Using a Renormalization Group to Create Ideal Hierarchical Network Architecture with Time Scale Dependency

1488 INVITED PAPER Special Secion on Froniers of Informaion Nework Science Using a Renormalizaion Group o Creae Ieal Hierarchical Nework Archiecure wih Time Scale Depenency Masaki AIDA a), Member SUMMARY This paper employs he naure-inspire approach o invesigae he ieal archiecure of communicaion neworks as large-scale an complex sysems. Convenional archiecures are hierarchical wih respec o he funcions of nework operaions ue enirely o implemenaion concerns an no o any funamenal concepual benefi. In conras, he large-scale sysems foun in naure are hierarchical an emonsrae orerly behavior ue o heir space/ime scale epenencies. In his paper, by examining he funamenal requiremens inheren in conrolling nework operaions, we clarify he hierarchical srucure of nework operaions wih respec o ime scale. We also escribe an aemp o buil a new nework archiecure base on he srucure. In aiion, as an example of he hierarchical srucure, we apply he quasi-saic approach o escribe user-sysem ineracion, an we escribe a hierarchy moel evelope on he renormalizaion group approach. key wors: large-scale an complex sysems, renormalizaion, aiabaic approximaion, local ineracion, hierarchical srucure 1. Inroucion Informaion an communicaion neworks are he worl s larges sysems in he erms of boh he number of evices connece an heir spaial exen. Also, by consiering environmenal changes such as he eepening of ies wih sociey an he iversificaion of applicaions, we can regar he neworks as large-scale an complex sysems. How can we esign an operae such large-scale an complex sysems appropriaely? Wha esign principles are require? Before saring concree iscussions, i is necessary o explain he sanpoin of his paper [1]. 1.1 Neworks as Large-Scale an Complex Sysems The mos well-known large-scale an complex sysem is our worl. The number of componens ha form his worl an heir iversiy reaily confirm ha i is he ulimae largescale an complex sysem. So why is his ulimae largescale an complex sysem, he worl, sable? We believe ha he worl will sill exis omorrow an ha he sun will rise omorrow jus like he pas. Even hough we know ha no prior sae is ever repeae exacly a he scale of aoms or elemenary paricles ha make up our worl, we believe ha he worl is sable. Such orerly behavior of he worl is creae hrough so-calle self-organizaion, synergy effec, or collecive phenomena of funamenal srucure. This Manuscrip receive Sepember 6, 2011. Manuscrip revise December 22, 2011. The auhor is wih he Grauae School of Sysem Design, Tokyo Meropolian Universiy, Hino-shi, 191-0065 Japan. a) E-mail: maia@s.mu.ac.jp DOI: 10.1587/ranscom.E95.B.1488 framework is ineresing an gives useful inenion o engineering. The quesion of where he sabiliy or orerly behavior of he worl comes from, probably correspons o he following quesions. Assuming ha Go creae he worl, whaholy secres (or gimmicks) wereuseahe Creaiono yiel he orerly behavior of he worl? Conversely, even if we assume ha Go oes no exis, wha are he gimmicks ha make us feel ha somehing is behin he orerly behaviors of he worl? This paper iscusses one par of a research projec ha is examining such gimmicks an focuses on he esignof informaion communicaion neworks as large-scale an complex sysems. In oher wors, he aim of his research is as follows. Engineering sysems are creae by humans, who consciously or unconsciously imiae he Creaion of he worl by Go. Our goal is o creae a esign approach ha can prouce large-scale complex sysems ha auonomously creae well-orere behavior. In his conex, we iscuss he nee for a nework archiecure base on a hierarchical srucure; is nework operaions exhibi ime scale epenencies. In aiion, we focus on he relaionship beween he user an he sysem as a ypical example of he hierarchy, an iscuss how o esign he hierarchy by using a renormalizaion group. 1.2 Where Does he Well-Orere Behavior of he Worl Come from? The quesion of wha are he gimmicks ha sabilize he worl can be answere in various ways. For example, one explanaion base on he anhropic principle is as follows. Firs of all, he sabiliy of he worl allows he emergence of inelligen life like human beings, an our exisence allows he worl s sabiliy o be iscusse. So, he quesion abou why he worl is sable can arise only in a sable worl, suggesing ha he quesion is some form of auology. Of course, we canno give a complee answer abou he gimmicks since he naural mechanisms are no compleely unersoo. However, since our purpose is no o unersan naure bu apply some form of gimmicks o engineering sysems, we can ry he currenly consiere gimmicks o evaluae heir usefulness for engineering. In his paper, we consier he following wo gimmicks. Acion hrough a meium (Local ineracion) [2] In physical sysems, here are wo conceps ha escribe he ineracions ha can occur beween wo objecs oc- Copyrigh c 2012 The Insiue of Elecronics, Informaion an Communicaion Engineers

AIDA: USING A RENORMALIZATION GROUP TO CREATE IDEAL HIERARCHICAL NETWORK ARCHITECTURE WITH TIME SCALE DEPENDENCY 1489 cupying ifferen posiions; acion a a isance an acion hrough a meium (local ineracion). The former yiels a moel in which wo wiely-separae objecs inerac irecly. The laer oes no permi he exisence of irec ineracion beween wiely-separae objecs; i assumes ha ineracion occurs only beween spaiallyajacen objecs, an he effec of ineracion is graually exchange beween he objecs. Moern physics suppors he acion hrough meium concep, so ineracion occurs locally. In such a moel, space is fille wih physical quaniies a all poins (which forms a fiel), an any variaion in he physical quaniy a a poin woul propagae hrough he fiel a finie spee. Renormalizabiliy (Reucing he egrees of freeom in ynamics) When aemping o observe a massive aggregaion of exremely small objecs ha inerac in complex ways, we can more easily comprehen he aggregae (or sysem) by reucing eiher he emporal or spaial resoluion (or boh), i.e., coarse graining ransformaion. In renormalizaion heory, complex sysems are unersoo by observing changes in a measurable aribue ienifie by he coarse graining ransformaion. The coarse graining ransformaion of observaions is calle he renormalizaion ransformaion. We consier a sysem ha exhibis large (or infinie) egrees of freeom a he microscopic scale. If he sysem is well escribe by small (or finie) egrees of freeom a he macroscopic (measurable) scale hrough renormalizaion ransformaion, he sysem is calle renormalizable. While he renormalizaion heory has many brillian successes in various fiels of physics, i mus be cusomize for each problem. Tha is, here is no general analyical meho ha can be freely applie in various fiels [3]. In he acion hrough a meium concep, an objec ineracs only wih is neighbors, a any insan. In he worl of acion a a isance, he acion of an objec insanly influences all places, incluing he en of he universe, an conversely he acion of any objec, regarless of is locaion, insanly influences he objec. In his siuaion, he componens of worl are associae wih each oher very srongly, which migh limi he flexibiliy of he worl. Therefore, he acion hrough a meium concep appears o be a key gimmick in proucing sable sysems, while ensuring he freeom of local acion. Even if we o no fully comprehen he aribues of micro-componens such as aoms or we o no unersan he complee mechanisms of naure, we can amire he orerly behavior of he worl. This means ha even if here are huge egrees of freeom when he worl is observe a he micro-scale, almos all egrees of freeom are missing a he human percepible macro-scale, an only a small number of macro parameers are neee o escribe he worl. This confirms he renormalizabiliy of he worl. Fig. 1 An example of hierarchical archiecure exhibiing ime scale epnency. 1.3 Relae Work The convenional archiecure has a hierarchical srucure wih respec o funcions of nework operaions, bu i migh no have any funamenal jusificaion, only implemenaion benefis. In conras, he large-scale sysems foun in naure exhibi hierarchies ha are space/ime scale epenen; hese hierarchies unerlie he orerly behavior of he sysems. In his paper, we assume ha he hierarchy concep is he key o esigning an operaing large-scale an complex sysems. In orer o apply his concep o engineering sysems, we aop he naure-inspire approach [1], [4]. Figure 1 shows an example of he hierarchical srucure of nework conrol mechanisms ha yiel operaions wih ime scale epenency. For example, TCP, a proocol of he ranspor layer, inclues funcions acing on wie range of ime scale. Winow flow conrol acs aroun roun rip ime, an exponenial backoff someimes acs aroun ozens of secons. These funcions migh be spli ino ifferen layers of ime scale. As anoher approach for esigning nework conrol mehos inspire by phenomena in naure, he bio-inspire approach has been acively suie [5] [7]. Reflecing he iversiy of biological phenomena, he bio-inspire approach covers a wie variey of applicaions of nework issues, bu he relevan echnology is self-organizaion o form auonomous srucures. The ypical example of selforganizaion in he bio-inspire approach is he reacion iffusion moel, which is base on he Turing paern. This moel emans ha he values of several parameers be une, bu his is ifficul o o in general neworks. In aiion, he ineracion beween wo ifferen sae variables is require, bu his yiels long convergence imes. Renormalizaion groups for communicaion neworks have been suie for evaluaing he scalabiliy of rouing proocols; his requires he inroucion of a renormalizaion ransformaion of he nework opology [8]. However, neworks wih hierarchical srucure have no been iscusse. The res of his paper is organize as follows. Secion 2 sars wih an overview of he nework archiecure

1490 base on our naure-inspire approach. Afer iscussing he hierarchical srucure wih spaial an emporal epenencies, we show a esign approach for he hierarchy layers. Also, we explain he quasi-saic approach, which escribes he ineracion beween user an sysem, as an example of he iner-hierarchy layer esign process. Secion 3 shows he esign of a hierarchical srucure base on he renormalizaion group. Afer inroucing he noion of renormalizaion, we explain he quasi-saic approach on he basis of he renormalizaion group. Secion 4 clarifies he srucure of he quasi-saic approach by using aiabaic approximaion an perurbaion of non-aiabaic effecs. We conclue his paper in Sec. 5. 2. Nework Archiecure Base on Naural Orer In his secion, base on wo gimmicks inrouce in Sec. 1.2, we briefly ouline a nework archiecure wih hierarchical srucure wih ime scale epenency. Nex, we inrouce concree examples of acion hrough a meium (local-acion heory) an renormalizaion. 2.1 Hierarchical Srucure wih Time Scale Depenen Nework Operaions Various sysems in naure exhibi well-orere behavior ue o heir hierarchical srucure wih spaial an emporal scale epenencies. In his paper, we assume ha he wo gimmicks inrouce in Sec. 1.2 are essenial for proucing he sabiliy an well-orere behavior of large-scale an complex sysems. Here, we briefly escribe he ouline of he relaionship beween he wo gimmicks an he hierarchical srucures of nework sysems. For simpliciy of iscussion, le us consier a oneimensional space. Le p(x, ) be a (ensiy) funcion of posiion x an ime. This funcion represens he sae or performance a each posiion, an x enoes he logical or physical posiion in he nework. We assume ha he change in he value of he ensiy funcion a each poin is cause only by he migraion of he quaniy consiere, he quaniy is never creae nor annihilae in he nework. The emporal evaluaion equaion, he maser equaion, is wrien as p(x, ) = w(x, r, ) p(x, )r + w(x r, r, ) p(x r, )r, (1) where w(x, r, ) is he ransiion rae per uni of ime, an is ransiion is from x o x + r a ime. Here, we inrouce he n-h orer momen of w(x, r, ) wih respec o ransiion r as c n (x, ) := r n w(x, r, )r, (2) an use Taylor expansion f (x r) = e r x f (x) of funcion f (x). The emporal evoluion of p(x, ) is given by infinie series of spaial erivaives as p(x, ) = n=1 ( 1) n n! n x n c n(x, ) p(x, ). (3) This is calle Kramers-Moyal expansion [9] Since he series on he righ sie of (3) conains spaial erivaives of infinie orer, he evoluion of p(x, ) a poin x is influence by he sae of p(x+r, ) a oher poins x+r, simulaneously. Noeha sincehis is rue for anyvalueof r, (3) inclues effecs such as acion a a isance. In orer o eliminae he effec of acion a a isance, le us consier he runcaion of he series a some finie orer. If he series is runcae (ha is, we can fin some n 0 such ha c n (x, ) = 0foralln > n 0 ), hen he series only inclues spaial erivaives of finie orer, an hus he emporal evoluion of p(x, ) is eermine only by informaion in he infiniesimally close neighborhoo of x. This correspons o acion hrough a meium. Therefore, accoring o he concep of he acion hrough meium, we woul be ealing wih moels base on parial ifferenial equaions or ifference equaions, ineviably. In nework sysems, acion hrough a meium or local ineracion is a noion use for convenience, an of course he range of local ineracion is no infiniesimal in he mahemaical sense. Local ineracion a a cerain ime scale requires he following hree facors: local informaion ha can be collece wihou egraaion in informaion freshness, neighborhoo (he corresponing spaial range), an auonomous ineracion wih he neighborhoo base only on he local informaion. Therefore, local ineracion can be efine in each ime scale an i migh no seem local if we observe i a microscopic scale (Fig. 2) One of mos elegan an myserious facs in naure is ha sysems having ifferen microscopic srucures occasionally exhibi he same macroscopic behaviors. This is referre o as he universaliy of naural phenomena [3] As an example of he universaliy, he iffusion equaion in Sec. 2.2 can cover various iffusion phenomena (for example, hea flow in solis, he ensiy of ink in he liqui, an he ensiy of gas in he air). The only ifference is foun in he value of a consan (he iffusion coefficien) an ifference in he microscopic srucure is reflece in he value of he iffusion coefficien. In his sense, we can recognize ha his ype of emporal evoluion equaion shows he ime scale ecomposiion of hierarchical sysems. Inee, his ecomposiion iself enables us o recognize ha he worl is sable. The form of he emporal evoluion equaion escribes he phenomena presen a he observe ime scale As shown in Sec. 2.3, x enoes he posiion in absrac parameer space. Generalizaion o inclue creaion an annihilaion is easy. However, if we inrouce hem now, we canno isinguish creaion/annihilaion from elepor, ha is a ypical non-local effec. Since he srucure of neworks is iscree, he ifferenial equaion becomes a ifference equaion. In his siuaion, he erm of higher-orer erivaive requires informaion of far isan componens even if i is finie-orer. A iscree moel base only on local informaion is iscusse in Sec. 2.2.

AIDA: USING A RENORMALIZATION GROUP TO CREATE IDEAL HIERARCHICAL NETWORK ARCHITECTURE WITH TIME SCALE DEPENDENCY 1491 Fig. 2 Hierarchical srucure of neworks wih emporal an spaial scale epenencies: The range of local ineracion is no local if we observe i a a more microscopic scale. an effecs from more finely granular srucures are reuce an represene as he value of he coefficien. In conras, he effecs from longer ime scales impacs he iniial an he bounary coniions of he equaion. From he above iscussion, in orer o compose he hierarchical nework archiecure wih ime scale epenencies, we nee o resolve he following wo issues: Designing acion rules for each layer of he hierarchy Le us consier, for a cerain ime scale, a conrol acion base only on local informaion where acions influence only he neighborhoo. In orer o esablish he concree conrol acion, we nee o evelop a framework of auonomous isribue conrol base on acion hrough a meium. Tha is, he acion rule in a cerain layer shoul be escribe by a parial ifferenial equaion. Unersaning he muual ineracion of layers in he hierarchy Le us consier a siuaion observing phenomena of shorer ime scale. In orer o unersan he muual influence beween acions a ifferen ime scales, we nee o know he appearance of he phenomena a longer ime scales. Tha is, he coefficiens of he parial ifferenial equaion, ha escribes he acion rule, shoul be eermine so as o reflec effecs of he unerlying layer. This proceure requires us o evelop a renormalizaion heory cusomize for neworks. In he following wo subsecions, we show examples of hese issues, respecively. 2.2 Local Ineracion an Auonomous Disribue Conrol Here, we explain he esign of an auonomous isribue mechanism for nework conrol, base on local ineracion using he iffusion phenomena as an example. Assuming he change in ensiy funcion p(x, ) occurs only wih coninuous flow, i.e., we can ignore creaion, annihilaion, an jump o oher posiion, hen p(x, ) saisfies he coninuous equaion, p(x, ) = J(x, ), (4) x where J(x, ) enoes a one imensional vecor represening he flow amoun of p(x, ) ha moves hrough x per uni of ime. In iffusion, he flow is from higher ensiy sie o lower ensiy sie, an flow srengh is proporional o he graien of he ensiy, so we have p(x, ) J(x, ) = κ, (5) x where κ is a posiive consan an is calle he iffusion coefficien. By subsiuing (5) ino (4), we have he emporal evoluion equaion of p(x, ) as follows: p(x, ) = κ 2 p(x, ) x 2. (6) This is he well-known iffusion equaion. Diffusion is a common phenomenon seen everywhere in naure. Surprisingly, an exremely wie variey of iffusion phenomena can be escribe by he iffusion Eq. (6), as explaine in he previous subsecion. The complex microscopic srucure characerisic of each phenomena is reuce, an he characerisics of each phenomenon are expresse by he small number of parameers (in his case, only one parameer). For he iniial coniion p(x, 0) = p 0 (x), (6) has he following soluion. p(x, ) = + N(x y, 2κ) p 0 (y)y, (7) where N(x,σ 2 ) is he ensiy funcion of he normal isribuion wih mean 0 an variance σ 2,hais, N(x,σ 2 ) = 1 x2 e 2σ 2. (8) 2πσ 2 The physical meanings of (7) are simple. The ensiy funcion a he iniial sae a each poin iffuses, over ime, in accorance wih he normal isribuion, an he soluion is he superposiion of he ensiy funcions. As seen in his example, from an engineering sanpoin, he behavior of sysems base on acion hrough a meium (local ineracion) can be associae wih he framework of auonomous isribue conrol [1], [2]. The sae of he enire sysem exhibis orerly behavior as escribe by he soluion (7) of he ifferenial Eq. (6), even hough all subsysems auonomously ac base only on heir local informaion, as (5), an noboy knows he informaion for he enire sysem. Applicaions of auonomous conrol using he iffusion phenomenon inclue raffic conrol for congesion avoiance an loa balancing sysems [2], [4], [10], [11]. The recipe of he framework of auonomous isribue conrol base on local ineracion is summarize in Fig. 3. If he behavior of subsysems is properly esigne a a microscopic scale, his framework allows us o inirecly conrol he behavior of he whole sysem a a macroscopic scale [2]. For he example of iffusion, we consiere a conrol mechanism ha harmonizes he nework sae by he

1492 Le us hink abou he behavior of sae ha he whole sysem has o have ((7) an (9) correspon). Moreover, le us fin he parial ifferenial equaion ha has he soluion ha provies such a behavior ((6) an (10) correspon). Le us ienify he local ineracion ha he parial ifferenial equaion escribes ((5) an (11) correspon). Finally, we esign he behavior of subsysems o replicae local ineracion. As a resul, even hough he auonomous acion of each subsysem is base only on is local informaion, he sae of he whole sysem exhibis he esire behavior as a soluion of he ifferenial equaion. Fig. 3 Recipe for esigning mechanisms of auonomous isribue conrol base on local ineracions. even when he neworks are given a iscree srucure. In orer o enable o apply his conrol mechanism o any nework opology, we have o enhance i so ha he local ineracion oes no epen on he coorinae sysem [12], [13]. Alernaively, if we resric he nework opology o a regular gri, we can apply Fourier ransformaion an efine a higher-orer erivaive. By using hem, oher ypes of srucure formaion mechanisms for he resrice neworks are possible [14]. 2.3 Quasi-Saic Approach as an Example of Creaing Hierarchical Archiecure Fig. 4 Renormalizaion ransformaion of iffusion phenomenon. smoohing effecs of iffusion. We can inrouce anoher mechanism ha prouces spaial paerns of a finie size. The following proceure is an applicaion of he recipe shown in Fig. 3. Firs, we efine a new funcion q(x, ) by using (7) as 2κe 2c 2κe 2c q(x, ) := p σ x, e 2c σ, (9) where c an σ are posiive consans. This funcion is obaine by he proceure ha makes he emporal evoluion of he soluion (7) of he iffusion equaion exponenial agains ime an simulaneously scales he spaial axis in accorance wih iffusion, as shown in Fig. 4. The limi isribuion is lim q(x, ) = N(x,σ 2 ). This ransformaion (9) is a sor of renormalizaion ransformaion an is iscusse in he nex secion. In accorance wih he approach shown in Fig. 3, we can obain he emporal evoluion equaion of q(x, ) an he corresponing local-acion rule, as ( ) q(x, ) = c x x + σ2 2 q(x, ), (10) x ( 2 J(x, ) = c x + σ 2 ) q(x, ). (11) x This conrol mechanism prouces a spaial srucure whose size epens on he value of parameer σ, an we can apply i, for example, o auonomous isribue clusering mechanisms in a hoc neworks. This mechanism has a esirable propery for applicaion o acual neworks. Since he emporal evoluion Eq. (10) conains up o he secon-orer erivaive, local ineracion requires only local informaion, Le us consier he effec ha arises beween ifferen layers of he hierarchy. The characerisic whereby he egrees of freeom of a sysem are reuce when he sysem is examine a a macroscopic scale is no special in iself, an we can fin many examples in naural an engineering sysems. Saisical muliplexing effec (economy of scale) in he esign of communicaion channels is one example. If we aggregae a lo raffic flows, he saisical effec ens o ecrease he relaive variaion aroun he average, an herefore he esigns ha use he average en o work well. In aiion o he saisical effecs, we woul like o ake cerain neworking effecs ino consieraion. The meaning of he neworking effecs is as follows. When we ry o unersan he characerisics of he enire sysem, one approach is o invesigae he eails of each componen of he sysem. This concep is calle reucionism. The neworking effec means he phenomena ha canno be unersoo hrough reucionism. Tha is, he characerisics of each componen are no he sole eerminan of he characerisics of he enire sysem, insea we mus consier he neworking effecs generae by componen ineracion. In his siuaion, he effecs creae by he characerisics of each componen become weak bu he neworking effec becomes ominan, an new non-rivial characerisics emerge a he macro-scale. One example of he above siuaion is he quasi-saic approach; i escribes he rery raffic generae by ineracion beween users an a sysem [15]. This approach has he following characerisics. Descripion of ineracion beween users an a sysem The response ime of he sysem increases uner congesion cause by an increase in inpu raffic. The increasing response ime riggers an increase in rery raffic from users, an he rery raffic worsens congesion. In unersaning he sysem behavior, he ineracion beween he users an he sysem is essenial, no heir iniviual characerisics. Decomposiion of users an sysem ynamics Since he sae ransiion rae of he sysem is exremely high in high-spee neworks compare o he ime-scales perceive by humans, we uilize he ifference in ime scales o ecompose he layers in he hierarchy. This proceure is a kin of renormalizaion ransformaion an is

AIDA: USING A RENORMALIZATION GROUP TO CREATE IDEAL HIERARCHICAL NETWORK ARCHITECTURE WITH TIME SCALE DEPENDENCY 1493 Fig. 5 Exene M/M/1 moel wih reries. λ 0 λ 0 + ɛ λ 0 + (i 1)ɛ λ 0 + iɛ λ 0 + (i + 1)ɛ 0 1 i i + 1 i + 2 μ μ μ μ μ Fig. 6 Sae ransiion rae iagram in which he rery rafficis proporional o he number of currenly acive cusomers. iscusse in subsequen secions. The simples moel ha aresses he ineracion beween users an a sysem is he exene M/M/1 moel ha inclues reries (Fig. 5). We assume ha he rae of rery raffic is proporional o he number of cusomers in he sysem, because he number of cusomers in he sysem increases uner congesion. As he mos primiive moel, le us consier a moel in which he rae of rery raffic isproporional o he number of cusomers who currenly sojourn in he M/M/1 sysem, he currenly acive cusomers. The sae ransiion rae iagram is shown in Fig. 6, where λ 0 is heraeofprimaryraffic (wihou reries), μ is he service rae, an ɛ ( 0) is a proporionaliy consan. This moel oes no have a seay sae when ɛ>0evenifɛ 1, so he inpu raffic wih reries iverges. Since he volume of rery raffic in acual sysems oes no iverge in normal operaions, he above moel fails o escribe acual raffic. Wha is wrong wih he above moel? The assumpion ha he rae of rery raffic is proporional o he number of currenly acive cusomers means ha he sysem spee is exremely slow, or he ime resoluion of cusomers responses is relaively high. In oher wors, cusomers can reac immeiaely in response o he presen sae of he sysem. However, in acual sysems, since he cusomers canno reac immeiaely, he moel epice in Fig. 6 is inappropriae. If he cusomers ime resoluion eviaes from he moel epice in Fig. 6, he rery raffic epens no only on he curren sae bu also on he pas sae. For example, by measuring he number of cusomers in he sysem a appropriaely chosen ime poins, we can esimae he average number of cusomers in he sysem for a cerain perio. We consier ha he rery raffic epens on he average number. If we consier he average of he pas n measuremens, he sae ransiion rae iagram can be expresse as an n- imensional Markov chain. However, since n 1 in highspee neworks, he sae space exploes an his moel becomes ifficul o calculae. The quasi-saic approach has been inrouce for resolving his problem; i can evaluae he inpu raffic raean Fig. 7 Graphic assessmen of sysem sabiliy. he sabiliy of he sysem. This approach is briefly summarize as follows. Firs, we inrouce ime scale T ha represens he ime scale ha maches he human response rae (for using he communicaion service). Nex, we consier iscree ime inervals ha are T long. Since he sysem spee is very high, T is very long for he sysem bu realisic for he cusomers. Thus, we regar ha he sysem is basically in equilibrium in each T, an any change in he sysem mainains he equilibrium (i.e., i is quasi-saic). Base on he above assumpion, we represen he inpu rae incluing rery raffic a iscree ime k as λ k. The inpu rae λ k+1 a iscree ime k + 1 is obaine from sum of he primary raffic raeλ 0 anhereryraffic rae eermine by he inpu rae λ k a iscree ime k,as λ k /μ λ k+1 = λ 0 + ɛ 1 λ k /μ, (12) where, he secon erm on he righ han sie enoes rery raffic, an a equilibrium, i is proporional o he average number of acive cusomers of M/M/1 [15]. This moel correspons o he high-spee limi of he sysem an allows us o exrac a simple relaion beween users an he sysems as a eerminisic moel. We efine ha he sysem is sable if he inpu raffic oes no iverge, ha is, lim k λ k <. Sabiliy can be iscusse graphically. In Fig. 7, λ k+1 = f (λ k ) shows (12). If here are inersecions of f (λ k ) an he line wih graien of 1, he sysem is sable uner cerain iniial coniions. If here is no inersecion, he sysem is insable. In acual sysems, since he spee of he sysems is high bu finie, he approach akes he ifference from he eerminisic moel ino consieraion as flucuaions (Fig. 8). Then, by choosing an appropriae quaniy X() harepre- sens he volume of inpu raffic, he emporal evoluion of X() obeys he following sochasic process, X() = g 1 (X) + g 2 (X)W(), (13) where g 1 (X) enoes he eerminisic change obaine from he infinie-spee limi of he sysem. W() is he Wiener process for escribing he ifference from he infinie-spee moel as flucuaions, an g 2 (X) enoes he srengh of he flucuaions. Changinghe perspecive,if p(x, ) is he probabiliy ensiy funcion of X(), he emporal evoluion equaion of p(x, ) is expresse as he following Fokker-Planck equaion [9],

1494 Fig. 8 Concep of he quasi-saic approach. ( ) p(x, ) = x g 1(x) + 2 x g 2(x) p(x, ). (14) 2 From he iscussion of he relaionship beween ifferen layers of he hierarchy, we obain a parial ifferenial equaion. The quasi-saic approach escribes he ineracion beween layers by using he ifference in ime scales an suppressing he eails of he microscopic srucure. Hereafer, we reconsier he quasi-saic approach from he viewpoin of renormalizaion. Fig. 9 Renormalizaion ransformaion of infinie 2-imensional laice. 3. Inroucion of a Renormalizaion Group an Is Applicaion o he Quasi-Saic Approach In he convenional approach o esigning neworks, we en o believe ha eaile informaion of sae will yiel precise conrol or an exac esign. However, in he hierarchical archiecure wih space/ime scale epenency, since he eails of he microscopic lower layer canno be recognize hrough macroscopic observaions, we nee o know wha kin of quaniies can be obaine a he macroscopic higher layer, sysemaically. Conversely, he quaniy obainable from higher layer is wha is essenial for escribing he relaionship beween ifferen layers. This is because he unobservable quaniies canno affec he higher layer, an canno be conrolle from he higher layer. In his secion, in orer o escribe he archiecure beween layers, we inrouce he noion of renormalizaion, apply i o he formulaion of he quasi-saic approach, an iscuss is physical meaning. 3.1 Renormalizaion Transformaion an Renormalizaion Group Renormalizaion was originally evelope in he fiel of quanum elecro ynamics in he 1940 s by Tomonaga, Schwinger, an Feynman [16]. Wilson clarifie is physical meaning an inrouce he renormalizaion group in he 1970 s [17]. Renormalizaion ransformaion is efine as he combinaion of coarse graining ransformaion an scaling. Le us consier wo examples. The firs one is he renormalizaion of iffusion (Fig. 4). The emporal evoluion of iffusion correspons o coarse graining ransformaion, in his case. As a secon example, le us consier an infinie Go boar. Each gri poin is occupie by a Go sone an is color is black wih probabiliy p or whie wih probabiliy 1 p. The problem is how o eermine is appearance from afar [18]. Firs, we aop he following rule o realize 2 2 subsampling. A black sone is se if he 2 2 gri inclues hree or more black sones. A whie sone if 2 2 gri inclues less han hree black sones. Whie is slighly favore because whie is he more visually prominen han black. This is a coarse graining ransformaion an yiels 1/4 simplificaion, an hen we apply scaling (Fig. 9). These wo rules form a renormalizaion ransformaion. The probabiliy ha he unifie gri is black afer applying he renormalizaion ransformaion jus once, R(p), is expresse as R(p) = p 4 + 4p 3 (1 p). (15) Here, we can fin p c such ha R(p c ) = p c an 0 < p c < 1, as p c = 1 + 13 ( 0.7676). (16) 6 This is he criical probabiliy. If p > p c, he boar looks black from afar an if p < p c i looks whie. Thus, by using he renormalizaion ransformaion, we migh be able o escribe wha can be observe from he macroscopic scale, sysemaically. The eaile value of p oes no affec he macroscopic observaion.

AIDA: USING A RENORMALIZATION GROUP TO CREATE IDEAL HIERARCHICAL NETWORK ARCHITECTURE WITH TIME SCALE DEPENDENCY 1495 Fig. 10 Q T in he rery raffic moel of (19). 3.2 Renormalizaion Transformaion of he Arrival Rae Incluing Rery Traffic Le us inrouce he renormalizaion ransformaion o he M/M/1 moel wih rery escribe in Sec. 2.3. We efine he inpu rae Λ(; T) a ime as Λ(; T) = λ 0 + ɛ Q T, (17) where ɛ is a posiive consan an Q T isameasureofhe average number of cusomers in he sysem, more specifically, Q T is he average wihin he human-percepible ime perio T immeiaely before he presen ime. So, rae Λ(; T) is given by he sum of he rae λ 0 for he primary rafficanheraeforhereryraffic, which is proporional o he average number of cusomers, in he pas perio. The following are wo concree examples of he average number of cusomers, Q T.LeQ() be he number of cusomers in he sysem a ime, an he firs example is Q T := 1 Q(s)s. (18) T T This moel means he rae of rery raffic a ime is proporional o he average number of cusomers in [ T, T). The secon example is Q T := 1 Q(s)e 1 T ( s) s. (19) T This moel means ha he rery from cusomers a s (< )occurs ranomly afer exponenial ime wih mean T (Fig. 10). Regarless of wheher we choose (18) or (19) as he efiniion of Q T, we can evelop a unifie iscussion. Hereafer, unless oherwise noe, he resuls are vali for boh cases. Ascribing o humans he abiliy o reac immeiaely correspons o he limi T 0. From lim Q T = Q( ), (20) T +0 an λ() :=Λ(; +0), we have λ() = λ 0 + ɛ Q( ). (21) This correspons o he sysem moel escribe by he sae ransiion rae of Fig. 6. Noe ha he inpu rae incluing he rery raffic an he number of cusomers in he sysem influence each oher. The variaion of he inpu rae irecly affecs he number of cusomers, an conversely he average number of cusomers affecs he inpu rae hrough (17). From his iscussion, if he human percepible ime T is change, he inpu rae changes hrough (17) which affecs he value of Q(). So, o be exac, he number of cusomers Q() mus be a quaniy ha epens on he human percepible ime T. In orer o ake he T-epenence of Q() ino consieraion, we inrouce he following renormalizaion ransformaion. Firs, we efine a coarse graining ransformaion. For α 1, we sar o consier he siuaion ha he human ime resoluion is lowere by 1/α. We call he corresponing ransformaion K α of he inpu rae he Kaanoff ransformaion. The concree form of K α for he average of (18) is K α (Λ(; T)) =Λ(; αt), = λ 0 + ɛ αt an ha for (19) is K α (Λ(; T)) =Λ(; αt). = λ 0 + ɛ αt at Q (α, s)s, (22) Q (α, s)e 1 αt ( s) s (23) Here, Q (α, ) represens he number of cusomers accoring o parameer α, his was change from Q() by he lowering of he human ime resoluion. Of course, Q (1, ) = Q(). To enable a unifie iscussion of boh cases of (18) an (19), we inrouce he following noaion for Q (α, ); For (18), Q := 1 α,β,t Q (α, βs)s, (24) T T an, for (19) Q := 1 α,β,t Q (α, βs)e 1 T ( s) s. (25) T Using his noaion, boh (22) an (23) are wrien in he same form as K α (Λ(; T)) = λ 0 + ɛ Q α,1,αt. (26) Nex, we inrouce he ajusmen of ime scale by 1/α imes, as S α (Λ(; T)) = λ 0 + ɛ Q 1,α,T/α. (27) Since his is merely a change of scale on he ime axis, he form of Q() = Q (1, ) is unchange. By combining he above wo ransformaions [3], we Accoring o his noaion, Q( ), which appears in (20) an (21), is expresse as Q (+0, ), if we assume α<1.

1496 To see explicily he effecs of iffereniaion wih respec o α, we change he expression of (29) ino a form ha simplifies invesigaion. Since S α is merely a change of scale on he ime axis, i is an ieniy ransformaion, as a ransformaion of Λ(; T). Therefore, (27) is expresse as S α (Λ(; T)) = λ 0 + ɛ Q 1,α,T/α = λ 0 + Q 1,1,T =Λ(; T). (34) Fig. 11 Renormalizaion ransformaion for he moel of (18). efine he renormalizaion ransformaion R α as, R α := S α K α. (28) The concree form of he renormalizaion ransformaion of he inpu rae Λ(; T) is enoe as R α (Λ(; T)) = {S α K α }(Λ(; T)) = λ 0 + ɛ Q α,α,t. (29) Figure 11 explains he proceures of he renormalizaion ransformaion for he case ha he average number of cusomers is chosen as (18). Incienally, he renormalizaion ransformaions form a semi-group, R 1 (Λ(; T)) =Λ(; T), (30) {R α R β }(Λ(; T)) = R αβ (Λ(; T)), (31) {R αβ R γ }(Λ(; T)) = {R α R βγ }(Λ(; T)), (32) ha is calle as he renormalizaion group. Hereafer we enoe Λ α (; T) := R α (Λ(; T)) for breviy. 3.3 Renormalizaion Group Equaion of Arrival Rae an he Quasi-Saic Approach Since we canno know he concree form of Q (α, ) fora general α, we canno eermine he average Q α,α,t in (29) an herefore he inpu rae Λ α (; T) canno be eermine for a general α. Here, we consier a special case of he renormalizaion ransformaion wih α 1, an invesigae Λ α (; T). This case means ha he spee of he sysem is much higher han ha of humans as expresse by percepible ime scale T. We assume he following renormalizaion group equaion, α Λ α(; T) = 0. (33) The physical meaning of his equaion is ha even if we lower he human ime resoluion furher, no new behavior emerges. From (33) an (34), we have α Λ α(; T) = α K α(λ(; T)) = ɛ Q α α,1,αt = 0. (35) This means he average Q α,α,t is unchange even if T becomes longer, an so we can recognize ha he average remains in a seay sae. 4. Reucion of Dynamics an Quasi-Saic Approach In his secion, we inrouce he aiabaic approximaion an show ha i leas o he same resul obaine by he renormalizaion. In aiion, we iscuss he escripion of non-aiabaic effecs an relaionship o he quasi-saic approach. 4.1 Aiabaic Approximaion an Renormalizaion Group Equaion Aiabaic approximaion was originally use in soli sae physics, an is base on he fac ha he nuclei of molecules an solis move much more slowly han elecrons. I approximaes he sae of elecrons by assuming ha he nucleus is saionary. This approach is applicable o sysems consising of very slow an very fas componens. Firs, we inrouce he aiabaic approximaion aken from [19]. Le us consier he following sysem. The sysem moves o relaxe sae q() = 0 if no exernal force exiss, an he srengh of he relaxaion is proporional o he ifference q() from equilibrium 0. When we apply exernal force F() o he sysem, we have q() = γq() + F(), (36) where q() is he porion of he inpu rae ha correspons o reries, q() :=Λ α (; s) λ 0, (37) an γ>0. The soluion of (36) is given as q() = ɛ 0 e γ( s) F(s)s. (38)

AIDA: USING A RENORMALIZATION GROUP TO CREATE IDEAL HIERARCHICAL NETWORK ARCHITECTURE WITH TIME SCALE DEPENDENCY 1497 We can recognize ha q() is he response o inpu F(). In general, q() epens on no only F() a he presen momen bu also he exernal force in he pas. If q() changes much more rapily han F(), we can recognize ha only he exernal force a he presen momen influences q(). For example, le he ime consan of F() be1/δ, anwese F() = ae δ where a is a consan. Subsiuing his ino (38) an execuing he inegraion, we obain q() = a ( e δ e γ). (39) γ δ Here we use he assumpion ha he change of q() ismuch faser han ha of F(), ha is γ δ, so q() a γ e δ 1 F(). (40) γ This siuaion means he ime consan of he sysem, 1/γ, is much smaller han he ime consan of he exernal force, 1/δ. This reamen is calle aiabaic approximaion. Noe ha he approximaion (40) is also obaine from (36) by seing q()/ = 0. Nex, we consier he relaionship beween he aiabaic approximaion an he renormalizaion group. Le us sar from (29), Λ α (; T) = λ 0 + ɛ Q α,α,t. (41) The aiabaic approximaion gives Λ α (; T) = λ 0 + 1 F(). (42) γ By comparing his wih (41), we have he slow exernal force as F() = γɛ Q α,α,t. (43) Thus, (36) becomes q() = γ { q() + ɛ } Q α,α,t, (44) an, from he aiabaic approximaion of (44), we have q() = ɛ Q α,α,t. (45) In aiion, by applying he aiabaic approximaion q()/ = 0again,wehave Q α,α,t = 0. (46) The physical meaning of his resul is ha he average number of cusomers is inepenen of ime. In oher wors, we can regar he average Q α,α,t is in a seay sae, he same as (35) in renormalizaion. 4.2 Perurbaion Expansion of Non-Aiabaic Effecs an Unersaning of he Quasi-Saic Approach Boh he renormalizaion group Eq. (33) an he aiabaic approximaion correspon o he limi of he siuaion ha he sysem spee is significanly higher han ha of he cusomers, an boh give he same resul. However, as shown in Fig. 8, our original goal is a sysem ha has high bu finie spee. Therefore, we shoul also ake non-aiabaic effecs ino consieraion. We inrouce he parameer δ ha represens he users spee an consier he slow variable Q α,α,t an he fas variable q(), as follows. Q = δ G ( α,α,t Q α,α,t, q), q() = γ q() + γɛ (47) Q α,α,t, where G(, ) is an unknown funcion. The human percepible ime scale is exremely long compare wih ha of he sysem. So, we inrouce he smallness parameer η := δ/γ = 1/T 1, where η represen he raio of users spee o he sysem spee. Nex, we se he ime consan of he sysem as 1/γ = 1. This proceure means he change of he uni of ime or he replace of (/γ), an we have Q = η G ( α,α,t Q α,α,t, q), q() = q() + ɛ (48) Q α,α,t. We nee o invesigae he asympoic behaviors, for, of he sysem ha inclue aiabaic an non-aiabaic effecs. To his en, we ake he perurbaive approach wih respec o he power of he smallness parameer in orer o escribe he small non-aiabaic effecs aroun he aiabaic approximaion. Firs, we consier he lowes orer of he perurbaion. We inrouce he noaion of he slow variable in (45) as Q := Q α,α,t, (49) for breviy. The aiabaic approximaion is hen expresse as q a () = ɛ Q, Q = η G ( Q, q a () ). (50) In orer o realize he higher orer correcion of nonaiabaic effecs aroun he aiabaic approximaion, we ake he following approach [20] [22]. Because he neural sabiliy of Q riggers he emergence of a secular erm (ha inclues he facor (η)) in perurbaion, perurbaion expansion canno be applie o Q. However, since ( Q /) is a small variable, we can apply perurbaion expansion o i. The perurbaion expansion of q() aroun q a () isof course possible. q() is epenen on ime only hrough Q. As long as he perurbaion is small (ha is, Q is a slow variable), here is an invarian manifol o which he

1498 rajecory of q() approaches for. This reamen was inrouce o eliminae he secular erm from he perurbaion expansion base on he renormalizaion group [20], an o reuce he egrees of freeom in evaluaion equaions ha escribe sysem ynamics [21]. In any case, he exisence of he invarian manifol is criically imporan in successfully applying his reamen, an i is, exacly, renormalizabiliy [22]. Accoring o he above iscussion, le us consier he following perurbaion expansions q() = q 0 () + ηq 1 () + η 2 q 2 () + η 3 q 3 () +, Q = v 0 () + ηv 1 () + η 2 v 2 () + η 3 v 3 () +. (51) Base on he aiabaic approximaion (50), q 0 () = q a (), an v 0 () = 0. (52) By using he expansion of q(), he higher-orer correcion of non-aiabaic effecs in ( Q / ) is expresse as Q = η G ( Q, q 0 () + ηq 1 () + O(η 2 ) ) = η G ( Q,ɛ Q ) ( ( G Q + η 2, q ) ) q 1 () q q=q 0 + O(η 3 ). (53) Therefore, we have v 1 () = G ( Q,ɛ Q ), (54) ( ( G Q v 2 () =, q ) ) q 1 (). (55) q q=q 0 Nex, we consier he higher-orer correcion of nonaiabaic effecs in q(). Because he ime epenency of q() occurs only hrough Q,werepresenq() = q( Q ) for convenience. From (44), we have Q By expaning his as, q( Q ) Q = q() + ɛ Q. (56) (ηv 1 () + O(η 2 )) Q ( q 0 ( Q ) + O(η)) = (q 0 () + ηq 1 () + O(η 2 )) + ɛ Q, (57) an by exracing he erms of he orer of η,wehave v 1 () q 0( Q ) Q = q 1 (). (58) Thus, we have q 1 () = G ( Q,ɛ Q ) q 0 ( Q ) Q = ɛ G ( Q,ɛ Q ). (59) We can summarize he resuls as Q = η G ( Q,ɛ Q ) ( ( G Q +η 2, q ) ) q 1 () + O(η 3 ), q q=q 0 q() = ɛ Q ηɛ G ( Q,ɛ Q ) + O(η 2 ). (60) From (60), here is no erm of he orer of η 0 in (q()/). This means he variaion of Q an q() are no observe a he ime scale of T 0 = 1, bu i oes appear a he ime scale of T 1. This resul correspons o he quasi-saic approach; he sysem is in he equilibrium sae when observe a he ime scale of T, an he sae changes very slowly keeping he equilibrium sae. Therefore, by inroucing he ime sep uni of T,weefine λ k :=Λ α (kt, T), k = 1, 2,..., (61) an he emporal evoluion of (61) can be escribe by (12). 4.3 Temporal Evoluion Equaion of he Number of Arriving Cusomers Incluing Reries In his subsecion, we aop he average number of cusomers in he sysem as (18) an efine he acual number of cusomers arriving uring [ T, ]asx(, T).IfT, ha is he limi of higher sysem spee, he observe inpu rae is equivalen o he inpu rae X(, T)/T =Λ α (; T). However, for a finie T, X(, T)/T Λ α (; T), in general. When we iscuss he ifference from he high spee limi as shown in Fig. 8, we shoul escribe X(, T)/T raher han Λ α (; T). The variaion of X(, T)/T occurs very slowly bu can be observe a he human perceivable scale. This observaion is equivalen o fas forwaring a vieo. Nex we eermine he eails of he unknown funcion G(, ) in (60), by using moel-specific characerisics of large-scale M/M/1. The infiniesimal variaion of X(, T) is efine as X(, T) := X( +, T) X(, T) = X( +, ) X( T +, ). (62) We assume ha he iming of rery raffic inpu is fully ranomize an i follows a Poisson process. In aiion, he large-scale sysem argee has a large value of primary raffic rae λ 0, an hus he Poisson isribuion is sufficienly close o he normal isribuion. Therefore, he number of arriving cusomers can be expresse, by using he Wiener process, as X( +, ) =Λ α (, T) + Λ α (, T)W() Of course, we can aop (19) alernaively.

AIDA: USING A RENORMALIZATION GROUP TO CREATE IDEAL HIERARCHICAL NETWORK ARCHITECTURE WITH TIME SCALE DEPENDENCY 1499 = ( ) ɛ X(, T)/(μT) λ 0 + 1 X(, T)/(μT) + λ 0 + ɛ X(, T)/(μT) 1 X(, T)/(μT) W(), (63) X(, T) X(, T) X( s +, ) = + W(). (64) T T Thus he infiniesimal variaion of X(; T) is obaine as ( ) X(, T) ɛ X(, T)/(μT) X(, T) = λ 0 + T 1 X(, T)/(μT) X(, T) ɛ X(, T)/(μT) + λ 0 + + T 1 X(, T)/(μT) W(). (65) In he form of Langevin equaion, (65) can be expresse as ( ) X(, T) X(, T) ɛ X(, T)/(μT) = λ 0 + T 1 X(, T)/(μT) X(, T) ɛ X(, T)/(μT) + λ 0 + + T 1 X(, T)/(μT) ξ(), (66) where ξ() is he whie Gaussian noise ha saisfies E[ξ()] = 0anE[ξ() ξ(s)] = δ( s), an ξ() obeys he sanar normal isribuion. This resul correspons o (13) an we can also express he resul in he form of (14) as p T (x, ) = ( λ 0 x + 2 x 2 X(, T) T λ 0 + + X(, T) T ) ɛ X(, T)/(μT) p T (x, ) 1 X(, T)/(μT) + ɛ X(, T)/(μT) 1 X(, T)/(μT) p T (x, ), (67) where p T (x, ) is he probabiliy ensiy funcion of X(, T). The valiiy of (65) an (67) was verifie by comparison agains simulaion resuls in [23]. 5. Conclusions In his paper, we have iscusse he esign of a nework archiecure ha aops he approach of reproucing he sabiliy an orer of naure. Our guiing principle is hierarchy wih ime scale epenency, an i inclues he local-acion heory an he renormalizaion group. We emonsrae he imporance of he renormalizaion group in hierarchical esign by using an example of ineracion beween cusomers an a sysem. The form of he emporal evoluion Eq. (67) escribes he phenomena of p T (x, ) observe a he human percepible macro-scale, an effecs from more finelygranular srucures reflecing sae ransiion of he sysem are reuce an represene as he value of he coefficien of (67). This is an example of hierarchical srucure shown in Sec. 2.3. In aiion, we clarifie he physical inerpreaion of he quasi-saic approach. Acknowlegmen This research is suppore by a Gran-in-Ai for Scienific Research (B) No. 21300027 (2009 2011) from he Japan Sociey for he Promoion of Science. References [1] M. Aia, C. Takano, an Y. Sakumoo, Challenges o new nework archiecures base on hierarchical srucure of ime scales, J. IEICE, vol.94, no.5, pp.401 406, May 2011. [2] C. Takano an M. Aia, Auonomous ecenralize flow conrol mechanism base on iffusion phenomenon as guiing principle: Inspire from local-acion heory, J. IEICE, vol.91, no.10, pp.875 880, Oc. 2008. [3] Y. Oono, H. Tasaki, an K. Higashijima, Expaning horizon of renormalizaion heory, Mahemaical Sciences, Saiensu-sha, Publishers (in Japanese), vol.35, no.4, pp.5 12, 1997. [4] M. Aia an C. Takano, Principle of auonomous ecenralize flow conrol an layere srucure of nework conrol wih respec o ime scales, Supplemen of he ISADS 2003 Conference Fas Absracs, pp.3 4, 2003. [5] G. Neglia an G. Reina, Evaluaing acivaor-inhibior mechanisms for sensors coorinaion, IEEE/ACM BIONETICS 2007, pp.129 133, Buapes, Hungary, Dec. 2007. [6] N. Wakamiya, K. Hyoo, an M. Muraa, Reacion-iffusion base opology self-organizaion for perioic aa gahering in wireless sensor neworks, Secon IEEE Inernaional Conference on Self- Aapive an Self-Organizing Sysems, pp.351 360, Venice, Ialy, Oc. 2008. [7] K. Leibniz an M. Muraa, Aracor selecion an perurbaion for robus neworks in flucuaing environmens, IEEE Nework, vol.14, no.3, pp.14 18, May/June 2010. [8] C. Consaninou an A. Sepanenko, Nework proocol scalabiliy via a opological Kaanoff ransformaion, 6h Inernaional Symposium on Moeling an Opimizaion (WiOPT 2008), pp.560 563, April 2008. [9] N.G. van Kampen, Sochasic Processes in Physics an Chemisry, 3r e., Norh Hollan, 2007. [10] C. Takano an M. Aia, Diffusion-ype auonomous ecenralize flow conrol for en-o-en flow in high-spee neworks, IEICE Trans. Commun., vol.e88-b, no.4, pp.1559 1567, April 2005. [11] M. Uchia, K. Ohnishi, K. Ichikawa, M. Tsuru, an Y. Oie, Dynamic an ecenralize sorage loa balancing wih analogy o hermal iffusion for P2P file sharing, IEICE Trans. Commun., vol.e93-b, no.3, pp.525 535, March 2010. [12] C. Takano, M. Aia, M. Muraa, an M. Imase, New framework of back iffusion-base auonomous ecenralize conrol an is applicaion o clusering scheme, IEEE Globecom 2010 Workshop on he Nework of he Fuure (FuureNe III), 2010. [13] C. Takano, M. Aia, M. Muraa, an M. Imase, Auonomous ecenralize mechanism of srucure formaion aaping o nework coniions, 11h IEEE/IPSJ Inernaional Symposium on Applicaions an he Inerne (SAINT 2011) Workshops (HEUNET 2011), Munich, Germany, July 2011. [14] T. Kubo, T. Hasegawa, an T. Hasegawa, Mahemaically esigning a local ineracion algorihm for auonomous an isribue sysems, 10h Inernaional Symposium on Auonomous Decenralize Sysems (ISADS 2011), 194 203, March 2011. [15] M. Aia, C. Takano, M. Muraa, an M. Imase, A suy of conrol

1500 plane sabiliy wih rery raffic: Comparison of har- an sof-sae proocols, IEICE Trans. Commun., vol.e91-b, no.2, pp.437 445, Feb. 2008. [16] J.D. Bjorken an S.D. Drell, Relaivisic Quanum Mechanics, McGraw-Hill, New York, 1964. [17] K.G. Wilson, The Renormalizaion Group an Criical Phenomena, Nobel Lecure, Dec. 1982. [18] T. Takea, Lecure Noe on TCSE, 2010, The Universiy of Elecro- Communicaions, Japan. web.mac.com/akea /Bonryu/Lecure files/tcse9 10.pf [19] H. Haken, Synergeics: An Inroucion. Nonequilibrium Phase Transiions an Self-Organizaion in Physics, Chemisry an Biology, Springer, 1978. [20] Y. Oono, Inroucion o Nonlinear Worl, Universiy of Tokyo Press (in Japanese), June 2009. [21] Y. Kuramoo, Chemical Oscillaions, Waves, an Turbulence, Springer-Verlag, New York, 1984. [22] T. Kunihiro, Reucion of iffusion equaions an envelopes Geomerical inerpreaion of he renormalizaion-group meho an consrucion of invarian manifols, Busuri (in Japanese), vol.65, no.9, pp.683 690, Sep. 2010. [23] K. Waabe an M. Aia, Flucuaions in quasi-saic approach escribing he emporal evoluion of rery raffic, 23r Inernaional Teleraffic Congress (ITC 2011), San Francisco, USA, Sep. 2011. Masaki Aia receive his B.S. an M.S. egrees in Theoreical Physics from S. Paul s Universiy, Tokyo, Japan, in 1987 an 1989, respecively, an receive he Ph.D. in Telecommunicaions Engineering from he Universiy of Tokyo, Japan, in 1999. In April 1989, he joine NTT Laboraories. From April 2005 o March 2007, he was an Associae Professor a he Faculy of Sysem Design, Tokyo Meropolian Universiy. He has been a Professor of he Grauae School of Sysem Design, Tokyo Meropolian Universiy since April 2007. His curren ineress inclue raffic issues in compuer communicaion neworks. He is a member of he IEEE an he Operaions Research Sociey of Japan.