Phase transition in a network model of social balance with Glauber dynamics
aa r X i v : . [ phy s i c s . s o c - ph ] J u l Phase transition in a network model of social balance with Glauber dynamics
Rana Shojaei and Pouya Manshour ∗ Physics Department, Persian Gulf University, Bushehr 75169, Iran
Afshin Montakhab † Physics Department, Shiraz University, Shiraz 71454, Iran
We study the evolution of a social network with friendly/enmity connections into a balanced stateby introducing a dynamical model with an intrinsic randomness, similar to Glauber dynamics instatistical mechanics. We include the possibility of the tension promotion as well as the tensionreduction in our model. Such a more realistic situation enables the system to escape from localminima in its energy landscape and thus to exit out of frozen imbalanced states, which are unwantedoutcomes observed in previous models. On the other hand, in finite networks the dynamics takes thesystem into a balanced phase, if the randomness is lower than a critical value. For large networks,we also find a sharp phase transition at the initial positive link density of ρ ∗ = 1 /
2, where thesystem transitions from a bipolar state into a paradise. This modifies the gradual phase transitionat a nontrivial value of ρ ∗ ≃ .
65, observed in recent studies.
I. INTRODUCTION
As Mark Buchanan discussed in his book,
The SocialAtom [1], we can think of people as elementary build-ing blocks (the atoms) of the social world. The inter-actions between such interdependent elements lead tothe emergence of macroscopic patterns such as cultures,wars, social classes, political parties, racial groups, etc.To understand such phenomena, one needs a new wayof thinking, which borrows concepts from physics, inparticular thermodynamics and statistical mechanics tostudy the macroscopic aspects of the human dynamicsin social networks. Indeed, the evolution of our worldis strongly ruled by social networks. In general, all po-litical, economical, social or military conflicts occur ina social network, which includes a set of elements likecountries, corporations or people that interact throughdifferent types of connections, such as friendship, hostil-ity, political treaties, trade, or sharing ideas [2–11].Avoiding distress and conflict is a completely naturalphenomenon in societies and interpersonal relationships,and almost all efforts are in the direction of tension re-duction, if the individual nodes behave rationally [12–14].In motivation psychology, Heider proposed a theory forattitude change, known as the balance theory [12]. Bytaking into account the relationship between three ele-ments includes Person (P), and Other person (O) withan object (X), known as the POX pattern, he postu-lated that only balanced triads are stable. The POX is“balanced” when P and O are friends, and they agreein their opinion of X. To reduce the stress in an imbal-anced triad, the individuals change their opinions so thatthe triad becomes balanced. Empirical examples of Hei-ders balance theory have been found in human and otheranimal societies [3, 15–19]. Cartwright and Harary de- ∗ [email protected] † [email protected] veloped Heider model and showed that a complete signedgraph with positive (agree) and negative (disagree) linksis balanced, if and only if it can be decomposed into twofully positive subgraphs that are joined by negative links,called bipolar state [13, 14].For many years, authors only considered static signednetworks. However, an important subject in the field ofbalance theory is the understanding of appropriate dy-namics that can more accurately address the evolution ofsocial networks, and explain how such a social balancedstate emerges [20–28]. In such models one usually consid-ers a complete network, i.e., all to all connections amongnodes with dynamic signed links. Each link changes itsstatus in order to reduce the local/global tension, if someconditions are satisfied. In a few works, continuous-valued links models have been investigated [22, 24], andhave shown that the probability of reaching a balancedstate in finite time tends to unity only for infinite systemsizes. The influence of asymmetry in networks were alsostudied [29]. Recently, the effect of memory on the evo-lution of the links has been investigated, which leads toa new glassy state in the networks [30].In an interesting work, Antal and colleagues intro-duced a dynamical model in complete networks with pos-itive/negative links, called Constrained Triad Dynamics (CTD) [20, 21]. By definition, a triad is balanced if it hasodd number of positive links. The update rule is as fol-lows: A randomly chosen link is flipped, i.e., changes itssign, if the total number of imbalanced triads, N imb , de-creases. If N imb remains conserved, then the chosen linkis flipped with probability 1 /
2, and otherwise no changesin the link sign is allowed. This dynamics always takesthe system into a more balanced situation. Two possibleoutcomes of such a dynamics are a balanced state or a jammed state . A jammed state is an unwanted outcomein finite size, where the system is trapped into an imbal-anced state, forever. They proved that the number ofsuch jammed states greatly exceeds the number of bal-anced states, and that the probability of reaching themvanishes as the system size increases. By introducing anenergy landscape, the properties of such jammed stateshave also been studied, extensively [31, 32]. Another re-sult of CTD dynamics is a nontrivial gradual phase tran-sition for the difference in sizes of the two final poles at ρ ≃ .
65, where ρ is the initial density of the positivelinks. They argued that this observed gradual transitionis not in agreement with analytical calculations.As mentioned above, one of the fundamental assump-tions in such dynamical models is the tendency to re-duce the tension between elements of a social network,i.e., the update rules always take the system into a morebalanced situation. But, this is not the case when wedeal with the real world. For example, dissatisfaction,discomfort, profit, pride, anger, or generally speaking so-cial anomalies, as well as random activity of each agent,are always present in social networks. In this article, weshow, via detailed numerical and analytical calculations,that one can overcome the difficulties observed in pre-vious models by introducing a more realistic dynamicalmodel that takes into account the possibility of reduc-tion as well as promotion of the tension among the socialagents. We find some interesting results. Our dynamicsis never trapped into a jammed state. In addition, thesystem undergoes a sharp phase transition from a bipolarstate into the paradise at ρ = 1 /
2, for large networks.
II. MODEL DEFINITION
In order to investigate how fluctuations in individualbehaviors affect the balance theory, and also to overcomethe observed shortcommings in previous studies, we in-troduce a dynamics as follows: We consider a fully con-nected network of size N , and use a symmetric conectiv-ity matrix s , such that s ij = ±
1. The positive sign rep-resents friendship, and the negative one represents hos-tility between two arbitrary nodes i and j . By definitionof a triad to be of type ∆ k if it contains k negative links,then ∆ and ∆ are balanced while ∆ and ∆ are imbal-anced. Such conditions for balanced/imbalanced triadsassert that a friend of my friend or an enemy of my enemyis my friend, and vice versa. For simplicity, one assumesthat everyone knows everyone else, i.e., the dynamics oc-curs on a complete graph, which is appropriate for smallreal-world networks. As in [21, 31], the total energy ofthe system is defined as: U = 1 N tri X i>j>k u ijk (1)where u ijk = − s ij s jk s ki , and the normalization factor N tri = (cid:0) N (cid:1) is the total number of triads in the network,so we have − ≤ U ≤
1. By this definition, we have u = − U = − p , defined as p = 11 + e β ∆ U ( t ) (2)where β is a control parameter, and represents the inverseof the disorder in the system, which may be considered asthe stochasticity in the individual behavior. Also, ∆ U ( t )indicates the change in the energy due to the link-flippingin every time step t . Fig. 1 shows all possible configu-rations of each triad due to an update. Such dynamicsis similar to that of Glauber dynamics used in simula-tions of kinetic Ising model [33]. It is important to notethat such dynamics provides a more realistic feature ofcreating or reducing tension at any given time while onaverage reducing tension for positive finite β . Further-more, since it allows for increase in the energy of thesystem, it could provide a natural mechanism to escapeout of local minima, i.e., the jammed states.We intend to study the dynamics of such a model, i.e., ρ ( t ) and U ( t ), for various initial configurations ρ andrandomness parameter β . We first provide mean-fieldanalytical results for some special cases and then considerthe model in more general conditions, numerically. FIG. 1. Four distinct configurations of elementary units of thenetwork. Solid lines show friendship links and dashed linesrepresent hostility links. The triads with energy − III. MEAN-FIELD APPROACH
Due to the large number of degrees of freedom in thesystem, finding exact time dependent equations for ourdynamics is inaccessible. Thus, we try to find a mean-field approximation for the rate equations of our dynam-ics, using the notations used in [21]. It is appropriateto work with quantity n i which is the density of triadsof type ∆ i , i.e., n i = N i /N tri , where N i is the num-ber of such triads. With this definition, the numberof positive links and the density of such links become L + = (3 N + 2 N + N ) / ( N − ρ = L + /L , respec-tively, where L = (cid:0) N (cid:1) is the total number of links and L + is the number of positive links. Thus, the energy, U ,and the density ρ can be written as follows: ρ = n + 2 n / n / U = − n + n − n + n (3)Another useful quantity is the density n + i ( n − i ) of triadsof type ∆ i that are connected to a positive (negative)link. The total number of positive links connected totriads of type ∆ i is (3 − i ) N i , and N + i = (3 − i ) N i /L + is the average number of such triads. Since, each linkis connected to N − n + i = N + i / ( N − n − i = N − i / ( N − n + i = (3 − i ) n i / (3 n + 2 n + n ) n − i = in i / ( n + 2 n + 3 n ) (4)Taking into account that ρ is the probability of findinga positive link, the probability of flipping a positive linkis π + = p + ρ , with p + = 11 + e β ∆ U + − (5)and of flipping a negative link is π − = p − (1 − ρ ), with p − = 11 + e β ∆ U − + (6)where ∆ U + − and ∆ U − + are the energy difference due tothe flipping a positive and a negative link, respectively.Thus, for an each update at step n , we have L + ( n + 1) − L + ( n ) = − π + + π − (7)Since each time step equals L updates, so one can simplyfind the rate equation for (average) ρ , as dρdt = − π + + π − (8)In each update, the energy difference due to the flip-ping of a positive link equals to 2( N +0 − N +1 + N +2 ) /N tri ,and similarly for the flipping of a negative link we have2( − N − + N − − N − ) /N tri . Thus we obtain U ( n + 1) − U ( n ) = 2 π + ( N +0 − N +1 + N +2 ) /N tri + 2 π − ( − N − + N − − N − ) /N tri (9)Therefore, we find the rate equation of the total energyas dUdt = π + ∆ U + − + π − ∆ U − + (10)where ∆ U + − = 6( n +0 − n +1 + n +2 ) and ∆ U − + = − n − − n − + n − ). In a similar way one can also find the rate equations for all triad densities, n i , as follows: dn dt = − π + n +0 + 3 π − n − dn dt = − π + n +1 − π − n − + 3 π + n +0 + 3 π − n − dn dt = − π + n +2 − π − n − + 3 π + n +1 + 3 π − n − dn dt = − π − n − + 3 π + n +2 (11)where Eqs. (8) and (10) can also be derived from Eqs.(3) and (11).For completely random flipping, β →
0, we have p + = p − = 1 /
2. Thus Eq. (8) becomes dρdt = 1 / − ρ . Simply,we find that ρ ( t ) = 1 / ρ − / e − t (12)which ρ ∞ = 1 / ρ , as expected. Dueto the uncorrelated nature of the dynamics at β = 0,the triad densities become n = ρ , n = 3 ρ (1 − ρ ), n = 3 ρ (1 − ρ ) , and n = (1 − ρ ) . By substitutingthese densities into ∆ U + − and ∆ U − + , using relations for n + i and n − i , we find∆ U + − = +24( ρ − / ∆ U − + = − ρ − / (13)Thus, Eq. (10) becomes dUdt = 24( ρ − / (14)One simply finds U ( t ) = − ρ − / e − t (15)which shows that U ∞ →
0, as expected for an uncorre-lated network. Also, we find a relation between U and ρ for an uncorrelated network as: U = − ρ − / (16)For the case of large β , we first assume that the systemremains uncorrelated during its early stages of the evolu-tion, as observed from simulations for large networks (seethe next section). Therefore, Eq. (13) holds for initialtime steps. Consequently, p − →
1, and p + →
0, and thus dρdt ≃ (1 − ρ ). We find the time behavior of ρ , as ρ ( t ) = 1 + ( ρ − e − t (17)and also for U as U ( t ) = − / ρ − e − t ) (18)This shows that for large t , the dynamics takes the sys-tem into a paradise state, i.e., ρ ∞ → U ∞ → − ρ .On the other hand, one can find stationary solu-tions of the rate equations, for any arbitrary β . FromEqs. (8), (10) and (11), we find π + = π − , n +0 = n − , n +1 = n − , n +2 = n − , and also ∆ U + −∗ = − ∆ U − + ∗ =+24( ρ ∞ − / . With a little algebra, one can obtain U ∞ = − ρ ∞ − / , and ρ ∞ = p −∗ p −∗ + p + ∗ (19)where p + ∗ = 11 + e β ( ρ ∞ − / ) p −∗ = 11 + e − β ( ρ ∞ − / ) (20)Note that Eq. (19) is a self-consistent equation for ρ ∞ .As it can be seen, the balanced ( U ∞ = −
1) stationary so-lution of the system only occurs for β → ∞ , with ρ ∞ = 1.However, as we will demonstrate later, another balancedsolution ( ρ ∞ = 1 /
2) also occurs for large β . We notehere that these analytical relations are obtained by tak-ing into account the mean-field approximation, where weassume the same dynamics for all triads of type ∆ i , andthe effects of correlations are averaged out. We will fur-ther show that for values of ρ near 1 /
2, correlation effectscannot be neglected, and our mean-filed results are notable to represent such regimes. Thus, our analytical re-sults are not exact. In the next section, we will provethis claim and also discuss the dynamics near this point,in more details.
IV. NUMERICAL RESULTS
In this section, we simulate our proposed model, onnetworks with different sizes of N , and different initialconditions of ρ . Note also that we take each time stepas L simulation updates, so that at every time step, onelink is updated, on average. At first, we focus on the fullyrandom case with β = 0. In Fig. 2(a) and 2(b), we plotthe time evolution of the energy, and the positive links’density, ρ , for different initial conditions of ρ = 0 .
0, 0 . .
5, 0 .
8, and 1 .
0. The network size is N = 128, here. For β = 0, and from Eq. (2), we have p = 1 /
2, and thus,the updates are fully random. As it can be seen, at large t , the system approaches into a completely random con-figuration with ρ ∞ = 1 / U ∞ = 0. The network isimbalanced and has equal number of positive and nega-tive links, as expected. Fig. 2(c) shows the correspondingtrajectory of the dynamics in U − ρ space for two initialconditions of ρ = 0 (with U = +1) and ρ = 1 (with U = − ρ = 1 /
2. Note that the dashed lines in (a), (b), and(c) show our analytical findings of Eqs. (12), (15), and(16), respectively, which are all in complete agreementwith our simulations.To investigate the case of complete order, i.e., β → ∞ ,Figs. 3 and 4 show the time evolution of U ( t ) and ρ ( t ) FIG. 2. The time evolution of (a) the total energy, U , and (b)the density of positive links, ρ , with different initial conditionsfor β = 0. (c) The trajectory of this dynamics in U − ρ space,for two initial conditions of ρ = 0, and 1. The network sizefor all plots is N = 128. Note that the dashed lines in (a),(b), and (c) show our analytical solutions of Eqs. (12), (15),and (16), respectively. for initial conditions of ρ ≤ / ρ > /
2, respec-tively. We see that the network always reaches a finalbalanced state of U ∞ = − t (see Figs. 3(a) and4(a)). As it is observed in Fig. 3(b), the final values ofpositive links’ density, ρ ∞ , are independent of the initialconditions for all ρ ≤ /
2. Indeed, the network reachesa bipolar states with nearly same-size poles. However,for ρ > /
2, Fig. 4(b) shows that ρ ∞ →
1, and the final
FIG. 3. The time evolution of (a) the total energy, U , and (b)the density of positive links, ρ , with different initial conditionsof ρ ≤ / β → ∞ . The dashed lines show our analyticalsolutions, i.e., Eqs. (17) and (18), which are not in agreementwith the simulations. The network size for all plots is N =128. state is paradise. To better understand the evolution ofthe system, in Fig. 5, we have also plotted U versus ρ .As it can be seen, the system finally approaches into oneof its attractors of ρ ∞ = 1 / ρ ∞ = 1, depending onthe corresponding initial conditions. Note here that inearly stages of the dynamics, the evolution of the systemcoincides on the trajectory of an uncorrelated dynamics(see Fig. 2(c)). The dashed lines in Figs. 3 and 4 showour analytical findings for large β . We observe that onlyfor ρ > /
2, a good agreement between our analytic cal-culations and our simulations exists, and for ρ ≤ / ρ ≤ / ρ > /
2, the conditions∆ U + − > U − + < dρ/dt = 0 holds. But, for the case of ρ < /
2, the aboveconditions also holds until ρ → /
2, at which p + ≃ p − ,and thus ρ is trapped into this value. It is worth to men-tion here that the reason that our rate equations cannot FIG. 4. The time evolution of (a) the total energy, U , and(b) the density of positive links, ρ , with different initial con-ditions of ρ > / β → ∞ . The dashed lines show ouranalytical solutions, i.e., Eqs. 17 and 18, which are in com-plete agreement with the simulations. The network size forall plots is N = 128. directly represent such a behavior is due to the devia-tion of the dynamics from its uncorrelated trajectory near ρ = 1 /
2, which is observed in simulations. As depicted inFig. 5, for ρ → /
2, the deviation from the uncorrelatedtrajectory (dashed line) increases, which indicates thatthe mean-field approximation is not applicable here, andthus our analytical results are not able to cover the caseof ρ ≤ / β = 0 and ∞ . Clearly, a finite value of randomness β isof more interest. For example, considering the fact that β = 1 /kT and the fact that the dynamics of the links canbe mapped to the spin dynamics in ferromagnetic (Ising)models, one can look for a possible phase transition atfinite β . To study the intermediate randomness level,we calculate the behavior of final values of energy, U ∞ for various β . Fig. 6(a) shows plots of U ∞ versus β for ρ = 0 .
6, for different network sizes. Interestingly, forlarge N , we find a nearly sharp transition from a random(imbalanced) state to an ordered (balanced) state, at β = β c . In fact, for β < β c , the dynamics takes the systeminto an imbalanced state with U = −
1, and the positivelink density satisfies the relation 16, independent of the
FIG. 5. The trajectory of the dynamics in U − ρ space, for β → ∞ with five different initial conditions. Note that rep-resentative points with U = − ρ → /
2, the deviation from the random trajectory increases.The network size is N = 128. initial conditions. However, for β ≥ β c , the system finallybecomes balanced, in the sense that ρ ∞ → / ρ ≤ / ρ > /
2, respectively. As it can be seen, β c → ∞ , as N → ∞ . It is important to note that ourmodel leads to a final balanced state, despite large β ,which is in contrary to the previous results where thesystem would be trapped in jammed states.Further, it is interesting to check the effect of the net-work size on final values of the order parameter in thebalanced phase, i.e., the average link value, δ = h s ij i =2 ρ ∞ − ρ . In this respect, we plotted inFig. 6(b), | δ | versus ρ for a typical β > β c (here β = 3).We find a sharp phase transition from bipolar state intoparadise one, at ρ = 1 / / < ρ ∞ <
1. It is worthmentioininghere that we find a sharp transition at ρ = 1 /
2, in com-parison with the gradual transition at ρ ≃ .
65, foundin previous studies [20, 21].
V. CONCLUSION
Balance theory has been used to study the solidarityand the stability of a social network. Much work hasbeen done in studying the static and dynamic aspectsof the balance theory. Almost all studied models takeinto account the assumption of tension reduction. How-ever, social agents do not always change their relationshipto reduce the tension. Also some previous models haveshown that an unwanted outcome may emerge, wherethe system is trapped in an imbalanced state (a jammedstate), forever. In this paper, we introduced a more re-alistic dynamical model by adding an intrinsic random-ness, denoted by 1 /β , into the social agents behaviors. FIG. 6. (a) The β dependency of large time behaviors of thetotal energy, U , for ρ = 0 .
6. Different network sizes areshown with different symbols. For large networks, a nearlysharp transition occurs at some values of β , which goes toinfinity as N → ∞ . (b) The order parameter, δ , defined asthe average link value, h s i , as a function of ρ , for differentnetwork sizes of N = 8, 16, 32, 64, 128, 256, and 512, at β = 3.There is a sharp transition at ρ = 1 / Our dynamics considers the possibility of both increas-ing and decreasing the tension, somehow that on average,the overall tension reduces. We find that it helps the sys-tem escape from the local minima in its energy landscape.Due to this feature, the dynamics is able to exit jammedstates and finally is able to find a balanced state. Thisshows that our dynamics can solve the undesirable find-ing in previous studies. In addition, we find that for acritical value of the randomness, 1 /β c , a transition oc-curs from an imbalanced into a balanced phase. Indeed,for β > β c , the system approaches into two possible bal-anced states: bipolar or paradise. For finite networks,we showed that a final bipolar state with positive linkdensity of ρ ∞ > / ρ = 1 /
2. Indeed, by passing throughthis point, the system transitions from a bipolar state of ρ ∞ = 1 / ρ ∞ = 1. This sharp transition can be compared with the gradual transitionin a nontrivial point of ρ ≃ .
65, observed in previousmodels.We also derived an analytical description for the timeevolution of the system, by applying a mean-field ap-proximation. Our analytical solutions are in completeagreement with our simulations, except for the case of ρ ≤ / β . We showed that this inconsistencyis the result of deviation of the dynamics from the un- correlated situation, as the system approaches ρ = 1 / ACKNOWLEDGMENT
PM would like to gratefully acknowledge the PersianGulf University Research Council for support of thiswork. AM also acknowledges support from Shiraz Uni-versity Research Council. [1] M. Buchanan,
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