Zero-range processes, in which particles hop between sites on a lattice, are
closely related to equilibrium networks, in which rewiring of links take place.
Both systems exhibit a condensation transition for appropriate choices of the
dynamical rules. The transition results in a macroscopically occupied site for
zero-range processes and a macroscopically connected node for networks.
Criticality, characterized by a scale-free distribution, is obtained only at
the transition point. This is in contrast with the widespread scale-free
real-life networks. Here we propose a generalization of these models whereby
criticality is obtained throughout an entire phase, and the scale-free
distribution does not depend on any fine-tuned parameter.Comment: 4 pages, 4 figure

A. G. Angel

B. Schmittmann

D. Mukamel

E. Levine

G. Grinstein

M. R. Evans

P. Bak

R. Dickman

S. N. Dorogovtsev

English

arXiv

Crossref

Critical phase in nonconserving zero-range processes and rewiring networks

Angel, A. G.

Evans, M. R.

Levine, E.

Mukamel, D.

Aberdeen University Research

arXiv:cond-mat/0503487v1  [cond-mat.stat-mech]  18 Mar 2005Critical phase in non-conserving zero-range processes and equilibrium networksA.G. Angel,1 M.R. Evans,1 E. Levine,2, 3 and D. Mukamel21School of Physics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK2Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel 761003Center for Theoretical Biological Physics, University of California at San-Diego, La Jolla, CA 92093Zero-range processes, in which particles hop between sites on a lattice, are closely related toequilibrium networks, in which rewiring of links take place. Both systems exhibit a condensationtransition for appropriate choices of the dynamical rules. The transition results in a macroscopicallyoccupied site for zero-range processes and a macroscopically connected node for networks. Criticality,characterized by a scale-free distribution, is obtained only at the transition point. This is in contrastwith the widespread scale-free real-life networks. Here we propose a generalization of these modelswhereby criticality is obtained throughout an entire phase, and the scale-free distribution does notdepend on any fine-tuned parameter.PACS numbers: 89.75.-k, 05.70.Ln, 05.40.-aMany driven, non-equilibrium models, reach a criticalor scale invariant steady state only when their dynami-cal parameters are fine-tuned to reach a phase transitionpoint. Examples include a wide range of systems suchas jamming in traffic [1], coalescence in granular gases[2], gelation in networks [3] and wealth condensation inmacroeconomics [4]. In all these systems one has a con-densation phase transition which we shall discuss in de-tail below. In other non-equilibrium models, for examplein driven lattice gases and sandpile models [5, 6], it hasbeen argued that scale invariance and power-law distri-butions are generic, or at least one may have scale-freedistributions across wide regions of the parameter spacerather than just at critical points. This phenomenon hasbeen termed self organized criticality.In recent years considerable attention has been givento the study of real-life networks. Networks, defined ascollections of nodes connected by links, are found in manyfields of study, ranging from molecular biology to socialcommunities and the Internet [7, 8]. With each node oneassociates a degree k which is the number of links con-nected to it. In general links may be directed or they maycarry a weight, however for our purposes we do not con-sider such features. It has been observed that very oftenreal-life networks are characterized by a degree distribu-tion p(k) which decays algebraically for large k [7, 8].These networks, termed scale-free networks, are indeedcritical, suggesting the existence of a mechanism whichdrives them to this state. Subsequently, dynamical pro-cesses for growing networks have been proposed whichresult in a critical distribution for a wide range of the dy-namical parameters. In these processes nodes and linksare continually added to the network with some prede-termined rates [3, 9, 10, 11]. On the other hand “equi-librium” networks [8, 12], whose dynamics constitutesrewiring processes with a fixed number of nodes, exhibita critical distribution only at a critical point. This tran-sition corresponds to condensation (also referred to asgelation) where a single node captures a finite fractionof the links. Note that although termed equilibrium net-works, their dynamics does not always obey detailed bal-ance, thus their steady state may not always be a thermalequilibrium one.Instructive insight has been gained into the condensa-tion transition through the analysis of simple interactingparticle systems [13, 14, 15]. These systems form fun-damental models which may be mapped onto particularapplications. For example, the zero-range process (ZRP)[16] is a particularly simple and exactly soluble modelin which each site µ of a lattice may contain an integernumber of particles nµ and particles hop to a neighbor-ing site with rate u(nµ). This model is closely related tothe equilibrium networks discussed above which undergorewiring dynamics [12]; in the following we will exploitthis relationship.Condensation in the ZRP will occur, for example, whenu(n) decays to some finite, large-n asymptotic value β,as u(n) ∼ β(1+b/n) with b > 2. The transition is simplyunderstood by considering p(n) the steady-state proba-bility that a site contains n particles. For a subcriticaldensity of particles one finds that p(n) decays exponen-tially with decay length dependent on the conserved par-ticle density ρ = N/L where N is the number of particlesand L is the number of sites. This p(n) describes the lowdensity, fluid phase. As the density is increased the de-cay length increases until at the critical density, ρc, it di-verges and one has a power-law distribution p(n) ∼ n−b,thus a critical fluid. Above ρc, in addition to the powerlaw, a piece of p(n) emerges centered about n = L(ρ−ρc);this piece represents the condensate [15]. Thus the su-percritical phase corresponds to a critical fluid coexistingwith a condensate. Only at criticality does one have apure power-law distribution.In the present work we investigate how a critical phasemay emerge in processes such as ZRPs and networkmodels. We shall see that the introduction of non-conservation in an appropriate fashion, modifies the con-densate phase into a scale-free phase by effectively sup-pressing the condensate and leaving the critical fluid. Inthis generalization of the ZRP particles are created atall sites at constant rate but are removed at a rate thatdepends non-linearly on the occupation number. Equiv-2alently in the network context links are created and de-stroyed. We begin by elucidating the mechanism for thegeneration of a critical phase within a generalized ZRP;later we will define an equilibrium network where thismechanism is also manifested.Consider a lattice of L sites upon which reside a num-ber of particles. With rate u(nµ) (probability per unittime) which depends on the occupation nµ of site µ, aparticle is transferred from site µ to another site. Weconsider a fully connected geometry, where the destina-tion site is chosen randomly from the other L − 1 sites.In addition to the hopping dynamics, particles are addedto site µ with a constant rate c, or removed with a ratea(nµ), which increases with the site occupation nµ. Thus,our choice for the dynamical rates is given byn,m → n− 1,m+ 1 with rate u(n) =(1 + bn)θ(n)n → n+ 1 with rate c =(1L)s(1)n → n− 1 with rate a(n) =(nL)k,where b, s and k are positive parameters, and θ(n) is theusual Heaviside step function. The dynamical rates areconveniently implemented by using a random-sequentialupdating scheme, whereby at each time step a site µ ischosen at random, and a hop, annihilation or creationevent may occur with relative probabilities given by therates (1). With b ≤ 2 the system is always found in asub-critical phase, where the particle number distribu-tion is exponential. We therefore restrict our discussionhereafter to the case b > 2.In Fig. 1 we compare the particle number distributionp(n) of our model with that of the ZRP with conservingdynamics. It is clearly seen that for this choice of param-eters, the creation/annihilation dynamics can selectivelydestroy any condensate and sustain the power-law dis-tribution, corresponding to the critical fluid. In whatfollows we analyze the model showing that this featureholds for an entire region in the parameter space.The steady state of the model is fully described by theprobability distribution P (n1, n2, . . . , nL) over all possi-ble configurations. In contrast to the conserving ZRP[16], the steady-state distribution of the model (1) doesnot factorize generally. However, we make the mean fieldapproximation that the steady state distribution doesfactorize, i.e. P (n1, n2, . . . , nL) →∏Li=1 p(ni). Due tothe fully-connected geometry we expect this approxima-tion to become exact in the limit L → ∞.Using this approximation, the steady-state masterequation is given by0 = [u(n+ 1) + a(n+ 1)] p(n+ 1)− [λ+ c] p(n) (2)−{[u(n) + a(n)] p(n)− [λ+ c] p(n− 1)} θ(n).Here the current λ is given byλ =∞∑n=1u(n)p(n) . (3)10 100 1000 10000n1e-091e-081e-071e-061e-050.00010.0010.01p(n)FIG. 1: Steady state distribution of the non-conserving ZRP(black solid line) and the conserving ZRP (grey dotted line).The data is from simulations run on a system of size L = 104,with b = 2.6, k = 3 and s = 1.96. In the conserving modelthe particle density was set to ρ = 4 > ρc. The peak athigh occupation number, which exists only in the conservingmodel, corresponds to the condensate.From (2) it follows thatp(n) =(λ+ c)n∏nm=1 [a(m) + u(m)]p(0) . (4)Note that this is not a closed solution as λ depends onp(n). The values of λ and p(0) should be set such thatboth the normalization condition 1 =∑∞n=0 p(n) and thecreation/annihilation balance conditionc =∞∑n=1a(n)p(n) (5)are obeyed. We now identify the three phases of themodel by determining the asymptotic, large L behaviorsof p(n) and λ which satisfy (3) and (5). The emergentphase diagram is summarised in Fig. 2. Deferring detailsto a later publication, we find the following results:Low-density phase, s > k — Rewriting (5) asLk−s =∑nkp(n) implies p(n) is a rapidly decreasingfunction of n, and the steady state density ρ is ≪ 1.Thus, p(1) ≃ ρ ∼ Lk−s and in the thermodynamic limitthe density goes to zero.For s < k the sum in (5) is controlled by the behaviorof p(n) at large n. We find the following regimes:High-density phase, s < k/(k + 1) — Herep(n) ∼ n−b exp[nLs−nk+1(k + 1)Lk], (6)thus p(n) is strongly peaked at n ∼ L1−s/k. This is thehigh density phase, where all sites are highly occupied.Note that the mean number of particles in the system,N ∼ L2−s/k is super-extensive.Critical phase, k > s > k/(k+1) — In this phase thesystem relaxes to the critical density, and p(n) takes analgebraic form. However, for large but finite systems two3High Densityks(a)DensityLowCritical(b)FIG. 2: Typical phase diagram of both ZRP and networkmodels in the k-s plane with b fixed, k,s,b are defined in (1).sub-phases are observed, distinguished by the finite-sizecorrections to the dominant power law.For k > s > kb/(k + 1)p(n) ∼ n−b exp[−nLx], (7)where x = (k − s)/(k − b + 1). This cuts off the powerlaw at n ∼ Lx. We refer to this as critical sub-phase (a).For kb/(k + 1) > s > k/(k + 1)p(n) ∼ n−b exp[nd(lnLL)kk+1−nk+1(k + 1)Lk], (8)where d = b − s(k + 1)/k. Here, on top of the algebraicpart, p(n) is weakly peaked at n ∼ Lk/(k+1)(lnL)1/(k+1).This peak will diminish as L → ∞. We refer to this ascritical sub-phase (b).In Fig. 3 we present typical data obtained from nu-merical simulations in the different phases and comparewith theoretical curves of p(n). We found that in allphases, starting from random initial configurations ofvarious densities, the system relaxes towards its expectedsteady state value. However, for the low-density phasethe time scales for full relaxation were prohibitive and wedo not present steady state data for this phase. In Fig. 3we also provide data for a one-dimensional (1d) variantof the model, where sites are arranged in a 1d array, andparticles are allowed to hop only to the right neighbor ofthe departure site. For the 1d geometry the mean-fieldapproximation is not expected to be exact even in thelimit L → ∞. Nevertheless, we find numerically that thethree phases discussed above exist also in the 1d model.We now apply the approach discussed above to equi-librium networks. To make the analogy with the ZRP,one identifies a site of the ZRP and its occupation num-ber with a node in the network and its degree, respec-tively. We define a network model which incorporatesboth rewiring and creation/annihilation dynamics, andshow how a proper choice of rates leads to the existenceof a critical phase, much like that of the non-conserving10 100 1000 10000n1e-091e-081e-071e-061e-050.00010.0010.010.11p(n)FIG. 3: Steady state distributions from simulations of theZRP model on a fully-connected lattice (+) and a 1d lattice(O), compared with the theoretical asymptotic curves. HereL = 5000 and b = 2.6, k = 3. Dashed curve is critical sub-phase (a) (s = 2); dotted curve is critical sub-phase (b) (s =1.2); full line is high-density phase (s = 0.4).ZRP, within which networks are scale free. Due to theintroduction of annihilation of links, there is no simplemapping from the ZRP to the network model since thiswould require keeping track of pairs of linked particlesin the ZRP. However the two systems are closely relatedand as we shall see share the same phase diagram.We consider a network of L nodes which are linkedtogether by an integer number, N/2, of undirected links(N is the number of particles in the corresponding ZRP).With rate u(nµ) = 1 + b/nµ one of the nµ links is dis-connected from node µ and is rewired to another ran-domly chosen node. This does not change the number oflinks in the network. With rate a(nµ) = (nµ/L)k one ofthe links connected to node µ is removed from the net-work. In addition, a new link is created between nodeµ and other randomly chosen node at a constant ratec = 1/Ls. Again the dynamics is conveniently imple-mented by choosing a node µ randomly at each time step,and changing the wiring with probabilities constructedfrom the relevant rates. The mean-field master equationfor the network model differs slightly from that of theZRP (2), and is given by0 = [a(n+ 1) + u(n+ 1) + Λ(n+ 1)] p(n+ 1)− [a(n) + u(n) + Λ(n)] p(n)θ(n) (9)− [λ+ 2c] p(n) + [λ+ 2c] p(n− 1)θ(n) .Here λ =∑u(n)p(n) as before, andΛ(n) =nN∞∑l=1a(l)p(l) = cnN. (10)The steady state solution to (9) is given byp(n) =(λ+ 2c)n∏nm=1 [Λ(m) + a(m) + u(m)]p(0) . (11)41 10 100 1000n1e-091e-081e-071e-061e-050.00010.0010.010.11p(n)FIG. 4: Steady state probability distributions from simula-tions of the network model in the critical phase (sub-phase(a)). Here b = 2.6, k = 3 and s = 2, with L = 1000 (circles),L = 2000 (squares) and L = 4000 (triangles).The main difference between this result and that of theZRP (4) lies in Λ(n) and its dependence on the to-tal number of links in the system. This complicatesslightly the analysis, however all phases persist includ-ing the critical phase. Thus the phase diagram in Fig. 2also describes the network model: the low-density andhigh-density phases characterize extremely sparse and ex-tremely dense networks; the critical fluid corresponds toa scale-free phase in the thermodynamic limit. Examplesof the degree distribution in the critical phase are given inFig. 4, where we present data obtained from simulationsof systems of increasing sizes. The power-law regime in-creases with system size.Further interesting observations are that the weakpeak of p(m) in critical sub-phase (b) may correspondto a number of highly connected nodes but is distinctfrom a condensate phase. This would correspond to alarge number of hubs in the network. Also, we note thatin the network model the suppression of events whichlink nodes to themselves has a considerable effect on theresults [19], by introducing a new cut-off into the proba-bility distribution which can become the dominant scale.A full analysis will be published elsewhere.Our main interest in the systems studied lies in theemergence of a critical phase which we have shown ex-ists for annihilation and creation indices k, s in the rangek > s > k/(k + 1). We conclude by comparing the crit-ical phases we have identified to the critical points ofthe corresponding conserving ZRP and network models.In the latter, the average particle/link density ρ is anexternal parameter. A power-law distribution of the oc-cupation number/degree is only obtained at ρ = ρc. Incontrast, in the non-conserving models we have studied,the steady-state density is set by the dynamics to be ρcand a power-law distribution is obtained throughout thecritical phase. In models exhibiting self-organized criti-cality [17], a critical phase is typically obtained only whenthe driving rate of the system vanishes with the systemsize [18], in order to ensure relaxation between stimuli.In comparison, in the present work the creation rate c ine.g. (1), which could be thought of as a driving rate forthe system, vanishes in the large L limit whereas u thehopping/rewiring rate does not vanish. Thus, there is aseparation of timescales in the dynamical processes. Onthe other hand, there are no avalanches or underlying ab-sorbing states which are features usually associated withself organized criticality.This study was supported by the Israel Science Foun-dation (ISF). Visits of MRE to the Weizmann Institutewere supported by the Albert Einstein Minerva Centerfor Theoretical Physics. A visit of DM to Edinburgh wassupported by EPSRC programme grant GR/S10377/01.[1] O. J. O’Loan, M. R. Evans and M. E. Cates Phys. Rev.E 58, 1404 (1998).[2] J. Eggers Phys. Rev. Lett. 83, 5322 (1999).[3] P. L. Krapivsky, S. Redner and F. Leyvraz Phys. Rev.Lett. 85, 4629 (2000)[4] Z. Burda et al (2002) Phys. Rev. E 65, 026102[5] B Schmittmann and R K P Zia 1995 Statistical Mechanicsof Driven Diffusive Systems ed C Domb and J L Lebowitz(Academic Press, New York)[6] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett.59, 381 (1987); P. Bak, How Nature Works (Copernicuspress, NY, 1996); H.J. Jensen, Self-Organised Criticality(CUP, 1998)[7] For recent reviews see R. Albert and A.-L. Barabási, Rev.Mod. Phys. 74, 47 (2002); M.E.J. Newman, SIAM Re-view 45, 167 (2003).[8] S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Net-works, (Oxford, OUP, 2003)[9] A.-L. Barabási and R. Albert, Science 286, 509 (1999).[10] P. L. Krapivsky and S. Redner Phys. Rev. E 63, 066123(2001)[11] G. Bianconi and A-L Barabási Phys. Rev. Lett. 86, 5632(2001)[12] S.N. Dorogovtsev, J.F.F. Mendes, A.N. Samukhin, Nucl.Phys. B 666, 396 (2003).[13] For a recent review see M. R. Evans and T. Hanney(2005) cond-mat/0501338[14] P. Bialas, Z. Burda, and D. Johnston, Nucl. Phys. B 493,505 (1997).[15] S.N. Majumdar, M. R. Evans, R. K. P. Ziacond-mat/0501055[16] M. R. Evans Braz. J. Phys. 30, 42 (2000).[17] A. Vespigniani, R. Dickman,M. A. Munoz, and S. Zap-peri, Phys. Rev. Lett. 81, 5676 (1998)[18] D. Sornette, A. Johansen and I. Dornic, J. Phys. I(France) 5, 325 (1995).[19] M. Boguna, R. Pastor-Satorras and A. Vespignani, Euro.Phys. J. B 38, 205 (2004)

Critical phase in non-conserving zero-range processes and equilibrium networks

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Critical phase in non-conserving zero-range processes and equilibrium
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Critical phase in non-conserving zero-range processes and equilibrium networks

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