The Degree Distribution of Thickened Trees

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1 Discrete Mathematics ad Theoretical Computer Sciece DMTCS vol. subm., by the authors, The Degree Distributio of Thickeed Trees Michael Drmota, Berhard Gitteberger, ad Alois Paholzer Istitute of Discrete Mathematics ad Geometry, TU Wie, Wieder Hauptstrasse 8-, A-4 Viea, Austria received April 8, revised..., accepted... We develop a combiatorial structure to serve as model of radom real world etworks. Startig with plae orieted recursive trees we substitute the odes by more comlex graphs. I such a way we obtai graphs havig a global tree-like structure while locally lookig clustered. This fits with observatios obtaied from real-world etworks. I particular we show that the resultig graphs are scale-free, that is, the degree distributio has a asymptotic power law. Keywords: scale free etworks, recursive trees, geeratig fuctios Cotets Itroductio The degree distributio of PORT s The degree distributio of thickeed PORT s 6 4 Examples 4. Substitutig by oe or two odes Thickeig with triagles Thickeig with triagles Itroductio There has bee substatial iterest i radom graph models where vertices are added to the graph succesively ad are coected to several already existig odes accordig to some give law. The so-called Albert-Barabási model see Albert ad Barabási jois a ew ode to a existig oe with probability proportioal to the degree. The idea behid is to model various real-word graphs like the iteret or social etworks. This research was supported by EU FP6-NEST-Adveture Programme, Cotract 8875 NEMO ad by the Austria Sciece Foudatio FWF NFN-Project S96 subm. to DMTCS c by the authors Discrete Mathematics ad Theoretical Computer Sciece DMTCS, Nacy, Frace

2 Michael Drmota, Berhard Gitteberger, ad Alois Paholzer It turs out that the Albert-Barabási model is ot uambigously defied. Oe rigorous approach is due to Bollobás ad Riorda 4 They itroduced a radom multigraph G m. For example, G is described i the followig way. Oe starts with a iitial ode with a loop. This meas that has degree. The at step k we add oe ode that is coected to j k with propability deg G k j k k if j < k, if j = k. Of course, after steps we have produced a radom multigraph with vertex set {,,..., } ad edges. Now the radom graph G m ca be costructed from Gm by idetifyig the odes {l m +, l m +,...,lm} l of G m to a ew ode l ad all edges withi the odes {l m +, l m +,...,lm} are ow loops of the ew ode l. Of course, this procedure results i a radom multigraph with vertex set {,,..., } ad m edges. It turs out that the degree distributio of G m satisfies a power law. The probability that a radomly chose ode of G m has degree d is asymptotically /d see Bollobás et al.. Graphs with this property are called scale-free. We ow use a slightly modified evolutio process that always leads to a labelled recursive tree. The process starts with the root that is labeled with. The iductively at step j a ew ode with label j is attached to ay previous ode of outdegree k with probability proportial to k +. These kids of trees are also called plae orieted recursive trees PORT s. This evolutio process is quite similar to the process that produces usual recursive trees. A usual recursive tree is a rooted tree wit odes where the odes are labeled with,,..., such that all successors of each ode have a larger label. I particular, the root has label, ad every path from the root to a leaf has strictly icreasig labels. As above, we ca cosider a recursive tree as the result of a evolutio process. The process starts as above with the root that gets label. Next, aother ode is attachted to the root that gets label ad i every step a ew ode is attachted to a already existig ode ad gets the ext label. The labels are the history of the tree evolutio. The PORT-model, where we attach a ew ode accordig to the degree-distributio of the already existig tree, ca be also see as a plaar versio of the recursive trees. Namely, if a ode of a plaar rooted tree has out-degree k, that is, it has degree d = k +, the there are precisely d ways to attach there a ew ode i order to get differet plaar trees. This explais the ame plae orieted recursive trees. We will idicate i the ext sectio that the degree distributio of these radom trees is scale-free as i the case of G m. I fact, we have lim p d = 4 dd + d +, where p d deotes the probability that a radom ode i radom PORT of size has degree d. We ow itroduce a substitutio process that creates radom graphs that have a global tree structure that is govered by plae orieted recursive trees. For every k let T k deote a o-empty set of labelled graphs with k + additioal ordered half edges. Now cosider the followig radom process. For every tree PORT T we substitute every ode

3 The Degree Distributio of Thickeed Trees v of outdegree k by a radomly chose graph of T k where the k + half edges are glued to the edge comig from the predecessor of v resp. the k successors of v correspodig to the give order of the half-edges. Further we relabel all odes i the ew graph G = GT i a way that is cosistet with the origial labellig. We deote the graphs that are obtaied by this process thickeed trees or more precisely thickeed PORT s. The idea behid this model is to simulate real etworks that are produced by a evolutio process followig, for example, the Albert-Barabási priciple ad have a global tree structure with local clusters. Of course, we make a strog simplificatio. Our model relies o a two-step-procedure. We first let a tree etwork evolve followig the Albert-Barabási priciple ad the replace the tree odes by radom clusters. Hece, the clusters are ot produced by a evolutio process. Nevertheless we thik that our model has several advatages ad ca be used to explai several properties that are obsered i practice: There is large flexibility i choosig the structure of local clusters ad, thus, ca be adapted to the situatio. The model is feasable for a aalytic treatmet. It ca be used to study aalytically the ifluece of local chages of the etwork to the global behaviour. The mai focus of our paper is the degree distributio. We show that uder atural coditios the resultig etwork is scale-free ad that the umber of odes of give degree satisfy a cetral limit theorem. The degree distributio of PORT s I this sectio we shortly preset a proof that PORT s are scale-free. I particular we show the followig property that is due to Mahmoud et al. 99, see also Bergero et al. 99 for similar results ad Kuba ad Paholzer 7 for geeralizatios. The reaso for presetig a proof is that the proof of a correspodig property for thickeed PORT s will work alog similiar lies. Theorem Let p d deote the probability that a radom ode i a radom PORT of size has degree d. The lim p 4 d = dd + d +. Furthermore, for every d let X d deote the umber of odes of degree d i radom PORT s of size. The X d satisfies a cetral limit theorem X d EX d V X d d N,, where EX d B, d = 4/dd + d + ad V X d ad Ba, b deotes the Beta-fuctio. B, d+4b, d 4B4, d Proof: The proof of Theorem is based o a geeratig fuctio approach. It is well kow that the geeratig fuctio yz = z y!

4 4 Michael Drmota, Berhard Gitteberger, ad Alois Paholzer of PORT s satisfies the differetial equatio y z = + yz + yz + = that reflects the recursive structure of PORT s. The solutio is ad, thus, oe has yz, yz = z =! z!! y =!.! We ow tur to the degree distributio. By obvious reasoig we have p d = EXd. Thus, we ca get the degree distributio with help of the distributio of X d. Let y d,k deote the umber of PORTS s of size with exactly k odes with degree d. The the probability geeratig fuctio Eu Xd is give by ad the double geeratig fuctio satisfies the differetial equatio Equivaletly we have yz, u =,k Eu Xd = y k y d,k uk y d z,k uk! = y Eu Xd z y z = y + u yd = + yd yu y y t + t d dt = z tu We first determie the expected value EX d. For this purpose we set yz, u Sz = u y EN d, z u=!. From we get that this fuctio satisfies the differetial equatio S z = Sz yz + yzd!

5 The Degree Distributio of Thickeed Trees 5 ad has solutio Sz = yz yz Recall that yz = z. Thus, it follows that yz Cosequetly, we have Sz = t t d dt = t t d dt. t t d dt = B, d + + O B, d + z + O = dd + d + yz z /. t t d dt z + O which implies with help of the trasfer-lemma of Flajolet ad Odlyzko Flajolet ad Odlyzko 99 that EX d Cosequetly, we get the degree distributio lim p d = lim 4 = dd + d + + O. EX d = 4 dd + d +. I order to prove the cetral limit theorem we go back to ad similarly expad the itegral y t + t d tu dt = t + t d tu dt =: Cu Dy, u Observe that Dy, u is asymptotically give by y Dy, u = + O y u. Hece traslates to Cu z = y + O y u Cu which ca be iverted to yz, u = Cu z Cu + O z Cu. satisfies a cetral limit theo- Cosequetly it follows compare with Flajolet ad Soria 99 that X d rem with mea ad variace EX d = C C + O ad y t + t d tu dt V X d = C C C C + C C + O. It is a easy exercise to compute C = /, C = B, d, ad C = B4, d.

6 6 Michael Drmota, Berhard Gitteberger, ad Alois Paholzer The degree distributio of thickeed PORT s We ow deal with a substitutio process that creates radom graphs that have a global tree structure that is govered by plae orieted recursive trees. Let us cosider the formal solutio y = yz, x, x, x,... of the differetial equatio y = k x k y k, where deotes differetiatio with respect to z. It is clear that y = yz, x, x, x,... ca be cosidered as a power series i z, x, x,... By costructio the coefficiet [z x k xk...] yz, x, x, x,... is exactly the umber of PORT s T of size ad k j odes of outdegree j j. For every k let T k deote a o-empty set of labelled graphs with k + additioal half edges ẽ, ẽ,...,ẽ k. Further, let t k z = G T k z G G! deote the expoetial geeratig fuctio of these graphs. We recall that we cosider the followig radom process. For every PORT T we substitute every ode v of outdegree k by a radomly chose graph of T k where the half edges ẽ, ẽ,..., ẽ k are glued to the edge comig from the predecessor of v resp. the k successors of v correspodig to the left-to-right order. Further we relabel all odes i the ew graph G = GT i a way that is cosistet with the origial labellig. Thus, the geeratig fuctio of the umber gz = g z! of the umbers g of graphs that are produced i this way is give by gz = yz, t z/z, t z/z,.... The graphs that are obtaied by this process are deoted here by thickeed trees or thickeed PORT s. Lemma Set Fz, y = The gz satisfies the fuctioal equatio y dt k t kz/z t k. Fz, g = z. Proof: We recall that y = yz, x, x,... satisfies the differetial equatio y = k x ky k. Hece, y dt k x ky k = z + c for some costat c. Of course, if we substitute x k by t k z/z the we immediately get the result. Note further that g = ad F, =. Thus, we ca fix c =.

7 The Degree Distributio of Thickeed Trees 7 Fig. : The substitutio set T I a similar way we ca deal with parameters. For example, fix a degree d ad let t d k z, u = G T k z G G! un dg, where N d G deotes the umber of odes i G of degree d icludig the half-edges ẽ,...,ẽ k. The the geeratig fuctio gz, u = yz, t d z, u/z, td z, u/z,... ecodes the distributio of odes of degree d of thickeed trees. Of course, the above lemma exteds to this case. It is ow coveet to itroduce the otatio T d z, y, u = z k Of course, for all d we have T d z, y, = k t kz/z y k. Lemma Set G d z, y, u = The gz, u satisfies the fuctioal equatio y G d z, g, u = z. t d k z, uyk. dt T d z, t, u. Example We just give a simple example. Suppose that T k cosists for every k of exactly two graphs, the first oe with oe ode ad the secod with two odes, where all half-edges ẽ,..., ẽ k that will be liked to the k subgraphs are o the secod ode. Figure depicts the set T. I this example we have for d t d k z, u = {z + z if k d uz + u z else,

8 8 Michael Drmota, Berhard Gitteberger, ad Alois Paholzer ad cosequetly T d z, y, u = The cases d = ad d = are similar. Our mai result is the followig theorem: + z + u yd. y Theorem Let T k be substitutio sets as described above so that the equatio ρ = dt T d ρ, t, has a uique positive solutio i the regio of covergece of T d z, y, u ad that the T d z, y, u ca be represeted as C z, y + C z, y y r y d+α u + O y r u T d z, y, u = y r, where r ad r are real umbers with < r r, α i a iteger, C z, y ad C z, y are power series that cotai z = ρ ad y = i their regios of covergece ad that satisfy C i ρ, for i =,, ad the O -term is uiform i a eighbourhood of z = ρ ad y =. Let p d deote the probability that a radom ode i a thickeed PORT of size has degree d. The the limits lim p d =: pd exist ad we have, as d, pd C d r+r +. Furthermore, for every d let X d deote the umber of odes of degree d i radom thickeed PORT s of size. The X d satisfies a cetral limit theorem where EX d X d EX d V X d d N,, ad V X d are both asymptotically proportial to. Remark. The coditios o the geeratig fuctio Tz, y, u ca be more of less iterpreted i the followig way. If we set u = the the sigularity / y r i Tz, y, essetially says that the set T k cosists k r graphs. Furthermore for k d + α there are k r r graphs with a vertex of degree d, compare also with the examples give i Sectio 4. Proof: The proof rus alog similar lies as that of Theorem. We start by ispectig the geeratig fuctio gz of all thickeed PORT s. For simplicity we assume that the substitutio sets T k are of a form that g > for sufficietly large, that is, we exclude, for example, the case that the umber

9 The Degree Distributio of Thickeed Trees 9 of odes of graphs i T k are all cogruet to modulo some iteger m >. i The it follows that gz < g z if z is ot cotaied i the positive real lie. We first observe that ρ > is the oly sigularity o the circle of covergece z ρ ad that gρ =, that is, gz is coverget at z = ρ. First it is clear that gz ca be aalytically cotiued startig with g = ad the fuctioal equatio Fz, g = z. However, if gz for some z cotaied i the regio of covergece of gz the we have F g z, gz = T d z, gz, = gz r C z, gz. Thus, we ca cotiue aalytically with help of the implicit fuctio theorem. Thus, if gz has a sigularity ρ ad if gρ is coverget the gρ =. Sice gz is mootoe ad aalytic it certaily reaches a value with gρ = where it has to be sigular. Further, ρ is characterized by the equatio Fρ, = ρ. Next we characterize the kid of sigularity of gz at z = ρ. By Lemma we have z = g t r C z, t dt = t r C z, t dt t r dt =: Gz Hz, g g C z, t Hece, by expadig /C z, t locally aroud t = we thus get Gz z = c z g r+ + O g or /r+ Gz z = g + O g. c z Sice Gρ = ρ ad C z, y is icreasig i z we ca represet Gz z/c z = Kz z/ρ. Furthermore, we ca ivert the relatio ad obtai gz = Kz /r+ ρ z /r+ + O z ρ /r+. Sice there are o other sigularities o the circle z ρ ad gz ca be aalytically cotiued to a larger rage despite at the poit z = ρ it follows from Flajolet ad Odlyzko 99 that g Kρ /r+ ρ Γ r+ r+ r+ Next we determie asymptotics o the average value EX d. Set Sz = ugz,. The it follows from Lemma that gz Sz = gz r C z, t t r+r t d+α dt. i We call this the aperiodic case. I the periodic case we have to deal with m sigularities o the boudary of the circle of covergece of gz which are all of the same kid..

10 Michael Drmota, Berhard Gitteberger, ad Alois Paholzer As i the proof of Theorem it follows that Sz = Kz r r+ z/ρ r r+ C z, t t r+r t d+α dt + O which proves that EX d r + Kρ C ρ, t t r+r t d+α dt. Thus, the limit pd = lim EX d / exists ad is asympotically give by pd = r + Kρ C ρ, t t r+r t d+α dt C d r+r + for some costat C >. Fially the proof that the limitig distributio is ormal is very similar to the correspodig proof of Theorem. We skip the details. 4 Examples 4. Substitutig by oe or two odes Let us cotiue the example precedig Theorem. Here we have for d : T d z, y, u = + z + yu y d. y Thus, Theorem applies with r = r = ad α =. The degree distributio pd is scale-free with tail pd C/d. A detailed computatio icludig the istaces d < also shows that the degree distributio is give as follows: pd = + dd + d + 4. Thickeig with triagles +, for d, p = 6, p = We cosider two examples where we substitute each ode of the origial ode by a triagle. More precisely, each ode of out-degree k is the substituted by a triagle with k+ half-edges ẽ,..., ẽ k attached to it. The ẽ is glued to the predecessor of the origial odes ad the other half-edges to the successors of the origial odes accordig to their aturally give order e.g. left to right i the case of plae trees. Let the parameter of iterest be the umber of odes of degree d. The we have t k z, u = a,k + a,k u + a,k u + a,k u z! where a i,k is the umber of cofiguratios triagles with k half-edges cotaiig i odes of degree d

11 The Degree Distributio of Thickeed Trees We first cosider the special case where the igoig edge of each triagle is separated from the outgoig edges. That meas that ẽ coects the predecessor of the triagle to a vertex of degree the edges are the ẽ ad two edges of the triagle while ẽ,..., ẽ k are coected to the other two vertices of the triagle. Clearly we have the t k z = k + z!. Whe focusig o the umber of odes of degree d, the it is easy to see that for d 4 we have k + if k < d a,k = k if k = d 4 k else. This holds because of the followig argumet. Let us label the odes of the triagle by,, where ẽ is attached to. The l of the edges ẽ,..., ẽ k are attached to ad l = k l to. The cofiguratio cotais at least oe ode of degree d if ad oly if l = d or l = d. Exactly these cases do ot cotribute to a,k. Moreover we get a,k =, a,k = { if k = d 4 or k < d else, This implies for d 4: Tz, y, u = z + uy 6 k k k + = z 6 ad a,k = { if k d 4 if k = d 4. d uy k + u y d 4 k= + yy d u + y y d 4 u y. Here Theorem applies with r =, r = ad α =. Hece the degree distributio pd is scalefree with tail pd C/d 4. A detailed computatio icludig the istaces d < 4 also shows that the degree distributio is give as follows: pd = d dd + d +, for d 4, p =, p =, p =. 4. Thickeig with triagles I the case of geeral triagles the edges ẽ,..., ẽ k ca be attached to all the vertices of the triagles. This gives, after somewhat tedious cosideratios, the followig result for t d k z, u. Lemma Let d >. The we have t d z k z, u = 6 ad k u,

12 Michael Drmota, Berhard Gitteberger, ad Alois Paholzer with a d k u = k+ k+ k+ k+ k+ This implies for d 4: T d z, u, v = z 6 if k d 4 k + k d k + k d + 4u if d k d 6 k+ + k k+ u + k + u if k = d 5 k + k d + + k + k d + u + k + u if k d 4 ad k d 7 k+k+ + k+ u + k+ [ y 4 + yy d 4 dy + d u u if k = d 7. y y d 5 d 5y d + 4u + d y 4 y d 7 u ]. Now Theorem applies with r =, r = 4 ad α =. Hece the degree distributio pd is scalefree with tail pd C/d 6. A detailed computatio icludig the istaces d < 4 also shows that the degree distributio is give as follows: pd = 6 d dd + d + d + d + 4, for d 4, p =, p = 5, p = 9 6. Refereces R. Albert ad A.-L. Barabási. Statistical mechaics of complex etworks. Rev. Moder Phys., 74: 47 97,. ISSN F. Bergero, P. Flajolet, ad B. Salvy. Varieties of icreasig trees. I CAAP 9 Rees, 99, volume 58 of Lecture Notes i Comput. Sci., pages Spriger, Berli, 99. B. Bollobás ad O. Riorda. The diameter of a scale-free radom graph. Combiatorica, 4:5 4, 4. ISSN B. Bollobás, O. Riorda, J. Specer, ad G. Tusády. The degree sequece of a scale-free radom graph process. Radom Structures Algorithms, 8:79 9,. ISSN P. Flajolet ad A. M. Odlyzko. Sigularity aalysis of geeratig fuctios. SIAM Joural o Discrete Mathematics, :6 4, 99. P. Flajolet ad M. Soria. Geeral combiatorial schemas: Gaussia limit distributios ad expoetial tails. Discrete Math., 4-:59 8, 99. ISSN -65X. M. Kuba ad A. Paholzer. O the degree distributio of the odes i icreasig trees. J. Combi. Theory Ser. A, 44:597 68, 7. ISSN H. M. Mahmoud, R. T. Smythe, ad J. Szymański. O the structure of radom plae-orieted recursive trees ad their braches. Radom Structures Algorithms, 4:5 76, 99. ISSN 4-98.

13 The Degree Distributio of Thickeed Trees B. Pittel. Note o the heights of radom recursive trees ad radom m-ary search trees. Radom Structures Algorithms, 5:7 47, 994. ISSN

arxiv: v1 [math.co] 5 Jan 2014

arxiv: v1 [math.co] 5 Jan 2014 O the degree distributio of a growig etwork model Lida Farczadi Uiversity of Waterloo lidafarczadi@gmailcom Nicholas Wormald Moash Uiversity ickwormald@moashedu arxiv:14010933v1 [mathco] 5 Ja 2014 Abstract