Emergence of disconnected clusters in heterogeneous complex systems
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec Emergence of disconnected clusters in heterogeneous complexsystems
István A. Kovács
1, 2, ∗ and Róbert Juhász Northwestern University, Department of Physics and Astronomy, Evanston, 60208, USA Wigner Research Centre for Physics,Institute for Solid State Physics and Optics, Budapest, 1121, Hungary (Dated: December 3, 2020)Percolation theory dictates an intuitive picture depicting correlated regions in com-plex systems as densely connected clusters. While this picture might be adequateat small scales and apart from criticality, we show that highly correlated sites incomplex systems can be inherently disconnected. This finding indicates a counter-intuitive organization of dynamical correlations, where functional similarity decou-ples from physical connectivity. We illustrate the phenomena on the example of theDisordered Contact Process (DCP) of infection spreading in heterogeneous systems.We apply numerical simulations and an asymptotically exact renormalization grouptechnique (SDRG) in , and dimensional systems as well as in two-dimensionallattices with long-ranged interactions. We conclude that the critical dynamics is wellcaptured by mostly one, highly correlated, but spatially disconnected cluster. Ourfindings indicate that at criticality the relevant, simultaneously infected sites typi-cally do not directly interact with each other. Due to the similarity of the SDRGequations, our results hold also for the critical behavior of the disordered quantumIsing model, leading to quantum correlated, yet spatially disconnected, magneticdomains. Introduction
Correlated clusters emerge in a broad range of systems, ranging from magnetic modelsto out-of-equilibrium systems [1–3]. Apart from the critical point, the correlation length ∗ Electronic address: [email protected] is finite, limiting the spatial separation between highly correlated sites, leading to spatiallylocalized, finite clusters. Although at the critical point the correlation length diverges, ourtraditional intuition is driven by percolation processes, indicating that highly correlatedcritical clusters remain connected, while broadly varying in their size. As an alternativescenario, we show that at the critical point, highly correlated clusters can break up intospatially disconnected regions in both classical and quantum systems. As a proof-of-conceptexample, we focus on the Disordered Contact Process (DCP), a simple (out-of-equilibrium)infection spreading model [4–6]. The critical behavior of the model is well understoodin the presence of disorder, at least at the level of statistical properties averaged over alarge numbers of samples [6–10]. The detailed simulation of individual samples is muchmore challenging, due to large dynamical fluctuations and extremely slow dynamics aroundcriticality [11, 12].As an alternative, efficient approach, the strong disorder renormalization group (SDRG)method provides asymptotically exact results for the critical contact process in the presenceof disorder, at least below d = 4 spatial dimensions [11–14]. Besides being computationallyefficient, the SDRG method also offers some counter-intuitive insights into the underlyingcorrelation structure. Namely, the SDRG predicts that highly correlated sites at long timescales (simultaneously infected individuals) typically do not know each other directly, onlyvia indirect connections through the rest of the contact network. According to the SDRG,only a few of these highly correlated, but essentially disconnected, clusters govern the large-scale behavior of the system. Observing these clusters in simulations is notoriously difficultdue to extremely slow dynamics, inducing anomalously large fluctuations and poor statisticalproperties. As a key step, here we show how to find the highly correlated sites efficientlyvia a quasi-stationary simulation. The simulated density profile is then found to be in agood agreement with the SDRG predictions, confirming that the asymptotic dynamics isgoverned by spatially disconnected clusters in stark contrast to traditional intuition. The disordered contact process
The contact process, in the most general case, is defined on a network given by anadjacency matrix A ij . As a special case, the network is often chosen to be a d-dimensionalhypercubic lattice with nearest-neighbour edges. The state of the system is given by a set ofbinary variables, n i = 0 , , characterizing the sites of the network. As the contact process isfrequently interpreted as a simple epidemic spreading model, sites with n i = 1 are referredas ’infected’, while sites with n i = 0 are ’healthy’ or ’susceptible’. The contact processis a continuous-time Markov process on this state space, specified by the rates of possible(independent) transitions, which are the following (for an illustration see Fig. 1). λ µ λ FIG. 1:
Illustration of allowed transitions in the contact process.
Allowed transitions ofan active (infected) site and two of its neighbours, connected by two links ( and ). Active(inactive) sites are depicted by red (white) dots. The corresponding transition rates are written onthe arrows. First, infected sites become spontaneously healthy with a rate µ i , which may be site-dependent. Second, infected sites attempt to infect their adjacent sites with rates λ ij , andthe trial is successful if the target site j was healthy. Again, the infection rates λ ij canvary from link to link. We will assume that the infection rate from site i to site j are thesame as that from site j to site i , i.e. λ ij = λ ji . This variant of the model is also knownas the susceptible-infected-susceptible (SIS) model [15]. This technical restriction, which isnecessary for the applicability of the SDRG method, is irrelevant from the point of view ofuniversal critical properties [16].The contact process exhibits a continuous, non-equilibrium phase transition betweena non-fluctuating (absorbing) phase in which all sites are healthy and a fluctuating phase,where the density of infected sites is non-zero [17]. For regular lattices and uniform transitionrates the transition falls into the robust universality class of directed percolation [18–21].If the transition rates are independent, random variables, the universality class is wellcharacterized in one, two- and three-dimensional regular lattices due to large-scale MonteCarlo simulations [6–10]. The observed critical behavior is in line with the results of thestrong-disorder renormalization group method [11–14]. According to this, the dynamicalscaling of average quantities is ultra-slow, characterized by power-laws of the logarithm oftime rather than the time itself, whereas the static critical exponents are also altered com-pared to the directed percolation class [11, 12, 22]. Most interestingly, the critical exponentsare independent of the form of the distribution of transition rates. Besides low dimensionalregular lattices, a similar behavior has been found in spatially embedded networks with long-range connections [23–25]. These networks consist of a regular, d -dimensional, hypercubiclattice and a set of long links, which exist with a probability p ij ∼ βd ( i, j ) − s , where d ( i, j ) isthe Euclidean distance between i and j . Note, that even for uniform transition rates thesenetworks contain a so-called topological disorder due to the random connectivity of sites bylong links. In the case s = 2 d , the topological dimension is finite and varies with β [26–28].As it was demonstrated in Refs. [23–25] for d = 1 and s = 2 , the contact process showsan ultra-slow scaling and the critical exponents vary with β , while an additional disorder inthe transition rates is irrelevant. According to a general scaling argument presented in Ref.[29], a qualitatively similar behavior is expected in higher dimensions for s = 2 d . Strong-disorder renormalization group
The strong-disorder renormalization group is the key method to understand the long-time behavior of the disordered contact process (DCP). For a general review and a detailedintroduction we refer the reader to Ref. [13, 14]. The first application of the method to theDCP was in Ref. [11, 12]. Next, we recapitulate the essential features of the method.The SDRG is a sequence of iterative steps operating on blocks of sites containing thelargest rate (either an infection rate or a healing rate) of the model, see Fig. 2. If the largestrate is an infection rate, λ ij , the block of sites i and j is merged to a cluster, characterizedby an effective healing rate: ln ˜ µ ij = ln µ i + ln µ j − ln λ ij + ln 2 . (1)This simplification is a good approximation if µ i and µ j are much smaller than λ ij . If thelargest parameter is a healing rate, µ i , site i is deleted, and effective infection rates aregenerated between all neighbors of site i : ln ˜ λ jk = ln λ ji + ln λ ik − ln µ i . (2) elimination b)a) maximum rulemaximum rulecluster formation l lmjil . kij m kij mlj ki lj k lj kk FIG. 2:
Illustration of the SDRG method.
Two types of reduction steps of the SDRG pro-cedure: formation of a cluster (a) and elimination of a site (b). The thickness of lines indicatesthe magnitude of the infection rate on the corresponding link. The link (a) and the site (b) to bedecimated is shaded in green. For details, see the text.
This approximation is justified under the condition λ ji , λ ik ≪ µ i . Apart from a one-dimensional lattice, it may happen that a newly formed cluster has, through its constituents,two infection rates to a third site. Similarly, if a new infection rate is generated, there maybe a pre-existing infection rate between those two sites. These situations are usually treatedby discarding the smaller infection rate, which is known as the maximum rule.The above steps are applied sequentially, lowering thereby gradually the number of de-grees of freedom, as well as the rate scale Ω = max { µ i , λ ij } . Regarding the set of clusterspresent in the system, the SDRG procedure can also be viewed as a special coagulation-annihilation process with extremal dynamics. The critical behavior of the DCP is gov-erned by the so called infinite-randomness fixed-point (IRFP) of the SDRG transformation[11, 12, 30, 31], at least for sufficiently strong initial disorder [8, 11, 12, 32–34]. As the IRFPis approached along the critical renormalization trajectory, the distribution of logarithmicrates broadens without limits, and both types of reduction steps become asymptotically ex-act. Furthermore, this feature also justifies the applicability of the maximum rule, and eventhe neglection of the ln 2 term in Eq. (1). In the numerical SDRG calculations we thereforeresorted to these approximations, which, as a further advantage, enable a computationallyvery efficient implementation of the SDRG procedure [35, 36].Following the SDRG transformation of the DCP at a given realization of random ratesdown to a rate scale Ω ≪ provides an effective DCP with a smaller set of degrees offreedom, which approximates well the dynamics of the original system for times t ≫ Ω − .Consider, for instance, the state of the system, evolved from a fully occupied initial state, atsome late time t . Applying the SDRG procedure down to rate scale Ω = t − , provides theset of sites with an O (1) occupation as the constituents of the clusters of the renormalizedsystem, while the sites eliminated during the procedure will have an almost zero occupationprobability.In the case of a finite sample, the true steady state is always the absorbing, empty latticestate. Accordingly, all clusters are removed during the SDRG procedure, the last one atsome rate Ω , the second last at Ω > Ω , in general, the n th cluster becomes irrelevantat Ω n > Ω n − . The inverse rates, Ω − , Ω − , . . . , are interpreted as the mean lifetimes ofthe corresponding clusters removed during the procedure. Clearly, the last one, Ω − , givesthe mean lifetime of the sample needed to be absorbed in the empty state. In finite butlarge systems in the vicinity of the critical point, the last few lifetimes such as Ω − and Ω − are typically well separated, meaning that they differ by orders of magnitude, andthis time-scale separation becomes more and more pronounced with increasing system size.Consequently, in typical samples there is a considerable time span, Ω − ≪ t ≪ Ω − , withinwhich practically no sites but the constituents of the lastly removed cluster are occupied.The structure of the lastly removed cluster can also be used to determine a sample-dependent pseudo-critical point: making a double sized system by glueing together twocopies of the original one, the onset of the active phase is indicated when the last cluster isdifferent from that of the original system [22]. Quasi-stationary simulation
The simulations were implemented for regular lattices in the following way. An occupiedsite ( i ) is randomly chosen and, with a probability µ/ ( µ + 1) it is made unoccupied, or, witha probability / ( µ + 1) , one of its n neighbors ( j ) is picked with a uniform probability andinfected with the probability λ ij , provided it was healthy. The time increment related tosuch a move is ∆ t = 1 /N s , where N s is the total number of infected sites.For non-regular networks, where the sites may have different coordination numbers, adifferent implementation is needed in order to correctly simulate the process. Here, besidesthe list of infected sites, also a list of active links is stored, which contains all directed linkswith an infected source site. The number of elements of this list is denoted by N e . Then, witha probability µN s / ( µN s + N e ) , a healing event occurs on one of the infected sites, chosenequiprobably, whereas, with the complementary probability, N e / ( µN s + N e ) , an infectionevent is attempted. In this case, an active link ( ij ) is chosen with a uniform probabilityfrom the corresponding list, and the target site ( j ) is infected with a probability λ ij , providedit was healthy. The time increment of such an elementary move is ∆ t = 1 / ( µN s + N e ) .We apply a quasi-stationary simulation with a reflecting boundary condition. This meansthat, at the point where there is only a single active site in the system, the healing eventis rejected. This way we only need to validate that the system reached the quasi-stationarystate, easily done by checking whether the mean density and its variance remain unchangedunder increasing the relaxation time. The total simulation time was typically , the firsthalf of which is left for relaxation, while the measurements are performed in the second half.First, in each sample, we determine an individual pseudo-critical point, which, for thesystem sizes we use, can significantly deviate from the ensemble average. Following themethod proposed in Ref. [37], we identify the pseudo-critical point with the maximum ofsusceptibility [38] defined as χ = N h ρ i − h ρ i h ρ i , (3)where N is the total number of sites, ρ denotes the global density of active sites, and h·i stands for the expected value in the quasi-stationary state.Having estimated the pseudo-critical point, we performed here quasi-stationary simula-tions for a longer time, , and measured the mean local densities (Figs. 3-6). In addition,we started a number of independent simulations and averaged over them, in order to avoidthe (rather improbable) possibility that the long-lasting activity is stuck at a cluster otherthan the one with the longest lifetime.For a satisfying agreement with the SDRG method for moderate system sizes, the strengthof the disorder needs to be sufficiently large, i.e. the distribution of the logarithmic infectionrates needs to be sufficiently broad. In practice, the infection rates were chosen from apower-law distribution, P < ( λ ) = λ /α , where the exponent α > controls the strength ofdisorder. !"! S a m p l e S a m p l e S a m p l e ABC main cluster secondary cluster rest of the distribution
FIG. 3:
Results in 1D systems.
Asymptotic probability of being active in the quasi-stationarysimulations as the function of the spatial coordinate for N = 1000 sites with periodic boundaryconditions and strength of disorder α = 1 . As illustrated by the purple and red clusters the SDRGis able to efficiently capture the activity profile even with a few clusters. The correlated clustersare disconnected fractals with a fractal dimension d f = √ ≈ . [30, 31], corresponding toa highly uneven activity profile. Gaps of strongly reduced density inside the clusters can be seenalready at small scales, indicating asymptotically disconnected clusters of activity. Comparison of the cluster structure with the simulations
The average order parameter of the DCP is related to the fraction of original sites com-prised by the clusters of the renormalized system, which decreases gradually as the SDRG
AB CD EF R eno r m a li z a t i on S i m u l a t i on Sample 1 Sample 2 Sample 3
FIG. 4:
Results in 2D systems.
A good agreement is found between the asymptotic densityprofile in the simulations and the largest clusters obtained by the SDRG method. In 2D the fractaldimension of the clusters is d f = 1 . [22], illustrated here for N = 200 × sites with periodicboundary conditions and strength of disorder α = 4 . The grayscale indicates the site probability ofbeing active in the simulations and the cluster size in the SDRG calculations, respectively. proceeds. Its critical scaling properties can then be inferred from its dependence on theinverse time scale Ω and inverse length scale (the number density of clusters) along the crit-ical trajectory of the SDRG [13, 14]. Instead of this, here we compare the spatial structureof SDRG clusters with the map of local occupancies obtained by simulations in individualsamples.Ideally, the comparison of the two methods should be done under the same circumstances:i) for the same random sample (set of random rates), ii) at the same time (given by Ω − in the SDRG method). The first requirement is, however, not the appropriate choice for afair comparison of the methods. The reason is that the SDRG transformation, due to itsapproximative nature at early stages, does not preserve the position of the critical point.0 AB CD EF R eno r m a li z a t i on S i m u l a t i on Sample 1 Sample 2 Sample 3
FIG. 5:
Results in 3D systems.
We again find a good agreement between the asymptoticdensity profile in the simulations and the clusters obtained by the SDRG method. In 3D the fractaldimension is d f = 1 . [35, 36], illustrated here for N = 50 × × sites with periodic boundaryconditions and strength of disorder α = 4 . For better visibility, only sites with at least of themaximum probability are indicated in the simulations, while we show only clusters of size abovethe square root of the size of the main cluster. For instance, a truly critical initial system will depart from the critical trajectory and, inorder to arrive at the critical IRFP, a compensatory initial shift in the control parameteris needed to be imposed. Therefore, rather than the initial point of the renormalizationtrajectory, its end point has to match the sample used in simulations. For this reason,the first requirement, i.e. the complete identity of rates must be relaxed, but a notion of“equivalence of the random environment” must still be preserved. Our approach to overcomethis controversy was the following. We chose the infection rates randomly, while kept the1
AB CD EF R eno r m a li z a t i on S i m u l a t i on Sample 1 Sample 2 Sample 3
FIG. 6:
Results in 2D with long-range interactions.
Here we show our results for algebraicallydecaying interaction probability with an exponent s = 4 , for N = 100 × sites with periodicboundary conditions. The number of short-cuts is N/ and the strength of disorder is α = 4 . Redlinks indicate long-range connections within the main cluster. The critical clusters are even morestrongly disconnected objects, with a formally zero fractal dimension [29]. healing rates constant. The latter can then serve as a control parameter and can be usedto compensate the shift induced in the SDRG. The amount of the shift is quantitatively apriori unknown, therefore it is natural to choose the critical point as a “common point” towhich the system can be tuned in both methods by using an indicator of criticality internalto that method. By this construction, the transition rates used in the two methods arenot completely identical, but the difference is only in the uniform healing rate, while therandom infection rates, which form the “random environment”, and which govern the shapingof SDRG clusters, are identical.Concerning the second requirement, we implemented the SDRG procedure up to the stageat which only one cluster remains in the system. Provided the lifetimes of clusters are well2separated, this corresponds to the time span between the lifetime of the second last and thelast cluster. The state of the system in this time span can be conveniently simulated, evenwithout knowing the above lifetimes, by the above described quasi-stationary simulation,which prevents the system from getting absorbed in the empty state.In Fig. 3, we illustrate for three random samples that the main SDRG cluster (purple)not only contains all sites of nearly maximal probabilities, but also captures the majority ofthe probability weight. For such, moderately sized, systems smaller clusters still play a role,as shown by probability distribution captured by the secondary cluster. Interestingly, boththe main and secondary clusters are fractured by deep gaps in the probability distribution,as signatures of asymptotically disconnected clusters. Overall, we see a good agreementalready for moderate system sizes as illustrated in Figs. 3-5 for d = 1 , and dimensionallattices as well as for a two-dimensional lattice with long-range interactions (Fig. 6). Discussion
In this paper, we studied the disordered contact process (DCP) in a random environ-ment. By applying the combination of quasi-static simulations and an efficient renormaliza-tion group method, we showed that the critical behavior of the DCP is dominated by onestrongly correlated cluster. Yet, as opposed to equilibrium systems, the governing clustersare predominantly disconnected objects, indicating that individuals who are infected at thesame time typically do not know each other directly. According to the SDRG method, suchdisconnected clusters emerge from a strong, positive feedback mechanism, in which remotesites can effectively infect each other over and over again for a prolonged amount of timethrough indirect paths in the contact network [39].Besides infection spreading, variants of the DCP have recently gained increasing interestalso in functional modeling of the brain [40]. Our results suggest that, as opposed to tra-ditionally expected functional ’blobs’, strongly correlated brain regions are not necessarilydirectly connected. In other words, functional connectivity might qualitatively deviate fromphysical connectivity. This expectation exists in addition to the challenge that in the braincorrelations do not necessarily decay with increasing physical distance [41]. We leave theextension of our results to dynamical models on experimentally obtained brain connectomedatasets for future studies.3
Acknowledgements
This paper was supported by the National Research, Development and Innovation Office- NKFIH under grants No. K128989 and No. K131458. IAK was supported by the ’Na-tion’s Young Talents Scholarship’, Ministry of National Resources, Hungary under GrantNo. NTP-NFTÖ-17-C-0247.
Author contributions statement
I.A.K initiated the project and performed the SDRG calculations. R.J. performed theMC simulations. All authors contributed to writing the manuscript.
Competing interests
The authors declare no competing interests.
Data availability
All data generated or analysed during this study are included in this published article. [1] Fortuin, C.M. & Kasteleyn, P.W. On the random-cluster model : I. Introduction and relationto other models.
Physica , , (4): 536-564 (1972).[2] Carey, R. & Isaac, E.D. Magnetic Domains And Techniques For Their Observation
The EnglishUniversity Press Ltd, London, (1966).[3] Mansfield, M.L. & Douglas, J.F. Shape characteristics of equilibrium and non-equilibriumfractal clusters.
AIP The Journal of Chemical Physics , , 044901 (2013).[4] Noest, A.J. New universality for spatially disordered cellular automata and directed percola-tion. Phys. Rev. Lett. , 90 (1986).[5] Noest, A.J. Power-law relaxation of spatially disordered stochastic cellular automata and di-rected percolation. Phys. Rev. B , 2715 (1988). [6] Moreira, A.G. & Dickman, R. Critical dynamics of the contact process with quenched disorder. Phys. Rev. E , R3090 (1996).[7] Vojta, T., Farquhar, A. & Mast, J. Infinite-randomness critical point in the two-dimensionaldisordered contact process. Phys. Rev. E , 011111 (2009).[8] Vojta, T. & Dickison, M. Critical behavior and Griffiths effects in the disordered contactprocess. Phys. Rev. E , 036126 (2005).[9] Vojta, T. Monte Carlo simulations of the clean and disordered contact process in three dimen-sions. Phys. Rev. E , 051137 (2012).[10] Vojta, T. Rare region effects at classical, quantum and nonequilibrium phase transitions. J.Phys. A , R143 (2006).[11] Hooyberghs, J., Iglói, F. & Vanderzande, C. Strong disorder fixed point in absorbing-statephase transitions. Phys. Rev. Lett. , 100601 (2003).[12] Hooyberghs, J., Iglói, F. & Vanderzande. C. Absorbing state phase transitions with quencheddisorder. Phys. Rev. E , 066140 (2004).[13] Iglói, F. & Monthus, C. Strong disorder RG approach of random systems. Phys. Rep. , 277(2005).[14] Iglói, F. & Monthus, C. Strong disorder RG approach–a short review of recent developments.
Eur. Phys. J. B , 290 (2018).[15] Barrat, A., Barthélemy, M. & Vespignani, A. Dynamical Processes On Complex Networks
Cambridge Univ. Press (Cambridge) (2008).[16] Juhász, R. Disordered contact process with asymmetric spreading.
Phys. Rev. E , 022133(2013).[17] Liggett, T.M. Stochastic Interacting Systems: Contact, Voter, And Exclusion Processes
Springer (Berlin) (2005).[18] Marro, J. & Dickman, R.
Non-equilibrium Phase Transitions In Lattice Models
CambridgeUniv. Press (Cambridge) (1999).[19] Ódor, G.
Universality In Nonequilibrium Lattice Systems
World Scientific (Singapore) (2008).[20] Ódor, G. Universality classes in nonequilibrium lattice systems.
Rev. Mod. Phys. , 663(2004).[21] Henkel, M., Hinrichsen, H. & Lübeck, S. Non-equilibrium Phase Transitions
Springer (Berlin)(2008). [22] Kovács, I.A. & Iglói, F. Renormalization group study of the two-dimensional randomtransverse-field Ising model. Phys. Rev. B , 054437 (2010).[23] Juhász, R. & Kovács, I.A. Infinite randomness critical behavior of the contact process onnetworks with long-range connections. J. Stat. Mech.
P06003 (2013).[24] Muñoz, M.A., Juhász, R., Castellano, C. & Ódor, G. Griffiths phases on complex networks.
Phys. Rev. Lett. , 128701 (2010).[25] Juhász, R., Ódor, G., Castellano, C. & Muñoz, M.A. Rare-region effects in the contact processon networks.
Phys. Rev. E , 066125 (2012).[26] Benjamini, I. & Berger, N. The diameter of long-range percolation clusters on finite cycles. Rand. Struct. Alg. , 102 (2001).[27] Coppersmith, D., Gamarnik, D. & Sviridenko, M. The diameter of a long-range percolationgraph. Rand. Struct. Alg. , 1 (2002).[28] Juhász, R. Competition between quenched disorder and long-range connections: a numericalstudy of diffusion. Phys. Rev. E , 011118 (2012).[29] Juhász, R., Kovács, I.A. & Iglói, F. Long-range epidemic spreading in a random environment. Phys. Rev. E , 032815 (2015).[30] Fisher, D.S. Random transverse field Ising spin chains. Phys. Rev. Lett. , 534 (1992)[31] Fisher, D.S. Critical behavior of random transverse-field Ising spin chains. Phys. Rev. B ,6411 (1995).[32] Neugebauer, C.J., Fallert, S.V. & Taraskin, S.N. Contact process in heterogeneous and weaklydisordered systems, Phys. Rev. E , 040101(R) (2006).[33] Fallert, S.V. & Taraskin, S.N. Scaling behavior of the disordered contact process. Phys. Rev.E , 042105 (2009).[34] Hoyos, J.A. Weakly disordered absorbing-state phase transitions. Phys. Rev. E , 032101(2008).[35] Kovács I.A. & Iglói, F. Infinite-disorder scaling of random quantum magnets in three andhigher dimensions. Phys. Rev. B , 174207 (2011).[36] Kovács I.A. & Iglói, F. Renormalization group study of random quantum magnets. J. Phys.:Condens. Matter , 404204 (2011).[37] Ferreira, S.C., Castellano, C. & Pastor-Satorras, R. Epidemic thresholds of the susceptible-infected-susceptible model on networks: A comparison of numerical and theoretical results. Phys. Rev. E , 041125 (2012).[38] Binder, K. & Heermann, D.W. Monte Carlo Simulation In Statistical Physics (Springer, Berlin)(2010).[39] Kovács, I.A.
Infinitely Disordered Critical Behavior In Higher Dimensional Quantum Systems (Ph.D. dissertation) Eötvös Loránd University (2013).[40] Moretti, P. & Muñoz, M.A. Griffiths phases and the stretching of criticality in brain networks.
Nat. Commun. , 2521 (2013).[41] Bowman, F.D. Brain imaging analysis. Annu. Rev. Stat. Appl.1