Localization transition, Lifschitz tails and rare-region effects in network models
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Localization transition, Lifschitz tails and rare-region effects in network models
G´eza ´Odor
Research Center for Natural Sciences, Hungarian Academy of Sciences andP. O. Box 49, H-1525 Budapest, Hungary (Dated: February 14, 2018)Effects of heterogeneity in the suspected-infected-susceptible model on networks are investigatedusing quenched mean-field theory. The emergence of localization is described by the distributionsof the inverse participation ratio and compared with the rare-region effects appearing in simulationsand in the Lifschitz tails. The latter, in the linear approximation, is related to the spectral density ofthe Laplacian matrix and to the time dependent order parameter. I show that these approximationsindicate correctly Griffiths Phases both on regular one-dimensional lattices and on small worldnetworks exhibiting purely topological disorder. I discuss the localization transition that occurs onscale-free networks at γ = 3 degree exponent. PACS numbers: 05.70.Ln 89.75.Hc 89.75.Fb
I. INTRODUCTION
Epidemic spreading in complex networks such as bi-ological populations and computer networks is of greatinterest, both for practical applications and from a funda-mental point of view [1–3]. Simple models, like the Con-tact Process (CP) [4, 5] has been introduced and studiedintensively by various techniques. They can also be con-sidered as simple models of information spreading in so-cial [6] or in brain networks [7]. In these models sites canbe infected (active) or susceptible (inactive). Infectedsites propagate the epidemic to all of their neighbors,with rate λ , or recover (spontaneously deactivate) withrate ν = 1. The susceptible-infected-susceptible (SIS) [8]model differs slightly from the CP, in which the branchingrate is normalized by k , the number of outgoing edges ofa vertex permitting an analytic treatment via symmetricmatrices. By decreasing the infection (communication)rate of the neighbors a continuous phase transition mayoccur at some λ c critical point from a steady state withfinite activity density ρ to an inactive one, with ρ = 0.The latter is also called absorbing, since no spontaneousactivation of sites is allowed. In case of the SIS λ c = 0 innetworks with a degree distribution decaying slower thanan exponential [9] . The transition type is continuousand belongs to the directed percolation universality class[10–13].In real systems various heterogeneities occur, that maycause deviations from the results of the homogeneousmodels. From the homogeneous system point of viewif the disorder varies rapidly both in space and time, itscontribution can be described by an increased tempera-ture or noise of the system [12]. In the quasi-static limit,when the variation of the heterogeneity is much slowerthe dynamics of the pure model we can consider it as aquenched disorder. It causes a memory effect, whose rel-evancy has been studied in quantum and nonequilibrium Note, that some recent studies debate this, see: [67–69] systems (see [14]).In networks, with finite topological dimension, definedas N ∝ r D , where N is the number of nodes within the(chemical) distance r , it was shown [15], that disordercan be relevant. Heterogeneities can induce arbitrarilylarge, rare-regions (RR), changing their state exponen-tially slowly as the function of their sizes, induce so calledGriffiths Phases (GP) [14, 16]. In these phases the dy-namics is slow and non-universal and at the phase transi-tion point it is even slower, logarithmic dynamical scalingmay occur. These heterogeneities can be explicit featuresof the interactions or maybe the result of the topology ofthe graph.Recent observations show generically slow time evo-lution in various system. For example in the workingmemory of the brain [17] or in recovery processes follow-ing virus pandemics [6, 18, 19] power-law type of timedependencies have been found, resembling of dynamicalcritical phenomena [20]. In social networks the occur-rence of generic slow dynamics was suggested to be theresult of the non-Markovian, bursty behavior of agentsin small world networks [19]. Very recently it has beenshown [21], that bursty dynamics can arise naturally,in network models as the consequence of power-law de-caying auto-correlations due to the collective behavior ofMarkovian variables. Disorder effects are stronger in quantum systems,where the thermal noise does not fade effects of thequenched noise. However, in several cases the criticalbehavior is dictated by an infinitely strong disorder fixedpoint, resulting in robust universality classes, that canbe observed even in classical models. In particular, thesame universal behavior occurs in disordered quantumIsing chains and the CP [22]. The dynamics of the CP farin the absorbing state, can be mapped to the quantum-mechanical one, described by the disordered Hamiltonianof the Anderson type (see [23]).Heterogeneous Mean Field (HMF) theory provides agood approximation in network models, when the fluc-tuations are irrelevant [24–26]. To describe quencheddisorder in networks the so-called Quenched Mean-Field(QMF) approximation is introduced [27–30] and hetero-geneities of the steady state are quantified by calculat-ing the Inverse Participation Ratio (IPR) of the princi-pal eigenvector of the adjacency matrix. Effects of thequenched disorder on the dynamical behavior of SIS haverecently been compared using QMF approximations indifferent network models. Numerical evidences have beenprovided for the relation of localization to RR effects, thatslows the dynamics [31, 32].
The success of this relation is the consequence of thefact that GP effects arise even in the active phase, wherelocalization of the steady state can be traced by theIPR value. Although for best understanding the effectsof dynamical fluctuations should also be taken into ac-count, such approaches, like renormalization group meth-ods (RG) [33–35] have some limitations. For example,strong disorder RG works around an infinite disorderfixed point, which is not always present, still Griffithssingularities can co-exist with the clean critical behavior[63]. Furthermore, this method cannot handle modelswith pure topological inhomogeneity.In this work I show that the QMF theory describeslocalization in the one-dimensional SIS model, withquenched disorder, in agreement with the expectationthat RR effects and GP should occur below the criticalpoint. I extend previous localization studies by consid-ering distributions of the IPR and eigenvalues, castingmore light on the localization transition of SIS in vari-ous complex networks. In particular, I investigate SISon scale-free (SF) networks, possessing P ( k ) ∝ k − γ de-gree distributions and provide numerical evidence for alocalization transition at γ = 3.Very recently Moretti and Mu˜noz [7] have investigatedhierarchical, brain networks by simulations and QMF ap-proximations. They gave a brief overview about the rela-tion of slow dynamics and Lifschitz tails in synchroniza-tion and spreading models [36]. Lifschitz tails have pro-vided valuable information in regular, equilibrium sys-tems about the Griffiths singularities (see for example[37]). In network models they have been studied in math-ematics literature mainly [38]. In graph theory there is agrowing interest in spectral properties of linear operators,mostly of the adjacency matrix or the graph Laplacian(see for example [39, 40]). In physics literature Samukhinet al. [41] provided analytical forms for the Laplacianspectrum of complex random networks and for the dy-namical two-point functions of random walks running onthem. They pointed out that the minimum degree ofvertexes, is important for the dynamics, which is relatedto the lower edge of the Laplacian. On the other handnumerical evidences have been shown that the spectralgaps at the lower edge describe well the slow-down of dy-namics due to disorder in models like the CP [42] or bysynchronization transition [43, 44]. This is based on thevalidity of linearization near the phase transition point.In this case the probability distribution at the lower tailof the Laplacian can be considered the density of states,the Lifschitz tail of the disordered network model. If it holds it enables us to describe the dynamics near thecritical point. In this study I calculate the lower tail dis-tributions of the Laplacian of the networks consideredand test how well does it describe the GP behavior ofthe SIS. II. SUMMARY OF EARLIER STUDIES:LOCALIZATION VERSUS RR EFFECTS
Starting from the master equation for state vectors ofsite occupancies, | P ( n ,n ,...,n N ) ( t ) i where n i = 0 or 1, onecan derive the QMF theory for the SIS model [28, 30].Although QMF neglects the dynamical correlations, itcan take into account heterogeneities of the network byconsidering the vector of infection probabilities ρ i ( t ) ofnode i at time tdρ i ( t ) dt = − ρ i ( t ) + λ (1 − ρ i ( t )) N X j =1 A ij w ij ρ j ( t ) . (1)Here A ij is an element of the adjacency matrix and w ij describes the possibility of weights attributed to theedges. For large times the SIS model evolves into a steadystate, with an order parameter ρ ≡ h ρ i i . This equationwith i ↔ j symmetric weights can be treated by a spec-tral decomposition on an orthonormal eigenvector basis.Furthermore the non-negativity of the B ij ≡ A ij w ij ma-trix involves a unique, real, non-negative largest eigen-value y M .For t → ∞ the system evolves into a steady state andthe infection probabilities can be expressed via B ij as ρ i = λ P j B ij ρ j λ P j B ij ρ j . (2)The order parameter (prevalence) ρ ≡ h ρ i i becomes fi-nite above an epidemic threshold λ c . In the QMF ap-proximation one finds λ c and ρ ( λ ) around it from theprincipal eigenvector. Using a Taylor expansion of ρ onecan solve Eq. (2) and find that the threshold is related tothe largest eigenvalue of B ij as: 1 /λ c = y M . The orderparameter near, above λ c can be approximated via ρ ( λ ) ≈ a ∆ + a ∆ + ... , (3)where ∆ = λy M − ≪ a j = N X i =1 e i ( y j ) / [ N N X i =1 e i ( y j )] (4)are functions of eigenvectors e ( y j ) of the largest eigen-values ( j = M, M − , M − , ... ) of B ij . This expressionis exact, if there is a gap between y M and y M − [45].It was proposed in [30] and tested on weightedBarabasi-Albert models [31] that the localization of ac-tivity in the active steady state can be characterized bythe IP R value, related to the eigenvector of the largesteigenvalue e ( y M ) as I ( N ) ≡ N X i =1 e i ( y M ) (5)This quantity disappears as ∼ /N in case of homoge-neous eigenvector components or remains finite if the ac-tivity is concentrated on a finite number of nodes. III. LIFSCHITZ TAILS IN NETWORK MODELS
Besides IPR calculation, that works in the activesteady state, some other way to check RR effects wouldbe desirable. The study the spectrum tail of the Lapla-cian will be introduced here in the hope of providing in-formation about GPs below the critical point of the SIS.The Laplacian matrix of a graph is defined as L ij = δ ij X l A jl − A ij , (6)which takes values − k i in the diagonal. The Laplacian is positive-semi-definite, i.e.: Λ i ≥ = 0. The smallestnon-zero eigenvalue Λ is called the spectral gap.Near the critical point, in the inactive phase we canlinearize the dynamical equation of SIS (1) as dρ i ( t ) dt = − ρ i ( t ) + λ X j B ij ρ j ( t ) . (7)We can rewrite it, using the weighted (symmetric) Lapla-cian matrix [46, 47] L ij = δ ij X l B jl − B ij , (8)which has the sums of weights in the diagonal, expressedby the Kronecker delta ( δ ij ), as follows dρ i ( t ) dt = " λδ ij X l B jl − ρ i ( t ) − λ X j L ij ρ j ( t ) . (9)A linear stability analysis can be performed above thecritical point, similarly to the synchronization process[43]. For the normal modes of the perturbations abovethe absorbing state we can write dρ i ( t ) dt = − λ X j L ij ρ j ( t ) . (10)By this approximation we replaced the diagonal elementsin Eq. (7), from − − λL ii , which increases the sponta-neous recovery rate ν of sites, pushing the system deeperinto the inactive phase. In spreading models it is knownthat the value of ν can modify non-universal quantities, shift λ c , but in the inactive phase, where this approach isapplied, it is not expected to induce relevant RR effects,it can make them weaker and harder to detect. However,I have confirmed this approximation in case of CP onnetworks with purely topological disorder.Using the spectrum of L ij one can make the eigenvalueexpansion ρ i ( t ) = X jl e − λ Λ l t f i (Λ l ) f j (Λ l ) ρ j (0) , (11)where f i (Λ l ) is i -th the component of the l -th eigenvectorof the Laplacian. The total density is determined by thelowest eigenvalues of the spectrum ρ ( t ) ∼ N X l =2 e − λ Λ l t (12)for any network. In finite systems there is always a finiteΛ > P (Λ) above Λ ,i.e. shift the numerically obtained distributions to zeroand express ρ ( t ) as the Laplace transform of P (Λ) in thecontinuum limit ρ ( t ) ∝ Z Λ M Λ d Λ P (Λ) e − λ Λ t , (13)where Λ M corresponds to the experimentally determinedend of tail value of the finite network. Note, that thecontrol parameter λ appears as a constant, which caninduce non-universal power-laws in the inactive GP.One can also take into account the original diagonalelements of (7), if one considers the CP instead of SIS,where the interactions are normalized by the degree as λ i /k i . In case of purely topological heterogeneities thelinearized, governing equation takes the form: dρ i ( t ) dt = − ρ i ( t ) + N X j =1 λk j A ij ρ j ( t ) . (14)thus the sum of non-diagonal elements: λk j A ij is con-stant: λ . The eigenvalue spectrum of the matrix L ′ ij = δ ij − λk j A ij is the linear combination of the normalizedLaplacian: L ′ ij = λk j L ij − δ ij ( λ −
1) for such models.Therefore, by performing a spectral analysis of L ′ ij wecan investigate the lower gap behavior. The penalty isthat we have non-symmetric matrices, which can be di-agonalized by slower algorithms. I have determined thisspectrum for uncorrelated random and generalized smallnetworks (for definition see later sections) and found tailsvery similar as that of SIS, except from the linear trans-formation.For comparison I calculated the Laplacian eigenvaluespectrum of the Erd˝os R´enyi (ER) [65] graph with N =10 nodes and h k i = 4 average degree. Averaging over2 . × random graph realizations and histogramming, ∆Λ i −3 −2 −1 P ( ∆ Λ i ) FIG. 1: (Color online) Lifschitz tail of the ER graph with h k i = 4 and N = 10 . The dashed line shows a numerical fitwith the form (15) as: 2400∆Λ / i exp( − . / (∆Λ / i ) . with the bin size δ Λ = 0 .
001 one can determine numer-ically the probability distribution P (Λ i ) in the region0 < Λ < .
6. The gap size due to the finite system wasΛ = 0 . i = Λ i − Λ . A goodfitting can be obtained with the cumulative distributionderived form [41] with the numerical factors shown inFig. 1. P (∆Λ) ≃ ∆Λ / e − a/ √ ∆Λ . (15)The Laplace transform of (15) predicts the long timeasymptotic behavior of the density decay ρ ( t ) ∼ e − ( ctλ ) / (16)which is a λ dependent stretched exponential time depen-dence. Numerical simulations of the disordered CP onER graphs have obtained indeed λ dependent stretchedexponential density decay behavior below the criticalpoint [61]. However, the validity of Eq. (15) is limited to t / / ln N << N = 5 × the effect of topological disorder canbe seen for very early times, otherwise exponential decayis observed.Contrary, for a power-law distributed P (Λ) the Laplacetransformation results in a power-law decaying density ρ ( t ) ∝ Z Λ M Λ d Λ Λ x e − λ Λ t ∝ t − λ ( x +1) , (17)which suggests a GP behavior for the model. Therefore,in the following sections I determine numerically P (Λ)for certain models and determine how well can the tailbehavior be fitted by a power-law form. By knowing dy-namical simulation results about the existence of GPsin these systems I test the predicting power of this ap-proach. Later, I apply the method to more difficult casesand try to support statements about existence of GPs inthem. IV. QMF OF THE ONE-DIMENSIONAL SISMODEL WITH QUENCHED INFECTION RATES
The CP on regular lattices with quenched infectionrates has been studied by many authors (for a recentoverview see [14]). First [48] showed, using the Har-ris criterion [4], that spatially quenched disorder (frozenin space) changes the critical behavior of the directedpercolation for
D <
4. Field theoretical RG [49] foundquenched disorder to be a marginal perturbation below
D < λ c GP behavior emerges. Very re-cently GP is reported in the five dimensional CP belowthe clean, mean-field critical point [63].Here I consider the one-dimensional SIS model withquenched disorder (QSIS), which exhibits i ↔ j symme-try in the governing Eq. (1). First I investigated the caseof uniformly distributed disorder, by putting symmetricweights, drawn from the distribution w i,i +1 ∈ (0 , N = 10 , ..., × . The probability distribution of(5): P ( I ( N )) is calculated by histogramming with thebin size: δI = 0 . P ( I ( N )) distributions do notsmear, but shift to slower values by increasing the size.In the N → ∞ limit one can extrapolate the mean values¯ P ( I ( N )) with a power-law, resulting in the asymptoticvalue I = 0 . q take a reduced value rλ , while the remaining frac-tion of the nodes take a value (1 − r ) λ : p ( λ i ) = (1 − q ) δ [ λ i − (1 − r ) λ ] + qδ ( λ i − rλ ) (18)has also been studied. For q = 0 . I ( N ) could be observed, so I used a strong disorderdistribution: p = 0 . r = 0 .
9. In this case the IPRvalues are larger than for uniform distribution but ex-trapolate roughly to the same I = 0 . λ c = 1 /y M . This extrapolates with a similar correc-tion to scaling as for I ( N ) to the value λ c = 0 .
548 +(0 . /N ) . . Naturally, this value is much smaller than < I > I P (I) FIG. 2: (Color online) Finite size scaling of the IPR resultsof the one-dimensional QSIS model. Mean values of IPRfor uniformly (bullets) and binary (squares) distributed dis-order. Dashed line shows an extrapolation to N → ∞ as:0 . . /N ) . . Dotted line: 0 . . /N ) . .Inset: distributions of I ( N ) in case of uniform distributeddisorder for various sizes. the true critical point of the model due to the nature ofapproximations made.Thus the IPR, defined in the supercritical phase, pre-dicts a localization in agreement with the known RReffects of CP in one-dimension. Note, that for left-right asymmetric disorder, when sites interact with theirright or left neighbors, the localization disappears in the N → ∞ limit in agreement with the recent results [64].For the QSIS model the lower tail of the Laplacianhas been determined numerically for N = 2 × . AsFig. 3 shows one can fit the tails with power-laws well.For uniform distribution P (Λ) ∼ Λ . , (19)suggesting a GP behavior with decay law ρ ( t ) ∝ t − . λ (20)similarly for the known result of the CP.In conclusion I demonstrated here that even for thislow dimensional model, where dynamical fluctuations arerelevant at the critical point the effect of quenched dis-order away from λ c can be well described via the QMFapproximation. V. LOCALIZATION TRANSITION ON AGENERALIZED SMALL WORLD NETWORKMODEL
In this section I show results of the QMF analysisdone on networks, which exhibit purely topological disor-der. I analyzed a generalized small-world (GSW) networkmodel [55–57, 59, 60], which exhibits finite D , defined asfollows. We add to a one-dimensional lattice (a ring) a −14 −12 −10 −8 −6 ln( ∆Λ i ) l n ( P ( ∆ Λ i )) QSIS L=10KQSISB1 L=20K~x ~x FIG. 3: (Color online) Lifschitz tails in 1D QSIS models.Squares: tail distribution of the N = 2 × QSIS with bi-modal random infection rates. Circles: tail distribution ofthe L = 10 QSIS with uniform random distribution of infec-tion rates. This curve is shifted by ln(10) both in x and y direction for better visibility. Solid line shows a power-law fit ∼ ln(∆Λ i ) . , dotted line : power-law fit ∼ ln(∆Λ i ) . . set of long-range edges of arbitrary, unbounded, length.The probability that a pair of sites separated by the Eu-clidean distance l is connected by an edge decays with l as P ( l ) ≃ βl − s (21)for large l and amplitude β . These networks interpolatebetween the quasi-one-dimensional network ( s = ∞ ) andthe mean-field limit ( s = 0). Recently, simulations of theCP provided numerical evidence for the emergence of GPin s ≥ h I ( N ) i de-fined on these GSW networks for sizes N = 10 , ... × .As Fig. 4 shows a clear localization occurs in the s = 2case with β = 0 . N → ∞ . For s = 2 and large β , where the CP simulations and the RG analysis werenot completely conclusive, a slow crossover to localiza-tion can be concluded using an extrapolation to the datapoints: I ( N ) = 0 . − . /N ) . (see inset ofFig. 4). Here, the unusually small crossover exponentexpresses the very slow change from small to large IPRin the infinite size limit. This result suggests that the GPof the SIS model may exist for any β in case of marginal( s = 2) GSW networks. In numerical simulations oneshould observe GP regions of shrinking size, becominginvisible for large β -s. Finally, for s = 1 one observes ahomogeneous steady state above the critical point.The L ij matrices have also been diagonalized for N ≤ × in case of s = 1 and s = 2 networks with β = 0 . = 0 eigenvalues I calculated theprobability distribution of the smallest 500 eigenvaluesof the spectrum gap: P (∆Λ i ) = P (Λ i − Λ ). For the N = 4 × networks the Lifschitz tail results are sum- < I > < I > β=3 FIG. 4: (Color online) Mean values of IPR of the SIS model onGSW networks. For s = 2, β = 0 . N → ∞ shows the localization of the principal eigenvector.For s = 1, β = 0 . s = 2, β = 3(squares) a slow crossover to localization seems to emerge. Onthe main plot abscissa is rescaled to allow better visibility ofthe finite size scaling. Inset: The crossover region magnified,rescaled and fitted with a power-law. −3 −2 −1 ∆Λ i −4 −2 P ( ∆ Λ i ) s=1s=2x FIG. 5: (Color online) Lifschitz tails on GSW graphs with N = 4 × . Bullets: s = 1, squares: s = 2. Dashed line:power-law fitting: ∼ ∆Λ . i . marized on Fig. 5. For s = 2 a power-law tail emergesclearly, which can be fitted well using the least squareserror method as ∼ ∆Λ . i , in agreement with the ex-pected GP behavior. Contrary, at s = 1 a deviation frompower-law behavior can be observed on the log.-log. plot,the P (∆Λ i ) curve grows faster than a simple power-law.Plotting s = 1 curves on lin.-log. scale an exponential ini-tial tail can be detected for (∆Λ i ) < .
1, slowing downlater in a network size dependent way. I P (I) FIG. 6: (Color online) Probability distribution of IPR of the m = 3 BA SIS model for sizes N = 10 , 5 × , 5 × and N = 10 (from left to right). VI. LOCALIZATION TRANSITION ONSCALE-FREE NETWORKS
Up to know I showed agreement and success of theQMF-IPR method by predicting the RR effects in agree-ment with the expectations. Now I point out some limita-tions. Problems arise for example in case of SIS model onBarabasi-Albert (BA) networks [58]. These networks aregenerated by a linear preferential attachment rule, start-ing from a small fully connected seed ( N ). At each timestep s , a new vertex (labeled by s ) with m edges is addedto the network and connected to an existing vertex s ′ ofdegree k s ′ with the probability Π s → s ′ = k s ′ / P s ′′
2) = 0 constraints. The networkis completed by connecting pairs of these stubs chosenrandomly to form edges, respecting k i and avoiding selfor multiple connections. A minimum degree k = 2 anda structural cutoff k c = N / was used to generate un-correlated connected networks with probability one. Theresult of this construction is a random network, whosedegrees are distributed according to P ( k ) without degreecorrelations.I generated the adjacency matrices for a large num-ber of UCM graph realizations for degree distributionswith γ = 4, 3 .
5, 3, 2 .
8, 2 . N = 10 , 2 × , 10 , 5 × , 10 ,2 × . The estimated threshold values λ c = 1 /y M tendto zero in the N → ∞ limit, in agreement with theoret-ical arguments for SIS: λ c ∼ /N / for 2 . < γ ≤ λ c ∼ /N / [2( γ − for γ > h λ c i ∼ /N . for γ = 2 . , h λ c i ∼ /N . for γ = 4 (see Fig. 7).The probability distributions of IPR values are alsocalculated and as the right inset of Fig. 7 shows theyconverge to a sharp peak at I = 0 for γ = 2 .
5, 2 . γ = 3 (see left inset of Fig. 7).For γ = 4, 3 . P ( I ( N )) are localized around I ≃ . γ = 2 . N →∞ h I ( N ) i = 0, thus no sign of localization appears.On the other hand, for γ = 4 the mean IPR remains finiteand a localized network with I ( N ) → . /N . scale, whichshows the leading order finite size scaling in the best way. < I > γ=2.5γ=3γ=4 (0.34/L) FIG. 8: (Color online) Mean values of IPR on UCM graphswith N = 10 , ..., × . Rhombuses: γ = 4 extrapolation N → ∞ results in I = 0 . γ = 2 . γ = 3 (squares) a crossover (a localization transition)emerges. At γ = 3 we can see a crossover towards eigenvector lo-calization. The distribution of h I ( N ) i is very wide hereas in case of the BA graph.The coefficients of the expansion a , a and a inEq. (3) disappear as ∼ (1 /N ) in case of γ ≥
3. Onthe other hand for γ < a decays slower than ∼ (1 /N ),while a and a are roughly zero, corresponding to a clearmean-field transition with β = 1. Such change has beenobserved in [31, 32] in accordance with the emergence ofRR effects.I have also studied the Lifschitz tail above and be-low the localization transition in a similar way as beforeon UCM graphs with N = 10 nodes. The spectrumgap grows by decreasing γ as: Λ = 2 . γ = 4,Λ = 2 . γ = 3 and Λ = 2 . γ = 2 .
5. Thisis in agreement with our expectations, because largergap means more entangled networks, in which epidemicspreads quickly. The lowest 500 eigenvalues are calcu-lated and histogrammed using bin sizes δ Λ = 0 . = 0 eigenvalue. The P (Λ i )distributions are shifted by − Λ helping us to recognizepossible power-laws on log.-log. plots. As Fig. 9 shows,in the localized phase ( γ = 4) a power-law distribu-tion seems to emerge indeed, characterized by P (∆Λ i ) =2 . i ) . . On the other hand in the delocal-ized phase, for γ = 2 .
5, one can observe a faster thanpower-law behavior, which can be fitted well with thestretched a exponential form: 5000 exp( − / (∆Λ) . ), inagreement with the asymptotic of Eq. (15), valid for un-correlated random networks. For comparison, a power-law fit assumption would lead to a large standard error ofthe regression coefficient: ǫ = 0 . γ = 3localization transition point, the tail behavior at small∆Λ deviates slightly away from a power-law, suggestingthe lack of GP phase, in agreement with the numericalsimulations of [62] done for the CP in BA networks. As-sumption of a power-law fit form provides: ǫ = 0 . −2 −1 ∆Λ i −2 −1 P ( ∆ Λ i ) ∆Λ ) ) x γ=2.5γ=3γ=4 FIG. 9: (Color online) Lifschitz tails of SIS on UCM graphsBullets: γ = 4, triangles: γ = 3, squares: γ = 2 . ∼ (∆Λ i ) . . Dottedline: least squares fitting with the stretched exponential form5000 exp( − / (∆Λ) . ). Unfortunately the differences observed between thepower-law and stretched exponential tail behaviors arerather small. This is probably due to the limitation ofcomputing high precision P (Λ) for large sizes. This putsa question mark on the applicability of the Lifschitz tailmethod in general. VII. CONCLUSIONS
Probability distributions of the inverse participationratio have been calculated in various network modelsexhibiting explicit or topological heterogeneities. Care-ful finite size scaling analysis pointed out the emergenceof localization in generalized small world and scale-freemodels. This method describes well the GP singulari-ties both in one-dimensional SIS with interaction disor-der and in GSW-s with topological heterogeneity. Lo-calization, appearing in the active phase signals GP sin-gularities there. Former dynamical simulations in caseof generalized small world networks [7, 15, 61] supportthis. In infinite dimensional systems like ER graphs orBA networks GP-s with slow dynamics have been shownto appear only in weighted models [15, 31, 61, 62, 71]. InSF models with pure topological disorder the simulationshave been concentrated on the location of the criticalpoint by calculating stationary quantities [9, 25, 26, 68]and visible GP effects have not been reported yet. Onlyloop-less BA trees showed non-trivial phase transition byvery extensive density decay simulations [62]. This justcorresponds to the localization point, thus one can expectmore rare-region effects for γ > λ c = 0 for SFand GSW networks one cannot deeply be in the inactivephase of SIS, where the method is expected to work inthe thermodynamic limit.Application of these methods to SF networks resultsin a localization transition at γ = 3 both for correlatedand uncorrelated graphs. This is in agreement with thevery recent simulation results, discussed in [68] and withthe threshold, where the degree fluctuations h k i divergein the HMF approximation [18] due to the strong hetero-geneities. The localization in the active phase suggestsdynamical RR effects for γ >
3, like in the models pre-sented in [69]. However, for SIS, where λ c = 0 is expectedto be in the thermodynamic limit, this implies a smearedphase transition, with an algebraic decaying density in atime window towards the active steady state value. Thisscenario is feasible, because subspaces of an infinite di-mensional graph can be RR-s with arbitrary topologicaldimensions exhibiting phase transition at different λ -s,as suggested in [62]. According to [70] for large γ -s hubssustain the epidemic processes instead of the the inner-most, dense core, thus one may expect that hubs play therole of RR-s here. In finite networks the smeared phasetransition may also look like multiple phase transitions.The success of the QMF method for describing GP be-havior is demonstrated here for SIS in basic network mod-els. However, the linearization [33–35] and the completeneglection of dynamical fluctuations [26] warn for limita-tions on this relatively fast method, especially when thestrong fluctuations override the localization effects. Theappearance of strong RR effects above the upper criticaldimension [63] supports, that QMF method is capable topredict exotic GP-s with off-critical, power-law singular-ities. Acknowledgments
I thank R. Juh´asz for useful discussions and S. C. Fer-reira, P. Van Mieghem for their comments. Supportfrom the Hungarian research fund OTKA (Grant No.K109577) and the European Social Fund through projectFuturICT.hu (grant no.: TAMOP-4.2.2.C-11/1/KONV-2012-0013) is acknowledged. [1] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. ,(2002) 47.[2] S. N. Dorogovtsev and J. F. F. Mendes, Evolution of net-works: From biological nets to the Internet and WWW ,(Oxford Univ. Press Oxford, 2003).[3] M. E. J. Newman,
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