Opinions, Conflicts and Consensus: Modeling Social Dynamics in a Collaborative Environment
J´anosT¨or¨ok,1 Gerardo I˜niguez,2 Taha Yasseri,1 Maxi San Miguel,3 Kimmo Kaski,2 and J´anosKert´esz4, 1, 2 1Institute of Physics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary 2Department of Biomedical Engineering and Computational Science, FI-00076 Aalto, Finland 3IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07071 Palma de Mallorca, Spain 4Center for Network Science, Central European University, H-1051 Budapest, Hungary (Dated: September 10, 2018) Information-communication technology promotes collaborative environments like Wikipedia where, however, controversiality and conflicts can appear. To describe the rise, persistence, and resolution of such conflicts we devise an extended opinion dynamics model where agents with differ- ent opinions perform a single task to make a consensual product. As a function of the convergence parameter describing the influence of the product on the agents, the model shows spontaneous sym- metry breaking of the final consensus opinion represented by the medium. In the case when agents are replaced with new ones at a certain rate, a transition from mainly consensus to a perpetual conflict occurs, which is in qualitative agreement with the scenarios observed in Wikipedia.
PACS numbers: 89.75.Fb, 89.65.-s, 05.65.+b
104 Dresden bombing 106 Japan 107 Anarchism Society represents a paradigmatic example of complex 8× 1.6× 1.4× 0 1 104 0 1 104 0 1 104 systems, where interactions between many constituents 3 × 3 × 9 ×
and feedback and other non-linear mechanisms result in S S S
M 4 0.8 0.7 emergent collective phenomena. The recent availabil- 0 0 0 t t t ity of large data sets due to information-communication technology (ICT) has enabled us to apply more quantita- 0 0 0 0.04 0.17 0.3 0.2 0.7 1.2 0.1 0.7 1.3 tive methods in social sciences than before. As a matter (a) (b) t ( 104 ) (c) × of fact, physicists play an increasing role in the study of social phenomena by applying physics concepts and tools FIG. 1. (Color online) Empirical controversiality measure to investigate them [1]. M [10] as a function of the number of edits t for three different conflict scenarios in Wikipedia, corresponding to: (a) single Social interactions are heavily influenced by the opin- conflict, (b) plateaus of consensus, and (c) uninterrupted con- ions of the members of the society. This is especially troversiality. Titles are the article topics. Inset: Theoretical true when complex tasks are to be solved by coopera- conflict measure S(t) of Eq. (2), allowing for a qualitative tion, as practiced throughout the history of mankind. In comparison of the model with the empirical results. this respect new technologies open up unprecedented op- portunities: by using the Internet and related facilities even remote members of large groups can work on the typical patterns of different categories of the so-called same task and achieve a higher level of synergy. Exam- edit wars have been identified [8–12]. Take for example ples include open software projects or large collaborative Fig. 1, where the evolution of a controversiality measure scientific endeavors like high-energy physics experiments. M based on mutual reverts and maturity of editors is However, it is unavoidable that in such cases differences shown for three different regimes of conflict. in attitudes, approaches, and emphases (in short, opin- Our aim in this Letter is to model controversiality and ions) occur. Then questions arise: How can a task be conflict resolution in a collaborative environment and, solved in a collaborative environment of agents having where possible, to qualitatively compare our results with diverse opinions? How do conflicts emerge and get re- different scenarios observed in the Wikipedia. One of the arXiv:1207.4914v2 [physics.soc-ph] 22 Nov 2012 solved? The understanding of these mechanisms may most developed areas of quantitative modeling in social lead to an increase in efficiency of value production in phenomena is opinion dynamics [1, 13, 14]. These models cooperative environments. A prime example of the lat- have much in common with those of statistical physics, ter is Wikipedia, a free, web-based encyclopedia project yet interactions are socially motivated. The problem with where volunteering individuals collaboratively write and these models is, similarly to evolutionary game theoretic edit articles on their desired topics. Wikipedia is par- ones [15], that results are usually evaluated by quali- ticularly well-suited also as a target for a wide range of tative subjective judgment instead of comparison with studies, since all changes and discussions are recorded empirical data, one exception being the study of elec- and made publicly available [2–7]. tions [16, 17]. The well-documented Wikipedia edit wars Recently, the controversiality and dynamical evolution can also be of use in this respect. of Wikipedia articles have been studied in detail and Our model is based on the bounded confidence (BC) 2 mechanism [18, 19], describing a generic case when 1 1 1 convergence occurs only upon opinion difference being A smaller than a given threshold. Here N agents are char- , 0.5 0.5 0.5 i acterized by continuous opinion variables xi∈[0, 1] and x their interactions are pairwise. The agents’ mutual influ- I II III 0 0 0 ence is controlled by the convergence parameter µT , only 0 0.25 0.5 0 0.25 0.5 0 0.25 0.5 t ( 103 ) if their opinions differ less than a given tolerance T , i.e. × for |xi−xj|<T we update as follows, FIG. 2. (Color online) Example evolutions of the agents’ opin- (xi, xj) 7→ (xi + µT [xj − xi], xj + µT [xi − xj]). (1) ions xi (light gray/blue) and the value A of the medium (dark gray/green) for three qualitatively different regimes, corre- The dynamics set by Eq. (1) has been studied exten- sponding to (A, µA)=(0.075, 0.2) in I, (0.075, 0.45) in II, and sively in the literature [1], initially by using the mean- (0.15, 0.7) in III. field approach of two-body inelastic collisions in granular gases [20, 21]. It leads to a frozen steady state which is characterized by nc∼1/(2T ) disjoint opinion groups. being replaced by new ones at a certain rate. For the This is caused by the instability in the initial opinion sake of simplicity we fix T =0.2 (leading in general to distribution near the boundaries. Also, nc increases as one large mainstream and two small extremist groups) T →0 in a series of bifurcations [22]. The BC mechanism and µT =0.5 (implying a fair compromise of opinions). has many extensions, such as vectorial opinions [23] and Fixed agent pool.— For finite N and if 0<A, µA<1 coupling with a constant external field [24]. the system always reaches consensus. Let i be the agent Let us now consider the case in which agents with dif- with the largest opinion xi, so that a discussion with ferent opinions have the task to form a consensual prod- any other agent may only lower the value of xi. Con- uct. For this we can couple BC with a medium consid- sider now the event in which agent i alone modifies the ered as the common product on which the agents should medium for a number of consecutive steps. If A+A
0.02 mainstream with opinion x0 and two extremists with µA opinions x− and x+. Second, N is large enough such 0.25 0.01 0.35 that the change in the opinions of the groups and corre- 0.45 spondingly the change in the probability distribution ρ 0.45
A 0
v 0 of the opinion of the medium is slow compared to a single edit. In this case the stationary master equation for ρA -0.01 can be written as
-0.02 P 0.1 0.3 0.5 0.7 0.9 0 = − ρA(A)Θ[dA,xi ] + ρA(Ai)Θ[dAi,xi ] ni, (3) i (a) x0 1 10 0.2 N ( 103 ) where ni is the relative size of group i∈{0, −, +}, × d =|A−x |− , and A =(A−µ x )/(1−µ ) is the 1 A,xi i A i A i A III 7 3 opinion of the medium from where it would jump to A 10 A 0.5 A 0.1 σ after the interaction with xi. For all values of A with µ II |x0−A|< , the mainstream group is moved towards A A 3 with probability n0ρA. The resulting velocity of the opin- I ion of the mainstream group is then 0 0 0 0.05 0.15 0.25 0.1 0.3 0.5 R x0+A ² 4 µ v0(x0) = V (µ )n0 ρ (A)(A − x0)dA, (4) (b) A τ ( 10 ) (c) A A x0−A A × where V (µA) is a positive constant. In Fig. 3(a) we show FIG. 3. (Color online) (a) Velocity v0 of the mainstream group at =0.075 as a function of its opinion x , for equal how v0 depends on µA. If n−=n+ the opinion of the A 0 extremist group sizes (solid lines) and for 25% more extremists mainstream group is stable at x0=1/2 for low values of at x (dashed line). (b) Phase diagram ( , µ ) with shading µ , due to the negative slope of v . Its point of stabil- − A A A 0 indicating the relaxation time τ. Points give parameter val- ity bifurcates as µA increases and the mainstream group ues for the examples in Fig. 2 and lines indicate boundaries will drift towards one of the extremes. As soon as the between different regimes, denoted by Roman numerals. (c) opinions of the extremists get within the tolerance of Order parameter σA for the transition between I and II at the medium some of them will move towards the main- A=0.075 (dashed line in (b)). stream. When enough extremists have converted to the mainstream group the velocity of the mainstream gets re- verted (see dashed line in Fig. 3(a)) and the mainstream reflects a bimodal distribution ρA corresponding to the group will head towards the other extreme. According broken symmetry. to our calculations, more than 25% population difference Regime III is characterized by converging extremist between extremists is needed for the reversal. This en- groups. As A and µA increase, the jump of the medium sures that consensus is reached after a few oscillations, is big enough so that in one step the extremists get within which makes the relaxation time independent of N. In its tolerance interval and start drifting inwards. Thus the Fig. 3(b) the numerical boundary between regimes I and step size must be ∆=µA(1/2−A)=1/2−2A, where 1/2 II is drawn at the marginal stability of the mainstream is the distance between the mainstream and the extrem- group. We note here that for some values of T the shape ist groups. The boundary µA=1−A/(1/2−A) is shown of the boundary between regime I and II is more compli- in Fig. 3(b) separating regime III from the rest. cated and may even include islands. Agent replacement.—In real systems the agent pool is After the last interaction with the extremists the opin- often not fixed in time as people come and go. We intro- ion of the medium and the mainstream group will re- duce the agent renewal rate pnew as the probability for an main in the vicinity of one of the extremes. In the ther- agent to be replaced by a new one with random opinion modynamic limit this leads to a symmetry breaking in before the interactions. In this section we fix µA=0.1 to the stationary state of regime II. Conversely, in regime reduce the number of parameters, focusing solely on N, I the relaxation time grows exponentially with the num- A and pnew. Intuitively it is then clear that for A<1/2 ber of agents, as a sequence of low probability (∝1/N) and pnew>0 the dynamics never converges to a station- events are needed for convergence. Thus for N→∞ we ary state, as opposed to the case of a fixed agent pool. find a stationary state where the opinion of the main- This is because for any A value there is a finite probabil- stream group is at 1/2. Small shifts are possible due to ity that a new agent enters with an opinion outside the differences in extremist populations, but since their ra- tolerance level of the medium, after which this new agent tio√ determines the opinion of A disturbances vanish as may change the value of A. 1/ N. Then it is safe to define the order parameter In the insets of Fig. 1 we display some examples of the σA as the standard deviation of the opinion of the main- time evolution of S for Npnew=4 and A=0.47, 0.46, 0.44. stream group. When N→∞ this tends to 0 for case I and As in [10] we can distinguish three qualitatively different increases for case II, as depicted in Fig. 3(c). The latter regimes as A decreases: (a) Single conflict, where S is 4
0.55 ists. The rate equations for the medium and extremist * ǫ A movement can be established and solved analytically to 0.5 N=1280 give the mean relaxation time as N=360 N=80 2 2 τ = cN [2e + e0(n − 1)]n − ee0(n − 1)(2 + n) , (5) 0.45 0.4
A (a) ǫ¯ ∗ ∗ A where e= −A, e0= −1/2, c is a constant depending ǫ 0.3 A A
0.4 r ǫ¯A on µA, and n denotes the integer part of e/e0 counting the ǫ¯A 0.2 number of steps the medium can make in one direction. 0.35 (b) (c) 0.1 The number of new agents per unit time is Npnew, so
0 we expect the transition point ¯A to be at 1=Npnewτ(¯A), 0.3 0.35 0.4 0.45 0.5 0.55 ǫ A 0.3 shown in Fig. 4 as a continuous line. Such result is in 0.1 1Np 10 100 considerable agreement with the numerical computation new of ¯A, with one single fit parameter. It also holds for other values of µA, deviations appearing only for µA FIG. 4. (Color online) Phase diagram (Npnew, A) in the case 0.01. We expect that in the pnew→0 limit ¯A≡0, as there of agent replacement. The transition point ¯A shows agree- is no state with permanent conflict in the fixed agent ment between numerical (points) and analytical (line) results. ∗ pool case. Furthermore, for pnew→∞ we have ¯ = , as Letters denote regimes of conflict and correspond to labels in A A this is the point above which no position of the medium Fig. 1. Inset: Conflict rate r as a function of for varying A allows for a conflict. The curves for different system sizes N at pnew=0.01. fall upon each other if the number of new agents per unit time is used as control parameter irrespective of the total number of agents. This means that consensus is as dominated by an initial increase and a prolonged peace vulnerable to many people as it is to few. signaled by long plateaus. (b) A series of small plateaus Overall, agent replacement for the non-trivial case of consensus separated by conflicts. (c) A continuous <∗ gives a transition between regime (a) represent- increase of S indicating a permanent state of war. We A A ing practically peace (dS/dt 1) and regime (c) repre- define a conflict as the period between two plateaus of senting continuous conflict (dS/dt ' 1). The transition S and denote the number of conflicts per unit time by can happen if many new agents enter the system either r. The two extreme regimes are both characterized by by increasing p or N. An analogue of this transition low values of r: in the peaceful regime where is large new A is indeed observed in Wikipedia, namely that a peace- there are few conflicts, while for small there is only A ful article can suddenly become controversial when more one never-ending conflict. These regimes are separated people get involved in its editing [6, 10]. by a region full of small conflicts. Summary.—A general question addressed by our ex- In the inset of Fig. 4 we show the variation of r with tended opinion formation model is the competition and A. As N increases, regimes (a) and (c) are indeed sep- feedback loop between direct agent-agent interactions arated by a thinning transition region (b) of many con- and the indirect interaction of agents with a ‘mean field’ flicts. A critical tolerance value between consensus and collectively created. We find that convergence is al- controversy regimes is then identified by a maximum in ways reached when the indirect interaction mechanism is the conflict density r and is denoted by ¯A. present, even in situations in which the agent-agent inter- We note that both ¯A and r(¯A) increase for larger action alone does not lead to it. We have also described N, but true divergence associated with critical behavior different dynamical regimes of approaching convergence, near a phase transition cannot be observed here due to finding in particular that the transition between regimes the condition r≤1. The transition point ¯A depends both I and II for a critical value of the convergence parame- on N and pnew, and its corresponding phase diagram is ter µA involves a symmetry-breaking mechanism for the shown in Fig. 4. The transition between peace and con- collectively created opinion A. In the case of agent re- flict can be derived from the matching of two timescales: placement we find a transition from a relatively peace- (i) the relaxation time of the system without agent re- ful situation to a perpetual state of conflict when the newal, and (ii) the time scale of agent renewal. If the rate of replacement is increased above a threshold. Such latter is too small no relaxation takes place and we have finding is in agreement with different conflict scenarios an ever-present conflict. Thus at the transition point observed in Wikipedia. This first step in comparing an both timescales should be equal. opinion model with real data calls to extend the model The relaxation time τ for a fixed agent pool can be by including networked interactions between agents, indi- calculated in regime III if A>0.25 (for details see Sup- vidual tolerances, heterogeneously-distributed times be- plementary Material). In this case A can make only few tween successive edits, and external events to enable a jumps up and down. Knowing the distribution of the more quantitative comparison between the model and opinion of the agents the task is to eliminate the extrem- empirical observations. 5
JT, TY, KK, and JK acknowledge support from EU’s [22] E. Ben-Naim, P. L. Krapivsky, and S. Redner, Physica FP7 FET Open STREP Project ICTeCollective No. D 183, 190 (2003). 238597, and JK also support by FiDiPro program of [23] S. Fortunato, V. Latora, A. Pluchino, and A. Rapisarda, TEKES. MSM acknowledges support from project FISI- Int. J. Mod. Phys. C 16, 1535 (2005). [24] J. C. Gonz´alez-Avella, M. G. Cosenza, V. M. Egu´ıluz, COS (FIS2007-60327) of MINECO. GI acknowledges and M. San Miguel, New J. Phys. 12, 013010 (2010). support from an STSM within COST Action MP0801 [25] G. Veser, F. Mertens, A. S. Mikhailov, and R. Imbihl, and is grateful to IFISC for hospitality. Phys. Rev. Lett. 71, 935 (1993). [26] H. Nishimori and N. Ouchi, Phys. Rev. Lett. 71, 197 (1993). [27] I. Z. Kiss, Y. Zhai, and J. L. Hudson, Phys. Rev. Lett. 88, 238301 (2002). [1] C. Castellano, S. Fortunato, and V. Loreto, Rev. Mod. Phys. 81, 591 (2009). [2] V. Zlati´c,M. Boˇziˇcevi´c,H. Stefanˇci´c,ˇ and M. 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Lett. 105, 158701 (2010). nally, let use define e≡A−A and e0≡A−1/2. ∗ [7] T. Yasseri, R. Sumi, and J. Kert´esz,PLoS ONE 7, We divide A∈[0.25, A] into small intervals within e30091 (2012). which the medium can only make a given number of [8] R. Sumi, T. Yasseri, A. Rung, A. Kornai, and J. Kert´esz, ∗ steps up and down. The first is A>A≥1/2, where the in Proceedings of the ACM WebSci’11, Koblenz, Ger- opinion of the medium is either stable or can only make many (2011) pp. 1–3. a jump up and down. In this case there is an inter- [9] R. Sumi, T. Yasseri, A. Rung, A. Kornai, and J. Kert´esz, | − | − in Privacy, Security, Risk and Trust (PASSAT), 2011 val A 1/2 <A 1/2 for the opinion of the medium at IEEE Third International Conference on and 2011 IEEE which it covers the whole opinion range. Then consensus Third International Conference on Social Computing is immediately reached and the contribution to τ is zero. (SocialCom) (2011) pp. 724–727. Thus the probability to have an oscillating state is [10] T. Yasseri, R. Sumi, A. Rung, A. Kornai, and J. Kert´esz, e PloS ONE 7, e38869 (2012). posc = = e/e0. (6) ∗ − 1/2 [11] B.-Q. Vuong, E.-P. Lim, A. Sun, M.-T. Le, H. W. Lauw, A and K. Chang, in Proceedings of the international con- Consensus is hindered by extremists. The number ference on Web search and web data mining, WSDM ’08 of extremists in this case is on average proportional to (ACM, New York, 2008) pp. 171–182. e(2 − µ )N and is distributed between the two extrem- [12] U. Brandes and J. Lerner, Information Visualization 7, A ist groups. Mainstream agents who are always within 34 (2008). [13] J. A. Holyst,K. Kacperski, and F. Schweitzer, in Annual the tolerance of the medium can be disregarded since Reviews of Computational Physics IX (World Scientific, they never change it. If the opinion of the medium is Singapore, 2001) Chap. 5, pp. 253–273. on one side of the middle and an extremist there is cho- [14] H. Xia, H. Wang, and Z. Xuan, International Journal of sen for editing, the agent will change its opinion towards Knowledge and Systems Science 2, 72 (2011). the medium since it lies within the tolerance level, thus [15] G. Szab´oand G. F´ath,Phys. Rep. 446, 97 (2007). leaving the group. Therefore the extremist group loses [16] A. T. Bernardes, D. Stauffer, and J. Kert´esz,Eur. Phys. one member if it is chosen repeatedly without choosing J. B 25, 123 (2002). [17] S. Fortunato and C. Castellano, Phys. Rev. Lett. 99, agents from the other extremist group. 138701 (2007). The problem can be traced back to the following ∗ [18] G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, urn model. Let us have M=eN/A balls and choose Adv. Complex Syst. 3, 87 (2000). m1∈[0,M] randomly from a distribution√ with mean [19] R. Hegselmann and U. Krause, J. Artif. Soc. Soc. Simul. around M/2 and standard deviation σ= M. Then put 5, 2 (2002). m1 balls into one urn and the rest into another. An urn [20] E. 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0.1 where c is a constant. N=1280 n=5 Let us now consider the case 0.5−e < <0.5, where N= 640 0 A 0.01 n=4 any initial position of A may lead to oscillations. If the N= 320 n=3 N= 160 initial position of A is such that |1/2−A|>1/2−A, then 10−3 the previous urn model applies giving τa=ceNe0. On the other hand, if |1/2 − A|<1/2 − A the medium needs N
τ/ two jumps from covering one extreme to the other, yet 10−4 the jump probabilities are not equal. Jumping from the middle is much less probable since there are fewer agents −5 10 n=0 n=1 n=2 left out. So the jump from the middle will be a bottleneck and determine the relaxation time τb, calculated from the 10−6 previous urn model with asymmetric jump probabilities 0.01 0.1 as τb=c(2e − e0)N. e This double oscillation happens with probability pro- portional to p =e−e0. When completed, the medium can FIG. 5. Scaled relaxation time τ/N as a function of e=∗ − , b A A only make one jump back and forth and we return to the for several values of the integer n and system size N. Points are numerical results and lines analytical calculations. case of τa. Thus the total relaxation time is
τ1 = τa + pbτbN = c(e − e0)(2e − e0)N. (8)
It is straightforward to generalize this reasoning, which We have to take into account that we spend some time for ne0<A<(n + 1)e0 and integer n gives in choosing already satisfied agents. The rate at which n we choose extremists varies from e/∗ to 1/N, where the X A − − − − latter clearly dominates and gives another N dependence τn = cN(e ne0)[(n + 1)e ne0] + c [ie (i 1)e0]e0 to the relaxation time. Since τ is defined in units of N i=1 = cN [2e2 + e2(n − 1)]n − ee (n − 1)(2 + n) . (9) edits, the convergence time is given by 0 0 Fig. 5 shows good agreement between numerical and an- alytical calculations of the relaxation time in the case of 2 τ0 = poscτosc = ce N, (7) a fixed agent pool.