HOMOGENIZATION OF DIFFUSION PROCESSES ON SCALE-FREE METRIC NETWORKS

Transcription

1 Elctroic Joural of Diffrtial Equatios, Vol. 26 (26), No. 282, pp. 24. ISSN: URL: or HOMOGENIZATION OF DIFFUSION PROCESSES ON SCALE-FREE METRIC NETWORKS FERNANDO A. MORALES, DANIEL E. RESTREPO Abstract. This work studis th homogizatio of diffusio procsss o scal-fr mtric graphs, usig wak variatioal formulatios. Th oscillatios of th diffusio cofficit alog th dgs of a mtric graph iduc itral sigularitis i th global systm which, togthr with th high complxity of larg tworks costitut sigificat difficultis i th dirct aalysis of th problm. At th sam tim, ths facts also suggst homogizatio as a viabl approach for modlig th global bhavior of th problm. To that d, w study th asymptotic bhavior of a squc of boudary problms dfid o a std collctio of mtric graphs. This papr prsts th wak variatioal formulatio of th problms, th covrgc aalysis of th solutios ad som umrical xprimts.. Itroductio A scal-fr twork is a larg graph with powr law dgr distributio, i.., P[dg(v) = k] k γ whr γ is a fixd positiv costat. Equivaltly, th probability of fidig a vrtx of dgr k, dcays as a powr-law of th dgr valu. Powr-law distributd tworks ar of oticabl itrst bcaus thy hav b frqutly obsrvd i vry diffrt filds such as th World Wid Wb, busiss tworks, uroscic, gtics, coomics, tc. Th currt rsarch o scal-fr tworks is maily focusd i thr aspcts: first, gratio modls (s [, 2]), scod, solid vidc dtctio of tworks with powr-law dgr distributio (s [25, 9, 5, 24]). Th third ad fial aspct studis th xtt to which th powr-law distributio rlats with othr structural proprtis of th twork, such as slf-orgaizatio (s [6, 9]); this is subjct of its dbat, s [6] for a comprhsiv survy o th mattr. This articl studis scal-fr tworks from a vry diffrt prspctiv. Its mai goal is to itroduc a homogizatio procss o th twork to rduc th origial ordr of complxity but prsrv th sstial faturs (s Figur ). I this way, th homogizd or upscald twork is rliabl for data aalysis whil, at th sam tim, ivolvs lowr computatioal costs ad lowr umrical istability. Additioally, th homogizatio procss drivs a atr ad mor structural pictur of th startig twork sic ucssary complxity is rplacd by th avrag asymptotic bhavior of larg data. Th phomo kow as Th Aggrgatio Problm i coomics is a xampl of how this typ of rasoig 2 Mathmatics Subjct Classificatio. 35R2, 35J5, 5C7, 5C82. Ky words ad phrass. Coupld PDE systms; homogizatio; graph thory. c 26 Txas Stat Uivrsity. Submittd Dcmbr 3, 25. Publishd Octobr 22, 26.

2 2 F. A. MORALES, D. E. RESTREPO EJDE-26/282 is implicitly applid i modlig th global bhavior of larg tworks (s [8]). Usually, homogizatio tchiqus rquir som assumptios of priodicity of th sigularitis or priodicity of th cofficits of th systm (s [6, ]), i tur this cas dmads avragig hypothss i th Csàro ss. Th rsultig twork has th dsird faturs bcaus of two charactristic proprtis of scal-fr tworks. O o had, thy rsmbl star-lik graphs (s [7]), o th othr had, thy hav a commuicatio krl carryig most of th twork traffic (s [7]). This articl, for th sak of clarity, rstricts th aalysis to th asymptotic bhavior of diffusio procsss o star-lik mtric graphs (s Dfiitio 2.2 ad Figur 2 blow). Howvr, whil most of th modls i th prxistt litratur ar cocrd with th strog forms of diffrtial quatios (s [3] for a gral survy ad [22] for th stochastic modlig of advctio-diffusio o tworks), hr w us th variatioal formulatio approach, which is a vry usful tool for upscalig aalysis. Mor spcifically, w itroduc th psudo-discrt aalogous of th classical statioary diffusio problm (K p) = f i Ω, p = o Ω, (.) whr K is th diffusio cofficit (s Dfiitio 2.6 ad Equatio (2.8) blow). Bcaus of th variatioal formulatio it will b possibl to attai a-priori stimats for a squc of solutios, a asymptotic variatioal form of th problm ad th computatio of ffctiv cofficits. Fially, from th tchiqu, it will b clar how to apply th mthod to scal-fr mtric tworks i gral. Throughout th xpositio th trms homogizd, upscald ad avragd hav th sam maig ad w us thm idistictly. This articl is orgaizd as follows, i Sctio 2 th cssary backgroud is itroducd for L 2, H -typ spacs o mtric graphs as wll as th strog form ad th wak variatioal form, togthr with its wll-posdss aalysis. Also a quick rviw of quidistributd squcs ad Wyl s Thorm is icludd to b usd mostly i th umrical xampls. I Sctio 3 w itroduc a gomtric sttig ad a squc of problms for asymptotic aalysis, a-priori stimats ar prstd ad a typ of covrgc for th solutios. I Sctio 4, udr mild hypothss of Csàro covrgc for th forcig trms, th xistc ad charactrizatio of a limitig or homogizd problm ar show. Fially, Sctio 5 is rsrvd for th umrical xampls ad Sctio 6 has th coclusios. 2. Prlimiaris 2.. Mtric Graphs ad Fuctio Spacs. W bgi this sctio by rcallig som facts for mbddigs of graphs. Dfiitio 2.. A graph G = (V, E) is said to b mbddabl i R N if it ca b draw i R N so that its dgs itrsct oly at thir ds. A graph is said to b plaar if it ca b mbddd i th pla. It is a wll-kow fact that ay simpl graph ca b mbddd i R 2 or R 3 (dpdig whthr it is plaar or ot) i a way that its dgs ar draw with straight lis; s [8] for plaar graphs ad [4] for o-plaar graphs. I th followig it will always b assumd that th graph is alrady mbddd i R 2 or R 3.

3 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES 3 K, F K (), F () v 2 K (), F () K, F v 3 K, F v K (), F () K, F v 4 K (), F () (a) (b) Figur. (a) dpicts a scal-fr twork. (b) dpicts a homogizatio of th origial twork. Dfiitio 2.2. Lt G = (V, E) b a graph mbddd i R 2 or R 3, dpdig o th cas. (i) Th graph G is said to b a mtric graph if ach dg E is assigd a positiv lgth l (, ). (ii) Th graph G is said to b locally fiit if dg(v) < + for all v V. (iii) If th graph G is mtric, th boudary of th graph is dfid by th st of vrtics of dgr o. Th st will also b domiatd as th st of boudary vrtics ad dotd by V := {v V : dg(v) = }. (2.) (iv) Giv a mtric graph w dfi its atural domai by Ω G := E it(). (2.2) Dfiitio 2.3. Lt G = (V, E) b a mtric graph, w dfi th followig associatd Hilbrt spacs (i) Th spac of squar itgrabl fuctios, or mass spac is dfid by L 2 (G) := E L 2 (), dowd with its atural ir product f, g L2 (G) := fg. E (ii) Th rgy spac of fuctios is dfid by H (G) := { f E H () : lim f(x) = lim f(x), x v, x x v, x σ for all vrtics v V ad all dgs, σ icidt o v }. (2.3a) (2.3b) (2.4a)

4 4 F. A. MORALES, D. E. RESTREPO EJDE-26/282 I th squl f(v) := lim{f(x) : x v, x }, whr E is ay dg icidt o v. W dow th spac with its atural ir product f, g H (G) := f g + f g. (2.4b) E E Hr dots th drivativ alog th dg E. (iii) Th spac H (G) is dfid by H (G) := { f H (G) : f(v) =, for all v V }, (2.5) dowd with th stadard ir product (2.4b). Rmark 2.4. Lt G b a mtric graph. (i) Not that th dfiitio of is ambiguous i Exprssio (2.4b). Such ambiguity will caus o problms sic th biliar structur of th ir product is idiffrt to th choic of dirctio (q, r) q r = ( q ) ( r). (ii) Whvr thr is d to spcify th dirctio of th drivat, w writ,v to idicat th dirctio poitig from th itrior of th dg towards th vrtx v o o of its xtrms. (iii) Notic that if th mtric graph G is coctd, th th Poicaré iquality holds, ad th ir product (f, g) f g, (2.6) E is quivalt to th stadard o (2.4b) i th spac H (G). (iv) Obsrv that th coditio of agrmt of a fuctio f H (G) o th vrtics of th graph G, dos ot cssarily imply cotiuity as a fuctio f : Ω G R. For if th dgr of a vrtx v V is ifiit ad th fuctio is cotiuous o v, th it follows that th covrgc f(v) := lim{f(x) : x v, x } is uiform for all th dgs icidt o v. Such a coditio ca ot b drivd from th orm iducd by th ir product (2.4b), although th fuctio f it() is cotiuous for all E. Dfiitio 2.5. Lt G = (V, E ) b a squc of graphs. (i) Th squc {G : N} is said to b icrasig if V V + ad E E + for all N. (ii) Giv a icrasig squc of graphs {G : N}, w dfi th limit graph G = (V, E) i th atural way i.., V := N V E := N E. I aalogy with mooto squcs of sts w adopt th otatio G := N G Strog ad wak forms of th statioary diffusio problm o graphs. Dfiitio 2.6. Lt G = (V, E) b a locally fiit mtric graph, F L 2 (G) ad h : V V R, dfi th diffusio problm F i Ω G. (2.7a) E ( K p) = E

5 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES 5 Hr, K L (Ω G ) is a ogativ diffusio cofficit. W dow th problm with ormal strss cotiuity coditios lim p(x) = p(v) for all p V V, (2.7b) x v, x ad ormal flux balac coditios h(v) + lim,v p(x) = for all v V V. (2.7c) x v, x E, icidt o v Hr,,v dots th drivativ alog th dg poitig away from th vrtx v. Fially, w dclar homogous Dirichlt boudary coditios p(v) = for all v V. (2.7d) A wak variatioal formulatio of this problm is giv by p H (G) : K p q = F q + E E v V V h(v)q(v) (2.8) for all q H (G). For th sak of compltss w prst th followig stadard rsult. Propositio 2.7. Lt G = (V, E) b a locally fiit coctd mtric graph such that V ad lt K L (Ω G ) b a diffusio cofficit such that K(x) c K > almost vrywhr i Ω G. Th Problm (2.8) is wll-posd. Proof. Clarly th fuctioals o th right-had sid of (2.8) ar liar ad cotiuous, as wll as th biliar form b(p, q) := E K p q of th lft-had sid. Additioally, K p 2 c K p 2 c p 2 H () = c p 2 H (G). E E E Th first iquality abov holds du to th coditios o K. Th scod iquality hods du to th Dirichlt homogous boudary coditios ad th coctdss of th graph G, which prmits th Poicaré iquality o th spac H (G) as discussd i Rmark 2.4-(iii). Thrfor, by th Lax-Milgram Thorm, Problm (2.8) is wll-posd Equidistributd squcs ad Wyl s Thorm. Th brif rviw of quidistributd squcs ad Wyl s thorm of this sctio will b applid, almost xclusivly i th umrical xampls blow, s Sctio 5. For a complt xpositio o quidistributd squcs ad Wyl s Thorm s [23]. Dfiitio 2.8. A squc {θ : N} is calld quidistributd o a itrval [a, b] if for ach subitrval [c, d] [a, b] it holds that #{i : θ i [c, d], i } lim = d c b a. (2.9) Thorm 2.9 (Wyl s Thorm). Lt { θ : N } b a squc o [a, b], th followig coditios ar quivalt: (i) Th squc {θ : N} is quidistributd i [a, b]. (ii) For vry Rima itgrabl fuctio f : [a, b] C lim i= f(θ i ) = b a b a f(θ) dθ.

6 6 F. A. MORALES, D. E. RESTREPO EJDE-26/282 Dfiitio 2.. Lt Ω = B(, ) R 2 ad lt f : Ω R b such that its rstrictio to vry sphr B(, ρ) with ρ < is Rima itgrabl. Th, w dfi its agular avrag by th avrag valu of f alog th sphr (B(, ρ)), i.., m θ [f] : [, ) R, m θ [f](t) := 2π 2π 3. Squc of problms f ( t cos θ, t si θ) ) dθ. (2.) I this sctio w aalyz th bhavior of th solutios {p : N} of a family of wll-posd problms o a vry particular icrasig squc of graphs {G : N}, dpictd i Figur Gomtric sttig ad th -stag problm. I th followig w dot by Ω, S th uit disk ad th uit sphr i R 2 rspctivly. Th fuctio F : Ω G R is such that F Ω L 2 (Ω ) for all N, {h : N} is a squc of ral umbrs ad th diffusio cofficit K L (Ω G ) is such that K( ) c K > almost vrywhr i Ω G. Dfiitio 3.. Lt {v : } b a quidistributd squc i S ad v := R 2. (i) For ach N dfi th graph G = (V, E ) as follows: V := {v : i }, E := {v v i : i }. (3.) (ii) For th icrasig squc of graphs {G : N}, dfi th limit graph G := N G as dscribd i Dfiitio 2.5. (iii) I th followig w dot th atural domais corrspodig to G, G by Ω G ad Ω rspctivly. (iv) For ay dg E w dot by v its boudary vrtx ad θ [, 2π] th dirctio of th dg. (v) From ow o, for ach dg = v v ad f : R a fuctio, it will b udrstood that its o-dimsioal paramtrizatio, is oritd from th ctral vrtx v to th boudary vrtx v. Cosqutly th drivativ quals,v. (vi) For ay giv fuctio f : Ω G R (or f : Ω R) w dot by f : (, ) R, th ral variabl fuctio f (t) := (f )(t cos θ, t si θ ) o th dgs E (or E rspctivly). Rmark 3.2. From th followig aalysis, it will b clar that it is ot cssary to assum that th squc of vrtics {v : N} of th graph is quidistributd or that th vrtics ar i S or v that th graph is mbddd i R 2. W adopt ths assumptios, maily to facilitat a gomtric visualizatio of th sttig. For th rst of this articl it will b assumd that {G : N} is th icrasig squc of graphs, with G its limit graph, as i th Dfiitio 3. abov. Nxt, w dfi th family of wll-posd problms to b studid, for ach N, ths ar p H (G ) : K p q = F q + h q(v ), (3.2) E E for all q H (G ). W d to aalyz th asymptotic bhavior of th squc of solutios {p : N}. O of th mai challgs is that th lmts of th

7 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES 7 G 5 v G v v 6 v 4 v 4 v 3 v 3 v v v v 2 v 8 v 2 v 7 v 5 (a) (b) v 5 Figur 2. (a) dpicts th stag 5 of th graph G. (b) dpicts a mor gral stag of th graph G. squc ar ot dfid o th sam global spac. Th fact that p () may ot b zro maks it impossibl to xtd this fuctio to H (G) dirctly, howvr it will play a ctral rol i th asymptotic aalysis of th problm Estimats ad dgwis covrgc. I this sctio w obtai stimats for th squc of solutios, svral stps hav to b mad as it is ot dirct to attai thm. W start itroducig coditios to b assumd from ow o. Hypothsis 3.3. (i) Th forcig trm F is dfid i th whol domai, i.., F : Ω G R ad M := sup E F L 2 () < +. (ii) Th squc { h : N } is boudd. (iii) Th prmability cofficit satisfis that K L (Ω G ), if x ΩG K(x) = c K > ad K = K() i.., it is costat alog ach dg E. Rmark 3.4. Not that Hypothsis 3.3-(ii) stats that th balac of ormal flux o th ctral vrtx is of ordr O(), i.., it scals with th umbr of icidt dgs. Lmma 3.5. Udr Hypothsis 3.3, th followig facts hold (i) Th squc {p () : N} R is boudd. (ii) Lt E b a dg of th graph G th, th squc { p () : E } R is boudd. Morovr, thr xists M such that p () M for all E ad N such that E. (iii) Suppos that th squcs { E (t )F (t)dt : N }, { h : N } ad { E K() : N } ar covrgt, th lim p () = lim E (t )F (t)dt h E K(). (3.3a)

8 8 F. A. MORALES, D. E. RESTREPO EJDE-26/282 For ay fixd dg E, it holds lim K() p () = L() := (t )F (t)dt K() lim σ E (t )F σ(t)dt h σ E K(σ). (3.3b) Morovr, th covrgc is uiform i th followig ss: for ach ɛ > thr xists N N such that, if > N ad E, th K() p () L() < ɛ. (3.3c) Proof. (i) Lt q H (G ) b th fuctio such that q() = ad q (t) = t for all E. Tst (3.2) with q, this yilds q() p = F q + h q(). (3.4) E K E Computig ad doig som stimats w obtai #E c K p () 3 E F L 2 () + h. Hc p () M + h. c K c K This provs th first part. (ii) Lt E b a fixd dg, lt N b such that E ad lt p b th solutio to (3.2). Lt q H (G ) b as i th prvious part ad tst (3.2) to obtai K()p () q() + = K()p () = F q + σ E, σ σ E, σ σ E, σ σ K K σ p σ q σ σ F q + h q(). σ 2 p q + q() σ E, σ K σ p () I th xprssio abov, itgratio by parts was applid to ach summad σ of th lft had sid, to obtai th scod quality. Now, rcallig that E K p () = h ad that K 2 p = F for ach E, th quality abov rducs to K()p () = (t )F (t)dt K() p (). (3.5) Hc, p () p () + F L2 () 2 M + h. (3.6) c K c K c K Choosig M > larg ough, th rsult follows.

9 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES 9 (iii) Lt q H (G ) b as i th prvious part, tstig (3.2) with it yilds (3.4), which is quivalt to p () K = F q + E h q(). E Now, lttig, Equality (3.3a) follows bcaus th hypothsis K() > c K for all E, implis that E K() c K >. For th covrgc of { p () : E }, lt i (3.5) to obtai (3.3b). For th uiform covrgc obsrv that (3.5) yilds K() p () L() = K() lim K L lim σ E (t )F σ(t)dt h σ E K(σ) K()p () σ E (t )F σ(t)dt h p (). σ E K(σ) Fially, choos N N such that th right had sid of th xprssio abov is lss tha ɛ > for all > N th, th trm of th lft-had sid is also domiatd by ɛ > for all > N ad E. Rmark 3.6. It is clar that i Lmma 3.5 part (iii), it suffics to rquir th mr xistc of th limit σ E lim (t )F σ(t)dt h, σ E K(σ) to attai th sam coclusio. Howvr, th hypothss i (iii) ar cssary to idtify th asymptotic problm ad to comput th ffctiv cofficits. Thorm 3.7. Lt F, {h : N} ad K satisfy Hypothsis 3.3 as i Lmma 3.5. Th: (i) Thr xists a costat M such that p H 2 () M for all E ad N such that E. (ii) For ach E thr xists p () H () such that p p () H (). Morovr, this covrgc is uiform i th followig ss: for ach ɛ > thr xists N N such that, if > N ad E, th p p () H () < ɛ. (3.7) (iii) Th fuctio p : Ω G R giv by p := p () is wll-dfid ad it will b rfrrd to, as th limit fuctio. Proof. (i) Fix E ad lt N b such that E. Sic p is th solutio of (3.2) it follows that K() 2 p = F L 2 () for all E, i particular p H 2 () with 2 p L2 () c K F L2 () c K M. O th othr had, sic p is absolutly cotiuous, th fudamtal thorm of calculus applis, hc p (x) = p () + x 2 p (t) dt = p () + x F (t) dt for all x. Thrfor, p (x) 2 = 2 p () 2 + 2x F 2 L 2 () 2M 2 + 2M 2. Whr M is th global boud foud i Lmma 3.5-(ii) abov. Itgratig alog th dg givs p L 2 () 2(M 2 + M 2 ). Nxt, giv that p (v) = for

10 F. A. MORALES, D. E. RESTREPO EJDE-26/282 all v E, rpatig th prvious argumt yilds p L2 () 2(M 2 + M 2 ). Fially, sic 2 p L 2() c K M, th rsult follows for ay M satisfyig ( M 2 4M ) c 2 M 2. K (ii) Fix E, du to th prvious part th squc {p : E } is boudd i H 2 (), th thr xists p () H 2 () ad a subsquc { k : k N} such that as k, p k p () wakly i H 2 () ad strogly i H (). Lt ϕ H () b such that quals zro o th boudary vrtx of. Lt q b th fuctio i H (G ) such that q = ϕ ad q σ (t) = ϕ()( t) (liar affi) for all σ E {}. Tst (3.2) with this fuctio to obtai K() p q + K σ p σ q = F q + σ E, σ σ E, σ σ σ F q + h q(). Itgratig by parts th scod summad of th lft-had sid yilds K() p ϕ K 2 σ p q ϕ() K σ p () = F ϕ + σ E, σ σ E, σ σ σ F ϕ + h q(). σ E, σ Sic p is a solutio of th problm, th abov xprssio rducs to K() p ϕ + K() p ()ϕ() = F q. (3.8) Equality (3.8) holds for all N, i particular it holds for th covrgt subsquc { k : k N}; takig limit o this squc ad rcallig (3.3b), w hav K() p () ϕ = F ϕ L()ϕ(). (3.9) Th (3.9) holds for all ϕ H () vaishig at v, th boudary vrtx of. Dfi th spac H() := {ϕ H () : ϕ(v ) = } ad cosidr th problm u H() : K() u ϕ = F ϕ L()ϕ() ϕ H(). (3.) By th Lax-Milgram Thorm th problm abov is wll-posd, additioally it is clar that p () H(), thrfor it is th uiqu solutio to (3.) abov. Now, rcall that {p : E } is boudd i H 2 () ad that th prvious rasoig applis for vry strogly H ()-covrgt subsquc, thrfor its limit is th uiqu solutio to (3.). Cosqutly, by Rllich-Kodrachov, it follows that th whol squc covrgs strogly. Nxt, for th uiform covrgc tst both Statmts (3.8), (3.9) with (p p () ) ad subtract thm to obtai c K p p () 2 H () K() p p () 2 ( L() K() p () )( p () p () () )

11 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES L() K() p () p p () H (). Th abov yilds p p () H () c K L() K() p (). Now, th uiform covrgc (3.7) follows from (3.3c), which cocluds th scod part. (iii) Sic p () () = lim p () for all E, th limit fuctio p is wlldfid ad th proof is complt. 4. Homogizd problm: a Csàro avrag approach I this sctio w study th asymptotic proprtis of th global bhavior of th solutios {p : N}. It will b s that such aalysis must b do for crtai typ of Csàro avrags of th solutios. This is obsrvd by th tchiqus ad th hypothss of Lmma 3.5, which ar cssary to coclud th local covrgc of {p : E }. Additioally, th typ of stimats ad th umrical xprimts suggst this physical magitud as th most sigificat o for global bhavior aalysis ad upscalig purposs. W start itroducig som cssary hypothss. Hypothsis 4.. Suppos that F, {h : N} ad K satisfy 3.3, ad (i) Th diffusio cofficit K : Ω G (, ) has fiit rag. Morovr, if K(E) = {K i : i I} ad B i := { E : K() = K i }, th K() = #(E B i ) E B i With s i > for all i I ad such that I i= s i =. (ii) Th forcig trm F satisfis #(E B i ) F E B i K i s ik i. (4.) F i, for i I. (4.2) Whr F i L 2 (, ) ad th ss of covrgc is poitwis almost vrywhr. (iii) Th squc { h : N } is covrgt with h = lim h. Rmark 4.2. (i) Not that if (i) ad (ii) i Hypothsis 4. ar satisfid, th E F = I i= #(E B i ) ( #(E B i ) E B i F ) I s i Fi. Hc, th squc {F : E} is Csàro covrgt. (ii) A familiar cotxt for th rquird covrgc (4.2) i Hypothsis 4. is th followig. Lt F b a cotiuous ad boudd fuctio dfid o th whol disk Ω ad suppos that for ach i I, th squc of vrtics {v : N, v v B i } is quidistributd o S. Th, by Thorm 2.9, for ay fixd t (, ) it holds that #(E B i) Dfiitio 2.. E B i F (t) i= m θ[f] i., th agular avrag itroducd i

12 2 F. A. MORALES, D. E. RESTREPO EJDE-26/ Estimats ad Csàro covrgc. Lmma 4.3. Lt F, {h : N} ad K satisfy Hypothsis 4.. Th (i) Th squc { E p : N } is boudd i H 2 (, ). (ii) Th squc { } p : N #(E B i ) E B i is boudd i H 2 (, ) for all i {,..., I}. Proof. (i) Tst (3.2) with p to obtai c K p 2 L 2 () K p 2 E E F p + h p (v ) E ( ) /2 ( ) /2 F 2 L 2 () p 2 L 2 () + h p (v ). E E (4.3) Sic p (v ) = for all E, it follows that p L 2 () p L 2 () ad p H () 2 p L 2 (). Hc, dividig th abov xprssio by givs p 2 H () E ( ) /2 ( ) /2 2 F 2 L 2 () p 2 h H () + 2 p (v ) E E 2 M ( ) /2 p 2 H c K () + C. E Hr C > is a gric costat idpdt from N. I th scod li of th xprssio abov, w usd that M = sup E F L 2 () < +, { h : N} ar boudd ad that {p (v ) : N} is covrgt (thrfor boudd) as statd i Lmma 3.5-(i). Hc, th squc x := ( E p 2 H () )/2 is such that x 2 2 M c K x + C for all N, whr th costats ar all o-gativ. Th {x : N} must b boudd, but this implis p H (,) p H (,) E E p H () E ( ) /2 p 2 H (). E Fially, rcallig th stimat c K 2 p L2 (,) E E K() p L2 (,)

13 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES 3 th rsult follows. (ii) Fix i {, 2,..., I} th #(E B i ) H 2 (,) E B i p F L 2 (,) M, E #(E B i ) E p H 2 (,). O th right-had sid of th xprssio abov, th first trm is boudd bcuas of Hypothsis 4.-(iii), whil th bouddss of th scod trm was show i th prvious part. Thrfor, th rsult follows. Bfor prstig th limit problm w itroduc som cssary dfiitios ad otatio Dfiitio 4.4. Lt K satisfy 4. ad I = #K(E). Th (i) For all i I dfi w i := ( cos( 2π I i), si( 2π I i)) S, w := v = ad V := {w i : i I}. (ii) For all i I dfi th dgs σ i := w w i ad E := {σ i : i I}. (iii) Dfi th upscald graph by G := (V, E). (iv) For ay ϕ H (G) ad N dot by T ϕ H (G ), th fuctio such that ( T ϕ ) agrs with ϕ σi whvr B i. This is summarizd i th xprssio T ϕ = ( T ϕ ) I ( ) := ϕσi. E i= E B i I th squl w rfr to T ϕ as th H (G )-mbddig of ϕ. (v) I th followig, for ay i I, w adopt th otatio i := σi. Similarly, for ay giv fuctio f : Ω G R ad dg σ i E, w dot by f i : (, ) R, th ral variabl fuctio f i (t) := (f σi ) ( t cos( 2π I i), t si( 2π I i)). Thorm 4.5. Lt F, {h : N} ad K satisfy Hypothsis 4.. Th (i) Th followig problm is wll-posd. I I p H (G) : s i K i i p i q = s i Fi q i + h()q(), (4.4) σ i i= for all q H (G). I th squl, w rfr to Problm (4.4) as th upscald or th homogizd problm ad its solutio p as th upscald or th homogizd solutio idistictly. (ii) Th squc of solutios {p : N} satisfis p p i #(E B i ) H, for i I. (4.5) (,) E B i (iii) Th limit fuctio p : Ω G R satisfis p p i #(E B i ) H (,), for i I ; E B i E p I i= i= s i p i H (,).

14 4 F. A. MORALES, D. E. RESTREPO EJDE-26/282 Proof. (i) It follows immdiatly from Propositio 2.7. (ii) Lt ϕ H (G) ad lt T ϕ b its H (G )-mbddig. Not th qualitis K p T ϕ ad E = = E I i= I i= K i F T ϕ = E B i #(E B i ) I i= K i p T ϕ #(E B i ) ( #(E B i ) ( #(E B i ) E B i p ) ϕ i, E B i F From th prvious obsrvatios, tstig (3.2) with T ϕ, givs I i= = #(E B i ) I i= #(E B i ) K i ( #(E B i ) ( #(E B i ) E B i p E B i F ) ϕ i ) ϕ i. )ϕ i + h ϕ(). (4.6a) (4.6b) (4.7) By Lmma 4.3-(ii) thr xist a subsquc { k : k N} ad a collctio {ξ i : i I} H 2 (, ) such that #(E k B i ) ξ i k p k E k B i wakly i H 2 (, ) ad strogly i H (, ) for i I. O th othr had, by Hypothsis 4.-(ii) th itgrad of th right-had sid i (4.7) is covrgt for all i {,..., I}. Bcaus of Hypothsis 4.-(i) th squcs { #(E Bi) : N} ar also covrgt for all i {,..., I}. Th, usig (4.7) for th subsquc k ad lttig k givs I I s i K i ξ i ϕ i = s i F i ϕ i + hϕ(). i= i= Not that sic p k H (G k ), w hav () #(E k B i ) = #(E k B j ) E k B i p k p k (), i, j {,..., I}. E k B j (4.8) I particular ξ i () = ξ j (); cosqutly th fuctio η H (G) such that η i = ξ i is wll-dfid. Morovr, (4.8) is quivalt to I I s i K i i η i ϕ = s i Fi ϕ + h()ϕ(). σ i σ i i= i=

15 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES 5 Sic th lattr variatioal statmt holds for ay ϕ H (G) ad η H (G), from th prvious part it follows that η p. Fially, th whol squc {p : N} satisfis (4.5) bcaus, for vry subsquc {p j : j N} thr xists yt aothr subsquc {p j l : l N} satisfyig th covrgc Statmt (4.5). This cocluds th scod part. (iii) Both coclusios follow immdiatly from th prvious part ad th uiform covrgc (3.7) show i Thorm 3.7. Rmark 4.6 (Probabilistic Flxibilitis of th Rsults). Cosidr th followig two radom variabls: (i) Lt X : E (, ) b a radom variabl of fiit rag {K i : i I} ad such that E[X = K i ] = s i for i I. Notic that by th Law of Larg Numbrs, with probability o it holds E B i X() s ik i. (4.9) (ii) Lt Y : E L 2 (, ) b a radom variabl such that sup E Y () L2 () < + ad such that Y () F i, for i I. (4.) #(E B i ) E B i Thrfor, Thorm 4.5 holds, wh rplacig K by X or F by Y or wh makig both substitutios at th sam tim. 5. Exampls I this sctio w prst two typs of umrical xprimts. Th first typ ar vrificatio xampls, supportig our homogizatio coclusios for a problm whos asymptotic bhavior is kow xactly. Th scod typ ar of xploratory atur, i ordr to gai furthr isight of th phomo s upscald bhavior. Th xprimts ar xcutd i a MATLAB cod usig th Fiit Elmt Mthod (FEM); it is a adaptatio of th cod fmd.m [2]. 5.. Gral Sttig. For th sak of simplicity th vrtics of th graph ar giv by v l := (cos l, si l) S, as it is kow that {v l : l N} is quidistributd i S (s [23]). Th diffusio cofficit hits oly two possibl valus: o ad two. Two typs of cofficits will b aalyzd, K d, K p a dtrmiistic ad a probabilistic o rspctivly. Thy satisfy K d : Ω G {, 2}, K d (v l v ) := {, l mod 3, 2, l mod 3. (5.a) K p : Ω G {, 2}, E[K p = ] = 3, E[K p = 2] = 2 3. (5.b) I our xprimts th asymptotic aalysis is prformd for K p big a fixd ralizatio of a radom squc of lgth, gratd with th biomial distributio /3, 2/3. Sic #K d (E) = #K p (E) = 2, it follows that th upscald graph G has oly thr vrtics ad two dgs amly w = (, ), w 2 = (, ), w = (, ) ad σ = w w, σ 2 = w 2 w. Also, dfi th domais Ω G := { v l v : l N, K(v l v ) = }, Ω 2 G := { v l v : l N, K(v l v ) = 2 },

16 6 F. A. MORALES, D. E. RESTREPO EJDE-26/282 whr K = K d or K = K p dpdig o th probabilistic or dtrmiistic cotxt. Additioally, w dfi p := p, p 2 := p. #(E B ) #(E B 2 ) E B E B 2 I all th xampls w us th forcig trms h = for vry N. Th FEM approximatio is do with lmts pr dg with uiform grid. I ach xampl w prst two graphics for valus of chos from {, 2, 5,, 5, }, basd o optical atss. For visual purposs, i all th cass th dgs ar colord with rd if K() = or blu if K() = 2. Also, for displayig purposs, i th cass {, 2} th dgs v l v ar labld with l for idtificatio, howvr for {5,, 5, } th labls wr rmovd sic thy ovrload th imag Vrificatio for xampls. Exampl 5. (A Rima itgrabl forcig trm). W bgi our xampls with th most familiar cotxt, as discussd i Rmark 4.2. Dfi F : Ω R, F (t cos l, t si l) := π 2 si(πt) cos(l). (5.2) Sic both squcs {v l : l N, l mod 3} ad {v l : l N, l mod 3} ar quidistributd, Thorm 2.9 implis F = m θ [F Ω G ] = F 2 = m θ [F Ω 2 G ] = m θ [F ]. Hr F, F2 ar th limits dfid i Hypothsis 4.-(ii). For this cas th xact solutio of th upscald Problm (4.4) is giv by p := p σ + p σ2 H (G), with p (t) = p 2 (t) =. For th diffusio cofficit w us th dtrmiistic o, K d dfid i (5.a). Tabl summarizs th covrgc bhavior. Tabl. Covrgc of solutios to Exampl 5.: K = K d. p p L 2 ( ) p 2 p 2 L 2 ( ) p p H ( 2) p 2 p 2 H ( 2) Exampl 5.2 (Probabilistic flxibilitis for Exampl 5.). This xprimt follows th obsrvatios i Rmark 4.6. I this cas X := K p, dfid i (5.b). Lt Z : N [, ] b a radom variabl with uiform distributio ad dfi Y : Ω G R, Y (t cos l, t si l) := π 2 si(πt) cos(l) + Z(l). (5.3) It is straightforward to show that X ad Y satisfy Hypothsis 4. ad by th Law of Larg Numbrs, thy also satisfy (4.9) ad (4.) rspctivly. Thrfor, Ȳ = m θ [F Ω G ] = Ȳ2 = m θ [F Ω 2 G ] = m θ [F ] = p = p 2 =. Tabl 2 is th summary for a fixd ralizatio of X (to kp th dg colorig cosistt) ad diffrt ralizatios of Y o ach stag. Covrgc is obsrvd ad, as xpctd, it is slowr tha i th prvious cas. This would also occur for diffrt ralizatios of X ad Y simultaously.

17 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES (a) Solutio p, =. (b) Solutio p, =. Figur 3. Solutios to Exampl 5.. Diffusio cofficit K d, s (5.a). Th solutios dpictd i (a) ad (b) o th dgs v l v ar colord with rd if K d (v l v ) = (i.., l mod 3), or blu if K d (v l v ) = 2 (i.., l mod 3). Forcig trm F : Ω R, s (5.2). Tabl 2. Covrgc for solutios to For Exampl 5.2: K = K p. p p L2 ( ) p 2 p 2 L2 ( ) p p H ( 2) p 2 p 2 H ( 2) Exampl 5.3 (A o-rima itgrabl forcig trm). For our fial thortical xampl w us a o-rima Itgrabl forcig trm. Morovr, th followig fuctio is highly oscillatory isid ach subdomai Ω G ad Ω2 G, ad it ca ot b s as Rima itgrabl wh rstrictd to ay of ths sub-domais. Lt F : Ω G R b dfid by { 4π 2 si(2πt) + ( ) l 6 (l l F (t cos l, t si l) := 2π ), l mod 3, π 2 si(πt) + ( ) l 6 (l l 2π ), l mod 3. (5.4) O th o had, both squcs {v l : l N, l mod 3} ad {v l : l N, l mod 3} ar quidistributd. O th othr had, both parts of th forcig trm, th radial ad th agular, ar Csàro covrgt o ach Ω i G for i =, 2. Th Csàro avrag of th agular summad is zro o Ω i G for i =, 2. I cotrast, th radial summad ca b s as Rima itgrabl sparatly o ach Ω i G for i =, 2; thrfor, by Thorm 2.9 its Csàro avrag is giv by F = m θ [F Ω G ]

18 8 F. A. MORALES, D. E. RESTREPO EJDE-26/ (a) Solutio p 2, = 2. (b) Solutio p 5, = 5. Figur 4. Solutios of Exampl 5.2 Fixd ralizatio of diffusio cofficit K p, s (5.b). Forcig trm Y : Ω G R, s (5.3), with Z : N [, ], radom variabl ad Z uiformly. Diffrt ralizatios for Y o ach stag. Th solutios dpictd i (a) ad (b) o th dgs v l v ar colord with rd if K p (v l v ) = (E[K p = ] = 3 ), or blu colord if K p(v l v ) = 2 (E[K p = 2] = 2 3 ). ad F 2 = m θ [F Ω 2 G ]; mor xplicitly, F (t) = (2π) 2 si(2πt), F2 (t) = π 2 si(πt). (5.5) For this cas th xact solutio p = p σ + p σ2 H (, ) of th upscald Problm (4.4) is giv by p (t) = si(2πt), p 2 (t) = si(πt). (5.6) 2 W summariz th covrgc bhavior i Tabl 3. Tabl 3. Covrgd of solutios to Exampl 5.3: K = K d. p p L2 ( ) p 2 p 2 L2 ( ) p p H ( 2) p 2 p 2 H ( 2) Numrical xprimts. I this sctio w prst two xampls, brakig th hypothss rquird i th thortical aalysis discussd abov. As thr is o kow xact solutio, w follow Cauchy s covrgc critrio for th squcs

19 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES (a) Solutio p 2, = 2. (b) Solutio p 5, = 5. Figur 5. Solutios of Exampl 5.3 Diffusio cofficit K d, s (5.a). Th solutios dpictd i (a) ad (b) o th dgs v l v ar colord with rd if K d (v l v ) = (i.., l mod 3), or blu if K d (v l v ) = 2 (i.., l mod 3). Forcig trm F : Ω G R s (5.4). { p i : N} with i =, 2. Howvr, w do ot sampl oly poits but itrvals of obsrvatio ad rport th avrags of th obsrvd data. Mor spcifically ɛ i := +5 j= 4 for i =, 2, =, 2,,. p j i pj i L 2 ( i), δ i := +5 j= 4 p j i pj i H ( i), Exampl 5.4 (A locally uboudd forcig trm). For our xprimt w us a variatio of Exampl 5.3, kpig th wll-bhavd radial part but addig a uboudd agular part, which is kow to b Csàro covrgt to zro. Cosidr th forcig trm F : Ω G R dfid by { 4π 2 si(2πt) + ( ) l l, l mod 3, F (t cos l, t si l) := π 2 si(πt) + ( ) l (5.7) l, l mod 3. Clarly, sup E F L 2 () = i.., Hypothsis 3.3-(i) is ot satisfid. It is ot hard to adjust th tchiqus prstd i Sctio 4. to this cas, wh th forcig trm is Csàro covrgt without satisfyig th coditio sup E F L2 () < ; howvr, th proprtis of dgwis uiform covrgc of Sctio 3.2 ca ot b cocludd. Cosqutly, w obsrv th followig covrgc bhavior. Exampl 5.5 (A forcig trm with uboudd frqucy mods). For our last xprimt w us a variatio of Exampl 5.3, kpig it boudd, but itroducig uboudd frqucis. Cosidr th forcig trm F : Ω G R dfid by { 4π 2 si(2πt l), l mod 3, F (t cos l, t si l) := π 2 (5.8) si(πt l), l mod 3.

20 2 F. A. MORALES, D. E. RESTREPO EJDE-26/282 Tabl 4. Covrgc of solutios to Exampl 5.4: K = K d. ɛ ɛ 2 δ δ (a) Solutio p 2, = 2. (b) Solutio p, =. Figur 6. Solutios to Exampl 5.4 Diffusio cofficit K d, s (5.a). Th solutios dpictd i (a) ad (b) o th dgs v l v ar colord with rd if K d (v l v ) = (i.., l mod 3), or blu if K d (v l v ) = 2 (i.., l mod 3). Forcig trm F : Ω G R s (5.7). Clarly, F vrifis Hypothsis 3.3, cosqutly Lmma 3.5 implis dgwis uiform covrgc of th solutios, howvr Hypothsis-(ii) 4. is ot satisfid. Thrfor, w obsrv that th whol squc is ot Cauchy, although it has Cauchy subsqucs as th Tabl 5 shows. Tabl 5. Covrgc of solutios to Exampl 5.5: K = K d. ɛ ɛ 2 δ δ

21 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES 2 (a) Solutio p 5, = 5. (b) Solutio p, =. Figur 7. Solutios of Exampl 5.5. Diffusio cofficit K d, s (5.a). Th solutios dpictd i (a) ad (b) o th dgs v l v ar colord with rd if K d (v l v ) = (i.., l mod 3), or blu if K d (v l v ) = 2 (i.., l mod 3). Forcig trm F : Ω G R s (5.8). It follows that this systm has mor tha o itral quilibrium. Cosqutly, a upscald modl of a systm such as this, should cotai ucrtaity which, i this spcific cas, rmais boudd du to th proprtis of th forcig trm F Closig obsrvatios. (i) Th authors trid to fid xprimtally a rat of covrgc usig th wll-kow stimat + log p i p i α i log p i p i log p i p i log p i p 2, i =, 2. i Th samplig was mad o th itrvals 5 j + 5, for =, 2,, 5 ad. Exprimts wr ru o all th xampls xcpt for Exampl 5.5. I o of th cass was solid umrical vidc dtctd that could suggst a ordr of covrgc for th phomo. (ii) Exprimts for radom variatios of th xampls abov wr also xcutd, udr th hypothsis that radom variabls wr subjct to th Law of Larg Numbrs. As xpctd, covrgc slowr tha its corrspodig dtrmiistic vrsio was obsrvd. This is importat for its applicability to upscalig tworks drivd from gam thory, s []. 6. Coclusios ad fial discussio Th prst work yilds svral accomplishmts ad also limitatios as w poit out blow. (i) Th mthod prstd i this papr ca b asily xtdd to gral scalfr tworks i a vry simpl way. First, idtify th commuicatio krl (s [7]). Scod, for ach od i th krl, rplac its umrous icidt low-dgr ods by th upscald ods togthr with th homogizd diffusio cofficits ad forcig trms, s Figur.

22 22 F. A. MORALES, D. E. RESTREPO EJDE-26/282 (ii) Th particular scal-fr twork tratd i th papr i.., th star-lik mtric graph, ariss aturally i som importat xampls. Ths com from th thory of th stratgic twork formatio, whr th agts choos thir coctios followig utilitaria bhavior. Udr crtai coditios for th bfit-cost rlatio affctig th actors wh stablishig liks with othr agts, th asymptotic twork is star-shapd (s [2]). (iii) Th scal-fr tworks ar frqut i may ral world xampls as alrady mtiod. It follows that th mthod is applicabl to a wid rag of cass. Howvr, importat typs of tworks ca ot b tratd th sam way for homogizatio, v if thy shar som importat proprtis of commuicatio. Th small-world tworks costitut a xampl sic thy ar highly clustrd, this fatur cotradicts th powr-law dgr distributio hypothsis. S [4] for a dtaild xpositio o th mattr. (iv) Th upscalig of th diffusio phomo is do i a hybrid fashio. O th o had, th diffusio o th low-dgr ods is modld by th wak variatioal form of th diffrtial oprators dfid ovr th graph, but igorig its combiatorial structur. O th othr had, th diffusio o th commuicatio krl will still dpd o both: th diffrtial oprators ad th combiatorial structur. This is a importat achivmt, bcaus it is cosistt with th atur of availabl data for th aalysis of ral world tworks. Typically, th data for ctral (or highly coctd) agts ar mor rliabl tha data for margial (or low dgr) agts. (v) Th ctral Csàro covrgc hypothss for data bhavior, statd i Lmma 3.5-(iii), as wll as thos cotaid i Hypothsis 4., to coclud covrgc hav probabilistic-statistical atur. This is o of th mai accomplishmts of th work, bcaus th hypothss ar mild ad adjust to ralistic scarios; ulik strog hypothss of topological atur such as priodicity, cotiuity, diffrtiability or v Rima-itgrability of th forcig trms (s []). This fact is furthr illustratd i Exampl 5.3, whr good asymptotic bhavior is obsrvd for a forcig trm which is owhr cotiuous o th domai Ω G of aalysis. (vi) A importat ad dsirabl cosquc of th data hypothss adoptd, is that th mthod ca b xtdd to mor gral scarios, as mtiod i Rmark 4.6, rportd i Subsctio 5.4 ad illustratd i Exampls 5.2, 5.4 ad 5.5. Morovr, Exampl 5.5 suggsts a probabilistic upscald modl for th commuicatio krl, to b xplord i futur work. (vii) A diffrt li of futur rsarch cosists i th aalysis of th sam phomo, but usig th mixd-mixd variatioal formulatio itroducd i [2] istad of th dirct o usd i th prst aalysis. Th ky motivatio i doig so, is that th mixd-mixd formulatio is capabl of modlig mor gral xchag coditios tha thos hadld by th dirct variatioal formulatio ad by th classic mixd formulatios. This advatag ca broad i a sigificat way th spctrum of ral-world tworks that ca b succssfully modld ad upscald. (viii) Fially, th prxistt litratur typically aalyss th asymptotic bhavior of diffusio i complx tworks, startig from fully discrt modls (.g., [5, 3]). Th psudo-discrt tratmt that w hav itroducd hr, costituts mor of a complmtary tha a altrativ approach. Dpdig o th availability of data ad/or samplig, as wll as th scal of itrst for a particular problm, it is atural to cosidr a bldig of both tchiqus.

23 EJDE-26/282 HOMOGENIZATION OF DIFFUSION PROCESSES 23 Ackowldgmts. Th authors ackowldg fiacial support from th Uivrsidad Nacioal d Colombia, Sd Mdllí through th projct HERMES This articl is basd o th work i that projct. Th authors also wish to thak Profssor Ma lgorzata Pszyńska from Orgo Stat Uivrsity, for allowig us to us th cod fmd.m [2] i th implmtatio of th umrical xprimts for Sctio 5. It is a tool of rmarkabl quality, fficicy, ad vrsatility that has provd to b a dcisiv lmt i th productio ad shapig of this work. Rfrcs [] A.-L. Barabási, R. Albrt; Emrgc of scalig i radom tworks, Scic 286 (999) [2] R. Kumar, P. Raghava, S. Rajagopala, D. Sivakumar, A. Tompkis, E. Upfal; Stochastic modls for th wb graph, FOCS: IEES Symposium o Foudatios of Computr Scic (2) [3] G. Brkolaiko, P. Kuchmt; Itroductio to Quatum Graphs, Mathmatical Survys ad Moographs, Vol 86, Amrica Mathmatical Socity, Providc, RI, 23. [4] A. Body, U. Murty; Graph Thory, Graduat Txts i Mathmatics, vol. 244, Sprigr, 28. [5] S. Bortoluzzi, C. Romualdi, A. Bisogi, G. A. Dailli; Disas gs ad itraclullar proti tworks, Physiol. Gom. 5 (23) [6] D. Ciorascu, P. Doato; A Itroductio to Homogizatio, Oxford Sris i Mathmatics ad its Applicatios, Vol 7, Oxford Uivrsity Prss, NY, 999. [7] E. Estrada; Th Structur of Complx Ntworks, Thory ad Applicatios, Oxford Uivrsity Prss, Nw York, 22. [8] I. Farý; O th straight li rprstatios of plaar graphs, Acta Sci. Math. (948) [9] M. Faloutsos, P. Faloutsos, C. Faloutsos; O powr-law rlatioships of th itrt topology, Computr Commuicatio Rviw 29 (999) [] U. Horug; Homogizatio ad Porous Mdia, Vol. 6 of Itrdiscipliary Applid Mathmatics, Sprigr-Vrlag, Nw York, 997. [] M. O. Jackso; Social ad Ecoomic Ntworks, Pricto Uivrsity Prss, Pricto, NJ, 28. [2] M. O. Jackso, F. Bloch; Th formatio of tworks with trasfrs amog playrs, Joural of Ecoomic Thory 33() (27) 83. [3] M. O. Jackso, D. Lópz-Pitado; Diffusio ad cotagio i tworks with htrogous agts ad homophily, Ntwork Scic () (23) [4] M. O. Jackso, B. W. Rogrs; Th coomics of small worlds, Joural of th Europa Ecoomic Associatio 2(3) (25) [5] M. O. Jackso, L. Yariv; Diffusio o social tworks, Écoomi Publiqu () (25) 3 6. [6] E. F. Kllr; Rvisitig scal-fr tworks, BioEssay 27 (25) doi:.2/bis [7] D.-H. Kim, J. D. Noh, H. Jog; Scal-fr trs: Th skltos of complx tworks, Physical Rviw E 7 (24) doi:.3/physrve [8] D. M. Krps; Microcoomic Foudatios I, Choic ad Comptitiv Markts, Pricto Uivrsity Prss, Pricto, NJ, 23. [9] L. Li, D. Aldrso, W. Wiligr, J. Doyl; A first-pricipls approch to udrstadig th itrt s routr-lvl topology, Computr Commuicatio Rviw 34(4) (24) 3 4. [2] F. Morals, R. Showaltr; Itrfac approximatio of Darcy flow i a arrow chal, Mathmatical Mthods i th Applid Scics 35 (22) [2] M. Pszyǹska; fmd.m: Tmplat for solvig a two poit boudary valu problm, Library, mpsz/cod/tachig.html (28 25). [22] J. M. Ramírz; Populatio prsistc udr advctio-diffusio i rivr tworks, Joural of Mathmatical Biology 65(5) (2) doi:.7/s [23] E. M. Sti, R. Shakarchi; Fourir Aalysis. A Itroductio, Pricto Lcturs i Aalysis, Pricto Uivrsity Prss, Pricto, NJ, 23.

24 24 F. A. MORALES, D. E. RESTREPO EJDE-26/282 [24] P. Tsaparas, L. M. Ramírz, O. Bodridr, E. V. Kooig, I. Jorda; Global similarity ad local divrgc i huma ad mous g coxprssio tworks, BMC Evol. Biol. 6 (26) 7. doi:.86/ [25] S. Wuchty; Scal-fr bhavior i proti domai tworks, Molcular Biology ad Evolutio 8 (2) Frado A. Morals Escula d Matmáticas, Uivrsidad Nacioal d Colombia, Sd Mdllí, Colombia addrss: famoralsj@ual.du.co Dail E. Rstrpo Escula d Matmáticas, Uivrsidad Nacioal d Colombia, Sd Mdllí, Colombia addrss: darstrpomo@ual.du.co