Deterministic and stochastic internet-style networks with a single link, and one or two user under information delay

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1 Deeminii ohai inene-yle neok ih a inle link one o o e ne infomaion elay Gabiela Miea Mihaela Neam Mailen Piea Dmi Opi ba In hi pape e ineiae he ynami of he Inene-Syle Neok ih elay in a inle link one o o e ne elay We eablih he exiene of he Hopf bifaion he nomal fom he ohai yem i aoiae o he eeminii moel he mean ale he qae mean ale of he aiable fo he lineaie ohai yem ae analye he la pa of he pape inle nmeial imlaion onlion Keyo eeminii ynami eonomi moel eonomi oh eaion po hman apial ohai ynami eonomi moel G I IN M heoy poie a naal fameok fo eelopin piin oneion onol mehanim fo he Inene Ue on he neok an be moele a playe in a oneion onol ame hey hooe hei aeie o in hi he flo ae Playe ae nonoopeaie in em of hei em fo neok eoe hae no peifi infomaion on ohe e aeie e em o iliy fo bih i ape in a iliy fnion may no be bone o ompenae fo hi one an eie a piin fnion popoional o he bih ae of a e in oe o peee he neok eoe o poie an inenie fo he e o implemen en-o-en oneion onol [] efl onep in h a non-oopeaie oneion onol ame i Manip eeie Sepembe 5 : Reie eion eeie Gabiela Miea i ih he We Unieiy of imioaa aly of onomi ine miniaion Pealoi S 6 5 imioaa Romania abielamiea@feaao Mihaela Neam i ih We Unieiy of imioaa aly of onomi ine miniaion Pealoi S 6 5 imioaa Romania oeponin aho o poie phone: ; fax: ; mihaelaneam@feaao Mailen Piea i ih he We Unieiy of imioaa aly of onomi ine miniaion Pealoi S 6 5 imioaa Romania mailenpiea@feaao Dmi Opi i ih We Unieiy of imioaa aly of Mahemai Infomai l V Paan 6 imioaa Romania opi@maho finin he Nah eqilibim eah playe minimie hi on o o maximie payoff ien all ohe playe aeie he non-oopeaie oneion onol ame inoe in [] i haaeie by a o fnion fo eah e ha i efine a he iffeene of piin iliy fnion he piin fnion i popoional o he qein elay expeiene by he e a he iliy fnion ha qanifie he e em fo bih belon o a boa la of ily ineain ily onae fnion In [7] [8] [9] he ohe Inene moel ae analye In hi pape e ill analye he iffeenial yem hih hape one inene neok ih a inle link ih inle e o e ne infomaion elay he e of he pape i oanie a follo In eion he exiene of a niqe eqilibim of he yem i eablihe We analye he exiene of he Hopf bifaion oniein a bifaion paamee In eion e analye he ieion abiliy of he Hopf bifaion In eion e analye he ohai moel ih elay aoiae fo inene yle fo a inle link ih a inle e In eion 5 e analye he neok moel ih a inle link o e abiliy of he Hopf bifaion In eion 6 fo ien ale of he paamee he nmeial imlaion ae ien In eion 7 onlion fe eeah ae an II H QUILIRIUM POIN ND H HOP IURCION OR SINGL LINK ND SINGL USR UNDR INORMION DLY he one inene neok ih inle link inle e ne infomaion elay i ien by: ' xɺ U x ɺ x x i he e flo ae > i he link apaiy i he qein elay i he elay beeen he e he link > he iliy fnion U x i Ie Volme 6

2 ame o be ily ineain iffeeniable ily U x onae U x x he fi eqaion fom epeen he ynami yem of a ame he o objeie fnion i ien by: J x x U x o yem he folloin affimaion hol: Popoiion : he eqilibim poin i x : x U x Wih epe o he anlaion x x x x yem beome: xɺ x x xɺ x x U x x 5 he lineaiaion of yem i: ɺ a 6 ɺ a U x he haaeii eqaion of 6: a e 7 nalyin he oo of he haaeii eqaion ih epe o e obain: Popoiion : he oo of eqaion 7 ae iffeeniable fnion ih epe o If he oo of eqaion 7 hae a neaie eal pa o eablih he exiene of he Hopf bifaion e poe: Popoiion : he haaeii eqaion 7 ha he oo i : a a 8 fo ien by: a a 9 Coniein in 7 eiin i ih epe o e e: e a e In eplain ih ien by 8 e obain: a o Re a a om e hae Re > he aboe analyi an be mmaie a follo: Popoiion : qaion 7 ha one Hopf bifaion poin a i ien by 9 III DIRCION ND SILIY O H HOP IURCION OR INRN NWORK WIH SINGL LINK WIH SINGL USR UNDR INORMION DLY In hi eion e y he ieion abiliy he peio of he bifain peioi olion in yem he meho e e i bae on he nomal fom heoy he ene manifol heoem inoe in [] [8] he noaional oneniene le µ hen µ i he Hopf bifaion ale fo yem Syem an be eien a []: x ax x ax ax 6 x O x x ' '' ''' a U x a U x a U x o µ [ ] ih C[ ] C e onie: Lµ i ien by: Le a a a O µ 5 o C [ ] C e efine: < 6 < R µ 7 µ Ie Volme 6

3 Syem an be eien a: µ R µ 8 [ ] o C [ ] C he ajn opeao µ of µ i efine a: < µ 9 o C [ ] R C [ ] R e efine a bilinea fom by: < > δ [ ] δ i Dia iibion In oe o eemine he Poinae nomal fom of he opeao µ e nee o allae he eieneo q of µ aoiae o he eienale i he eieneo q of µ aoiae o he eienale i We an eaily eify ha: q exp [ ] exp i he eieneo of aoiae o he eienale he eieneo of aoiae o i ien by: q exp [ ] exp exp exp 5 Uin e an eify ha < q q> < q q> In he folloin e ill follo he iea e he noaion in [] Le: q > Re q 6 < hen 7 a a q q I ia a exp q q a I a exp exp exp exp exp om 8 e obain: D a a exp a a a a exp a D a D a exp exp exp exp D exp exp exp 8 9 Ie Volme 6

4 exp a exp exp exp a a Uin he heoy of he nomal fom [] e hae he folloin fomla: i C ReC µ Re ImC µ Im β ReC No e an ae he main el of hi eion: Popoiion 5: In he fomla µ eemine he ieion of he Hopf bifaion: if µ > < hen he Hopf bifaion i peiial biial he bifain peioi olion exi fo > < ; β eemine he abiliy of he bifain peioi olion: he olion ae obially able nable if β < > ; eemine he peio of he bifain peioi olion: he peio ineae eeae if > < IV H MHMICL SOCHSIC MODL WIH DLY SSOCID O H INRN SYL NWORKS WIH DLY Le Ω P be a ien pobabiliy pae R be a ale Wiene poe efine on Ω hain inepenen aionay Ga inemen ih min he ymbol enoe he mahemaial expeaion he ample ajeoie of ae onino no iffeeniable hae infinie aiaion on any finie ime ineal [5] We ae ineee in knoin i he effe of he noie pebaion on yem he ohai iffeenial eqaion ih elay i: ' x U x x σ x x x x σ x σ > σ > i he ala Wiene poe x x x ae he omponen of he poe x x on he pobabiliy pae Lineaiin aon he eqilibim x yiel he linea iffeenial eqaion ih elay: y y y C y y y ae ien by C ia σ σ Uin he meho fom [6] [] e analye he fi he eon momen of he olion fo ih epe o Popoiion 6: o yem he momen of he olion i ien by: y y y 5 he haaeii fnion fo 5 i: h a e 6 If he oo of eqaion h hae a neaie eal pa he eqaion h ha one Hopf bifaion poin a i ien by 9 5 If e enoe by y y hen ae ien by i ien by 8 7 i ien by 9 i x iy 8 o examine he abiliy of he eon momen of y fo he linea ohai iffeenial eqaion ih elay e e Io le o ie he ohai iffeenial of y y y y y be he oaiane maix of he poe y o ha R aifie: Le R { y y } R y { y y y C y y C } C R C ia σ σ R R C R R 9 om Rij yi y j i j e e: Popoiion 7: he iffeenial yem 9 i ien by: Ie Volme 6

5 R R a σ R R σr R R a σ σ R R R he haaeii fnion of i ien by: l a σ σ a σσ a Poof: Syem eie fom 9 ih σ σ e ien by Le Rij e Kij i j K ij ae onan Replain R ij in ein he oniion ha he yem e obain hol aep noniial olion e e l om e hae: Popoiion 8: he haaeii eqaion l i ien by: b b b b b e b σ a b a b a σ a σ σ σ σ a b σ σ σ σ a σ σ b If he haaeii eqaion l i ien by: b b b b b If σ σ aify ineqaliie: b > b b > b b > 5 b b b b b > hen he oo of he eqaion hae a neaie eal pa If σ σ aify 5 b b < hen he ale i a Hopf bifaion : σ σ bb σ a bb b bb b b a b b b bb i a poiie eal oo of he eqaion: 6 σσ 6 b b b b b b b b 7 If e enoe by: M ij Rij i j hen M M M exp σ exp a σ i i he olion of eqaion 8 9 x iy 5 i V NLYSIS O NWORK MODL WIH SINGL LINK ND WO USR UNDR INORMION DLY he inene neok ih inle link o e ne infomaion elay i ien by: x U x x U x 5 x x x x x ae he e flo ae > i he link apaiy i he qein elay i he elay beeen he e he he link > i he iliy fnion U x i ame o be he ily ineain iffeeniable ily onae U x U x x o yem 5 he folloin affimaion hol: Popoiion 9: he eqilibim poin i x x : x x U x Wih epe o he anlaion x x x x yem 5 beome: Ie Volme 6

6 U U 5 he lineaiaion of yem 5 i: 5 : x x x x U he haaeii fnion of 5 i ien by: e h 5 nalyin he oo of he eqaion h ih epe o e obain: Popoiion : If he eqaion h ha oo ih a neaie eal pa If exi ien by: a 55 i a poiie eal oo of he eqaion: 6 56 o ha fo [ eqaion h ha oo ih a neaie eal pa Coniein in h eiin i ih epe o e e: e e 57 Le: Re i M Im i N 58 om el: Popoiion : If he eqaion h ha one Hopf bifaion poin a i ien by55 We y he ieion abiliy peio of he bifain peioi olion in yem 5 he meho e e i bae on he nomal fom heoy he ene manifol heoem inoe in [] [8] he noaional oneniene le µ ih µ hen µ i he Hopf bifaion ale fo yem 5 Syem 5 an eien a:!!!! O O 59 x U x U IV o µ ] [ ih ] [ C C We onie L µ 6 i ien by: 6 Le!!!! µ 6 o ] [ C C e efine: µ µ µ o ] [ C C he ajn opeao µ of µ i efine a: µ o e efine a bilinea fom by: > < ϑ η 6 Ie Volme 6 5

7 η δ [ ] δ i Dia iibiion In oe o eemine he Poinae nomal fom of he opeao µ e nee o allae he eieneo q of µ aoiae o he eienale i he eieneo µ i We an eaily eify ha q of aoiae o eienale q exp [ ] exp exp 6 he eieneo of µ aoiae o eienale i ien by: q exp [ ] exp f e e e e exp f e e 65 Uin 6 e an eify ha < q q> < q q>< q q> < q q> Nex e ill follo he iea e he noaion in [] Le < q > Re q 66 hen q q q q he eo ae ien by: 7 exp om el: 7 he paamee C µ ien by ih 7 he ohai pebaion of fo he yem 5 i ien by: x U x σ x x x U x σ x x 7 x x x σ x σ i > i o he ohai iffeenial yem ih elay a imila y an be one VI NUMRICL SIMULION In ha follo e onie U x ln x Uin Maple fo: he eqilibim poin i x o hi ale e hae Ie Volme 6 6

8 67 85 β 59 µ 9 he limi yle i peiial he olion ae obially able he peio ineae he obi x i ien in i obi i ien in i obi x x in i he obi in i o σ σ in he le ohai meho he fie i5 i6 i7 peen he obi x x x x of he yem i5 he obi x i he obi x i6 he obi x i he obi i7 he obi x x i he obi x x o σ σ e obain 7 8 he fie i 8 fi9 fi ho he obi M M M i8 he obi M i he obi Ie Volme 6 7

9 Nmeial meho fo oinay iffeenial eqaion ae alo ien in [] i9 he obi M VII CONCLUSIONS In hi pape e hae examine he eeminii moel fo a neok ih a inle link ih o e ih elay he ime elay i eemine fo hih a Hopf bifaion ake plae he ieion he abiliy of he Hopf bifaion ae analye he ohai moel i aoiae o he eeminii moel o hi moel he mean ale he qae mean ale of he lineaie ohai ae analye I i poe ha hee i a ale of he elay fo hih a Hopf bifaion ake plae he heoeial el ae alo jifie by he nmeial imlaion n analyi of he neok moel ih a inle link mliple e ill be one in a fe pape i he obi M In ha follo e onie U x ln x he eqilibim poin i x 5 x 5 9 o hi ale e hae 89 7 β 5 µ 8 7 he limi yle i biial he olion ae obially nable he peio eeae he obi x i ien in i obi x i ien in i he obi x in i i he obi x i he obi x i he obi x CKNOWLDGMN he eeah a one ne he Gan ih he ile he qaliaie analyi nmeial imlaion fo ome eonomi moel hih onain eaion opion CNCSIS-UISCU Romania an No 85/8 RRNCS [] lman aşa Jimene N Shimkin Compeiie oin in neok ih polynomial o I anaion on omai Conol ol 7 pp 9-96 ; [] lpan aşa iliy-bae oneion onol heme fo inene-yle neok ih elay I ol ; [] S loy K all Pomoin he e of en-o-en oneion onol in he inene I/CM anaion on Neokin ol 7 no pp ; [] D Haa N D Kaainoff Y H Wan heoy appliaion of Hopf bifaion Cambie Unieiy Pe Cambie 98; [5] P Kloeen Plaen Nmeial Solion of Sohai Diffeenial qaion Spine Vela elin 995; [6] J Lei M C Makey Sohai Diffeenial Delay qaion Momen Sabiliy ppliaion o Hemaopoiei Sem Cell Relaion Syem SIM J ppl Mah 67 pp ; [7] G Miea - Inene oneion onol moel Poeein of he 9 h WSS Inenaional Confeene on Mahemai Compe in ine onomi hae Romania pp58-6 8; [8] G Miea M NeamŃ D Opiş - Uneain ohai faional ynamial yem ih elay ppliaion Lambe aemi Pblihin ; [9] G Miea D Opi - Neimak-Sake flip bifaion in a iee-ime ynami yem fo Inene oneion Poeein of he h WSS Inenaional Confeene on Mahemai Compe in ine onomi Pae Ceh Repbli pp86-9 9; [] G Miea M NeamŃ MPiea D Opiş - Deeminii ohai Inene-Syle Neok ih elay Poeein of he h WSS Inenaional Confeene on pplie Infomai Commniaion loene Ialy pp9- ; [] N Sihi M Neam DOpi - Deeminii ohai ynami fim moel ih ealh hman apial amlaion Nmeial imlaion Poeein of he h WSS Inenaional Confeene on pplie Infomai Commniaion loene Ialy pp9- ; [] S Solohkin - -e o he Sole of nional Diffeenial qaion hoh he Web Inefae Poeein of he h WSS Inenaional Confeene on pplie Infomai Commniaion loene Ialy pp5-8 Ie Volme 6 8