Persistence in a Random Bond Ising Model of Socio-Econo Dynamics
aa r X i v : . [ q -f i n . GN ] S e p Persistence in a Random Bond Ising Model of Socio-EconoDynamics
S. Jain ∗ Information Engineering, The Neural Computing Research Group,School of Engineering and Applied Science,Aston University, Birmingham B4 7ET, U.K.
T. Yamano † Department of Physics, Ochanomizu University,2-1-1 Otsuka, Bunkyo-ku Tokyo 112-8610, Japan (Received September 07 2007)
Abstract
We study the persistence phenomenon in a socio-econo dynamics model using computer simu-lations at a finite temperature on hypercubic lattices in dimensions up to 5. The model includesa ‘social’ local field which contains the magnetization at time t . The nearest neighbour quenchedinteractions are drawn from a binary distribution which is a function of the bond concentration, p . The decay of the persistence probability in the model depends on both the spatial dimensionand p . We find no evidence of ‘blocking’ in this model. We also discuss the implications of ourresults for possible applications in the social and economic fields. It is suggested that the absence,or otherwise, of blocking could be used as a criterion to decide on the validity of a given model indifferent scenarios. PACS numbers: 05.20-y, 05.50+q, 75.10.Hk, 75.40.Mg, 89.65.Gh, 89.75.-kKeywords: Econophysics, Sociophysics, Non-Equilibrium Dynamics, Ising Models, Persistence ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The persistence problem is concerned with the fraction of space which persists in its initial( t = 0) state up to some later time t . The problem has been extensively studied over thepast decade for pure spin systems at both zero [1-4] and non-zero [5] temperatures.Typically, in the non-equilibrium dynamics of spin systems at zero-temperature, the systemis prepared initially in a random state and the fraction of spins, P ( t ), that persists in thesame state as at t = 0 up to some later time t is studied. For the pure ferromagnetic Isingmodel on a square lattice the persistence probability has been found to decay algebraically[1-4] P ( t ) ∼ t − θ , (1)where θ ∼ .
22 is the non-trivial persistence exponent [1-3]. Derrida et al [4] have shownanalytically that for the pure 1d Ising model θ = 3 /
8. The actual value of θ depends onboth the spin [6] and spatial [3] dimensionalities; see Ray [7] for a recent review.At criticality [5], consideration of the global order parameter leads to a value of θ global ∼ . never flip. As a result, P ( ∞ ) > r ( t ) = P ( t ) − P ( ∞ ) . (2)Note that for the five dimensional pure Ising model without any disorder blocking has alsobeen observed at T = 0 [3]. At finite temperature there is no evidence of blocking [12].As well as theoretical models, the persistence phenomenon has also been studied in a widerange of experimental systems and the value of θ ranges from 0 .
19 to 1 .
02 [13-15], dependingon the system. A considerable amount of the recent theoretical effort has gone into obtainingnumerical estimates of θ for different models [1-11]. Recently, it has been found that thebehaviour of the random Ising ferromagnet at zero temperature on a Voronoi-Delaunaylattice [16] is very similar to the behaviour on the diluted ferromagnetic square lattice [8,9].2n this work we add to the knowledge and understanding regarding persistence by presentingthe simulation results for the behaviour of a recently proposed spin model which appearsto reproduce the intermittency observed in real financial markets [17]. In the next sectionwe discuss the model in detail. In Section III we give an outline of the method used andthe values of the various parameters employed. Section IV describes the results and theconsequent implications for using the model in a financial or social context. Finally, inSection V there is a brief conclusion. II. THE MODEL
Yamano [16] has proposed a ‘minimalist’ version of the Bornholdt model [18]. We study thepersistence phenomenon in the former. In this model one has N market traders, denoted byIsing spins S i ( t ) , i = 1 . . . N , located on the sites of a hypercubic lattice, L d = N . The actionof the i th trader of buying or selling a share of a traded stock or commodity at time step t corresponds to the spin variable S i ( t ) assuming the value +1 or −
1, respectively. Hence, ateach time step, a given trader will be either buying or selling. A local field, h i ( t ), determinesthe dynamics of the spins and, hence, the action of the trader. We follow [16] and assumethat h i ( t ) = d X j =1 J ij S j ( t ) − α | N X j =1 S j ( t ) /N | , (3)where the first summation runs over the nearest neighbours of i only, α > P ( J ij ) = (1 − p ) δ ( J ij + 1) + pδ ( J ij − , (4)where p is the concentration of ferromagnetic bonds. The case p = 1 / ± J Edwards-Anderson spin-glass [11]. Each agent is updated according to the followingheat bath dynamics: S i ( t + 1) = +1 with q = [1 + exp ( − h i ( t ) /T )] − , − − q , (5)where q is the probability of updating and T is temperature. The first term on the righthand side in equation (3) contains the influence of the neighbours and the second termreflects the external environment. The balance between the two terms determines whether3n agent buys or sells. If h i ( t ) = 0, the agent is equally likely to buy or sell as q = 1 / h i ( t ) >
0, then q > / i is more likely to buy than sell. Similarly,if h i ( t ) <
0, then we have q < / α and T are tunable parameters in our model. The values we select for these are determined by therequirement that the model should be able to reproduce, at least qualitatively, some aspectof actual behaviour observed in a real market. In this model the return is defined in termsof the logarithm of the absolute value of the magnetization, M ( t ) = P Nj =1 S j ( t ) /N , that isReturn ( t ) = ln | M ( t ) | − ln | M ( t − | (6)A key stylised fact observed in real financial markets is the intermittent or ‘bursty’ behaviourin the returns [19]. Simulations [16] in spatial dimensions ranging from d = 1 to d = 3confirm that the above model is able to reproduce the required intermittent behaviourin the returns; see, for example, figure 1 in Yamano [16]. It should be emphasised thatintermittency is only observed for certain values of the tunable parameters, α and T . Thetemperature, T int , is defined as the temperature at which intermittency is observed in thereturns. Although α = 4 . T int depending on both d and L .The values of T int as determined in [16] were: d = 1 , T int = 3 . , L = 10001; d = 2 , T int =2 . , L = 101; d = 3 , T int = 2 . , L = 21. A couple of points should be emphasised at thisstage. Simulations are performed only on hypercubic lattices where the linear dimension L is an odd value. This requirement ensures that the right hand side of equation (6) is welldefined. The simulations discussed in the present work were performed in spatial dimensionsranging from d = 1 to d = 5. For each value of d , we fine tune the parameters α and T int so that intermittent behaviour is observed in the returns as discussed above. We foundthat although α = 4 . T int depends on d (and also on L ) at whichintermittency is observed. Our values for L and T int are listed in Table 1. Note that thevalues of T int for d = 1 , , t either buying or selling continuously since t = 0. Later, wewill also suggest a possible interpretation within the context of sociophysics of the model.4 imension L T int L of the lattices used in the simulations. The couplingparameter α = 4 . T int as given above. III. METHODOLOGY
As mentioned in the previous section, for each spatial dimension d we first fine tune thetemperature to reproduce intermittent behaviour in the returns. As can be seen from Table 1,the temperature T int ( d ) decreases with d . For a given dimension, all subsequent simulationsare performed at that temperature. Averages over at least 100 samples for each run wereperformed and the error-bars in the following plots are smaller than the data points.The value of each agent at t = 0 is noted and the dynamics updated according to equation(3).At each time step, we count the number of agents that still persist in their initial ( t = 0)state by evaluating n i ( t ) = ( S i ( t ) S i (0) + 1) / . (7)Initially, n i (0) = 1 for all i . It changes to zero when an agent changes from buying to sellingor vice vera for the first time. Note that once n i ( t ) = 0, it remains so for all subsequentcalculations.The total number, n ( t ), of agents who have not changed their action by time t is then givenby n ( t ) = X i n i ( t ) . (8)A fundamental quantity of interest is P ( t ), the persistence probability. In this problemwe can identify P ( t ) with the density of agents continuously buying or selling without5 l n P ( t ) tFigure 1p = 0.1p = 0.2p = 0.3p = 0.4p = 0.5 FIG. 1: Here we plot ln P ( t ) versus t for d = 1 over the range 0 . ≤ p ≤ .
5. The straight line,which is a guide to the eye, has a slope of − . interruption since the start [1], P ( t ) = n ( t ) /N, (9)where N = L d is the total number of agents present. IV. RESULTS
We now discuss our results. We restrict ourselves to 0 ≤ p ≤ . p = 0 .
5. It should be noted that we tried various different fits (exponential, power-law, stretched-exponential, etc.) for our data. We will not discuss the fits we discarded asunsatisfactory.In figure 1 we show a semi-log plot of the persistence probability against time t for arange of bond concentrations 0 < p ≤ . d = 1. It’s clear from the plot that the datacan be fitted to P ( t ) ∼ e − γ ( p ) t , (10)where we estimate γ ( p ) ∼ .
56 from the linear fit. Note that γ ( p ) ≈ γ , independently of p .6 l n P ( t ) tFigure 2p = 0.1p = 0.2p = 0.3p = 0.4p = 0.5 FIG. 2: A semi-log plot of the data for d = 2. We see that here, in contrast to figure 1 for d = 1,the slopes are dependent on the bond concentrations. The linear fit shown is that for p = 0 . − . Figure 2 displays the results for d = 2. Although, just as for d = 1, there is evidence forexponential decay, this time it would appear that the value of the parameter γ ( p ) depends on p . For p = 0 . γ ( p = 0 . ∼ .
35. The results for the three-dimensional case areshown in figure 3. Here we see clear evidence that even the qualitative nature of the decaydepends on the bond concentration. For p = 0 . p = 0 . d = 4 are verysimilar to those for d = 3 and we will not present them here. Instead, in figure 4 we show alog-log plot of the persistence against time for d = 5. The decay of P ( t ) is seen to be heavilydependent on the concentration of ferromagnetic bonds. For low values of p ( ≤ . θ ∼ . p the decay would appear not to be a power-law but also not exponentialin it’s nature. We note that even though we are working with a model containing disorder,no ‘blocking’ is observed in the simulations. This is probably because we are working at a7 l n P ( t ) tFigure 3p = 0.1p = 0.2p = 0.3p = 0.4p = 0.5 FIG. 3: A plot of ln P ( t ) against t for d = 3 for the same bond concentrations as earlier. Thestraight line, which is a guide to the eye, has a slope of − .
39 and indicates that the decay for p = 0 . p = 0 . finite temperature and is in agreement with the earlier work on the pure Ising model in highdimensions [3,12]. V. CONCLUSION
To conclude, we have presented the results of extensive simulations for the persistence be-haviour of agents in a model capturing some of the features found in real financial markets.Although the model contains bond disorder, we do not find any evidence of ‘blocking’ .This is believed to be because of thermal fluctuations. The persistence behaviour appearsto depend on both the spatial dimensionality and the concentration of ferromagnetic bonds.Generally, whereas in low dimensions the decay is exponential, for higher dimensions andlow values of p we get power-law behaviour.The initial model was developed in an economic context. Power law persistence in this8 l n P ( t ) ln tFigure 4 p = 0.1p = 0.2p = 0.25p = 0.29p = 0.3p = 0.4p = 0.5 FIG. 4: Here we display the data for d = 5 and selected bond concentrations as a log-log plot.Clearly, the behaviour depends crucially on the value of p . For low ( p ≤ .
3) values the decay ispower-law. The straight line shown has a slope of − . case means the existence of traders who keep on buying or selling for long durations. Fur-thermore, the presence of ‘blocking’ would be highly unrealistic for modelling the dynamicsbecause the traders would have access to only a finite amount of capital. Hence, the absenceof blocking would suggest that our model is not an unreasonable starting point for furtherdevelopment.One can also interpret the model in a social context. Here the value S i ( t ) = +1 or − cknowledgments We thank the referee for his comments on the manuscript and for bringing [12] to ourattention. TY thanks L. Pichl for allowing him to use his CPU (mona) at InternationalChristian University, Japan, and acknowledges support by JSPS Grant-in-Aid
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