Dynamics of directed graphs: the world-wide Web Bosiljka Tadi´c Joˇzef Stefan Institute, P.O. Box 3000, 1001 Ljubljana, Slovenia We introduce and simulate a growth model of the world-wide Web based on the dynamics of 1 outgoing links that is motivated by the conduct of the agents in the real Web to update outgoing 0 links(re)directingthemtowardsconstantlychangingselectednodes. Emergentstatisticalcorrelation 0 between the distributions of outgoing and incoming links is a key feature of the dynamics of the 2 Web. Thegrowth phaseischaracterized bytemporalfractalstructureswhicharemanifested inthe n hierarchical organization oflinks. Weobtainquantitativeagreement withtherecentempirical data a in the real Web for the distributions of in- and out-links and for the size of connected component. J In afully grown networkof N nodeswe studythestructureofconnected clusters ofnodes thatare 7 accessible along outgoing links from a randomly selected node. The distributions of size and depth 1 of the connected clusters with a giant component exhibit supercritical behavior. By decreasing the control parameter—average fraction β of updated and added links per time step—towards ] h βc(N) < 10% the Web can resume a critical structure with no giant component in it. We find a c different universality class when the updates of links are not allowed, i.e., for β ≡0, corresponding e to thenetwork of science citations. m - t a t s I. INTRODUCTION large-scalemeasurementsintheworld-wideWebthatin- . t volved 108 nodes reveal [5] that: (a) Both the distribu- a m tionsofincominglinksanddistributionofoutgoinglinks Understanding the evolution of complex networks follow a power-law behavior with different exponents; - [1–13] remains a challenging problem of modern statisti- d (b) The distribution of size of connected component— calphysics. Modelingthedynamicstructureofthesenet- n number of linked nodes in the web crawls—also exhibits works has significant practical applications, for instance o scalingbehavior;and(c)TheWebshowsaveryintricate c in the design of search and control algorithms and in [ predicting emergence of new phenomena in information structureofconnections,withalargestronglyconnected networks [5]. An evolving network consists of increasing component in the center. 2 v number of nodes, which are connected by links accord- Recently a great deal of effort was devoted to under- 2 ing to certain rules specific to each network. Although standemergenceofthescale-freestructureoflinksinvar- 4 the spatial distribution of nodes in a networkis random, ious complex networks [10–13]. An important ingredient 4 a different structure can be observed when the distribu- ofthedynamics—preferentialattachment[10]wasshown 1 tion of links is considered. In this respect two major to lead to the power-law distribution for incoming links 1 0 groups [9] are: (1) random networks, with the number (the outgoing links are fixed by the rules of the model 0 of links attached to a node fluctuating around an aver- [10]). Themodelofpreferentialattachmentsforincoming t/ age value, and (2) networks with hierarchical structure linkshasbeengeneralizedtoincludeinitialattractiveness a of links, manifested in the power-law decay of the dis- of nodes in Ref. [11], where the authors proved analyti- m tributions of node ranks. The structure of connections callytheexistenceofthescalingregionforlargeevolution - has immediate impact on the accessibility of nodes and times. Recently it was demonstrated in Ref. [13] how a d n on functional stability of the network. For instance, the giant connected component can occur in the model of o hierarchically connected networks are shown [10] to be random directed graphs assuming arbitrary and statisti- c robust to large failure rates, however, they are vulnera- callyindependentdistributionsofincomingandoutgoing : v ble to removal of a few ’key nodes’. The inhomogeneous links. The predictions of the randomgraphtheory, how- i structure ofconnectionshas been foundin metabolic cy- ever,show substantial quantitative differences compared X cles [1], in various social networks [2], including the sci- to the properties of the real world-wide Web [13]. r a ence citation network [3], and the Internet [4] and the In this work we simulate growth and response of a di- world-wide Web [5–8]. rected graph with the dynamic rules that are motivated The world-wide Web is not a separate physical net- byprominentfeaturesofthe world-wideWeb. These dy- works. Rather it is a subset of the Internet that uses namicrulesyieldthestatisticallycorrelateddistributions a specific protocol (hypertext transfer protocol) and di- of outgoing and incoming links. Another important fea- rected links to access a variety of data, representing tureofthemodelisthatthelinksbetweenpairsofnodes an example of distributed client/server computing [14]. are not fixed in time, but may vary on the time scale of Basedonthesepropertiesahierarchicalstructureoflinks thenetwork’sevolution(updatesoflinks). Thisproperty, emergesthatispeculiartotheworld-wideWeb. Recently thatmaybefoundinsomemetaboliccycles,isnotshared 1 withmanyothersocialnetworks,whosephysicallinksare ues, however, considerably increases computation time. either fixed or vary on much slower time scale. We ar- (k) Preferential update (k1) and preferential at- gue that the (at present high) frequency of updates of tachment (k2). These properties representa paradigm the outgoing links, which is peculiar to the agents in the of social behavior, for instance preferential attachments Web, is essential for the observed inhomogeneous struc- aredrivenby“popularity”ofanode,asdiscussedinear- tureofconnections. Wefindaconsistentagreementwith lier models ofevolvingsocialnetworks. Inaddition, here the recent empirical data on the real world-wide Web we have that not all nodes are getting updated at ev- both in the structure of links and in the response of the ery time step, rather only a few nodes update at a time. network. Moreover,some ofthe nodesupdate outgoinglinks more The paper is organized as follows: In Sec. II we in- frequently than others. This features can be formulated troduce the growth rules of the network and determine in terms of probabilities as follows: Where do the links theemergentrankdistributionsofoutgoingandincoming come from? We assume that apart from the new node, links. InSec.IIIwestudythetemporalfractalstructures updatingattimetoccurswithlargerprobabilityatmost in the growth phase of the network and determine the active nodes (preferential activity), i.e., an outgoing link correspondingdistributions ofthe first-returnlinks. Sec. from the node k ≤i appears with the probability IV is devoted to investigation of the connected compo- αM +q (k,t) out nentsonthefullygrownnetworkandfractalpropertiesof Prob1= . (1) (1+α)M ∗i noise. Ashortsummaryandthe discussionoftheresults is given in Sec. V. A new link goes to the node n (whose age is t−n) with the probability αM +q (n,t) II. GROWTH MODEL AND RANK in Prob2= , (2) DISTRIBUTIONS (1+α)M ∗i i.e., a “popular” node attracts even more links. Here We suggest the following simplified set of rules to in- q (n,t) andq (n,t) are the dynamically varying num- out in corporatebasic features pertinent to the realworld-wide ber of outgoing and incoming links, respectively, at the Web: (i) Directed nature of linking. The world- nodenatcurrenttimet. Itisassumedthatatthetimeof wide Web represents a directed graph with nodes corre- additionofanodeitothenetworkq (i,i)=q (i,i)= out in sponding to Web pages, and arcs corresponding to hy- 0. NotethatthesecondruleinEq.(2)isformallyequiva- perlinks between pages [5,13]. (j) Growth and rear- lenttothemodelofpreferentialattachmentforincoming rangements at unique time scale. Ateachtime unit links [11]. However, in our model the role of the param- t a new node i = t is added to the network (growth) eter α in this equation is precisely determined through and a number M(t) of new links are distributed among the rule in Eq. (1), which regulates outgoing links: (1) the nodes (update of links) following two rules specified For α ≡ 1/(1+β) < 1 a fraction (1−α)M of outgoing below. A fraction f (t) ≡ αM(t) of new links are out- links in Eq. (1) refers to updates at earlier nodes; (2) 0 going links from the new added node i = t, whereas the For α=1 (equivalent to β =0) no updates are allowed. rest f (t) ≡ (1−α)M(t) are the updated links at other New links originate only from the new added node; (3) 1 nodes in the network. Hence, the relevant parameter in Formally one can apply Eqs. (1-2) for α strictly larger themodelistheratioofupdatedandaddedlinksateach than unity, however, in this region β becomes negative, time step, i.e., β ≡f (t)/f (t)=(1−α)/α, which is in- suggestingthatthenumberoflinksinthe networkisnot 1 0 dependent of the actual number of links M(t). Thus, in conserved. Therefore only the incoming links (2) can be the model M(t) represents a net increment of the num- counted correctly when α is in the interval ∞ > α > 1; ber of outgoing links at time step t. Variations in M(t) (4) In the mathematical limit α → ∞, or in practical are, in principle, not restricted, being caused by adding caseswhen α≫q¯/M,the effects ofthe dynamicalquan- new connections between old nodes (resulting in the in- tities q (k,t) and q (n,t) become negligible, and we out in creaseofthe number oflinks), and/orbyremovingsome recover the case of fully random network. earlier links (causing decrease of the number of links). In what follows we would like to demonstrate that the We assume that variations in M(t) are such that we can features(i), (j), (k1)and(k2)explainedaboveareessen- define anaveragevalue M ≡M(t), whichcanbe consid- tialforthebehaviorobservedintheWeb. Earliermodels eredasaconstantinfirstapproximation. Inpractice,the [10,11]concentrated only on the preferential attachment number of nodes and the number of links in the network rule (k2), while largely neglecting properties (i), (j) and increaseswithtime, so thatreasonablevalues forthe av- (k1). The most complete account of the preferential at- erage M are positive. For consistency, we keep M = 1 tachment model is given in Ref. [11]. On the other side, throughout this work. Different values of M do not af- therandomgraphmodel[13]consideredthe directedna- fect the universal properties in the scaling region (i.e., ture of network (i), while neglecting the details of the for large evolution times). Taking much larger M val- linking rules (j), (k1) and (k2). 2 UsingtherulesinEqs.(1)-(2)wegrowalargenetwork pattern is shown in Fig. 2 (top panel). Clustering in the of N = 106 nodes. We measure the ranking of nodes in patterns indicate frequently active nodes, whereas voids this network by the distributions of outgoing links and of various sizes indicate lesser activity at corresponding incoming links, P(q ) and P(q ), respectively, which nodes. Visuallyheterogeneouspicturesuggeststhatboth out in are shown in Fig. 1 for parameter β = 3, corresponding pattern of origins of links and pattern of targets have to α = 0.25 in Eqs. (1-2). For comparison we also sim- fractal structure. The fractality of these patterns can be ulate the distributions of outgoingand incoming links in quantified,forinstance,bycomputingthedistributionof the case of fully random directed network. For a range time intervals between two successive linking at a given of values of their arguments the cumulative distributions node(returntime). Thedistributionoftimeintervals∆t forfiniteβ valuesexhibitapower-lawbehavioraccording between two successive linking to a selected node in the to networkP(∆t)measuredforthenetworkof10,000nodes andthe parameterβ =3 is giveninFig.2 (lowerpanel). P(qout)∼qo−u(tτout−1) ; P(qin)∼qi−n(τin−1) . (3) The algebraic decay of the distribution is a signature of the underlying fractal structure of the pattern. The dis- ByincreasingtheparameterαinEqs.(1)-(2)intherange tribution of time intervals between two successive links (0,1), corresponding to decrease of the relative fraction fromaselectednodealsoshowsapower-lawtailforlarge of updated links β in the interval (∞,0), the slopes of ∆t. At small intervals ∆t the pattern is nearly random. the distributions increase. The scaling exponents τin The two distributions coincide, indicating mutual corre- and τout given in the inset to Fig. 1 are parametrized lations in the patterns, for time intervals ∆t>100 (that by α. Our numerical results for τin agree with the ex- are accessible for large evolution times t). To emphasize act solution [11] τin = 2+α for the incoming links. For therelationtoFig.1werecallthatthenumberoflinksof the distribution of outgoing links no analytical results either kind at a node accumulates with time. Hence, the are available. Our numerical results in the inset to Fig. regionsof large time intervals between successivelinking 1 can be well approximated with the linear dependence at a node, ∆t in Fig. 2, correspond to small number of τout ≈2+3α. Therefore, the difference between the two links at that node, q in Fig. 1, and vice versa. exponents ∆ ≡ τ − τ ≈ 2α persists for any finite out in 0 < α < 1. Comparing these results with the available empiricaldataweestimatetheparameterα=0.22±0.1. IV. DYNAMIC CRITICAL STATES Here we used the following values as the experimental estimates for the exponents τin = 2.16±0.1, obtained The scaling behavior of the rank distributions can be as the average value from the data in [5] and last refer- examinedintermsofthepotentialdynamiccriticalstates ence in [10], and τout = 2.62±0.1 from data in [5], by bymeasuringthestatisticsoftriggeredavalanches[16]on fitting only the straight part of the curves while avoid- thenetwork. Weconsiderananalogoftheavalanchesize ing the noisy tails and the curvature at small q. It is in the networks: the size of a cluster of nodes which are clear that these values are not definitive. More careful physically accessible along directed links starting from a fits of the data and further measurements are necessary random node in the network. A network is grown using in order to reduce the experimental error bars. Hence, the rules in Eqs. (1)-(2). We then select a random node the approximative value of the control parameter β in andmakealistofnodeswhichareconnectedbyoutgoing the currentstate of the Web is β ≈3, i.e., in the average linksfromthatnode. Inthenextstepwemakeanewlist three updated links come to one added link at each evo- following outgoing links from the nodes on the previous lutionstep. Wenowdiscussthepropertiesofthenetwork list,keepingonlythenodeswhicharedifferentinthetwo for this particular value of the control parameter β. lists,andsoon. Theprocessendsupwhennonewnodes can be reached—the list is empty. The number of differ- entlistsbefore anempty listoccurscanbe recognizedas III. DYNAMIC FRACTAL STRUCTURES the depth of the connected component. Technically, we preserveranksq andq foreachnodefromthegrowth in out Inthedynamicalsystemswithlarge-scaleorganization phaseand,inthecaseofsmallnetworks,theexactphysi- theoccurrenceofalgebraicallydecayingdistributionscan callinksbetweenthenodes. Inlargenetworksonlyranks be linked to the formation of certain fractal structures arepreservedandlinksaresearchedusingtheruleinEq. [15] in the system. In the case of complex evolving net- (1). For a large ensemble of networks the results are ex- works, however, the existence of such structures is less pected to be statistically the same. In this way the size clear, due to the spatial randomness of the graph. To of a cluster of linked nodes represents a response of the show that the power-law decay of the distributions of system on the random excitation. The process is remi- node ranks, shown in Fig. 1, is associated with certain niscent of the invasion percolation on a random graph, fractal structures on the network we examine the tem- where a new edge j, which already does not belong to poral pattern of linking between nodes. A part of the the graph, is invaded along an outgoing hyperlink i→j 3 from the node i if the node rank q (i) exceeds unity. τ decreasesandthepeakeventuallydisappearsatacrit- out s The probability to add the edge j to the graph is given ical value [20] of the parameter β =β ≈0.081. Beyond c by the preferencerule inEq.(1). Ina morefamiliarcase thecriticalvalueofthe controlparameterthescalingbe- oftheinvasionpercolationinphysicalwettingonthelat- havior of the size and depth of connected components is tice [17] that probability is given by the least resistance entirely lost. In the critical state the scaling exponents rule. To our knowledge the problem of invasion percola- are (cf. inset to Fig. 1): τ ≈ 2.925, τ = τ = 1. in s d tion on random graphs has not been studied so far [20]. The distribution of outgoing links shows very sharp de- The differential distribution of size of connected clus- cay, the exponent τ is difficult to measure. It should out ters of nodes is given in Fig. 3 for different values of the be stressed that the absence of the peak in the critical parameterβ. Thedistributionofdepthofthe sameclus- state suggests that no giant component can be formed. ters is given in the inset to Fig. 3. In the case β =3 we Rather, the network consists here of many small groups find that small clusters follow a power-law distribution of well interlinked nodes and percolating directed links with the exponentτ =2.79±0.12,comparablewith the between these groups. s measured [5] value τ = 2.52. (Note that in the case of Itshouldbestressedthatthespeciallimitofourmodel s Internet the corresponding exponent is estimated [4] as β = 0 (i.e., α = 1), corresponding to the original model τI = 1.90.) In Fig. 3 the occurrence of the peak in the of preferential attraction of incoming links proposed by s distributions at large clusters S ≈ 102 indicates that a Barabasi, Albert and Jeong [10], belongs to an entirely 0 largenumberoftheavalancheshaveapproximatelysame differentclassofscalingbehaviorasregardsthedistribu- size. This isasignatureofthe existenceofalargesubset tion of size and depth of connected clusters: these dis- of strongly connected nodes (a giant component): Once tributions are flat between lower and upper cutoffs (see a‘crawl’entersthesubsetofstronglyconnectednodes,it Fig. 3). The only distribution which enjoys a power law explores it entirely. By taking a larger number of nodes in this limit is the distribution of incoming links, which N the position of the peak moves towards larger values hastheexact[11]exponentτ =3(seealsoinsettoFig. in approximately as S ∼N/7. 1). Inthiscase(β =0)updatesatalreadyexistingnodes 0 BeforediscussingthedistributionsinFig.3,wedemon- areno longerpossible. Thisfeature—freezingofthe out- stratethatthenetworkwiththehierarchicalstructureof going links—as well as quantitative disagreement in the links is characterized by a fractal noise. In the search of exponent γ for distribution of incoming links, makes the a connectedcluster describedabovewe examinedetailed modelofRef.[10]inappropriateasa modelofthe world- variation of the number of nodes added to the cluster wideWebdynamics. Arealisticnetworkwithfrozenout- at each step of investigation. We obtain the noisy sig- goinglinksisthenetworkofscientificcitations[3],where nal shown Fig. 4. Properties of the signal vary with the physical links, corresponding to the cited references in control parameter of the dynamics β. Considering the already published papers, remain fixed in time. number of steps as a total elapsed time of investigation, In conclusion, we have demonstrated that certain the Fourier spectrum of the signal is shown in the top salient features of the dynamics of the world-wide Web panel in Fig. 4. It shows a region of correlated behav- require more careful modeling, compared to models of ior with power-law decay between the upper cut off at a generic complex evolving network with preferential at- high frequencies and lower cut off (due to finite size of traction of links [10] and models of random graphs with the network). In addition, the spectrum in the case of rewiring [13]. The dynamic structure and functioning supercritical network exhibits a peak at a characteristic of the world-wide Web is deeply rooted in the activity frequency f , which is absent in the case of the critical of the agents who are creating the outgoing links. The 0 parameter β =β (N) (cf. Fig. 4). hierarchical structure of outgoing links, which is docu- C mentedbymeasurementsinthe realworld-wideWeb[5], is related to the bias activity of the agents in the rule V. DISCUSSION AND CONCLUSIONS (1) ofour model. A randomselection ofthe active agent (see curve (c) in Fig. 1) fails to describe the distribution In the theory of critical states [16,19], appearance of of outgoing links in the real Web. Temporal variation the peak in the distribution of avalanches, such as the of the outgoing links inside the network, i.e., updates of distributions shown in Fig. 3, indicates that the system links,hastwofoldconsequencesontheglobalstructureof is supercritical. In the critical state the absence of any the network: (i) When the updates of the outgoing links characteristic scale is manifested in the purely algebraic are allowed (i.e., when β strictly larger than zero), the decay of the distribution of cluster size until a cut-off, structureofincoming linksqualitativelychanges,the ex- which depends on the size of the network N. On the ponent τ (β)<3. The distribution of incoming links in in other hand, the form of distribution in Fig. 3 suggests our model coincides with the analytical results of Doro- that a critical point exists that can be reached by vary- govtsev et al. [11] in the scaling region. In addition, the ingarelevantparameterofthedynamics[19]. Infact,by distribution of size and depth of connected clusters be- decreasingtheparameterβ weseeinFig.3thattheslope comeshierarchicalwhenβ >0,inagreementwithresults 4 foundinthe realWeb,whichisnotthe casewhennoup- ofthecontrolparameterisactiveintherealWeb. Inthis dates are allowed (β = 0). (ii) Varying the frequency respectthecurrentstateoftherealWebcanberegarded of updates β implies changes in the Web structure, both as a transient rather than a stationary state. in the outgoing and incoming links. This has immediate Finally, our results support the conclusion that in the impact on the accessibility of nodes. The correlations directed graphs two growth rules are necessary to de- between the outgoing and incoming links suggests that scribe the dynamics of the outgoing and incoming links, thelocalstructureofthenetworkisqualitativelydifferent respectively. Inthecaseoftheworld-wideWebthestatis- compared to the case without updates. Here we did not tically correlateddistributions of outgoing and incoming study in detail the probability that two nodes are linked links appearasa fundamentalfeature ofthe evolutionof (clustering coefficient). Rather we studied the physical the Web. This conclusion may serve as a starting point properties of the network which make the background for the future modeling of the real world-wide Web in of the observed behavior: fractal temporal patterns and terms of the master equations. number of nodes that join the cluster at one time unit. By comparison of the simulated results and the data obtained in the real world-wide Web we obtained a sys- tematic agreementboth for the distributions of outgoing and incoming links and for the size of connected com- ponents when a single control parameter β is fixed to β = 3 within error bars of the data. To our knowledge [1] P.M. Gleiss, P.F. Stadler, A. Wagner, and D.A. Fell, e- no previous model claimed to describe the world-wide print cond-mat/0009124. Web achieved such degree of consistency. This makes us [2] S. Wasserman and K. Faust, Social Network Analysis, believe that the present simplified model takes into ac- Cambridge University Press, Cambridge (1994). count some basic features of the dynamics of Web. It [3] S. Redner, Eur. Phys. J. B 4, 131-134 (1998); M. E. J. wouldbe interestingtoestimate β bydirectly measuring Newman, e-printcond-mat/0007214. [4] M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comp. the average number of updated links in the world-wide Comm.Rev.29,251(1999);G.Caldarelli,R.Marchetti, Web relative to the number of added links originating and Pietronero, Europhys.Lett., 52, 386 (2000). from each added node. [5] A. Broder, , R. Kumar, F. Maghoul, P. Ragha- The structure of links shown by the distributions in van, R. Sridhar, R. Stata, A. Tomkins, and Figs. 1 and 2 is closely related to the evolution of the J.Wiener, Graph structure in the web, available as number of links at a given node and to other local prop- http://www.almaden.ibm.com/cs/k53/www9.final/ . erties of the network. A detailed study of these prop- [6] B.A. Huberman and L.A. Adamic, Nature 401, 131 erties within the present model requires additional work (1999). that remains to be done in the future. Here we expect, [7] B.A. Huberman, P.L.T. Pirolli, J.E.Pitkow, and R.M. based on the analytical results for the incoming links of Lukose, Science, 280, 95-97 (1998). Ref. [11], that in our model the average number of links [8] S.LawrenceandC.L.Giles,Nature400,107-109(1999). < q (i,t) > at a node i will decay in time t ≫ i as [9] L.A.N.Amaral,A.Scala,andH.E.Stanley,e-printcond- in < q(i,t) >∼ (i/t)−γin, where the exponent γ is given mat/0001458. in [10] R. Albert, H. Jeong, and A.-L. Barab´asi, e-print cond- by the exact scaling relation [11]γ =1/(τ −1), lead- in in mat/0008064; A.-L. Barab´asi, R. Albert, and H. Jeong, ingtoγin =1/(1+α)≈0.86. Assumingthatthedensity Physica A 272, 173 (1999); A.-L. Barab´asi and R. Al- oftheoutgoinglinksatagivennodealsoexhibitsscaling bert, Science 286, 509 (1999). behavior,asFigs.1-2suggest,wecanpredictaslowerde- [11] S.N. Dorogovtsev, J.F.F. Mendes, and A.N. Samukhin, cay for the average number of outgoing links at a given Phys. Rev.Lett. 85, 4633 (2000). node. Theexpectedexponentisγout ≈1/(1+3α)≈0.6. [12] P.L.Krapivsky,S.Redner,andF.Layvraz,e-printcond- The simulation results that we reported here suggest mat/0005139; A. Vazquez, e-print cond-mat/0006132; that the emergent structure of links in the world-wide S.N. Dorogovtsev, J.F.F. Mendes, and A.N. Samukhin, Web is strongly related to the updating policy of the e-print cond-mat/0009065; S.N. Dorogovtsev and J.F.F. Mendes, e-printcond-mat/0009090. agents: whoupdatesandhowoften. Itremainstounder- [13] M.E. Newman, S.H. Strogatz, and D.J. Watts, e-print stand the potentially more intricate reverse effect: how cond-mat/0007235. the amount of information currently stored in the Web [14] R. Morelli, Java, Java, Java—Object-Oriented Problem influences the conduct of the agents, with implications Solving, Prentice Hall, UpperSaddleRiver, (2000). on self-tuning of the control parameter β. In our model [15] H.-O. Peitgen, H. Ju¨rgens, and D. Saupe, Chaos and the structure of the network at β > βc(N) appears to Fractals—New Frontiers of Science, Springer-Verlag, be supercritical, where βc(N) < 0.1 in a large network. New York (1992). Therefore, it is plausible to expect that the evolutionary [16] For a recent review of exact results on dynamic critical selected values of β will be much smaller than the cur- systemsseeD.Dhar,The Abbelian Sandpile and Related rentlyestimatedvalueβ ≈3,ifthescenarioofself-tuning Models, Physica A 263, 4 (1999). 5 8000 [17] G. Grimmett, Percolation, Second edition, Springer- Verlag, Berlin (1999). [18] For random percolation on graphs see D. S. Calaway et 7000 al.,e-print cond-mat/0007300 and references therein. [19] An example of dynamic criticality in the presence of a e m 6000 controlparameterisstudiedanalyticallyandnumerically Ti inB.Tadi´c,andD.Dhar,Phys.Rev.Lett. 79,1519-522 (1997); B. Tadi´c, Phys.Rev.E 59, 1452 (1999). 5000 [20] Our results suggest that 0 < βc(N) < 0.1 varying with N, but it is strictly larger than zero for any finite N ac- 4000 cessibleinthesimulations.Anestimateofβc byemploy- 1000 1200 1400 1600 1800 2000 ingdifferentsystemsizesandafinite-sizescalinganalysis Node remains out of thescope of this paper. 10−1 b 10−2 ACKNOWLEDGMENTS ∆t)10−3 a P( This work was supported by the Ministry of Science and Technology of the Republic of Slovenia. I am grate- 10−4 ful to Deepak Dhar, Vyatcheslav Priezzhev and A´lvaro Corral for their response and for the constructive criti- 10−5 100 101 102 103 104 cism on this work. ∆t 100 3 FIG.2. Toppanel: Partofthetemporalpatternoflinking 2.5 10−1 inthegrowth phaseofnetworkwithN =10,000nodes. One 1 link per time step is considered. Dots represent nodes from − 2 τ which the link originates, whereas crosses are target nodes. Lower panel: Distribution of time intervals ∆t between two 10−2 1.5 successive linking (a) from a given node, and (b) to a given node. The two patterns become correlated for the time in- 1 P(q)10−3 0 0α.5 1 ttherevasclsal∆intginregthioenraonfgtehe10r0an<k∆ditst<rib3u,t0i0o0n,sc2o0r0re0sp>onqd>ing20t0o in Fig. 1. Distributions are averaged over 100 samples and logarithmically binned. Both distributions are normalized to 10−4 the total number of time steps. Note that the events with ∆t=0thatcorrespondtotheoutgoinglinksfromnewadded b nodes, contributingto the distribution (a), are not shown. 10−5 d c a 10−6 100 101 102 103 q FIG.1. Cumulativedistributionsofoutgoinglinks(a)and incoming links (b) for the network of N = 1,000,000 nodes and average ratio β =3 of updated respectiveto added links per time unit. For comparison we have included the corre- spondingdistributions(c)and(d)inthecaseoffullyrandom directed graph. Fitted slopes of the straight sections of the curves(a) and(b)arecompatible with thescaling exponents in Eq. (3) as τout = 2.75 and τin = 2.25, respectively. Inset: Variation of the scaling exponents τout−1 (diamonds) and τin−1 (triangles) with the control parameter α≡1/(β+1) in the physical range (0,1). Dotted line: exact solution for thecase of incoming links from Ref. [11]. 6 100 100 100 10−1 10−1 10−1 P(d) 10−2 n(f) 10−2 P(s)10−2 10−3100 1d01 102 ββ==01.081 10−3 102 103 104 f [Index] 10−3 40 n(t) 100 101 102 103 s e s d o of n20 er b FIG.3. Differential distribution of size of connected clus- m u ters in the network grown with the rules of Eqs. (1)-(2) for N different values of the control parameter β = 3 (diamonds), 1 (circles), 0.081 (triangles), and zero (crosses). We employ 0 1000 1200 1400 1600 1800 2000 1,000 avalanches in each of 100 samples of the network with Cumulative number of steps t N = 1,000 nodes. Inset: Corresponding distributions of depth of theconnected clusters. FIG. 4. (Lower panel) Number of nodes n(t) added to a connected cluster at one investigation step vs cumulative number of steps t. Size of one connected cluster is repre- sentedbythesurfaceenclosedbetweentwoconsecutivedrops of the signal n(t) to the base line n(t) = 1. Parameters are: N = 1,000 and β = 1. (Top panel) Fourier spec- trum of the same signal. Also shown is the spectrum of the corresponding signal at the critical value of the parameter βc(N) = 0.081. Data are logarithmically binned. Straight lines are power-law fits with the slopes φ = 1.45±0.06 (for β=1) and φ=1.01±0.06 (for β=0.081). 7