Expulsion from structurally balanced paradise
EExpulsion from structurally balanced paradise
Krzysztof Malarz ∗ and Maciej Wo(cid:32)loszyn † AGH University of Science and Technology, Faculty of Physics andApplied Computer Science, al. Mickiewicza 30, 30-059 Krak´ow, Poland (Dated: July 7, 2020)We perform simulations of structural balance evolution on a triangular lattice using the heat-bathalgorithm. In contrast to similar approaches — but applied to analysis of complete graphs — thetriangular lattice topology successfully prevents the occurrence of even partial Heider’s balance.Starting with the state of Heider’s paradise, it is just a matter of time when the evolution of thesystem leads to an unbalanced and disordered state. The time of the system relaxation does notdepend on the system size. The lack of any signs of balanced state was not observed in earlierinvestigated systems dealing with structural balance.
Keywords: Heider balance; heat-bath algorithm; zero critical temperature
INTRODUCTION
The structural balance [1] (also termed as the Heiderbalance [2]) has been attracting attention of physicistsfor at least the past fifteen years. The considered topolo-gies include triangular lattices [3], complete graphs [4–9],and complex networks [10–13]; for review, see Refs. 14and 15. In all those cases, the model describes dynamicsof negative or positive links which represent hostile orfriendly attitudes among actors decorating nodes of anunderlying network. The available scenarios of actors’attitudes in a single triad are presented in Figure 1. Thetriads shown in Figures 1a and 1c are balanced in Heidersense as they obey the following rules: • friend of my friend is my friend, • friend of my enemy is my enemy, • enemy of my friend is my enemy, • and enemy of my enemy is my friend. ++ +(a) ++ − (b) + − − (c) −− − (d) Figure 1: Heider’s triads corresponding to balanced(the first and the third from the left) and imbalanced(the second and the fourth) states. Continuous bluelines and dashed red lines represent friendly and hostilerelations, respectively.The configurations presented in Figures 1b and 1d areimbalanced (i.e., do not obey the rules given above) andresult in actors’ feelings of discomfort known as cognitivedissonance [16]. In order to relieve this tension actorsshould change their attitudes to others by switching un-friendly or amicable relations into opposite ones. Such process may be realized for two connected triads usingthe links dynamics given by x ij ( t + 1) = sign (cid:2) x im ( t ) x jm ( t ) + x in ( t ) x jn ( t ) (cid:3) , (1)where x ab symbolizes friendly ( x ab = +1) or hostile( x ab = −
1) relation among the actors a and b (see Fig-ure 2 for two examples of system evolution towards theHeider balance, from an imbalanced state at time t to abalanced state at t + 1 after single link flip x ij ( t + 1) = − x ij ( t )). ijm nx ij x in x jn x im x jm t ijm nx ij x in x jn x im x jm t + 1(a) ijm nx ij x in x jn x im x jm t ijm nx ij x in x jn x im x jm t + 1(b) Figure 2: Examples of configuration of signed links x im , x in , x jm , x jn which influence the link x ij in the nexttime step according to Equation (1) in the deterministiccase, i.e. for T = 0.We note that when the system of N links is composedonly of balanced triads (presented in Figures 1a and 1c)then the system energy defined as U = − (cid:80) i,j,k x ij x jk x ki L (2) a r X i v : . [ phy s i c s . s o c - ph ] J u l is exactly equal to U = −
1, which allows for easy detec-tion of system balance without the triad-by-triad inspec-tion.Very recently, the deterministic evolution according toEquation (1) was enriched by introducing the thermalnoise simulated by Glauber [7] or heat-bath [8, 9] dy-namics. In the latter case, the first order phase transitionfrom (at least partially) balanced (with − ≥ U ≥ U ∗ > ≥ (cid:104) x ij (cid:105) ≥ x ∗ > U = 0 [8]) and disordered (withspatially averaged value of links strength (cid:104) x ij (cid:105) = 0 [9])state was observed for a complete graph. The transitiontakes place at the critical temperature T ∗ : for T < T ∗ ordered and balanced system states are observed, whilefor T > T ∗ the time evolution drives the system to im-balanced and disordered states.In this letter we show that keeping the thermal noise,but using the triangular lattice topology instead of a com-plete graph results in total vanishing of the ordered andbalanced phases in the system. METHODS
The non-deterministic version of Equation (1) reflect-ing the presence of thermal noise and using the heat-bathmethod [17] may be written as x ij ( t + 1) = (cid:40) +1 with probability p, − − p ) , (3a)where p = exp( c )exp( c ) + exp( − c ) (3b)and c = x im ( t ) x jm ( t ) + x in ( t ) x jn ( t ) T , (3c)while m and n are the common neighbours of nodes i and j (see Fig. 2), and T stands for temperature.We apply Equation (3) to find the time-evolution of N = 3 L links on a triangular lattice of L nodes assum-ing the periodic boundary conditions. A single MonteCarlo time step takes 3 L attempts to modification of x ij performed synchronously on the whole system. RESULTS
In Figure 3 we present the time evolution of the aver-age value of links strengths (cid:104) x ij (cid:105) and the system energy U for various values of the social temperate T and thesystem size L . The angle brackets (cid:104)· · · (cid:105) indicate the aver-aging procedure performed over all 3 L links, and all theresults are also averaged over one hundred simulations. h x i j i U t T = Figure 3: Time evolution of (a) the average value oflinks strengths (cid:104) x ij (cid:105) and (b) energy U for various valuesof social temperature T and N = 3 L edges, where L = 20 (lines) and L = 100 (symbols). Initially, all linksvalues are set to +1. The results are averaged over 100simulations.Initially, the Heider paradise is assumed, i.e., at t = 0 ev-ery link is set to x ij = +1. Neither the system size L , northe assumed temperature T prevents reaching the unbal-anced and disordered state with (cid:104) x ij (cid:105) = 0 and U = 0.However, the time between the start of simulation andthe first flip of any link value from +1 to − U = 4.Such probability is proportional to the number of links asthe changes may happen independently and at any place,which means that τ = 1 + exp(4 /T )3 L (4)and tends to infinity in the limit of very low tempera-tures, T → + , as shown in Figure 4 where both theabove dependence (lines) and the results of simulation(points) are presented. Thus for T = 0 . L = 100 nolink switching from +1 to − t = 10 (i.e. until N t = 3 · trials). To analyze the systemevolution in the low temperature limit we repeat our sim-ulations with the initial state where 5% of the links areset to x ij = −
1. For such starting point also systems keptin low temperature ( T = 0 .
1) reach unbalanced U = 0and disordered (cid:104) x ij (cid:105) = 0 state (see Figure 5).We assume that the system reaches the stationary statewhen U and |(cid:104) x ij (cid:105)| are smaller than ε = 10 − . Thetime τ needed for reaching the stationary state increaseswith decreasing the temperature T independently on theassumed system size L (see upper part of Figure 4). The -8 -6 -4 -2 ττ T L = Figure 4: (a) Time τ of reaching the stationary stateand (b) time τ of the first link switching from x ij = +1to x ij = −
1. The results are averaged over R = 100simulations. The error bars are smaller than size of thesymbols and they are of the order of 1 / √ R . Linesindicate the theoretical dependence τ = [1 + exp(4 /T )] / (3 L ).deviations from this statement are observed only for verysmall system sizes ( L <
50) and only in low temperatures(
T < . , , , and for triadspresented in Figures 1a to 1d, respectively. The resultsprove to be independent on system size L and tempera-ture T , and we will discus this issue shortly in Conclu-sions. CONCLUSIONS
In contrast to the stochastic evolution of the systemwith hostile and friendly attitudes among actors on acomplete graph [9] we show that the triangular latticetopology successfully prevents the occurrence of even par-tial Heider balance .Starting at the state of paradise ( ∀ i, j : x ij ( t = 0) =+1) it is just a matter of time τ , when the thermallydriven evolution (governed by the heat-bath algorithm)of the system leads to an unbalanced and disorderedstate. In contrast to the previously obtained results[18, 19] the relaxation time τ does not depend on the sys- h x i j i U t T = Figure 5: Time evolution of (a) the average value oflinks strengths (cid:104) x ij (cid:105) and (b) energy U for various valuesof social temperate T ( N = 3 L edges, L = 20 (lines)and L = 100 (symbols)). Initially, 95% links values areset to +1, hence (cid:104) x ij ( t = 0) (cid:105) = 0 .
9. The results ofevolution averaged over 100 simulations are presented. f t Figure 6: Time evolution of the fractions f of triadspresented in Figures 1a to 1d for L = 100 and T = 0 . tem size .The probabilities distribution of various triads pre-sented in Figures 1a to 1d changes from (1 , , ,
0) at t = 0 to (cid:0) , , , (cid:1) at t → ∞ . The latter is in agree-ment with probabilities (cid:18) k (cid:19) p k (1 − p ) − k of k = 3, 2, 1, and 0 successes in three Bernoulli trials,when the probability of success is equal to p = . Theresult accentuates the randomness of the final link valuesdistribution and emphasizes the system disorder for anypositive temperature.In summary, we found that introducing noise to thesystem of hostile and friendly attitudes on triangular lat-tice leads to the disordered and imbalanced state inde-pendently on the system size and for any positive tem-perature. In other words, the critical temperature of thesystem is zero. None of the earlier studies devoted to theproblem of structural balance, and conducted for varioustopologies and different schemes of updating the link val-ues, have shown complete lack of any signs of balancedstate.Our results prove that at least in certain cases the be-havior of the system does not have to follow what wasso far believed to be a general tendency towards a globalstructural balance. Signatures of such possibility were al-ready provided, for instance in studies of bilayer networks[13]. ∗ † , 277–293 (1956).[2] F. Heider, “Attitudes and cognitive organization,” TheJournal of Psychology , 107–112 (1946).[3] K. Malarz, M. Wo(cid:32)loszyn, and K. Ku(cid:32)lakowski, “Towardsthe Heider balance with a cellular automaton,” PhysicaD , 132556 (2020).[4] T. Antal, P. L. Krapivsky, and S. Redner, “Dynamicsof social balance on networks,” Physical Review E ,036121 (2005).[5] K. Ku(cid:32)lakowski, P. Gawro´nski, and P. Gronek, “The Hei-der balance: A continuous approach,” International Jour-nal of Modern Physics C , 707–716 (2005).[6] S. A. Marvel, J. Kleinberg, R. D. Kleinberg, and S. H.Strogatz, “Continuous-time model of structural balance,”Proceedings of the National Academy of Sciences ,1771–1776 (2011).[7] R. Shojaei, P. Manshour, and A. Montakhab, “Phasetransition in a network model of social balance withGlauber dynamics,” Physical Review E , 022303(2019).[8] F. Rabbani, A. H. Shirazi, and G. R. Jafari, “Mean-field solution of structural balance dynamics in nonzerotemperature,” Physical Review E , 062302 (2019).[9] K. Malarz and J. A. Ho(cid:32)lyst, “Comment on ‘Mean-fieldsolution of structural balance dynamics in nonzero tem-perature’,” (2020), arXiv:1911.13048.[10] P. Gawro´nski and K. Ku(cid:32)lakowski, “Heider balance in hu-man networks,” AIP Conference Proceedings , 93–95(2005).[11] M. E. J. Newman and J. Park, “Why social networks aredifferent from other types of networks,” Physical ReviewE , 036122 (2003). [12] E. Estrada and M. Benzi, “Walk-based measure of bal-ance in signed networks: Detecting lack of balance insocial networks,” Physical Review E , 042802 (2014).[13] P. J. G´orski, K. Ku(cid:32)lakowski, P. Gawro´nski, and J. A.Ho(cid:32)lyst, “Destructive influence of interlayer coupling onHeider balance in bilayer networks,” Scientific Reports , 16047 (2017).[14] K. Ku(cid:32)lakowski, “Some recent attempts to simulate theHeider balance problem,” Computing in Science & Engi-neering , 80–85 (2007).[15] M. J. Krawczyk, M. Wo(cid:32)loszyn, P. Gronek,K. Ku(cid:32)lakowski, and J. Mucha, “The Heider bal-ance and the looking-glass self: Modelling dynamics ofsocial relations,” Scientific Reports , 11202 (2019).[16] L. Festinger, A Theory of Cognitive Dissonance (Stan-ford University Press, Stanford, 1957).[17] K. Binder, “Applications of Monte Carlo methods to sta-tistical physics,” Reports on Progress in Physics , 487–559 (1997).[18] T. Antal, P. L. Krapivsky, and S. Redner, “Social bal-ance on networks: The dynamics of friendship and en-mity,” Physica D , 130–136 (2006).[19] P. Gawro´nski, P. Gronek, and K. Ku(cid:32)lakowski, “The Hei-der balance and social distance,” Acta Physica PolonicaB36