Nucleation in scale-free networks
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Nucleation in scale-free networks
Hanshuang Chen , Chuansheng Shen , , Zhonghuai Hou , ∗ and Houwen Xin Hefei National Laboratory for Physical Sciences at Microscales and Department of Chemical Physics,University of Science and Technology of China, Hefei, 230026, China Department of Physics, Anqing Teachers College, Anqing 246011, China (Dated: November 1, 2018)We have studied nucleation dynamics of the Ising model in scale-free networks with degree distri-bution P ( k ) ∼ k − γ by using forward flux sampling method, focusing on how the network topologywould influence the nucleation rate and pathway. For homogeneous nucleation, the new phaseclusters grow from those nodes with smaller degree, while the cluster sizes follow a power-law distri-bution. Interestingly, we find that the nucleation rate R Hom decays exponentially with the networksize N , and accordingly the critical nucleus size increases linearly with N , implying that homoge-neous nucleation is not relevant in the thermodynamic limit. These observations are robust to thechange of γ and also present in random networks. In addition, we have also studied the dynamicsof heterogeneous nucleation, wherein w impurities are initially added, either to randomly selectednodes or to targeted ones with largest degrees. We find that targeted impurities can enhance thenucleation rate R Het much more sharply than random ones. Moreover, ln( R Het /R Hom ) scales as w γ − /γ − and w for targeted and random impurities, respectively. A simple mean field analysis isalso present to qualitatively illustrate above simulation results. PACS numbers: 89.75.Hc, 64.60.Q-, 05.50.+q
I. INTRODUCTION
Complex networks describe not only the pattern dis-covered ubiquitously in real world, but also provide aunified theoretical framework to understand the inher-ent complexity in nature [1–4]. Many real networks, asdiverse as ranging from social networks to biological net-works to communication networks, have been found tobe scale-free [5], i.e., their degree distributions follow apower-law, P ( k ) ∼ k − γ . A central topic in this field hasbeen how the network topology would influence the dy-namics taking place on it. Very recently, critical phenom-ena in scale-free networks (SFNs) have attracted consid-erable research interest [6], such as order-disorder transi-tions [7–10], percolation [11–14], epidemic spreading [15],synchronization [16, 17], self-organized criticality [18, 19],nonequilibrium pattern formation [20], and so on. Thesestudies have revealed that network heterogeneity, charac-terized by diverse degree distributions, makes the criticalbehaviors of SFNs quite different from those on regularlattices. However, most previous studies focused on eval-uating the onset of phase transition in different networkmodels. There is little attention paid to the dynam-ics/kinetics of phase transition, such as nucleation andphase separation in complex networks.Nucleation is a fluctuation-driven process that initi-ates the decay of a metastable state into a more stableone [21]. A first-order phase transition usually involvesthe nucleation and growth of a new phase. Many impor-tant phenomena in nature, including crystallization [22],glass formation [23], and protein folding [24], etc., are ∗ Electronic address: [email protected] associated with nucleation. Despite its apparent impor-tance, many aspects of nucleation process are still un-clear and deserve more investigations. The Ising model,which is a paradigm for many phenomena in statisticalphysics, has been widely used to study the nucleationprocess. Despite its simplicity, the Ising model has madeimportant contributions to the understanding of nucle-ation phenomena in equilibrium systems and is likely toyield important insights also for nonequilibrium systems.In two-dimensional lattices, for instance, shear can en-hance the nucleation rate and at an intermediate shearrate the nucleate rate peaks [25], a single impurity mayconsiderably enhance the nucleation rate [26], and theexistence of a pore may lead to two-stage nucleation andthe overall nucleation rate can reach a maximum level atan intermediate pore size. Nucleation pathway of Isingmodel in three-dimensional lattice has also been studiedusing transition path sampling approach [27]. In addi-tion, Ising model has been frequently used to test the va-lidity of classical nucleation theory (CNT) [28–32]. How-ever, all these studies are limited to regular lattices inEuclidean space. Since many real systems can be mod-eled by complex networks, it is thus natural to ask howthe topology of a networked system would influence thenucleation process of Ising model. To the best of ourknowledge, this has never been studied in the literatureso far.Although the main motivation of the present study isto address a fundamental problem in statistical physics,it may also be of practical interest. For example, ourstudy may help understand how public opinion or beliefchanges in the social context [33], where binary spinscan represent two opposite opinions and the concept ofphysical temperature corresponds to a measure of noisedue to imperfect information or uncertainty on the partof the agent. Another example is functional transition inreal biological networks, such as the transition betweendifferent dynamical attractors in neural networks [34],wherein the two states of the Ising model may correspondto a neuron being fired or not. Other examples includegene regulatory networks, wherein genes can be on or off,corresponding to the two states of Ising model [35].In the present work, we have studied nucleation pro-cess of the Ising model in SFNs. Since nucleation is anactivated process that occurs extremely slow, brute-forcesimulation is prohibitively expensive. To overcome thisdifficulty, we adopt a recently developed forward fluxsampling (FFS) method to obtain the rate and path-way for nucleation [36]. For homogeneous nucleation, wefind that the nucleation begins with nodes with smallerdegree, while nodes with larger degree are more stable.We show that the nucleation rate decays exponentiallywith the network size N , and accordingly the critical nu-cleus size increases linearly with N , implying that ho-mogeneous nucleation can only occur in finite-size net-works. Comparing the results of networks with different γ and those of random networks, we conclude that net-work heterogeneity is unfavorable to nucleation. In addi-tion, we have also investigated heterogeneous nucleationby adding impurities into the networks. It is found thatthe dependence of the nucleation rate on the number ofrandom impurities is significantly different from the caseof targeted impurities. These simulation results may bequalitatively understood in a mean-field manner.The rest of the paper is organized as follows. In Sec.II,we give the details of our simulation model and the FFSmethod applied to this system. In Sec.III, we presentthe results for the nucleation rate and pathway. We thenshow, via both simulation and analysis, that the system-size effect of the nucleation rate and heterogeneous nu-cleation. At last, discussion and main conclusions areaddressed in Sec.IV. II. MODEL AND SIMULATION DETAILSA. The networked Ising model
The Ising model in a network comprised of N nodes isdescribed by the Hamiltonian H = − J X i
FFS method has been used to calculate rate constants,transition paths and stationary probability distributionsfor rare events in equilibrium and nonequilibrium systems[25, 26, 36, 38–40]. This method uses a series of interfacesin phase space between the initial and final states to forcethe system from the initial state A to the final state B ina ratchet-like manner. An order parameter λ ( x ) is firstdefined, where x represents the phase space coordinates,such that the system is in state A if λ ( x ) < λ and state B if λ ( x ) > λ m , while a series of nonintersecting interfaces λ i (0 < i < m ) lie between states A and B , such thatany path from A to B must cross each interface withoutreaching λ i +1 before λ i . The transition rate R from A to B is calculated as R = ¯Φ A, P ( λ m | λ ) = ¯Φ A, Y m − i =0 P ( λ i +1 | λ i ) , (2)where ¯Φ A, is the average flux of trajectories crossing λ in the direction of B . P ( λ m | λ ) = Q m − i =0 P ( λ i +1 | λ i )is the probability that a trajectory crossing λ in thedirection of B will eventually reach B before returningto A , and P ( λ i +1 | λ i ) is the probability that a trajectorywhich reaches λ i , having come from A , will reach λ i +1 before returning to A . For more information about FFS,please turn to Ref.[41]. III. RESULTSA. Homogeneous Nucleation: Rate and pathway
To begin, we first consider homogeneous nucleation inBarab´asi–Albert scale-free network (BA-SFN) whose de-gree distribution follows a power-law P ( k ) ∼ k − γ withthe scaling exponent γ = 3 [5]. We define the orderparameter λ as the total number of up spins in the net-works. We set N = 1000, the average degree h k i = 6, T = 2 . h = 0 . λ = 130 and λ m = 880, where T is lower than the critical temperature T c ≃ .
36. Thespacing between interfaces is fixed at 3 up spins, but thecomputed results do not depend on this spacing. During
FIG. 1: (Color online) Snapshots of nucleation in a BA scale-free network with N = 100 and h k i = 2 at four differentstages. Up-spins and down-spins are indicated by red squaresand black circles, respectively. the FFS sampling, we perform 1000 trials at each inter-face, from which at least 100 configurations are stored inorder to investigate the statistical properties of the en-semble of nucleation pathway. We obtain ¯Φ A, = 1 . × − M Cstep − spin − and P ( λ m | λ ) = 4 . × − ,resulting in R Hom = 5 . × − M Cstep − spin − fol-lowing Eq.2. Such a nucleation rate is very low such thata brute-force simulation would be very expensive.From the stored configurations at each interface, onecan figure out the details of the nucleation pathway. Fig-ure 1 illustrates schematically four stages of a typicalnucleation pathway. Clearly, the new phase (indicatedby squares) starts from nodes with smaller degree, whilenodes with larger degree are more stable. This pictureis reasonable because nodes with larger degrees need toovercome more interfacial energies. Figure 2(a) plotsthe average degree of network nodes in the new phase, h k new i , as a function of the order parameter λ . As ex-pected, h k new i increases monotonously with λ . On theother hand, it is observed that the formation of largeclusters of new phase is accompanied with the growthand coalescence of small clusters. Interestingly, we findthat the size N c of new phase clusters follows a power lawdistribution at early stages of nucleation, P ( N c ) ∼ N − αc with the fitting exponent α ≃ .
44, as shown in Fig.2(b).With the emergence of a giant component of new phase,the tail of the distribution is elevated, but the size distri-bution for the remaining clusters still follows power-law.The underlying mechanism of such phenomenon is stillan open question for us.To determine the critical size of the nucleus, λ c , wecompute the committor probability P B , which is theprobability of reaching the thermodynamic stable statebefore returning to the metastable state. The depen-dence of P B on λ is plotted in Fig.3(a). As commonly P ( N c ) N c =193,
5, giving the critical nucleus size λ F F Sc = 474. The committor distribution at λ F F Sc ex-hibits a peak at 0 .
5, of which 70% of spin configurationshave P B values within the range of 0 . . λ is a proper order pa-rameter.Note that conventionally the nucleation threshold λ c is usually estimated by using CNT [42–44]. One cancalculate the free energy change along the nucleationpath, ∆ F ( λ ), by using methods like umbrella sampling(US) [45]. According to CNT, ∆ F will bypass a max-imum at λ = λ USc , and the nucleation rate is given by ν exp( − β ∆ F c ), where ν is an attempt frequency. Herewe have computed ∆ F by using US, in which we haveadopted a bias potential 0 . k B T ( λ − ¯ λ ) , with ¯ λ beingthe center of each window. As shown in Fig.3(b), themaximum in ∆ F occurs at λ USc = 451, giving a free-energy barrier of ∆ F c ≃ . k B T . Clearly λ USc givesa fairly good estimation of λ F F Sc . To calculate the nu-cleation rate, however, one has to obtain the attemptfrequency ν , which is not a trivial task. If we justset ν = 1, we obtain a CNT prediction of a rate of2 . × − M Cstep − spin − , which is 9 orders of magni-tude faster than that computed from FFS method. Thislevel of disagreement in nucleation rate corresponds to anerror in the free-energy barrier of about 24%. Since theaccurate value of ν is generally unavailable, we will useFFS method to calculate the nucleation rate throughoutthis paper. In addition, real nucleation pathway cannot
400 450 500 550 6000.00.20.40.60.81.0 100 200 300 400 500-20020406080 0.5 0.6 0.7 0.8 0.9 1.0-175-150-125-100-75-50 380400420440460480500520 P B P B () P B ( )=0.5(a) F C =91.4 F USC =451(b) l n R H o m c h lnR Hom (c)
FFSc
USc
FIG. 3: (Color online) (a) The committor probability P B asa function of λ ; The inset plots the committor distributionat λ F F Sc . (b) The free energy ∆ F as a function of λ , inwhich the maximum in ∆ F occurs at λ USc . (c) The logarithmof homogeneous nucleation rate ln R Hom (left axis), and thecritical size of nucleus, λ F F Sc and λ USc (right axis), obtainedby FFS method and US method, respectively, as functions of h . Other parameters are the same as Fig.2. be obtained by conventional US method due to the useof a biased potential.We have also investigated how the nucleation rate andthreshold depend on the external field h . In Fig.3(c),ln R Hom , λ F F Sc and λ USc are plotted as functions of h .The error-bars are obtained via 20 different network re-alizations and 10 independent FFS samplings. As ex-pected, ln R Hom increases monotonously with h , and λ F F Sc and λ USc both decrease with h . For large h , thedifference between λ F F Sc and λ USc becomes small. If h is large enough, one expects that the free-energy barrierwill disappear, and nucleation will be not relevant. B. Homogeneous Nucleation: System-size effects
According to Fig.3, one finds that nearly half of thenodes must be inverted to achieve nucleation. This meansthat for a large network, nucleation is very difficult tooccur. An interesting question thus arises: How the nu-cleation rate and threshold depend on the network size?To answer this question, we have performed extensive simulations to calculate R Hom and λ c for different net-work size N . In particular, besides the BA-SFNs, we havealso considered different network types, including SFNswith other scaling exponent γ and homogeneous randomnetworks (HoRNs) [46]. The networks are generated ac-cording to the Molloy-Reed model [47]: Each node is as-signed a random number of stubs k that are drawn froma specified degree distribution. This construction elimi-nates the degree correlations between neighboring nodes.We note here that the exponent γ can be a measure ofdegree heterogeneity of the network, i.e., the smaller γ is, the degree distribution is more heterogeneous. In aHoRN, each node is equivalently connected to other h k i nodes, randomly selected from the whole network, suchthat no degree heterogeneity exists. By comparing the re-sults in SFNs with different γ as well as that in HoRNs,one can on one hand, check the robustness of the systemsize effects, and on the other hand, investigate how thedegree heterogeneity affects the nucleation process.Figure 4 shows the simulation results. All the param-eters are the same as in Fig.2, expect that N varies from N = 500 to N = 3000. Interestingly, both ln R Hom and λ c show very good linear dependences on the sys-tem size, i.e., ln R Hom ∼ − aN and λ c ∼ bN with a and b being positive constants. Obviously, in the thermody-namic limit N → ∞ , we have R Hom → λ c → ∞ .This means that nucleation in these systems is not rele-vant in the thermodynamic limit, and only finite-size sys-tems are of interest. As shown in Fig.4, for the networktypes considered here, qualitative behaviors are the same.Quantitatively, with increasing γ , the line slope becomessmaller, R Hom becomes larger and λ c gets smaller. Sincelarger γ corresponds to more homogeneous degree dis-tribution, these results indicate that the degree hetero-geneity is unfavorable to nucleation. This is consistentwith the nucleation pathway as shown in Fig.2: In a het-erogenous network, those hub nodes are difficult to flip,making the nucleation difficult.In the following, we will show that the system-size ef-fects can be qualitatively understood by CNT and simplemean-field (MF) analysis. According to CNT, the forma-tion of a nucleus lies in two competing factors: the energycost of creating a new up spin which favors the growth ofthe nucleus, and an opposing factor which is due to thecreation of new interfaces between up and down spins.The change in the free energy may be written as [42–44]∆ F ( λ ) = − hλ + σλ, (3)where σ denotes the effective interfacial free energy,which may depend on T , h , and N . Since interfacialinteractions arise from up spins inside the nucleus anddown spins outside it, one may write σ = 2 JK out byneglecting entropy effects (zero-temperature approxima-tion), where K out is the average number of neighboringdown-spin nodes that an up-spin node has. Using MFapproximation, one has K out = h k i (1 − λ/N ). Insertingthis relation to Eq.3 and maximizing ∆ F with respect to
500 1000 1500 2000 2500 3000-600-500-400-300-200-1000500 1000 1500 2000 2500 300002505007501000125015001750 (b) l n R H o m N SFN: =2.55 SFN: =3.0 SFN: =5.0 HoRN (a) c N FIG. 4: (Color online) (a) The logarithm of homogeneousnucleation rate ln R Hom and (b) the critical size of nucleus λ c as functions of the network size N in SFNs with different γ and in HoRNs. Other parameters are the same as Fig.2. λ , we have λ MFc = J h k i − h J h k i N, (4)and the free-energy barrier∆ F MFc = ( J h k i − h ) N J h k i . (5)Clearly, both λ MFc and ∆ F MFc linearly increase with N if other parameters are fixed. Therefore, the linear rela-tionships shown in Fig.4 are essentially analogous to thebehavior of a mean-field network. Quantitatively, how-ever, the MF analysis fails to predict the line slopes inFig.4. This can be understood because the approxima-tions are so crude, wherein important aspects such asnetwork heterogeneity and entropy effects have not beenaccounted for. A rigid analysis is not a trivial task andbeyond the scope of the present work. C. Heterogeneous nucleation
In practice, most nucleation events that occur in na-ture are heterogeneous, i.e., impurities of the new phaseare initially present. It is well known that impurities canincrease nucleation rate by as much as several orders ofmagnitude. In our model, impurities are introduced by fixing some nodes in up-spin state. We are interestedin how the number w of impurity nodes and the way ofadding impurities would affect the nucleation rate. Thefirst way of adding impurities we use is that impuritynodes are selected in a random fashion. Figure 5(a) givesthe simulation results of ln (cid:16) R Het R Hom (cid:17) as a function of w indifferent network models, where R Het are the rates of het-erogeneous nucleation. As expected, nucleation becomesfaster in the presence of random impurities no matterwhich kind of network model is applied. It seems that inFig.5(a) all data collapse and exhibit a linear dependenceon w , with the fitting slope 3 .
33. This means that eachadditional random impurity can lead to the increase ofthe rate by more than one order of magnitude. For thesecond way, we select w nodes with most highly degree asthe impurity nodes, termed as targeted impurities. Strik-ingly, such a targeted scheme is much more effective inincreasing the nucleation rate than random one, as shownin Fig.5(b). For example, for SFNs with γ = 3, one sin-gle targeted impurity can increase the rate by about 36orders of magnitude.As in Sec.III B, below we will also give a MF analy-sis of the heterogeneous nucleation, which qualitativelyagrees with the simulation results. Each impurity nodecontributes an additional term to the free energy bar-rier, which can, under zero-temperature approximation,be written as the product of − J and the expecteddegree of the impurity node. For random impurities,each impurity node has an expected degree h k i , yieldingthe term − J h k i . Thus, the resulting free-energy bar-rier of the heterogenous nucleation becomes ∆ F Hetc =∆ F Homc − J h k i w, where ∆ F Homc is the free-energy bar-rier of homogeneous nucleation. According to CNT, oneobtains ln (cid:18) R Het R Hom (cid:19) = 2 J h k i k B T w. (6)Therefore, nucleation with random impurities is alwaysfaster than without impurity, and ln (cid:16) R Het R Hom (cid:17) should varylinearly with w . The theoretical estimate of the slope isgiven by 2 J h k i /k B T = 4 .
63, approximately consistentwith the simulation one. Given the simple nature of theabove approximation the agreement is satisfactory. Forthe targeted way, a similar treatment to the former casecan also be executed, except that h k i should be replacedby h k i w , where h k i w is the average degree of the w tar-geted nodes. After simple calculations, we can obtain h k i w = h k i ( Nw ) γ − . This leads to a free-energy barrier,∆ F Hetc = ∆ F Homc − J h k i N γ − w γ − γ − , andln (cid:18) R Het R Hom (cid:19) = 2 J h k i k B T N γ − w γ − γ − . (7)Compared with Eq.6, besides the presence of an addi-tional size-dependent factor of N γ − , the w -dependentfactor becomes w γ − γ − rather than w . For homogeneous l n ( R H e t / R H o m ) w(b) targeted way l n ( R H e t / R H o m ) w l n ( R H e t / R H o m ) w SFN: =2.55SFN: =3.0SFN: =5.0HoRN (a) random way
FIG. 5: (Color online) ln (cid:16) R Het R Hom (cid:17) as a function of the numberof impurity nodes w . (a) corresponds to the case of randomimpurities, while (b) and (c) to the case of targeted impurities.(c) plots ln (cid:16) R Het R Hom (cid:17) vs w in double logarithmic coordinates.All dotted lines are drawn by linear fitting. Other parametersare the same as Fig.2 except for h = 0 . network, i.e. γ → ∞ , one obtains N γ − → w γ − γ − → w , and Eq.7 is thus equivalent to Eq.6. InFig.5(c), we plot the simulation results of ln (cid:16) R Het R Hom (cid:17) asa function of w in double logarithmic coordinates, wherelinear dependences are apparent, in agreement with theresult of Eq.7. The fitting slopes µ and intercepts κ for γ = 2 . , . , . µ = 0 . , . , . κ = 2 . , . , .
08, respectively, while our ana-lytical estimates are µ = γ − γ − = 0 . , . , .
75, and κ = log (cid:18) J h k i N γ − k B T (cid:19) = 2 . , . , .
42, respectively.Simulations and analysis give the same trends of the val-ues of µ , κ and γ . IV. DISCUSSION AND CONCLUSIONS
Our investigations of system-size effects of nucleationin SFNs and HoRNs have revealed that the nucleation rate is size-dependent, and it decreases exponentiallywith the network size, resulting in that nucleation onlyoccurs at finite-size systems. As we already know, suchsystem-size dependence does not exist in two-dimensionalregular lattices [26]. We have also studied nucleation ofIsing system in regular networks where each node is con-nected to its k -nearest neighbors (here we only considerthe case of sparse networks, i.e., k ≪ N ), and foundthat the rate is almost independent of network size (re-sults not shown). Therefore, nucleation process in SFNsand HoRNs are quite different from that in regular lat-tices or networks. Such differences may originate fromthe infinite-dimensional properties of SFNs and HoRNs,wherein the average path distance is rather small, render-ing the system’s behavior analogous to that of a mean-field network. An interesting situation arises when oneconsiders Watts-Strogatz small-world network, which isconstructed by randomly rewiring each link of a regularnetwork with the probability p [48]. With the incrementof p from 0 to 1, the resulting network changes from aregular network to a completely random one. As men-tioned above, for the nucleation process, no system-sizeeffects exist for p = 0, while system-size dependence ex-ists for p = 1. One may naturally ask: How does thecrossover happens when p changes from 0 to 1, and whatis the physical significance of such a transition? Thisquestion surely deserves further investigations and maybe the content of a future publication.In summary, we have studied homogeneous and hetero-geneous nucleation of Ising model in SFNs, by using FFSmethod. For homogeneous nucleation, we find that theformation of new phase starts from nodes with smallerdegree, while nodes with higher degree are more stable.Extensive simulations show that the nucleation rate de-creases exponentially with the network size N , and thenucleation threshold increases linearly with N , indicat-ing that nucleation in these systems are not relevant inthe thermodynamic limit. For heterogeneous nucleation,target impurities are shown to be much more efficient toenhance the nucleation rate than random ones. SimpleMF analysis is also present to qualitatively illustrate thesimulation results. Our study may provide valuable un-derstanding how first-order phase transition takes placein network-organized systems, and how to effectively con-trol the rate of such a process. Acknowledgments
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