Exact results for the extreme Thouless effect in a model of network dynamics
R.K.P. Zia, Weibin Zhang, Mohammadmehdi Ezzatabadipour, Kevin E. Bassler
aa r X i v : . [ phy s i c s . s o c - ph ] S e p epl draft Exact results for the extreme Thouless effect in a model of net-work dynamics
R.K.P. Zia , Weibin Zhang , , Mohammadmehdi Ezzatabadipour , and Kevin E. Bassler , , Center for Soft Matter and Biological Physics, Department of Physics, Virginia Polytechnic Institute and StateUniversity, Blacksburg, VA 24061, USA Department of Physics, University of Houston, Houston, Texas 77204, USA Texas Center for Superconductivity, University of Houston, Houston, Texas 77204, USA Department of Mathematics, University of Houston, Houston, Texas 77204, USA
PACS – Probability theory, stochastic processes, and statistics
PACS – Fluctuation phenomena, random processes, noise, and Brownian motion
PACS – Statistical mechanics of model systems
Abstract – If a system undergoing phase transitions exhibits some characteristics of both firstand second order, it is said to be of ‘mixed order’ or to display the Thouless effect. Such atransition is present in a simple model of a dynamic social network, in which N I/E extremeintroverts/extroverts always cut/add random links. In particular, simulations showed that h f i ,the average fraction of cross-links between the two groups (which serves as an ‘order parameter’here), jumps dramatically when ∆ ≡ N I − N E crosses the ‘critical point’ ∆ c = 0, as in typicalfirst order transitions. Yet, at criticality, there is no phase co-existence, but the fluctuations of f are much larger than in typical second order transitions. Indeed, it was conjectured that, in thethermodynamic limit, both the jump and the fluctuations become maximal, so that the systemis said to display an ‘extreme Thouless effect.’ While earlier theories are partially successful,we provide a mean-field like approach that accounts for all known simulation data and validatesthe conjecture. Moreover, for the critical system N I = N E = L , an analytic expression for themesa-like stationary distribution, P ( f ), shows that it is essentially flat in a range [ f , − f ],with f ≪
1. Numerical evaluations of f provides excellent agreement with simulation data for L . L , we find f → p (ln L ) /L , though this behavior begins to set in only for L > . For accessible values of L , we provide a transcendental equation for an approximate f which is better than ∼
1% down to L = 100. We conjecture how this approach might be usedto attack other systems displaying an extreme Thouless effect. Introduction. –
Phase transitions are dramatic oc-currences of collective behavior in systems with large num-bers of degrees of freedom ( N ). They are ubiquitous, whilenearly all of life on earth depends on their existence (e.g.,the ice-water-vapor transitions). Based on the works ofBoltzmann, Gibbs, and Ehrenfest, textbook treatmentsfocus mostly on first and second order transitions, empha-sizing on the different characteristics. Typically, an orderparameter (OP) is defined so that it is finite as N → ∞ (the thermodynamic limit) and its dependence on the con-trol parameters (CP) displays different behaviors in thevarious phases. As the CPs are varied across these tran-sitions (e.g., at the water-vapor transition across 100 ◦ Cunder 1 atm pressure or the Curie point for ferromagneticsystems), the OP or its derivative suffers a discontinuity. The Lenz-Ising system [1], with external field and temper-ature as CPs, is arguably the simplest theoretical modelwhich is known to display both of these transitions [2, 3].Many characteristics of these transitions are commonlyaccepted. Though the OP is singular (discontinuous) ata first order transition, its derivatives remain finite (oneither side of the transition). Since these derivatives areassociated with the fluctuations of the OP in the system,the implication is that ‘normal’ Gaussian fluctuations (as
N → ∞ ) prevail, along with the notion of finite correlationlengths. At the transition itself, the system may displayphase co-existence (e.g., water and steam at 100 ◦ C), if thesystem is constrained so that the OP is forced to be a valuewithin the discontinuity. Deep inside each phase, the fluc-tuations and the correlation lengths are finite , taking onp-1.K.P. Zia, et al. the values on either side of the transition. By contrast, asa CP crosses a second order transition, the OP remainscontinuous, but its derivative displays a discontinuity. Of-ten, this discontinuity is infinite, diverging with some non-rational exponent (critical exponent). In other words, thefluctuations of the OP and the correlation length typi-cally become ‘anomalously divergent.’ Finite size scalingis a well-established method [4] that displays clearly howthe OP behaves as a function of both N and the CPs.Studying one-dimensional Ising models with long-rangeinteractions, Thouless [5] found that some systems do notfollow such ‘standard behavior.’ In particular, the OPmay jump discontinuously and large fluctuations exist atcriticality. Many systems displaying such ‘mixed-ordertransitions’ have been found [6–20]. Most recently, theterm ‘extreme Thouless effect’ has been coined [21, 22] forsystems in which both the discontinuity and the fluctu-ations are maximal (e.g., the magnetization in an Ising-like model jumping from − XIE ) [23, 24]. Forthis simple model, the only CPs are N I/E , the num-bers of introverts/extroverts, with L ≡ ( N I + N E ) / ≡ N E − N I as natural alternatives. Meanwhile, thefraction of cross-links, f , plays the role of an OP. In simu-lations with L = 100, a dramatic jump in f was observedwhen ∆ crosses the ‘critical’ value ∆ c = 0, giving the im-pression of a first order transition. Yet, at criticality, thefluctuations of f are non-Gaussian and large (comparableto the jump in magnitude), more typical of second ordertransitions. The extreme Thouless effect is based on ex-trapolations with data on systems with L . L → ∞ limit. Theoreticalarguments have been put forth [25], suggesting that thejump approaches unity with O (cid:16)p /L (cid:17) corrections. Thisletter is devoted to a fresh analytic approach, providingexact results which agree well with all simulation data. Inparticular, the approach is found to be quite subtle, follow-ing more closely the solution of a transcendental equationfor L ’s accessible to our computers and converging ontothe asymptote p (ln L ) /L at hopelessly large L ’s. Thenext section is a brief summary of the XIE model, fol-lowed by some details of the novel analysis. Given theseinsights on
XIE , we speculate in a final section on possi-ble avenues for research in other systems that display anextreme Thouless effect.
A simple model (
XIE ) of dynamic networks andthe extreme Thouless effect. –
In a typical socialnetwork, links between individuals are dynamic, as newones are created while others are cut. At any time, thetopology is completely specified by the adjacency matrix, A , an element of which, a ij , is unity or zero dependingon the presence or absence of a link between individuals i and j (and so, a ij = a ji ). Thus, A ( t ) represents an evo-lution trajectory of the network and resembles an Ising model on a square lattice (with spins 2 a ij − κ . The stochastic evo-lution of our model involves choosing a random individualand, if it has more than κ links, it cuts one of its existinglinks. If it has κ or fewer links, it adds a link to a randomindividual not already connected to it. Despite the appar-ently random nature of this dynamic network, the systemsettles into a (non-equilibrium) stationary state, with theprobability for finding A , P ∗ ( A ), which differs consider-ably from the Erd˝os-R´enyi distribution [26, 27]. Apartfrom minor fluctuations, everyone is more or less contentwith the ‘status quo.’ Exploring the next simplest system,we considered just two subgroups (e.g., I’s and E’s) with κ , [27] and discovered a number of surprising properties,including anomalously large fluctuations in X , the totalnumber of links between the subgroups. As expected, wefind the phenomenon of ‘frustration,’ where some individ-uals are not content with the ‘status quo.’A remarkable simplification of such two-subgroup net-works emerges when we set the κ ’s at extreme values:zero and infinity. Coined the XIE model, an I/E al-ways attempts to cut/add links, so that the stationarystate has no I-I links and all E-E links. Labeling a ij sothat all indices for the I’s are smaller than those for theE’s, we see that A is a 2x2 block matrix with frozen I-I and E-E blocks. Only the I-E block remains dynamic,representing the incident matrix of a bipartite graph: N .This N now plays the role of a rectangular ( N I × N E )Ising-type model. Meanwhile, X is just the sum over allits elements and so, the average fraction of cross-links, h f i ≡ h X i /N I N E , plays the role of magnetization . Re-markably, detailed balance is restored and the Boltzmann-like stationary distribution, P ∗ ( N ), was found analytically[23]. Not surprisingly, the ‘Hamiltonian’ H ≡ − ln P ∗ in-volves long-range and multi-spin interactions, so that itis a gargantuan challenge to find analytically the ‘parti-tion function’, averages of observable quantities, or the fulldistribution P ( f ) ≡ P { N } δ ( f − X/N I N E ) P ∗ ( N ). Onthe other hand, it is straightforward to perform MonteCarlo simulations. Employing systems with L = 100, h f i was discovered to jump from 0 .
14 to 0 .
86 when ∆changes from − c , h f i = 0 . f ( t ) resemblesthat of an unbiased random walk, confined between ‘softwalls’ at approximately 0 .
21 and 0 .
79. In other words, Unlike the Ising model, the CPs here are the sizes of the system( N I,E ), though it is possible to introduce new CPs that correspondto the magnetic field and temperature. p-2xtreme Thouless effectthe distribution P ( f ) resembles a wide mesa, so that thevariance of f is O (1), instead of the typical O (1 /L ) in aLandau theory . Such a combination (discontinuous OPand anomalously large fluctuations) is the signature of aThouless effect. The simplest mean-field analysis consistsof replacing every matrix element in H ( N ) by its aver-age µ ∈ [0 ,
1] and obtaining a Landau-like ‘free energy’ F ( µ ; N E , N I ), so that h f i is identified by the minimum of F . Now, F = µ ln ( N I /N E ) at the lowest order in L , sothat its minimum is 0 (or 1) for ∆ < > L → ∞ , h f i jumps from 0 . .
81 when ∆ goes from − constants, away from the extremes.In subsequent studies [24,25,28], progress in simulationsand theory indicate otherwise: The jumps in f trend to-wards 0 and 1 for larger systems, while the ‘walls’ in a critical system are found to approach these extremes. Re-lying on data with various ∆ and L ’s up to 1600, roughscaling plots of h f i hint at anomalous behavior, thoughit was difficult to arrive at reliable critical exponents. Onthe theoretical front, a self consistent mean-field (SCMF)theory was developed, focusing on the degree distributionsof the two subgroups: ρ I,E ( k ). The agreement with datawere quite good, for all non-critical systems [24]. For the critical system however, though qualitatively correct, thepredictions are far from ideal. (See Fig. 3 below.) Never-theless, it was argued [25] that the ‘walls’ in this systemshould approach the extremes as O (cid:16)p /L (cid:17) . Theoretical studies, exact results and compar-isons with simulations. –
Here, we present a freshperspective on the
XIE model, exploiting the ideas of theSCMF theory [24] in a different context. Instead of keep-ing only N I,E fixed and letting a self-consistency conditionto determine X , we consider ‘cross sections’ of the criti-cal system ( N I = N E = L ) with fixed X , or f = X/L .Such systems are similar to the lattice gas version of theIsing model [29] in which the total magnetization is con-strained. Clearly, simulations can be easily carried out forsuch ensembles. We show next how our new perspectiveleads to significant progress on the theoretical front.Focusing on the steady state and to be specific, we con-sider an I with k links and degree distribution in a fixed X ensemble: ρ I ( k ; X ). If chosen (with probability 1 / L ),the I will cut one of its links, unless k = 0. Thus, ρ I (0; X )will play a crucial role. An I with k − and chooses to add a link to our particular I. The proba-bility for these choices are, respectively, ( L − k + 1) / (2 L ) For example, the Landau free energy for the Ising model is N f ( m ), with f = τm + um . As a result, far from criticality, τ > m and the variance scale as 1 / N . But at criticality,these scale as 1 / √N . In two dimensions, N = L , giving us the 1 /L in the text. and 1 / ( L − ℓ ), where ℓ is the number of I’s already linkedto this E. In general, ℓ is a stochastic variable, but in thespirit of mean field theory, we replace it by h ℓ i = X/L . Toemphasize, this is just a constant in a fixed X ensemble.Thus, we find ρ I explicitly by balancing gain and loss: ρ I ( k ; X ) = L ! ( L − X/L ) − k Z ( X ) ( L − k )! (1)where Z ( X ) = L ![ L (1 − f )] L L X q =0 [ L (1 − f )] q q ! (2)From the partial sum of an exponential series, it is clearthat the limits of L → ∞ and f → f is bounded from 0, we can prove that the sumapproaches exp [ L (1 − f )] as L → ∞ .Armed with ρ I (0; X ) = 1 /Z ( X ), we return to the orig-inal critical system, in which X wanders over most of itsallowed values. As noted previously [23], that X essen-tially performs an unbiased random walk (between ‘softwalls’) can be understood as follows. When X is not closeto 0 or L , the I’s have many links to cut and the E’s canadd links to many unconnected I’s. Thus, choosing any in-dividual (with equal probability) will result in X changingby unity, so that P ( X ) = P ( X − X -8 -7 -6 -5 -4 -3 -2 P Fig. 1: Comparing XIE simulation with theoretical calcula-tions: probability distribution P of number of cross links X .Results are shown for N I = N E = 100 (red), N I = 101 , N E =99 (green), and N I = 99 , N E = 101 (blue). In each case, thedashed color line is the theory result and the thick black lineenveloping it is the result of the corresponding numerical sim-ulations. Focusing on f < . − ρ I (0; X )] P ( X ) ∼ = P ( X −
1) instead. In other words,as X wanders towards 0, the chances of it being ‘repelled’increases, hinting at the notion of the ‘wall.’ Of course, bysymmetry, similar results can be obtained for the f ≃ ρ E ( L ; X ), thep-3.K.P. Zia, et al. probability that extrovert is fully connected. Imposingthe symmetric balance equation [1 − ρ I (0; X )] P ( X ) =[1 − ρ E ( L ; X − P ( X − P ( X ) ∝ X Y Ξ=1 − ρ E ( L ; Ξ − − ρ I (0; Ξ) (3)In Fig.1, we illustrate how remarkably well this pre-diction agrees with simulation data of the L = 100 case.Further, it is straightforward to generalize these consider-ations to the N E = N I cases, e.g., by studying ρ I ( k ; X ) ∝ [ N I (1 − f )] − k / ( N E − k )! In Fig. 1, we see that the re-sults for ∆ = ± P ( X ), we proceed to find theposition of the ‘wall’ analytically. First, let us proposea natural place to call ‘the edge of the mesa’: the steep-est decent as P drops from the ‘plateau’ into the ‘plain.’These are also the inflection points of P : one near f = 0and the other, near f = 1 as L → ∞ . Denoting theformer by X (while the latter is just L − X by sym-metry), we see that it maximizes the gradient, Q ( X ) ≡ P ( X ) − P ( X − X , X can be defined asthe value for which | Q ( X ) − Q ( X − | is smallest. Letus approximate this by Q ( X ) ∼ = Q ( X − X = 0, we have Q ( X ) = P ( X ) [1 − P ( X − /P ( X )] ∼ = P ( X ) ρ I (0; X ), so that the condition for X reduces to asuccinct one: Z ( X ) − Z ( X − ∼ = 1 (4)In Fig. 2, we see the excellent agreement between simula-tion data (circles with error bars) and predictions from Eq.(4) (crosses). Of course, we notice the small discrepanciesand ascribe them to the error inherent in our mean fieldapproximation (replacing the stochastic ℓ by its averagevalue X/L ).This approach also allows us to analyze the asymp-totic behavior of f = X /L as L → ∞ . It is clearthat the largest terms in the sum in Eq. (2) occur aroundˆ q = L (1 − f ) and that the summand is well approximatedby a Gaussian: exp n − ( q − ˆ q ) / q o . Thus, the termsare effectively zero for q − ˆ q & √ ˆ q ≃ √ L . Meanwhile,the sum extends beyond ˆ q by Lf . Thus, for f > p /L (which will turn out to be satisfied), we can extend thesum to infinity and replace it by exp [ L (1 − f )]. Fur-ther, in this limit, Eqn. (4) is just L − ∂ f Z | f = 1so that (cid:2) e f (1 − f ) (cid:3) − L − e f f = p L/ π . Assum-ing f ≪
1, dropping O (1 /L ) contributions, and letting f + ln (1 − f ) = − f / ... , we find a transcendentalequation for φ ≡ f L = X /L φ + ln φ = ln L / π (5)Though it is tempting to conclude that, to leading order, φ ∼ O (cid:0) ln L (cid:1) and f → p (ln L ) /L , such an estimatefails to fit the data for L <
100 1000 L f L φ / l n L Fig. 2: Position of the (smaller) inflection point f vs. sys-tem size L . Black circles with error bars are simulation data.Red crosses are theoretical predictions. Blue dashed line is anapproximate f from solving a transcendental equation. Insetshows true asymptotics, setting in around 10 . we keep the 2 π , as ln 2 π is comparable to ln 2000. In-stead, when Eq. (5) is solved numerically, the resultant f ’s appear to provide an increasingly tight upper boundto the data (dash line in Fig.2). Our conclusions are clear:While the true asymptotics of f is p (ln L ) /L , this be-havior does not set in for the L ’s we can access in sim-ulations. Fortunately, for such L ’s, Eq. (5) provides rea-sonable bounds while Eq. (4) agrees with data at the 1%level for L as small as 100. To appreciate how large L must be before the true asymptotic form sets in, we showin the inset of Fig.2 a plot of φ/ ln L , from the solutionof Eq. (5), against ln L , up to L = 10 . Even at 10 ,this quantity misses unity by about 2% ! Needless to say,we should not expect to see simulations to confirm thisasymptotic form in our lifetimes.We end this section with the resolution of another issuein the XIE model: the disagreement between the SCMFprediction and data in the degree distributions of the crit-ical L = 100 case (e.g., blue crosses and black circles for ρ I ( k ) in Fig. 3). Let us focus on an I again and notethat ρ I ( k ) = P X ρ I ( k ; X ) P ( X ). With expressions (1)and (3), we have a new prediction for ρ I ( k ). Thoughsomewhat cumbersome, it is simple to carry out the sumnumerically. Plotted as red crosses in Fig. 3, we againfind remarkably excellent agreement with data. Conclusions and Outlook. –
Since its discovery[23], the extraordinary variability in X , the number oflinks between an equal number of extreme introverts andextroverts, has remained a theoretical puzzle. In this let-ter, we presented a new perspective and an associated ap-proximation scheme which proved successful in solving thispuzzle. Unlike earlier approaches, we considered ensem-p-4xtreme Thouless effect k -9 -8 -7 -6 -5 -4 -3 -2 -1 ρ I Fig. 3: Introvert edge degree distribution ρ I ( k ) for the N I = N E = 100 case. The Black circles are the simulation data. Redcrosses are predictions from the present theory. Blue crossesare results from an earlier theory (SCMF). bles with fixed X , much like Ising models with conservedmagnetization. We are motivated to take this approachby two observations. One is the success of the SCMF the-ory [24] for all but the critical system. The other is thatcorrelations between the microscopic variables a ij appearto be minimal [30]. Thus, the conjecture is that, despitethe presence of long-range and multi-spin interactions in H , the large variations in X for the critical system are not in conflict with the applicability of a mean field treat-ment. Such a conjecture leads us to to approximate thestochastic ℓ with its average X/L and to the subsequentsuccesses. Further along these lines, we believe that anyobservable quantity O (which has a limited variability ina fixed X ensemble) will display an extreme Thouless ef-fect. The reasoning is that its statistics will be ‘carried’by P ( X ), so that its average will suffer a maximal dis-continuity across criticality while it variability will also bemaximal at criticality.These considerations dispel another ‘rule of thumb’in phase transitions: an intimate connection betweenthe large fluctuations of the OP and sizable correlationsamong the microscopic variables. Here, we see that thislink is severed in XIE , if only through some ‘conspiring’interactions in H . In this spirit, we believe a simple lessoncan be learned by considering the following. If we startwith a non-interacting Ising model, then the distributionof the total magnetization M = P s i is just the binomial: P ( M ) ∝ (cid:0) NN + (cid:1) , with N ± ≡ ( N ± M ) /
2. If we now im-pose a ‘Hamiltonian’ of the form H M = − ln N + ! − ln N − !,then the resultant P ( M ) is completely flat , so that thevariability in M is maximal . Nevertheless, regardless ofthe apparent existence of ‘long-range and multi-spin’ in- This system is precisely the one studied in ref. [16], arrived atfrom a minimal model of spin dynamics. Unlike
XIE , it is triviallysolvable, since it effectively reduces to a statistical mechanical systemwith a single variable, M . X P x Fig. 4: Probability distribution P of number of cross links X for different values of β : Black shows simulation results forthe critical value β = 1, blue the results above criticality at β = 0 .
99, and red the results below criticality at β = 1 . teractions in H M , correlations between the spins are ‘triv-ial’ (e.g., h s ij i being just 1/3 for all i = j , in contrast topower law decays in the critical region of the ordinary Isingmodel). Moreover, we can play the game of statisticalmechanics further, by adding temperature and magneticfield to a Boltzmann factor: exp − β [ H M − HM ]. Then,we can expect an extreme Thouless effect at the ‘criti-cal point’ ( β = 1 , H = 0). The same can be done forthe XIE model by multiplying a temperature-like param-eter, β , to that ‘Hamiltonian’, H . Illustrated in Fig. 4,preliminary results show that P ( X ) displays the expectedfeatures: single-peaked for β = 0 .
99 (‘above criticality’)and bimodal for β = 1 .
01 (‘below criticality’). Work isin progress to explore these ideas in a systematic way, aswell as more realistic social networks (e.g., with genericnumbers for preferred contacts rather than 0 and ∞ ). Webelieve these studies are valuable not only for further un-derstanding of the XIE model, but also for providing in-sight into the Thouless effect (extreme or more generic) inother systems, as well as painting a more complete pictureof the subtle characteristics of phase transitions in general. ∗ ∗ ∗ We thank D. Dhar and B. Schmittmann for illuminat-ing discussions, and F. Greil for his efforts during the ini-tial phases of this project. This research is supported bythe US National Science Foundation, through grant DMR-1507371.
REFERENCES[1]
Ising E. , Zeitschrift f¨ur Physik , (1925) 253.[2] Onsager L. , Phys. Rev. , (1944) 117. p-5.K.P. Zia, et al. [3] McCoy B. and
Wu T. T. , The Two-Dimensional IsingModel (Harvard University Press) 1973.[4]
Vladimir P. , Finite Size Scaling and Numerical Simula-tion of Statistical Systems (World Scientific) 1990.[5]
Thouless D. J. , Phys. Rev. , (1969) 732.[6] Poland D. and
Scheraga H. A. , The Journal of Chem-ical Physics , (1966) 1456.[7] Fisher M. E. , The Journal of Chemical Physics , (1966) 1469.[8] Yuval G. and
Anderson P. , Physical Review B , (1970) 1522.[9] Aizenman M., Chayes J., Chayes L. and
Newman C. , Journal of Statistical Physics , (1988) 1.[10] Blossey R. and
Indekeu J. , Physical Review E , (1995) 1223.[11] Kafri Y., Mukamel D. and
Peliti L. , Physical ReviewLetters , (2000) 4988.[12] Toninelli C., Biroli G. and
Fisher D. S. , Physicalreview letters , (2006) 035702.[13] Schwarz J., Liu A. J. and
Chayes L. , EPL (Euro-physics Letters) , (2006) 560.[14] Bizhani G., Paczuski M. and
Grassberger P. , Physi-cal Review E , (2012) 011128.[15] Whitehouse J., Costa A., Blythe R. A. and
EvansM. R. , Journal of Statistical Mechanics: Theory and Ex-periment , (2014) P11029.[16] Fronczak A., Fronczak P. and
Krawiecki A. , Phys.Rev. E , (2016) 012124.[17] Fronczak A. and
Fronczak P. , Phys. Rev. E , (2016)012103.[18] Choi W., Lee D. and
Kahng B. , Physical Review E , (2017) 022304.[19] Juh´asz R. and
Igl´oi F. , Physical Review E , (2017)022109.[20] Alert R., Tierno P. and
Casademunt J. , Proceedingsof the National Academy of Sciences , (2017) 201712584.[21]
Bar A. and
Mukamel D. , Phys. Rev. Lett. , (2014)015701.[22] Bar A. and
Mukamel D. , Journal of Statistical Mechan-ics: Theory and Experiment , (2014) P11001.[23] Liu W., Schmittmann B. and
Zia R. K. P. , EPL (Eu-rophysics Letters) , (2012) 66007.[24] Bassler K. E., Liu W., Schmittmann B. and
Zia R.K. P. , Phys. Rev. E , (2015) 042102.[25] Bassler K. E., Dhar D. and
Zia R. K. P. , Journalof Statistical Mechanics: Theory and Experiment , (2015) P07013.[26] Liu W., Jolad S., Schmittmann B. and
Zia R. K. P. , Journal of Statistical Mechanics: Theory and Experiment , (2013) P08001.[27] Liu W., Schmittmann B. and
Zia R. K. P. , Journalof Statistical Mechanics: Theory and Experiment , (2014) P05021.[28] Liu W., Greil F., Bassler K. E., Schmittmann B. and
Zia R. K. P. , Modeling interacting dynamic net-works: III. Extraordinary properties in a population ofextreme introverts and extroverts arxiv 1408.5421 (2014).[29]
Yang C. N. and
Lee T. D. , Phys. Rev. , (1952) 404.[30] Ezzatabadipour M., Zhang W., Bassler K. E. and
Zia R. K. P. , Fluctuations and correlations in a model ofextreme introverts and extroverts (to be published).[31]
Ferrenberg A. M. and
Swendsen R. H. , Phys. Rev. Lett. , (1988) 2635.(1988) 2635.