZGiven two graphs G1=(V,E) and G2=(W,F) then a graph product G12 = G1 X G2 =(X,H) has X=V*W where * is the Cartesian product (i.e. X is the set of ordered pairs (v,w) where v  V and w  W). H depends on the particular product; there will be a rule which specifies which (v1,w1) and (v2,w2) join."u S  (X ( ( ( ( F  oCartesian Product V=(v1,v2,& ..,vm) and W=(w1,w2,& & ,wn) J={((vi,wj),(vk,wl))| definition) Cartesian product [vi=vk & (wj,wl)  F] OR [(vi,vk)  E & wj=wl]  ( ( ( (   (   (  (   (     Kronecker Product  V=(v1,v2,& ..,vm) and W=(w1,w2,& & ,wn) J={((vi,wj),(vk,wl))| definition) Tensor (Knonecker) product [(vi,vk)  E & (wj,wl)  F] Associative Adjacency matrix of GH is direct product of the adjacency matrices of G and H.   ( ( ( (   (   ( % <     !: fKronecker..Exampleb Our Models*In our model we suppose there exists some graph at time t and this leads to another at time t+1. Each vertex at time t survives and gives rise to a new offspring vertex. Each edge at t survives. Some subset of the possible edges between the  new and  old vertices are added. Zc Gt+1=Fi(Gt)H `Formally we have that Gt+1 is a function of Gt, where Gt  G the set of all simple graphs. The index i on Fi specifies which member of our family of models is being applied.z  !2BT,3B Our ModelsGiven a network Gt=(Vt,Et) we generate Gt+1=(Vt+1,Et+1) in the following way:- Each edge (u,v) is replaced by pP3P      m>AF  Our ModelsWe fix the presence or absence of the edges labelled ,  and (indicated by 0 and 1). \5' B5"  Our Models> Our Models eMerger of Two Graphs   dFundamental Theorem ?Theoremj Thus we may investigate Fit(Z) where Z=(V={0,1},E={(0,1)} is the single edge graph. Then apply to any G0`i9 LO Our Models g Our Models0,1,4 & 5 are Kronecker products 6 is Cartesian 2 is Comb 7 is Strong 3 is Non-standard }! Model 1&Model 5i,The  Knonecker Models  Since the adjacency matrix of the Knonecker product of two graphs is the Knonecker product of the adjacency matrices we can exploit this for models 0,1,4 & 5,"  Lj,The  Knonecker Models  The adjacency matrices for the four models 0,1,4,5 are respectively. Only that for model 1 is interesting which is essentially the bitwise AND. "Model 3%Model 6h#  j%  l'  n)  PModel 2;  =  ?  A  '6N.B& ..Binary RepresentationIf at time t we have a set of nodes {v1,v2,& .,vn} where each vi is a binary string then at time t+1 we have set of nodes {v11,v10,v21,v20,& & ..,vn1,vn0} where vi becomes vi1 and is the parent of a new node vi0. All our models can be specified in terms of the nodes as binary strings and logical operations defining the edges. BGZZ&    ;          # y.BBinary Representation!Model 1. Kronecker product of  h Properties1. Chromatic Number 2. Number of Vertices and Edges 3. Distance Structure 4. Degree Distribution 5. Automorphism 6. Generating all graphs as subgraphs.( BP  &lAModel 7  Number of Vertices & Edges@No of nodes Vt doubles No of edges Et so n)PP        3kCThe Distance StructureDenote distance (shortest path) between vertices u and v by d(u,v),the diameter (max distance) by D(G), and the number of pairs of vertices with distance x as Nt(x). We demonstrate our methods wrt Model 2. If u in Vt then u0 and u1 are the resulting offspring and parent vertices.Z72P= YADThe Distance Structure Model 2 u & v in Vt and d(u,v)=d then d(u0,u1)=1, d(u0,v0)=d+2, d(u0,v1)=d(u1,v0)=d+1, d(u1,v1)=d. We can then derive Nt+1(0)=2Nt(0); Nt+1(1)=Nt(0)+Nt(1); Nt+1(2)=2Nt(1)+Nt(2) & Nt+1(k)=Nt(k-2)+2Nt(k-1)+Nt(k) for k>2  !! ,EThe Distance Structure Model 2We have also number of distances Lt =4t(V0)2+2tV0 and the total distance Lt*=4tL0*+22t-1(N0(0))2-(22t-1-2t-1)N0(0) From this we derive an expression for the average distance dt=Lt*/Lt->c+t for large t which we can also obtain more directly by considering a random pair of vertices at time t+1 in terms of those at t. In fact we can also show that asymptotically the variance of the average distance is f(G0)+3t/2.#                 @      , DDegree Distribution *,1 & 5~Adjacency matrix of G H is Kronecker product of the adjacency matrices of G and H. i.e. here so degrees are direct products of giving 2t 1 s, tCi 2i s and 2t 2t  s respectively.Xh> <T /CXDegree Distribution Model 6ZStarting from a single edge we have just a hypercube at time t so all nodes of degree 2t $[WY*Degree  Dist Model 3 A node u of degree d gives rise to an offspring u0 of degree (x+1) and a survivor u1 of degree (2x+1). Thus (1) >(2,3)->(3,4,5,7)->(4,5,6,7,8,9,11,15)-> (5,6,7,8,9,9,10,11,12,13,15,16,17,19,23,31) totals 1,5,19,65,211,& & ,3n-2n,& .d3"X  E Age CullingSuppose now that we associate with each node an index which we regard as its age. If at time t node u in V is age s then at time t+1 u1 is of age s+1 and u0 is of age 0. We suppose that after reproduction a node of age Q+1 dies (i.e. is deleted from the graph) BlYModel 6 Age CulledZ  p*  q+  r,  s-  t.  u/  v0  w1  x2  y3  z4  {5  |6  }7  ~8  FModel 2This is the easiest case to treat since the graph grows trees on each original individual. Here we can start with a single node GModel 2Now when we cull at any given age Q+1 we obtain copies of all the trees  of ages 0,1,2,3,& & ,Q If there are nit trees of age i at time t then we get ni,t+1=ni-1,t + nQ,t i=1,2,& Q so we have nt+1= Ant where nt is the column vector of the ni,t s. m'>(B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  H>Tree  Dist Model 2 age-cap=3  0The matrix A is similar to a Leslie matrix L with guaranteed survival except that ai,j=lQ-i,Q-2i+j e.g. {rate  max root of 4= 3 + 2 +  +1, (4)} T   ( ( (   S ] Leslie MatrixIDistribution of age(i,j) edges*It is convenient to introduce a direction `!Distribution of age(i,j) edges a"Distribution of age(i,j) edgeset+1=Let where H\Distribution of age(i,j) edgesWe can prove that the system is irreducible (though if =0 need to refine the state space to exclude (i,i) states). Essentially a generalisation of the Lesley matrix notion. Eigenvalues related to generalisations of the  golden ratio (tribonacci, etc.) 7 b7.E 3 <H ReferencesSouthwell & Cannings, Some models of graph reproduction; 1 Pure Reproduction. (to appear) AM. 2 Age Capped Vertices (to appear) AM 3 Game Based Reproduction (submitted) AM Jordan & Southwell. Further properties of reproducing graphs (submitted) AM AM http://www.scirp.org/journal/am/x ZF$'7 !H  G  IApplied Mathematics Editor in Chief & .CC Editorial Board includes Mark Broom David Greenhalgh http://www.scirp.org/journal/am/ Z, ' ^ Other ResultsSFor age-culled, degree distribution diameter average distanceDTQ[ Further Workv1) Culling by degree 2) Stochastic/non-synchronous 3) Conflict with neighbours to determine survivor &/or reproductionJKLMN+q,r-s.t/u0v1w2x3y4z5{6|7}8~9:;/   + , -./0123@ACJKLMNOQbc e"f#%')+,-./0123456789:>@BDghijklmnopqrstuvw{F|G}P~   0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> f(    6  `}  T Click to edit Master title style! !  0  `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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E f Tp s *A ? ?    Vp <   4H p 0޽h ? 33___PPT10i.qqЖL+D=' = @B +$  0 ##Et$ -#(  tx t c $6 `}   x t c $ `  8 ] yt]  t B\]D Bu1$  t B< ]D Bu0$@   xt `B t 0D>  t B&  Bv1$  t B* Bv0$`B  t 0D>`B t 0D>^2 t 6^2 t 6^2 t 65^2 t 658   {tp `B Ft 0D>m  `B Gt 0D>V V `B Ht 0D>  `B It 0D>   Jt B02   Bu1$ Kt Bd7   Bv1$ Lt B;   Bu0$ Mt BdA x Bv0$f2 Pt 6I w f2 Qt 6I w f2 Rt 6  f2 St 6  8 3 c ) |tp  `B Vt 0D>  `B Yt 0D>" "  Zt BG3 c ,J  Bu1$ [t BLS D Bv1$ \t B\Rc J  Bu0$ ]t BW8 )  Bv0$`B ^t 0D>  `B _t 0D> " f2 `t 6  f2 at 6 8 f2 bt 6 8; f2 ct 6 ; X8 3 ]) zt]3 )`B ft 0D>  `B gt 0D>`B ht 0D>** jt B^3 ],D Bu1$ kt BddS D Bv1$ lt Bi]D Bu0$ mt Bn8) Bv0$`B nt 0D> `B ot 0D> "^2 pt 6 ^2 qt 68^2 rt 685^2 st 6 5u tt 0u-t 1 ut <ylt8S 3u vt 0,} i  6 wt <F S-  58 F  t  `B ~t 0D> } } `B t 0D>  `B t 0D>   t B F -  Bu1$ t B$   Bv1$ t B  F -  Bu0$ t BD   Bv0$ t <(  2H t 0޽h ? 33___PPT10i.qqЖL+D=' = @B +~  0 $, (  r  S , `}   r  S  `  XB  0D>\ \ t XB  0D> 5XB  0D> 5 XB  0D>3bbt   BH? 8  Bu1$  BԨVO Bu0$XB  0D>\ 5t XB  0D>\ bt XB  0D> ] ] =XB  0D> 6 XB  0D>j 6jXB  0D> cc=  BIn _ 0 Bv1$  Bu Bv0$XB  0D> ] 6=XB  0D> ] c=  <غ :  D w1$  <   Bw0$RB  s *D>   RB  s *D>  IX2  09X2  0 9 X2  0# } X2 ! 0)X2 " 0 # } X2 # 0 ) X2 $ 09wX2 % 0# } wX2 & 0)w ) <2    w * <L ~ v + <   uXB , 0D) Z H  0޽h ? 33___PPT10i.zpL+D=' = @B +   0    .@$ (  r  S  `}     TA ?TI  8 7I 7IZB  s *D7F XIZB  s *DXIIZB  s *D7 F ZB   s *D J xF  % 5v  , TB  c $D vN  % 6v   % 5vTB  c $D s vTB  c $D % s TB  c $D s  TB  c $D 6vxF  % 5v  C TB  c $D vN  % 6v    % 5vTB ! c $D s vTB " c $D % s TB # c $D s  TB $ c $D 6vF 7I % C TB & c $D7F XITB ' c $DXIITB ( c $D7 F TB ) c $D J LB *@ c $DF 7 LB +@ c $D Z LB , c $D,S LB - c $D CS  . <J TG J HUUH  0޽h ? 33___PPT10i.R+D=' = @B +;  0 RJ(  r  S  `}     <`= (  TA ?'2  H  0޽h ? 33___PPT10i.h+D=' = @B +  0  V(  r  S  `}     s *` S"  `  H  0޽h ? 33___PPT10i.zP a+D=' = @B +# ~ 0 ""EE7"(  x  c $$ `}   x  c $ `  F ]  ]  <]D Bu1$  < ]D Bu0$N     ZB  s *D>   <  Bv1$   < Bv0$ZB   s *D>ZB   s *D>X2   0X2  0X2  05X2  05nF    p ZB  s *D>m  ZB  s *D>V V ZB  s *D>  ZB  s *D>    <   Bu1$  <    Bv1$  <t%   Bu0$  <x* x Bv0$`2  0I w `2  0I w `2  0  `2  0  nF 3 c )  p  ZB  s *D>  ZB   s *D>" "  ! </3 c ,J  Bu1$ " <85S D Bv1$ # <t:c J  Bu0$ $ <?8 )  Bv0$ZB % s *D>  ZB & s *D> " `2 ' 0  `2 ( 0 8 `2 ) 0 8; `2 * 0 ; 0F 3 ]) + ]3 )ZB , s *D>  ZB - s *D>ZB . s *D>** / <tG3 ],D Bu1$ 0 <LS D Bv1$ 1 <Q]D Bu0$ 2 <\V8) Bv0$ZB 3 s *D> ZB 4 s *D> "X2 5 0 X2 6 08X2 7 085X2 8 0 5u 9 0(_-t 1 : <blt8S 3u ; 0f i  6 < <DjF S-  5F F  =   ZB > s *D>} } ZB ? s *D> ZB @ s *D>  A <HmF -  Bu1$ B <xr  Bv1$ C <w F -  Bu0$ D <p}  Bv0$ E <  2H  0޽h ? 33___PPT10i.qqЖL+D=' = @B +$  0 PP$(  r  S  `}   r P S \ `  H  0޽h ? 3380___PPT10.v8@w   0   '| (  |x | c $ `}    |0 NA ? ?P   F ]  | "   | <]D Bu1$  | <]D Bu0$N    |  ZB | s *D> | <H  Bv1$ | < Bv0$ZB | s *D>ZB | s *D>X | 00 x | <ȶ   8 ,  &|, TB | c $D2F% `2 | 0,%`2  | 0R`2 !| 0%  t8 6   '|6   "| 0 R 9  where  #|2 NA ? ?6     %| <   Nis the Kronecker product  H | 0޽h ? 33___PPT10i.qqЖL+D=' = @B +  0 @ %6(  r  S , `}   nF 3 c )  PdZ ZB  s *D>  ZB  s *D>" "   <3 c ,J  Bu1$  <S D Bv1$   <`c J  Bu0$   <8 )  Bv0$ZB   s *D>  ZB  s *D> " `2   0  `2  0 8 `2  0 8; `2  0 ; x  <d?^& X  00 x  < M  x  <`~   0 TA ? ?   x  <d RB  s *D5^2  6Ps^2  6c X2  0]FX2  0C x  <b I  x ! <t  W  ^F 6   " 6   # 0t R 9  where ~ $ s *A ? ?6      % < % {  NIs the Kronecker product  H  0޽h ? 33___PPT10i.~q`#M+D=' = @B +$  0 p$(  r  S   `}   r  S    `   H  0޽h ? 3380___PPT10.EP  0 P(  r  S 4  `}   x  c $\      0 NA ? ?c       0 TA ? ?c # q    f  s *A ? ? )  f  s *A ? ?c   H  0޽h ? 3380___PPT10.P+6fe  0 |t (  x  c $  `}   x  c $  `   0F 3 ])  " ZB  s *D>  ZB  s *D>ZB  s *D>**  <% 3 ],D Bu1$   <* S D Bv1$   </ ]D Bu0$   <,5 8) Bv0$ZB   s *D> ZB  s *D> "  <4<  J  0Not a standard graph productH  0޽h ? 33___PPT10i.qqЖL+D=' = @B +   0   0f (  r  S B  `}    0 NA ? ?    nF     ZB  s *D>m  ZB  s *D>V V ZB  s *D>  ZB  s *D>     <E    Bu1$   <K    Bv1$   <P    Bu0$   <T  x Bv0$`2   0I w `2  0I w `2  0  `2  0  x  <\ l^S X  00 x  <` ?  & l8 }F }FTB  c $DSC`2  0}F`2  0  <Te  g  where  0 NA ? ? :PD      <i R  9  ,is the Cartesian productH  0޽h ? 33___PPT10i.~q@+D=' = @B + % 0   @Z (    0(p  8Model 6  2 X2  0,<LB  c $D=m X2  0m } 8 V G V G`B   0D>+ , V0@ V G V G`B  0D>V`B   0D> `B   0D>j j   Bv  Bu1$   6{ -`G Bv1$  B4 ] V  Bu0$  6 0 Q@ 8 Bv0$f2  6 f2  6 6 f2  6 S6 f2  6S  <p % Jv  1-cubeH  0޽h ? 33___PPT10u..:>+ , VN V G  V GZB  s *D>VZB  s *D> ZB  s *D>j j  <Ԑ  Bu1$  0 -`G Bv1$  <l ] V  Bu0$  0L 0 Q@ 8 Bv0$`2  0 `2  0 6 `2  0 S6 `2  0S  <@ 6 Model 6  < b I  2-cubeH  0޽h ? 33___PPT10u..:>+ , VN V G  V GZB  s *D>VZB  s *D> ZB  s *D>j j  <  Bu1$  0 -`G Bv1$   <X ] V  Bu0$ ! 0 0 Q@ 8 Bv0$`2 " 0 `2 # 0 6 `2 $ 0 S6 `2 % 0S & <  Model 6 ' <$  x 3-cubeH  0޽h ? 33___PPT10u..:>+ , VN V G 6 V GZB 7 s *D>VZB 8 s *D> ZB 9 s *D>j j : <  Bu1$ ; 0x -`G Bv1$ < <@ ] V  Bu0$ = 0 0 Q@ 8 Bv0$`2 > 0 `2 ? 0 6 `2 @ 0 S6 `2 A 0S B <h Model 6 C <xU< 4-cubeH  0޽h ? 33___PPT10u..:><XB 8 0D>tXB 8 0D>QtQ 8 B`  Bu1$  8 B9ZA(  Bv1$  8 B Bu0$  8 B ,  Bv0${  8 6i  5X 8 0j 2zF }F 8 pTB 8 c $DSC`2 8 0}F`2 8 0R 8 s *X99?"  8 <", |  5 8 <(%"# | A$G$ 8 <\*A  CG&G$ 8 <T0R 9  where 8 63 r  k 5 JG$O$G$  8 <: g  @,Is the  Comb product H 8 0޽h ? 33___PPT10i.Dz#+D=' = @B + > 0 hl(  hl h 0BMM 4 X2 h 0,< h <F}Zd 0LB h c $D=m f2 h 0"`m }  h <J}m 1 d 1F F 5  h  ZB  h s *D>} } ZB  h s *D> ZB  h s *D>   h <PF   Bu1$ h <S * ' Bv1$ h <Y F  Bu0$ h <t]  Bv0$ h <c 5 5 h <g,p Model 2H h 0޽h ? 33___PPT10u..:>} } ZB x s *D> ZB x s *D>  x <(F  [  Bu1$ x <\   Bv1$ x <ć F [  Bu0$ x <8  Bv0$ x <  5 x <Ȗ,p Model 2H x 0޽h ? 33___PPT10u..:>} } ZB  s *D> ZB  s *D>   <lF w   Bu1$   <  Bv1$ ! < F V   Bu0$ " <D a  Bv0$ # <   5 $ <,p Model 2H  0޽h ? 33___PPT10u..:>} } ZB 5 s *D> ZB 6 s *D>  7 <!F ;  Bu1$ 8 <& H > Bv1$ 9 <+ F   Bu0$ : </ ' ( Bv0$ ; <6 K 5 < <:,p Model 2H  0޽h ? 33___PPT10u..:> `}   r  S >6 `Y  H  0޽h ? 33___PPT10i.~q+D=' = @B +   0  @(  r  S  L `}   r  S L `  F ,   d TB  c $D2F% `2  0,%`2  0R`2  0%     <Oo fV  0   <T?& 1LB  @ c $DLB   c $Ds LB   c $Ds s LB  c $DLB  c $DF LB  c $DF wF   <ZLM3 n  <^^N  = ^  0_  4G[V={x in {0,1}n},E={(x,y) s.t. (xi,yi) n.e. (0,0)} V5   P H  0޽h ? 3380___PPT10./z@T@  0 <4`(  r  S Xn `}   x  c $0o     0 NA5 /? ? 2 / H  0޽h ? 3380___PPT10. /8 0  88LL7(  lB  D?Line 2k| lB  D?Line 31k| 1lB  D?Line 4 k| lB  D?Line 5 0A lB  D?Line 61 1lB  D?Line 7 lB  D?Line 8 P a lB   D?Line 9 nB   F?Line 10!k{ 1nB   F?Line 11!k{ 1nB   F?Line 12 P ` nB   F?Line 13nkk nB  F?Line 14{ { ~ nB  F?Line 15 0@ nB  F?Line 16 0@ nB  F?Line 17V 00 nB  F?Line 18 @@f nB  F?Line 19! 1nB  F?Line 20! 1nB  F?Line 21! !nB  F?Line 22n nB  F?Line 23 ~ nB  F?Line 24 nB  F?Line 25 P ` nB  F?Line 26 P ` nB  F?Line 27U P P nB  F?Line 28 ` ` e 2  X?Oval 29? T`  2  X?Oval 30O  T`  2  X?Oval 31y> T`  2   X$?Oval 32yN  T`  2 ! XT?Oval 33?^ T`  2 " XԷ?Oval 34O ^ T`  2 # XT?Oval 35>O T`  2 $ X?Oval 36N O T`  2 % XT?Oval 37~ ?  T`  2 & X?Oval 38~ O  T`  2 ' XT?Oval 39o> T`  2 ( X?Oval 40oN  T`  2 ) XT?Oval 41f _  T`  2 * X?Oval 42f o  T`  2 + X?Oval 43W ^  T`  2 , X(?Oval 44W n  T`  2 - X!?Oval 45  ^ T`  2 . X*?Oval 46 ^ T`  2 / X4?Oval 47  O T`  2 0 X=?Oval 48 O T`  2 1 XG?Oval 49~   T`  2 2 XP?Oval 50~   T`  2 3 XZ?Oval 51o   T`  2 4 Xc?Oval 52o  T`  2 5 Xm?Oval 53e $   T`  2 6 Xv?Oval 54e 4  T`  2 7 X?Oval 55V # ~  T`  2 8 X?Oval 56V 3  T`   9 N?Text Box 64[ c Invariant `   : : # 0e0e?Rectangle 73  lChromatic number`   nB ; F?Line 74y y2 < X?Oval 75[   T`  2 = X?Oval 76[  T`  2 > X8?Oval 77L   T`  2 ? X?Oval 78L  T`   @ Nd?Text Box 79L _0df    A Np?Text Box 80!L _1df    B N?Text Box 81L  _2df    C N?Text Box 82 Mj  _3df    D N?Text Box 83y _ ` _4df    E N?Text Box 84! _  _5df    F N?Text Box 85  _6df    G N?Text Box 86 n 2 j  _7df   2 H X0o?Oval 87<wE  T`  2 I X$#o?Oval 88   T`   J N-?Text Box 89   f Goes up by 1 `   2 K X6o?Oval 90s  T`    L N8?Text Box 91V   HGoes up but doesn t more than double%%` %  H  0޽h ? 33___PPT10i.ŞP+D=' = @B +G   0   HG (  Hr H S 4F `}   F ?  H T ZB H s *D>C ZB H s *D> ZB H s *D> ZB H s *D> p   H <4I?& Bu1$  H <hN   Bv1$  H <4S ? & Bu0$  H <X  w  Bv0$ZB  H s *D>C ZB H s *D>C  H0 NA6 0? ?%   0 XB H 0D>   H 0@^ R  rBSince model 7  contains model 6 ""XB H 0D> ` C8  H <c M S#Equality achieved by complete graph$$H H 0޽h ? 3380___PPT10.D  0  x;(  xx x c $i `}   ~ x s *j    x0 NA  ? ?2G    x0 NA ? ? 799     x <n 3 M9for Models 1,2,3,4,5. For model 3 second term is linear. H x 0޽h ? 33___PPT10i.qqЖL+D=' = @B +  0 (  r  S  `}   r  S  `  X  0A?ZH  0޽h ? 3380___PPT10.K*  0 `*(  `r ` S ą `}   x ` c $   H ` 0޽h ? 3380___PPT10.O(  0 ?7 h(  hr h S T `}   r h S , `  XB h 0D>nXB h 0D>]7^XB h 0D>e h BL/ Bu1$  h B Bv1$  h Bt/ Bu0$  h B4 Bv0$  h B 2H h 0޽h ? 3380___PPT10.$  0 l$(  lr l S  `}   r l S Ю `  H l 0޽h ? 3380___PPT10.PxF  0 ` (  r  S X `}   x  c $    0 NA l? ?2-S  l  0 NA ? ?      <2   F2 0 1 5f   s *A ?? ? m ?H  0޽h ? 33___PPT10i.1zf^+D=' = @B +  0  \(  \r \ S  `}   r \ S  `  nF  [ \ p ZB \ s *D>m  ZB \ s *D>V V ZB \ s *D>  ZB \ s *D>    \ <   Bu1$  \ <d  [ Bv1$  \ <$   Bu0$  \ <$ D Bv0$`2  \ 0I w `2 \ 0I w `2 \ 0  `2 \ 0  H \ 0޽h ? 33___PPT10i.~`㮛+D=' = @B +   0  `\(  `r ` S $ `}  $ r ` S $ ` $ 0F 3 ]) `   ZB ` s *D>  ZB ` s *D>ZB ` s *D>** ` <$3 ],D Bu1$  ` <$S D Bv1$  ` <\$]D Bu0$  ` <#$8) Bv0$ZB  ` s *D> ZB  ` s *D> "H ` 0޽h ? 33___PPT10i.+D=' = @B +_ 0 @ X_(  XX X 0A? rR X s *Az  X 00$   ; frequency 2  f X 0"`: X 04$k 8degree 2 f X 0"`pLB X c $D:  X 0P9$ Tg 0 Degree Distribution Model 3  X <=$e2 L C/(1) after 18 updates, nodes of degree < 5,000   X <DA$>  ?+(1) After 18 updates, nodes of degree <500H X 0޽h ? 3380___PPT10.CT0  0 0(  x  c $|D$ `}  $ x  c $$E$ ` $ H  0޽h ? 3380___PPT10.1zf^   0  P\(  \r \ S I$ `}  $ r \ S J$ ` $ F @  \  ZB \ s *D>+ , VN @  \ @ ZB \ s *D>VZB \ s *D> ZB  \ s *D>j j  \ <O$L Bu1$  \ 0XU$-` Bv1$  \ <Z$]  Bu0$  \ 0^$0 Q@  Bv0$`2 \ 0 `2 \ 0 6 `2 \ 0 S6 `2 \ 0S \ <d$ #  =)Grows as hypercube with pure reproductionH \ 0޽h ? 3380___PPT10.0 { 0   `` (  `l ` 0m$MM 4 X2 ` 0,<LB ` c $D=m X2 ` 0m }  ` 0p$  10 2 ` 0lt$~ a  11 2 ` 0x$q FModel 6 with age-cap=1 2F V G  ` V GZB  ` s *D>+ , VN V G  ` V GZB  ` s *D>VZB  ` s *D> ZB ` s *D>j j ` <x|$ Bu1$ ` 0$-`G Bv1$ ` <L$] V  Bu0$ ` 0,$0 Q@ 8 Bv0$`2 ` 0 `2 ` 0 6 `2 ` 0 S6 `2 ` 0SRB `@ s *D CRB `@ s *D3 C ` <($YM@ AgesH ` 0޽h ? 33___PPT10i.:>+ , VN V G   V GZB  s *D>VZB  s *D> ZB  s *D>j j  <\$ Bu1$  0d$-`G Bv1$  <$] V  Bu0$  0\$0 Q@ 8 Bv0$`2  0 `2  0 6 `2  0 S6 `2  0SRB @ s *D CRB @ s *D3 C  <$YM@ AgesH  0޽h ? 33___PPT10i.:>+ , VN V G  V GZB  s *D>VZB  s *D> ZB  s *D>j j  < $ Bu1$  0P$-`G Bv1$  <$] V  Bu0$  0$0 Q@ 8 Bv0$`2  0 `2  0 6 `2  0 S6 `2  0SH  0޽h ? 33___PPT10i.:>+ , VN @ G  @ GZB  s *D>VZB  s *D> ZB  s *D>j j  <' Bu1$  0L '-`G Bv1$  <'] "  Bu0$  0'0 Q@ 8 Bv0$`2  0 `2  0 6 `2  0 S6 `2  0SH  0޽h ? 33___PPT10i.:>+ , VN @ G  @ GZB  s *D>VZB  s *D> ZB  s *D>j j  <\5' Bu1$  0:'-`G Bv1$  <?'] (  Bu0$  0C'0 Q@ 8 Bv0$`2  0 `2  0 6 `2  0 S6 `2  0SH  0޽h ? 33___PPT10i.:>+ , VN @ G  @ GZB  s *D>VZB  s *D> ZB  s *D>j j  <_' Bu1$  0e'-`G Bv1$  <i']   Bu0$  0xn'0 Q@ 8 Bv0$`2  0 `2  0 6 `2  0 S6 `2  0SH  0޽h ? 33___PPT10i.:>+ , VN @ G  @ GZB  s *D>VZB  s *D> ZB  s *D>j j  <<' Bu1$  0'-`G Bv1$   <Դ']  Bu0$ ! 0'0 Q@ 8 Bv0$`2 " 0 `2 # 0 6 `2 $ 0 S6 `2 % 0SH  0޽h ? 33___PPT10i.:>+ , VN @ G  @ GZB  s *D>VZB  s *D> ZB  s *D>j j  <' Bu1$  0,'-`G Bv1$   <h'] "  Bu0$ ! 0('0 Q@ 8 Bv0$`2 " 0 `2 # 0 6 `2 $ 0 S6 `2 % 0SH  0޽h ? 33___PPT10i.:>+ , VN @ G  @ GZB  s *D>VZB  s *D> ZB  s *D>j j  <4( Bu1$  0d(-`G Bv1$   <"(] *  Bu0$ ! 0'(0 Q@ 8 Bv0$`2 " 0 `2 # 0 6 `2 $ 0 S6 `2 % 0SH  0޽h ? 33___PPT10i.:>(Cz *  10 2   0B(C` p*  10 2f2   0"`-="f2  0"` f2  0"` M ] f2  0"`0 @ f2  0"`   0F(' AModel 6 age-cap=1 2F @ G  : GZB  s *D>+ , VN @ G  @ GZB  s *D>VZB  s *D> ZB  s *D>j j  <L( Bu1$  0Q(-`G Bv1$  <$V(] "  Bu0$  0[(0 Q@ 8 Bv0$`2  0 `2  0 6 `2   0 S6 `2 ! 0SH  0޽h ? 33___PPT10i.:>+ , VN @ G   @ GZB   s *D>VZB   s *D> ZB   s *D>j j   <e( Bu1$   0(-`G Bv1$   <(] "  Bu0$   0(0 Q@ 8 Bv0$`2   0 `2   0 6 `2   0 S6 `2 !  0SH   0޽h ? 33___PPT10i.:>+ , VN @ G *( @ GZB +( s *D>VZB ,( s *D> ZB -( s *D>j j .( <( Bu1$ /( 0(-`G Bv1$ 0( <D(]  Bu0$ 1( 0$(0 Q@ 8 Bv0$`2 2( 0 `2 3( 0 6 `2 4( 0 S6 `2 5( 0SH ( 0޽h ? 33___PPT10i.:>+ , VN @ G *0 @ GZB +0 s *D>VZB ,0 s *D> ZB -0 s *D>j j .0 <* Bu1$ /0 0*-`G Bv1$ 00 <*]   Bu0$ 10 0D$*0 Q@ 8 Bv0$`2 20 0 `2 30 0 6 `2 40 0 S6 `2 50 0SH 0 0޽h ? 33___PPT10i.:>+ , VN @ G *8 @ GZB +8 s *D>VZB ,8 s *D> ZB -8 s *D>j j .8 <0L* Bu1$ /8 0c*-`G Bv1$ 08 <g*]  Bu0$ 18 0m*0 Q@ 8 Bv0$`2 28 0 `2 38 0 6 `2 48 0 S6 `2 58 0SH 8 0޽h ? 33___PPT10i.:>+ , VN @ G !@ @ GZB "@ s *D>VZB #@ s *D> ZB $@ s *D>j j %@ <P* Bu1$ &@ 0X*-`G Bv1$ '@ <*] (  Bu0$ (@ 0Ĭ*0 Q@ 8 Bv0$`2 )@ 0 `2 *@ 0 6 `2 +@ 0 S6 `2 ,@ 0SH @ 0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB   s *D>    <5[F w   Bu1$   <*  Bv1$   <* F V   Bu0$   <D* a  Bv0$  <*   5  <*V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB   s *D>    <*F w   Bu1$   <*  Bv1$   <* F V   Bu0$   <D+ a  Bv0$  <,+   5  < +V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB   s *D> ZB   s *D>    <+F w   Bu1$  <+  Bv1$  <!+ F V   Bu0$  < '+ a  Bv0$  <-+   5  </+V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB   s *D> ZB   s *D>    <<@+F w   Bu1$  <DE+  Bv1$  <$J+ F V   Bu0$  <P+ a  Bv0$  <T+   5  <TX+V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB   s *D> ZB   s *D>    <|g+F w   Bu1$  <W[  Bv1$  <o+ F V   Bu0$  <hu+ a  Bv0$  <|+   5  <Y[V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB  s *D>   < +F w   Bu1$  <[[  Bv1$  <+ F V   Bu0$  <+ a  Bv0$  <`+   5  < +V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB  s *D>   <D+F w   Bu1$  <+  Bv1$  <+ F V   Bu0$  <+ a  Bv0$  <x+   5  <p`[V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB  s *D>   <@+F w   Bu1$  <p+  Bv1$  <$- F V   Bu0$  <- a  Bv0$  < -   5  <`[V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB  s *D>   <5-F w   Bu1$   <4:-  Bv1$ ! <>- F V   Bu0$ " <4D- a  Bv0$ # <I-   5 $ <M-V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB  s *D>   <v-F w   Bu1$   <4{-  Bv1$ ! <0- F V   Bu0$ " <- a  Bv0$ # <-   5 $ <X-V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB  s *D>   <-F w   Bu1$   <D-  Bv1$ ! <- F V   Bu0$ " <D- a  Bv0$ # <-   5 $ <-V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB  s *D>   <h-F w   Bu1$  <-  Bv1$  <D. F V   Bu0$  <. a  Bv0$  < .   5  <p.V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB  s *D> ZB  s *D>   <1.F w   Bu1$  <7.  Bv1$  <;. F V   Bu0$  <A. a  Bv0$  <F.   5  <J.V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB - s *D> ZB . s *D>  / <Ȋ.F w   Bu1$ 0 <X.  Bv1$ 1 <. F V   Bu0$ 2 <4. a  Bv0$ 3 <.   5 4 <.V= %Model 2 age-cap=2H  0޽h ? 33___PPT10i.:>} } ZB -  s *D> ZB .  s *D>  /  <.F w   Bu1$ 0  <4.  Bv1$ 1  <. F V   Bu0$ 2  <. a  Bv0$ 3  <.   5 4  <.V= %Model 2 age-cap=2H   0޽h ? 33___PPT10i.:>0V   0 F F a  +( - ;ZB ,( s *D>} } ZB -( s *D> ZB .( s *D>  /( <A0F w   Bu1$ 0( <DF0  Bv1$ 1( <K0 F V   Bu0$ 2( < P0 a  Bv0$ 3( <W0   5 4( <Z0sZ %Model 2 age-cap=2H ( 0޽h ? 33___PPT10i.:>} } ZB (0 s *D> ZB )0 s *D>  *0 <0F w   Bu1$ +0 <,0  Bv1$ ,0 <0 F V   Bu0$ -0 <Ȣ0 a  Bv0$ .0 <0   5 /0 <$0V= %Model 2 age-cap=2H 0 0޽h ? 33___PPT10i.:>6 XB 0 0D>   XB 0 0D>   XB 0 0D>c   0 Bt02  i+1  0 B0 .  j+1  0 B02 0  0 B|0 j  0XB  0 0D>6  XB 0 0D>6  XB 0 0D> XB 0 0D>XB 0 0D>S C6 XB 0@ 0D>%  XB 0@ 0D>  XB 0@ 0D> e RB 0 s *D>**  0 <0?F& i 0 <40b FI  jXB 0 0D>S ** H 0 0޽h ? 33___PPT10i.1zf^+D=' = @B +% " 0 <4l(  lr l S 0 `}  0 x l c $0   0  l0 NA ? ?'  0H l 0޽h ? 33___PPT10i.O+D=' = @B +/ # 0 F>p(  pr p S x0 `}  0 x p c $<0   0  p0 NA ? ?] C  0 p0 NA ? ?6 dw   0` p c $A ?? `   H p 0޽h ? 33___PPT10i.P+D=' = @B +$  0 $(  r  S  0 `}  0 r  S 0 ` 0 H  0޽h ? 3380___PPT10.o[ 0  ll1g(  0lB 0 D?Line 2  | @ `  0S .Group 3" )tB 0 D?Line 4  2 0 PD8?Oval 5 `  T`  2 0 PL 8?Oval 6 `  T`   0 `8o?Rectangle 7@ `  T`   |   0S .Group 8"P  0 `0o?Rectangle 9  T`  vB  0 F?Line 10 * 2  0 R'8?Oval 11S H  T`  2  0 R)8?Oval 12S X  T`  |B  0 L?Line 13F ) |B 0B L?Line 14 vB 0 F?Line 15 * 2 0 R<98?Oval 16S H  T`  2 0 RA8?Oval 17S X  T`  2 0 XxK833?Oval 18c G  T`  2 0 XU833?Oval 19qW T`  vB 0 F?Line 20F ) vB 0B F?Line 21 vB 0 F?Line 22 * 2 0 R$_8?Oval 23S H  T`  2 0 Rh8?Oval 24S X  T`  2 0 R$r8?Oval 25c G  T`  2 0 R{8?Oval 26qW T`  ~ jV 2 i 0S 0Group 27" QvB 0 F?Line 28 v? vB 0 F?Line 29\ O vB 0 F?Line 30/ " uvB 0 F?Line 31 v? *vB  0 F?Line 32  vB !0 F?Line 33/ 8 uvB "0 F?Line 34\ v? vB #0B F?Line 35/  vvB $0 F?Line 36/ v@ v2 %0 R8?Oval 37 H^  T`  2 &0 R8?Oval 38 Hn  T`  2 '0 R8?Oval 39 X]  T`  2 (0 R8?Oval 40 f m  T`  2 )0 X8?Oval 41 ?  T`  2 *0 X8?Oval 42L + T`  2 +0 X8?Oval 43\ 8  T`  2 ,0 X8?Oval 44" }  T`   -0 b8o?Rectangle 45jV 2 i T`  2 .0 X8?Oval 46" }  T`  ~ 7`  /0S 0Group 47"h`2 00 Xh8?Oval 48W T`  2 10 X8?Oval 49 g T`  2 20 Xh8?Oval 50   T`  2 30 X9?Oval 515 :  T`  2 40 X9?Oval 525 W  T`  2 50 X9?Oval 53tS   T`  2 60 X 9?Oval 54S   T`  2 70 X|*9?Oval 55tC T`  2 80 X39?Oval 56t F  T`  2 90 X|=9?Oval 57: s  T`  2 :0 XF9?Oval 58w   T`  2 ;0 X|P9?Oval 59C T`  vB <0B F?Line 60* vB =0B F?Line 61  vB >0 F?Line 62*vB ?0B F?Line 63W:vB @0B F?Line 64:vB A0 F?Line 65: vB B0 F?Line 66  vB C0B F?Line 67  c vB D0 F?Line 68 *c vB E0B F?Line 69*c c vB F0B F?Line 70c * vB G0 F?Line 71 c gF 2 H0 X]9?Oval 72tB T`  2 I0 Xg9?Oval 73W T`  2 J0 Xp9?Oval 74 g T`  2 K0 Xz9?Oval 75B T`  2 L0 X9?Oval 76   T`  2 M0 X9?Oval 77w   T`  2 N0 X9?Oval 785 :  T`  2 O0 X 9?Oval 79: s  T`  2 P0 X$9?Oval 805 W  T`  2 Q0 X9?Oval 81t E  T`  2 R0 X$9?Oval 82tR   T`  2 S0 X9?Oval 83R   T`   T0 b$9o?Rectangle 847`  T`  zB U0 R>?Line 85V V0 N9?Text Box 86, a1f    zB W0 R>?Line 87  X0 NX9?Text Box 88m  a1f    zB Y0 R>?Line 89F Z0 N9?Text Box 90/ a1f    nB [0@ F?Line 91 m nB \0 F?Line 92E m nB ]0@ F?Line 93E  nB ^0 F?Line 94 p nB _0@ F?Line 95mHnB `0 F?Line 963vnB a0 F?Line 97 mE nB b0@ F?Line 98  nB c0 F?Line 99HmpB d0 H?Line 100v2`pB e0 H?Line 101 p% pB f0 H?Line 102  pB g0 H?Line 103  pB h0@ H?Line 104 E pB i0 H?Line 105  pB j0 H?Line 106 ? pB k0 H?Line 107  pB l0 H?Line 108s  pB m0@ H?Line 109  pB n0 H?Line 110 pB o0@ H?Line 111 mpB p0@ H?Line 112pB q0 H?Line 113`pB r0@ H?Line 114 `pB s0 H?Line 115 %pB t0 H?Line 116 % pB u0 H?Line 117 ]mpB v0 H?Line 118pB w0 H?Line 119fpB x0 H?Line 120  pB y0@ H?Line 121 m pB z0@ H?Line 122 " pB {0 H?Line 123  pB |0@ H?Line 124 pB }0 H?Line 125 mpB ~0@ H?Line 1262 0 Z;?Oval 127 "(  T`  2 0 Z9?Oval 128 2(  T`  2 0 Z;?Oval 129 `  T`  2 0 Z;?Oval 130e T`  2 0 Z(;?Oval 131e" T`  2 0 Z1;?Oval 132 ?  T`  2 0 Z;;?Oval 133 \  T`  2 0 ZD;?Oval 134 ?s  T`  2 0 ZN;?Oval 135`v T`  2 0 ZW;?Oval 136 B  T`  2 0 Za;?Oval 137 `s  T`  2 0 Zj;?Oval 138   T`  2 0 Z t;?Oval 139E   T`  2 0 Z};?Oval 140 @  T`  2 0 ZH;?Oval 141   T`  2 0 ZȐ;?Oval 142v@ T`  2 0 ZH;?Oval 1432+ T`  2 0 Zȣ;?Oval 144 S  T`  2 0 ZH;?Oval 145   T`  2 0 Zȶ;?Oval 146mI T`  2 0 ZH;?Oval 147f/ T`  2 0 Z;?Oval 148+ T`  2 0 ZH;?Oval 149vC T`  2 0 Z;?Oval 150 %(  T`   7`  0S 2Group 151"h`2 0 Z;?Oval 152W T`  2 0 Z;?Oval 153 g T`  2 0 Z;?Oval 154   T`  2 0 Z<?Oval 1555 :  T`  2 0 Z<?Oval 1565 W  T`  2 0 Z<?Oval 157tS   T`  2 0 ZX <?Oval 158S   T`  2 0 Z)<?Oval 159tC T`  2 0 ZX3<?Oval 160t F  T`  2 0 Z<<?Oval 161: s  T`  2 0 Z@ <?Oval 162w   T`  2 0 ZXO<?Oval 163C T`  xB 0B H?Line 164* xB 0B H?Line 165  xB 0 H?Line 166*xB 0B H?Line 167W:xB 0B H?Line 168:xB 0 H?Line 169: xB 0 H?Line 170  xB 0B H?Line 171  c xB 0 H?Line 172 *c xB 0B H?Line 173*c c xB 0B H?Line 174c * xB 0 H?Line 175 c gF 2 0 Z\\<?Oval 176tB T`  2 0 Z(e<?Oval 177W T`  2 0 Z0n<?Oval 178 g T`  2 0 Zw<?Oval 179B T`  2 0 Z<?Oval 180   T`  2 0 Z<?Oval 181w   T`  2 0 Z<?Oval 1825 :  T`  2 0 Z<?Oval 183: s  T`  2 0 Z<?Oval 1845 W  T`  2 0 Z<?Oval 185t E  T`  2 0 Z<?Oval 186tR   T`  2 0 Z<?Oval 187R   T`   0 d<o?Rectangle 1887`  T`   0 d<o?Rectangle 189 c T`  |B 0 T>?Line 19088  0 P4<?Text Box 191pe  a1f    pB 0 H?Line 192  r pB 0@ H?Line 193 0  pB 0 H?Line 194r  ( pB 0 H?Line 195 3  pB 0@ H?Line 196 upB 0 H?Line 197~ pB 0@ H?Line 198 z 0 pB 0 H?Line 199 z pB 0 H?Line 200u  pB 0 H?Line 201  +pB 0 H?Line 202  +pB 0@ H?Line 203  3 +pB 0 H?Line 2048~ pB 0 H?Line 205( 3  pB 0 H?Line 206( z z pB 0 H?Line 207C 0 0 pB 0 H?Line 208 @ pB 0 H?Line 2093  a pB 0 H?Line 210  pB 0 H?Line 211a 3 pB 0 H?Line 212 CpB 0 H?Line 213 pB 0@ H?Line 214r 3 ( pB 0 H?Line 215 3 pB 0 H?Line 216 ( pB 0@ H?Line 217 3 r pB 0 H?Line 218 ~ ( pB 0 H?Line 219 pB 0 H?Line 220 ( pB 0 H?Line 221 ] ( pB 0@ H?Line 222( M pB 0@ H?Line 223  pB 0@ H?Line 224 pB 0 H?Line 225  pB 0 H?Line 226   pB 0 H?Line 227m +pB 0 H?Line 228~ pB 0 H?Line 2290 pB 0 H?Line 230  @ 6pB 0@ H?Line 231d" pB 0@ H?Line 232G}  G2 0 Z<?Oval 233 m U  T`  2 0 Z=?Oval 234 } U  T`  2 0 Z<?Oval 235  a 8 T`  2 0 Z=?Oval 236P  T`  2 0 Z=?Oval 237m  T`  2 0 Z(=?Oval 238 @  T`  2 0 Z1=?Oval 239  ]  T`  2 0 Zl;=?Oval 240E @  T`  2 0 ZE=?Oval 241H   T`  2 0 Z@N=?Oval 242 C8 T`  2 0 ZW=?Oval 243E  a  T`  2 0 Z@a=?Oval 244 P   T`  2 0 Zj=?Oval 245r 0  T`  2 0 Z@t=?Oval 246   T`  2 0 Z}=?Oval 247 M  T`  2 0 Z@=?Oval 248  T`  2 0 Z=?Oval 249 3 X T`  2 0 Z@=?Oval 250 8 T`  2 0 Z=?Oval 251 `   T`  2 0 Z@=?Oval 252 n v T`  2 0 Z=?Oval 253 0  T`  2 0 Z@=?Oval 254@ X T`  2 0 Z=?Oval 255  T`  2 1 Z@=?Oval 256 pU  T`  2 1 Z=?Oval 257d ]  T`  2 1 Z=?Oval 258] 7 T`  2 1 Zp=?Oval 259 M q  T`  2 1 Z=?Oval 260  ] q  T`  2 1 Z>?Oval 261~  m  T`  2 1 Z =?Oval 262 a  T`  2 1 Z>?Oval 263 C4  T`  2 1 Z>?Oval 264  T`  2  1 Z(>?Oval 265p T`  2  1 Z1>?Oval 266T& T`  2  1 Zl;>?Oval 267 P  T`  2  1 ZE>?Oval 268P t T`    1 d@N>o?Rectangle 269  T`  |B 1@ T>?Line 270 d pB 1 H?Line 2719 wwpB 1 H?Line 272 2 1 ZX>?Oval 273 If  T`  2 1 Zdb>?Oval 274 II T`  2 1 Zk>?Oval 275 ZI T`  2 1 Zu>?Oval 276 I T`  2 1 ZTx>?Oval 277I T`  2 1 ZD>?Oval 278YI T`  2 1 Zt>?Oval 279 I T`  2 1 Z>?Oval 280 9I T`   1 d>o?Rectangle 281 = T`   1 Pd>?Text Box 282 WN  a1f    |B 1 T>?Line 283 =  1 P|>?Text Box 284 =  a6f    pB 1 H?Line 285 YpB 1 H?Line 286 ^ pB 1 H?Line 287 vpB  1 H?Line 288 ^^vpB !1@ H?Line 289X>XpB "1@ H?Line 290>pB #1 H?Line 2918 pB $1 H?Line 2928 >> 2 %1 Z>?Oval 293 !9  T`  2 &1 Z@>?Oval 294 ! T`  2 '1 Z>?Oval 295 | T`  2 (1 Z@>?Oval 296 l T`  2 )1 Z>?Oval 297v! T`  2 *1 Z>?Oval 298,! T`  2 +1 ZD??Oval 299 1 T`  2 ,1 Z??Oval 300 [ T`  2 -1 Z??Oval 301H| T`  2 .1 Z$??Oval 302H1 T`  2 /1 Z'??Oval 303+l T`  2 01 Z`+??Oval 304vl T`  2 11 Z8??Oval 305 [f  T`  2 21 Z,B??Oval 306 lf  T`   31 dK?o?Rectangle 307  T`  |B 41 T>?Line 308   51 PXV??Text Box 309   a1f    pB 61 H?Line 310 BpB 71 H?Line 311*BpB 81 H?Line 312BgBpB 91 H?Line 313 pB :1 H?Line 314`''pB ;1 H?Line 315  pB <1 H?Line 316 pB =1 H?Line 317' pB >1 H?Line 318'BpB ?1 H?Line 319 dd5 pB @1 H?Line 320S d pB A1 H?Line 321 5 pB B1 H?Line 322 d5 pB C1 H?Line 323S  pB D1 H?Line 324 5 pB E1 H?Line 325S pB F1 H?Line 326**pB G1 H?Line 327::pB H1 H?Line 328*:pB I1 H?Line 329*:pB J1 H?Line 330:B2 K1 ZO??Oval 331Tp T`  2 L1 Zo??Oval 3327p T`  2 M1 Z|y??Oval 333% G  T`  2 N1 Z??Oval 334 G6  T`  2 O1 Z|??Oval 335 G  T`  2 P1 Z؎??Oval 336Gp T`  2 Q1 Z@??Oval 337Wp T`  2 R1 Z??Oval 338 hp T`  2 S1 Z@??Oval 339`G T`  2 T1 Z??Oval 340`T T`  2 U1 Z??Oval 341`7 T`  2 V1 Z0??Oval 3427&  T`  2 W1 Z ??Oval 343T&  T`  2 X1 ZP??Oval 344q T`  2 Y1 Z??Oval 345`W T`  2 Z1 Z??Oval 346` h T`  2 [1 ZDA?Oval 347W&  T`  2 \1 ZA?Oval 348 h&  T`  2 ]1 ZA?Oval 349S 7  T`  2 ^1 Z$A?Oval 350 7c  T`  2 _1 Z'A?Oval 351S   T`  2 `1 Z`+A?Oval 352 c  T`  2 a1 Z8A?Oval 353 T`  2 b1 Z,BA?Oval 354d T`  2 c1 ZKA?Oval 355`t T`  2 d1 ZUA?Oval 356`d T`   e1 dXAo?Rectangle 357JF  T`  |B f1 T>?Line 358F  V  g1 PTjA?Text Box 359 7M  a1f    |B h1 T>?Line 360ncc~ i1 PsA?Text Box 361:u a1f    |B j1 T>?Line 362J6  k1 P8~A?Text Box 363p   a3f     l1 JXA?Text Box 364,Ys qDegree cap = 6  2f   H 0 0޽h ? 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Gene Networks@ GenealogiesBiological/Social Networks7Individuals <-> Vertices, Nodes Relationships <-> Edges`GraphdA graph G=(V,E) consists of a set V={1,2,& n} of vertices (nodes, points, individuals) and a set E={(i,j), i,j  V} of edges (joins, lines, relationships). We deal here only with simple, undirected graphs i.e. there are no self-edges (i,i), no multiple edges, and edges have no direction i.e. (i,j)=(j,i). (3n xd|8_Reproducing Graphs_We investigate a class of models in which graphs reproduce, corresponding to a growing social/relationship network. We concentrate here initially on models without mortality, then make a few comments on age-specific-mortality. We have also studied models in which vertices are eliminated according to degree and/or payoff in a game against neighbours.aGraph ProductsZGiven two graphs G1=(V,E) and G2=(W,F) then a graph product G12 = G1 X G2 =(X,H) has X=V*W where * is the Cartesian product (i.e. X is the set of ordered pairs (v,w) where v  V and w  W). H depends on the particular product; there will be a rule which specifies which (v1,w1) and (v2,w2) join."u S  (X ( ( ( ( F  oCartesian Product V=(v1,v2,& ..,vm) and W=(w1,w2,& & ,wn) J={((vi,wj),(vk,wl))| definition) Cartesian product [vi=vk & (wj,wl)  F] OR [(vi,vk)  E & wj=wl]  ( ( ( (   (   (  (   (     Kronecker Product  V=(v1,v2,& ..,vm) and W=(w1,w2,& & ,wn) J={((vi,wj),(vk,wl))| definition) Tensor (Knonecker) product [(vi,vk)  E & (wj,wl)  F] Associative Adjacency matrix of GH is direct product of the adjacency matrices of G and H.   ( ( ( (   (   ( % <     !: fKronecker..Exampleb Our Models*In our model we suppose there exists some graph at time t and this leads to another at time t+1. Each vertex at time t survives and gives rise to a new offspring vertex. Each edge at t survives. Some subset of the possible edges between the  new and  old vertices are added. Zc Gt+1=Fi(Gt)H `Formally we have that Gt+1 is a function of Gt, where Gt  G the set of all simple graphs. The index i on Fi specifies which member of our family of models is being applied.z  !2BT,3B Our ModelsGiven a network Gt=(Vt,Et) we generate Gt+1=(Vt+1,Et+1) in the following way:- Each edge (u,v) is replaced by pP3P      m>AF  Our ModelsWe fix the presence or absence of the edges labelled ,  and (indicated by 0 and 1). \5' B5"  Our Models> Our Models eMerger of Two Graphs   dFundamental Theorem ?Theoremj Thus we may investigate Fit(Z) where Z=(V={0,1},E={(0,1)} is the single edge graph. Then apply to any G0`i9 LO Our Models g Our Models0,1,4 & 5 are Kronecker products 6 is Cartesian 2 is Comb 7 is Strong 3 is Non-standard }! Model 1&Model 5i,The  Knonecker Models  Since the adjacency matrix of the Knonecker product of two graphs is the Knonecker product of the adjacency matrices we can exploit this for models 0,1,4 & 5,"  Lj,The  Knonecker Models  The adjacency matrices for the four models 0,1,4,5 are respectively. Only that for model 1 is interesting which is essentially the bitwise AND. "Model 3%Model 6h#  j%  l'  n)  PModel 2;  =  ?  A  '6N.B& ..Binary RepresentationIf at time t we have a set of nodes {v1,v2,& .,vn} where each vi is a binary string then at time t+1 we have set of nodes {v11,v10,v21,v20,& & ..,vn1,vn0} where vi becomes vi1 and is the parent of a new node vi0. All our models can be specified in terms of the nodes as binary strings and logical operations defining the edges. BGZZ&    ;          # y.BBinary Representation!Model 1. Kronecker product of  h Properties1. Chromatic Number 2. Number of Vertices and Edges 3. Distance Structure 4. Degree Distribution 5. Automorphism 6. Generating all graphs as subgraphs.( BP  &lAModel 7        !"#$%&'()*+,-.1234Number of Vertices & Edges@No of nodes Vt doubles No of edges Et so n)PP        3kCThe Distance StructureDenote distance (shortest path) between vertices u and v by d(u,v),the diameter (max distance) by D(G), and the number of pairs of vertices with distance x as Nt(x). We demonstrate our methods wrt Model 2. If u in Vt then u0 and u1 are the resulting offspring and parent vertices.Z72P= YADThe Distance Structure Model 2 u & v in Vt and d(u,v)=d then d(u0,u1)=1, d(u0,v0)=d+2, d(u0,v1)=d(u1,v0)=d+1, d(u1,v1)=d. We can then derive Nt+1(0)=2Nt(0); Nt+1(1)=Nt(0)+Nt(1); Nt+1(2)=2Nt(1)+Nt(2) & Nt+1(k)=Nt(k-2)+2Nt(k-1)+Nt(k) for k>2  !! ,EThe Distance Structure Model 2We have also number of distances Lt =4t(V0)2+2tV0 and the total distance Lt*=4tL0*+22t-1(N0(0))2-(22t-1-2t-1)N0(0) From this we derive an expression for the average distance dt=Lt*/Lt->c+t for large t which we can also obtain more directly by considering a random pair of vertices at time t+1 in terms of those at t. In fact we can also show that asymptotically the variance of the average distance is f(G0)+3t/2.#                 @      , DDegree Distribution *,1 & 5~Adjacency matrix of G H is Kronecker product of the adjacency matrices of G and H. i.e. here so degrees are direct products of giving 2t 1 s, tCi 2i s and 2t 2t  s respectively.Xh> <T /CXDegree Distribution Model 6ZStarting from a single edge we have just a hypercube at time t so all nodes of degree 2t $[WY*Degree  Dist Model 3 A node u of degree d gives rise to an offspring u0 of degree (x+1) and a survivor u1 of degree (2x+1). Thus (1) >(2,3)->(3,4,5,7)->(4,5,6,7,8,9,11,15)-> (5,6,7,8,9,9,10,11,12,13,15,16,17,19,23,31) totals 1,5,19,65,211,& & ,3n-2n,& .d3"X  E Age CullingSuppose now that we associate with each node an index which we regard as its age. If at time t node u in V is age s then at time t+1 u1 is of age s+1 and u0 is of age 0. We suppose that after reproduction a node of age Q+1 dies (i.e. is deleted from the graph) BlYModel 6 Age CulledZ  p*  q+  r,  s-  t.  u/  v0  w1  x2  y3  z4  {5  |6  }7  ~8  FModel 2This is the easiest case to treat since the graph grows trees on each original individual. Here we can start with a single node GModel 2Now when we cull at any given age Q+1 we obtain copies of all the trees  of ages 0,1,2,3,& & ,Q If there are nit trees of age i at time t then we get ni,t+1=ni-1,t + nQ,t i=1,2,& Q so we have nt+1= Ant where nt is the column vector of the ni,t s. m'>(B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  H>Tree  Dist Model 2 age-cap=3  0The matrix A is similar to a Leslie matrix L with guaranteed survival except that ai,j=lQ-i,Q-2i+j e.g. {rate  max root of 4= 3 + 2 +  +1, (4)} T   ( ( (   S ] Leslie MatrixIDistribution of age(i,j) edges*It is convenient to introduce a direction `!Distribution of age(i,j) edges a"Distribution of age(i,j) edgeset+1=Let where H\Distribution of age(i,j) edgesWe can prove that the system is irreducible (though if =0 need to refine the state space to exclude (i,i) states). Essentially a generalisation of the Lesley matrix notion. Eigenvalues related to generalisations of the  golden ratio (tribonacci, etc.) 7 b7.E 3 QExample. Model 1, cull at age 3sOffspring joined to neighbours of parents. Next slide shows progress through time omitting isolated vertices.tZt<H ReferencesSouthwell & Cannings, Some models of graph reproduction; 1 Pure Reproduction. (to appear) AM. 2 Age Capped Vertices (to appear) AM 3 Game Based Reproduction (to appear) AM Jordan & Southwell. Further properties of reproducing graphs (to appear) AM AM http://www.scirp.org/journal/am/x ZF$'7 !H  G  IApplied Mathematics Editor in Chief & .CC Editorial Board includes Mark Broom David Greenhalgh http://www.scirp.org/journal/am/ Z, ' ^ Other ResultsSFor age-culled, degree distribution diameter average distanceDT[ Further Workv1) Culling by degree 2) Stochastic/non-synchronous 3) Conflict with neighbours to determine survivor &/or reproduction/   + , -./0123@ACJKLMNOQbc e"f#%')+,-./0123456789:>@BDghijklmnopqrstuvw{F|G}P~$  0 ##Et$ -#(  tx t c $x `}   x t c $'g%  8 ] yt]  t B ]D Bu1$  t BT%]D Bu0$@   xt `B t 0D>  t B|*  Bv1$  t B/ Bv0$`B  t 0D>`B t 0D>^2 t 6^2 t 6^2 t 65^2 t 658   {tp `B Ft 0D>m  `B Gt 0D>V V `B Ht 0D>  `B It 0D>   Jt Bl7   Bu1$ Kt B<   Bv1$ Lt B4B   Bu0$ Mt BxG x Bv0$f2 Pt 6I w f2 Qt 6I w f2 Rt 6  f2 St 6  8 3 c ) |tp  `B Vt 0D>  `B Yt 0D>" "  Zt BM3 c ,J  Bu1$ [t BSS D Bv1$ \t BpXc J  Bu0$ ]t B]8 )  Bv0$`B ^t 0D>  `B _t 0D> " f2 `t 6  f2 at 6 8 f2 bt 6 8; f2 ct 6 ; X8 3 ]) zt]3 )`B ft 0D>  `B gt 0D>`B ht 0D>** jt Bd3 ],D Bu1$ kt B0jS D Bv1$ lt Bo]D Bu0$ mt Bt8) Bv0$`B nt 0D> `B ot 0D> "^2 pt 6 ^2 qt 68^2 rt 685^2 st 6 5u tt 0|-t 1 ut <lt8S 3u vt 0 i  6 wt <F S-  58 F  t  `B ~t 0D> } } `B t 0D>  `B t 0D>   t Bp F -  Bu1$ t B   Bv1$ t B䔳  F -  Bu0$ t BX   Bv0$ t <<  2H t 0޽h ? 33___PPT10i.qqЖL+D=' = @B +2  0 2(  r  S ` `}   r  S r `  RB  s *DԔ" RB  s *DԔ"- RB  s *DԔ"Zc H  0޽h ? 3380___PPT10.uYG~$  0 0t$(  tr t S  `}   r t S h `  H t 0޽h ? 3380___PPT10.DOW`?r (  H Q L q( Q1  !"#$%՜.+,04     On-screen ShowUniversity Of SheffieldH [cG' kArial Arial BlackSymbolTimes New Roman 新細明體Default DesignAdobe Acrobat DocumentMicrosoft Equation 3.0Reproducing GraphsGene Networks GenealogiesBiological/Social NetworksGraphReproducing GraphsGraph ProductsCartesian Product Kronecker ProductKronecker..Example Our Models Gt+1=Fi(Gt) Our Models Our Models Our Models Our ModelsMerger of Two Graphs Fundamental TheoremTheorem Our Models Our ModelsModel 1Model 5The “Knonecker” ModelsThe “Knonecker” ModelsModel 3Model 6 Slide 28 Slide 29 Slide 30 Slide 31Model 2 Slide 33 Slide 34 Slide 35 Slide 36N.B…..Binary RepresentationBinary Representation Properties Slide 40Model 7Number of Vertices & Edges Slide 43The Distance StructureThe Distance Structure Model 2The Distance Structure Model 2Degree Distribution *,1 & 5Degree Distribution Model 6Degree “Dist” Model 3 Slide 50 Age CullingModel 6 Age Culled Slide 53 Slide 54 Slide 55 Slide 56 Slide 57 Slide 58 Slide 59 Slide 60 Slide 61 Slide 62 Slide 63 Slide 64 Slide 65 Slide 66 Slide 67 Slide 68Model 2Model 2 Slide 71 Slide 72 Slide 73 Slide 74 Slide 75 Slide 76 Slide 77 Slide 78 Slide 79 Slide 80 Slide 81 Slide 82 Slide 83 Slide 84 Slide 85 Slide 86 Slide 87$Tree “Dist” Model 2 age-cap=3 Leslie MatrixDistribution of age(i,j) edgesDistribution of age(i,j) edgesDistribution of age(i,j) edgesDistribution of age(i,j) edges Example. Model 1, cull at age 3 Slide 95 ReferencesApplied MathematicsOther Results Further Work  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titlesc_$ ChrisChris