A Supplementary Material

Opinions, Conflicts and Consensus: Modeling Social Dynamics in a Collaborative Environment

Abstract

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 different 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 symmetry 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:
89.75.Fb, 89.65.-s, 05.65.+b

Society represents a paradigmatic example of complex systems, where interactions between many constituents and feedback and other non-linear mechanisms result in emergent collective phenomena. The recent availability of large data sets due to information-communication technology (ICT) has enabled us to apply more quantitative methods in social sciences than before. As a matter of fact, physicists play an increasing role in the study of social phenomena by applying physics concepts and tools to investigate them Castellano et al. (2009).

Social interactions are heavily influenced by the opinions of the members of the society. This is especially true when complex tasks are to be solved by cooperation, as practiced throughout the history of mankind. In this respect new technologies open up unprecedented opportunities: by using the Internet and related facilities even remote members of large groups can work on the same task and achieve a higher level of synergy. Examples include open software projects or large collaborative scientific endeavors like high-energy physics experiments. However, it is unavoidable that in such cases differences in attitudes, approaches, and emphases (in short, opinions) occur. Then questions arise: How can a task be solved in a collaborative environment of agents having diverse opinions? How do conflicts emerge and get resolved? The understanding of these mechanisms may lead to an increase in efficiency of value production in cooperative environments. A prime example of the latter is Wikipedia, a free, web-based encyclopedia project where volunteering individuals collaboratively write and edit articles on their desired topics. Wikipedia is particularly well-suited also as a target for a wide range of studies, since all changes and discussions are recorded and made publicly available Zlatić et al. (2006); Capocci et al. (2006); Wilkinson and Huberman (2007); Kittur et al. (2007); Ratkiewicz et al. (2010); Yasseri et al. (2012a).

Recently, the controversiality and dynamical evolution of Wikipedia articles have been studied in detail and typical patterns of different categories of the so-called edit wars have been identified Sumi et al. (2011a, b); Yasseri et al. (2012b); Vuong et al. (2008); Brandes and Lerner (2008). Take for example Fig. 1, where the evolution of a controversiality measure based on mutual reverts and maturity of editors is shown for three different regimes of conflict.

Figure 1: (Color online) Empirical controversiality measure  Yasseri et al. (2012b) as a function of the number of edits for three different conflict scenarios in Wikipedia, corresponding to: (a) single conflict, (b) plateaus of consensus, and (c) uninterrupted controversiality. Titles are the article topics. Inset: Theoretical conflict measure of Eq. (2), allowing for a qualitative comparison of the model with the empirical results.

Our aim in this Letter is to model controversiality and conflict resolution in a collaborative environment and, where possible, to qualitatively compare our results with different scenarios observed in the Wikipedia. One of the most developed areas of quantitative modeling in social phenomena is opinion dynamics Castellano et al. (2009); Hołyst et al. (2001); Xia et al. (2011). These models have much in common with those of statistical physics, yet interactions are socially motivated. The problem with these models is, similarly to evolutionary game theoretic ones Szabó and Fáth (2007), that results are usually evaluated by qualitative subjective judgment instead of comparison with empirical data, one exception being the study of elections Bernardes et al. (2002); Fortunato and Castellano (2007). The well-documented Wikipedia edit wars can also be of use in this respect.

Our model is based on the bounded confidence (BC) mechanism Deffuant et al. (2000); Hegselmann and Krause (2002), describing a generic case when convergence occurs only upon opinion difference being smaller than a given threshold. Here agents are characterized by continuous opinion variables and their interactions are pairwise. The agents’ mutual influence is controlled by the convergence parameter , only if their opinions differ less than a given tolerance , i.e. for we update as follows,

(1)

The dynamics set by Eq. (1) has been studied extensively in the literature Castellano et al. (2009), initially by using the mean-field approach of two-body inelastic collisions in granular gases Ben-Naim and Krapivsky (2000); Baldassarri et al. (2002). It leads to a frozen steady state which is characterized by disjoint opinion groups. This is caused by the instability in the initial opinion distribution near the boundaries. Also, increases as in a series of bifurcations Ben-Naim et al. (2003). The BC mechanism has many extensions, such as vectorial opinions Fortunato et al. (2005) and coupling with a constant external field González-Avella et al. (2010).

Let us now consider the case in which agents with different opinions have the task to form a consensual product. For this we can couple BC with a medium considered as the common product on which the agents should work collectively. In this case the medium has also a convergence parameter and tolerance , such that the opinion represented by the medium can be modified by agents being dissatisfied with it. If we update . Conversely, agents tolerating the current state of the common product () adapt their view towards the medium, . Thus agents can interact directly with each other and indirectly through the medium, and a complex dynamics governed by competition of these local (direct) and global (indirect) interactions emerges. Such an interplay is also present in many other systems, ranging from surface chemical reactions Veser et al. (1993) and sand dunes Nishimori and Ouchi (1993) to arrays of chaotic electrochemical cells Kiss et al. (2002).

All opinions are first initialized uniformly at random and the original BC algorithm is run until opinion groups are formed. Then pairs of agent-agent and agent-medium interactions are performed in each time step , with agents being selected uniformly at random. If all agents fall within the tolerance level of the medium the dynamics is frozen and we call such stable state consensus. The cumulative amount of conflict or controversy in the system is defined as the total sum of changes in the medium,

(2)

This quantity is analogous to the empirical controversiality measure as it sums up the actions of dissatisfied agents Yasseri et al. (2012b).

In what follows we will analyze two versions of this model: (i) with the agent pool fixed, and (ii) with agents being replaced by new ones at a certain rate. For the sake of simplicity we fix (leading in general to one large mainstream and two small extremist groups) and (implying a fair compromise of opinions).

Fixed agent pool.— For finite and if the system always reaches consensus. Let be the agent with the largest opinion , so that a discussion with any other agent may only lower the value of . Consider now the event in which agent alone modifies the medium for a number of consecutive steps. If the medium is moved towards the opinion of the agent by a finite amount . Finally, after a finite number of steps when falls within the tolerance level of the medium, will be lowered by a finite amount larger than . In this way we have devised an event of finite probability where is decreased, which in turn leads to a shrunken interval for the available opinion pool. Thus the convergence to consensus is secured and the relaxation time can be defined, which, however, may be astronomical for large .

Figure 2: (Color online) Example evolutions of the agents’ opinions (light gray/blue) and the value of the medium (dark gray/green) for three qualitatively different regimes, corresponding to in I, in II, and in III.

If the tolerance is large, however, consensus may be quickly reached in a finite number of unidirectional steps. By decreasing a limiting case is reached, where the opinion of the medium starts to oscillate between the points and . The change in the opinion of the medium should be the distance between such points, so for a given the limit of oscillatory behavior is . From now on we are interested in the non-trivial case .

We observed three different scenarios (see Fig. 2) for the dynamics depending on the values of and : In case I the opinion of the mainstream group fluctuates for a long time around a stable value. This state is characterized by an astronomical relaxation time. In case II the opinion of the mainstream group oscillates between the vicinities of the extremists, with independent of . In case III the extremists converge as groups towards the mainstream opinion.

The transition between regimes I and II can be described with stability analysis. We use the following assumptions: First, there are three opinion groups, one mainstream with opinion and two extremists with opinions and . Second, is large enough such that the change in the opinions of the groups and correspondingly the change in the probability distribution of the opinion of the medium is slow compared to a single edit. In this case the stationary master equation for can be written as

(3)

where is the relative size of group , , and is the opinion of the medium from where it would jump to after the interaction with . For all values of with , the mainstream group is moved towards with probability . The resulting velocity of the opinion of the mainstream group is then

(4)

where is a positive constant. In Fig. 3(a) we show how depends on . If the opinion of the mainstream group is stable at for low values of , due to the negative slope of . Its point of stability bifurcates as increases and the mainstream group will drift towards one of the extremes. As soon as the opinions of the extremists get within the tolerance of the medium some of them will move towards the mainstream. When enough extremists have converted to the mainstream group the velocity of the mainstream gets reverted (see dashed line in Fig. 3(a)) and the mainstream group will head towards the other extreme. According to our calculations, more than 25% population difference between extremists is needed for the reversal. This ensures that consensus is reached after a few oscillations, which makes the relaxation time independent of . In Fig. 3(b) the numerical boundary between regimes I and II is drawn at the marginal stability of the mainstream group. We note here that for some values of the shape of the boundary between regime I and II is more complicated and may even include islands.

Figure 3: (Color online) (a) Velocity of the mainstream group at as a function of its opinion , for equal extremist group sizes (solid lines) and for 25% more extremists at (dashed line). (b) Phase diagram with shading indicating the relaxation time . Points give parameter values for the examples in Fig. 2 and lines indicate boundaries between different regimes, denoted by Roman numerals. (c) Order parameter for the transition between I and II at (dashed line in (b)).

After the last interaction with the extremists the opinion of the medium and the mainstream group will remain in the vicinity of one of the extremes. In the thermodynamic limit this leads to a symmetry breaking in the stationary state of regime II. Conversely, in regime I the relaxation time grows exponentially with the number of agents, as a sequence of low probability () events are needed for convergence. Thus for we find a stationary state where the opinion of the mainstream group is at 1/2. Small shifts are possible due to differences in extremist populations, but since their ratio determines the opinion of disturbances vanish as . Then it is safe to define the order parameter as the standard deviation of the opinion of the mainstream group. When this tends to 0 for case I and increases for case II, as depicted in Fig. 3(c). The latter reflects a bimodal distribution corresponding to the broken symmetry.

Regime III is characterized by converging extremist groups. As and increase, the jump of the medium is big enough so that in one step the extremists get within its tolerance interval and start drifting inwards. Thus the step size must be , where is the distance between the mainstream and the extremist groups. The boundary is shown in Fig. 3(b) separating regime III from the rest.

Agent replacement.—In real systems the agent pool is often not fixed in time as people come and go. We introduce the agent renewal rate as the probability for an agent to be replaced by a new one with random opinion before the interactions. In this section we fix to reduce the number of parameters, focusing solely on , and . Intuitively it is then clear that for and the dynamics never converges to a stationary state, as opposed to the case of a fixed agent pool. This is because for any value there is a finite probability that a new agent enters with an opinion outside the tolerance level of the medium, after which this new agent may change the value of .

In the insets of Fig. 1 we display some examples of the time evolution of for and . As in Yasseri et al. (2012b) we can distinguish three qualitatively different regimes as decreases: (a) Single conflict, where is dominated by an initial increase and a prolonged peace signaled by long plateaus. (b) A series of small plateaus of consensus separated by conflicts. (c) A continuous increase of indicating a permanent state of war. We define a conflict as the period between two plateaus of and denote the number of conflicts per unit time by . The two extreme regimes are both characterized by low values of : in the peaceful regime where is large there are few conflicts, while for small there is only one never-ending conflict. These regimes are separated by a region full of small conflicts.

In the inset of Fig. 4 we show the variation of with . As increases, regimes (a) and (c) are indeed separated by a thinning transition region (b) of many conflicts. A critical tolerance value between consensus and controversy regimes is then identified by a maximum in the conflict density and is denoted by .

Figure 4: (Color online) Phase diagram in the case of agent replacement. The transition point shows agreement between numerical (points) and analytical (line) results. Letters denote regimes of conflict and correspond to labels in Fig. 1. Inset: Conflict rate as a function of for varying at .

We note that both and increase for larger , but true divergence associated with critical behavior near a phase transition cannot be observed here due to the condition . The transition point depends both on and , and its corresponding phase diagram is shown in Fig. 4. The transition between peace and conflict can be derived from the matching of two timescales: (i) the relaxation time of the system without agent renewal, and (ii) the time scale of agent renewal. If the latter is too small no relaxation takes place and we have an ever-present conflict. Thus at the transition point both timescales should be equal.

The relaxation time for a fixed agent pool can be calculated in regime III if (for details see Supplementary Material). In this case can make only few jumps up and down. Knowing the distribution of the opinion of the agents the task is to eliminate the extremists. The rate equations for the medium and extremist movement can be established and solved analytically to give the mean relaxation time as

(5)

where , , is a constant depending on , and denotes the integer part of counting the number of steps the medium can make in one direction.

The number of new agents per unit time is , so we expect the transition point to be at , shown in Fig. 4 as a continuous line. Such result is in considerable agreement with the numerical computation of , with one single fit parameter. It also holds for other values of , deviations appearing only for . We expect that in the limit , as there is no state with permanent conflict in the fixed agent pool case. Furthermore, for we have , as this is the point above which no position of the medium allows for a conflict. The curves for different system sizes 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 vulnerable to many people as it is to few.

Overall, agent replacement for the non-trivial case gives a transition between regime (a) representing practically peace () and regime (c) representing continuous conflict (). The transition can happen if many new agents enter the system either by increasing or . An analogue of this transition is indeed observed in Wikipedia, namely that a peaceful article can suddenly become controversial when more people get involved in its editing Ratkiewicz et al. (2010); Yasseri et al. (2012b).

Summary.—A general question addressed by our extended opinion formation model is the competition and feedback loop between direct agent-agent interactions and the indirect interaction of agents with a ‘mean field’ collectively created. We find that convergence is always reached when the indirect interaction mechanism is present, even in situations in which the agent-agent interaction alone does not lead to it. We have also described different dynamical regimes of approaching convergence, finding in particular that the transition between regimes I and II for a critical value of the convergence parameter involves a symmetry-breaking mechanism for the collectively created opinion . In the case of agent replacement we find a transition from a relatively peaceful situation to a perpetual state of conflict when the rate of replacement is increased above a threshold. Such finding is in agreement with different conflict scenarios observed in Wikipedia. This first step in comparing an opinion model with real data calls to extend the model by including networked interactions between agents, individual tolerances, heterogeneously-distributed times between successive edits, and external events to enable a more quantitative comparison between the model and empirical observations.

JT, TY, KK, and JK acknowledge support from EU’s FP7 FET Open STREP Project ICTeCollective No. 238597, and JK also support by FiDiPro program of TEKES. MSM acknowledges support from project FISICOS (FIS2007-60327) of MINECO. GI acknowledges support from an STSM within COST Action MP0801 and is grateful to IFISC for hospitality.

Appendix A Supplementary Material

Derivation of the relaxation time.—We calculate the relaxation time for a fixed agent pool if . We use a mean-field approach and assume that in the ensemble average the distribution of agents is homogeneous along , since the relaxation time is also defined as an ensemble average. We consider the initial opinion of the medium to be random and uniformly distributed, and without loss of generality starting from . Finally, let use define and .

We divide into small intervals within which the medium can only make a given number of steps up and down. The first is , where the opinion of the medium is either stable or can only make a jump up and down. In this case there is an interval for the opinion of the medium at which it covers the whole opinion range. Then consensus is immediately reached and the contribution to is zero. Thus the probability to have an oscillating state is

(6)

Consensus is hindered by extremists. The number of extremists in this case is on average proportional to and is distributed between the two extremist groups. Mainstream agents who are always within the tolerance of the medium can be disregarded since they never change it. If the opinion of the medium is on one side of the middle and an extremist there is chosen for editing, the agent will change its opinion towards the medium since it lies within the tolerance level, thus leaving the group. Therefore the extremist group loses one member if it is chosen repeatedly without choosing agents from the other extremist group.

The problem can be traced back to the following urn model. Let us have balls and choose randomly from a distribution with mean around and standard deviation . Then put balls into one urn and the rest into another. An urn is chosen with probability . If we chose the same urn as in the previous turn, we remove a ball from this urn. Then, the average time one needs to empty an urn is for large .

We have to take into account that we spend some time in choosing already satisfied agents. The rate at which we choose extremists varies from to , where the latter clearly dominates and gives another dependence to the relaxation time. Since is defined in units of edits, the convergence time is given by

(7)

where is a constant.

Figure 5: Scaled relaxation time as a function of , for several values of the integer and system size . Points are numerical results and lines analytical calculations.

Let us now consider the case , where any initial position of may lead to oscillations. If the initial position of is such that , then the previous urn model applies giving . On the other hand, if the medium needs two jumps from covering one extreme to the other, yet the jump probabilities are not equal. Jumping from the middle is much less probable since there are fewer agents left out. So the jump from the middle will be a bottleneck and determine the relaxation time , calculated from the previous urn model with asymmetric jump probabilities as .

This double oscillation happens with probability proportional to . When completed, the medium can only make one jump back and forth and we return to the case of . Thus the total relaxation time is

(8)

It is straightforward to generalize this reasoning, which for and integer gives

(9)
(10)

Fig. 5 shows good agreement between numerical and analytical calculations of the relaxation time in the case of a fixed agent pool.

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