- Regular article
- Open Access
Modeling social dynamics in a collaborative environment
© Iñiguez et al.; licensee Springer 2014
- Received: 17 March 2014
- Accepted: 18 July 2014
- Published: 24 September 2014
Wikipedia is a prime example of today’s value production in a collaborative environment. Using this example, we model the emergence, persistence and resolution of severe conflicts during collaboration by coupling opinion formation with article editing in a bounded confidence dynamics. The complex social behavior involved in editing articles is implemented as a minimal model with two basic elements; (i) individuals interact directly to share information and convince each other, and (ii) they edit a common medium to establish their own opinions. Opinions of the editors and that represented by the article are characterised by a scalar variable. When the pool of editors is fixed, three regimes can be distinguished: (a) a stable mainstream article opinion is continuously contested by editors with extremist views and there is slow convergence towards consensus, (b) the article oscillates between editors with extremist views, reaching consensus relatively fast at one of the extremes, and (c) the extremist editors are converted very fast to the mainstream opinion and the article has an erratic evolution. When editors are renewed with a certain rate, a dynamical transition occurs between different kinds of edit wars, which qualitatively reflect the dynamics of conflicts as observed in real Wikipedia data.
- social dynamics
- mathematical modeling
- bounded confidence
- opinion dynamics
- social conflict
Cooperative societies are ubiquitous in nature , yet the cooperation or the mutual assistance between members of a society is also likely to generate conflicts . Potential for conflicts is commonplace even in insect species  and so is conflict management through policing and negotiation in groups of primates , . In human societies cooperation goes further not only in its scale and range, but also in the available mechanisms to promote conflict resolution , . Collaborative and conflict-prone human endeavors are numerous, including public policy-making in globalized societies , , open-source software development , teamwork in operating rooms , and even long-term partnerships . Moreover, information communication technology opens up entirely new ways of collaboration. With such a diversity in system size and social interactions between individuals, it seems appropriate to study this phenomenon of social dynamics in the framework of statistical physics , , an approach benefiting greatly from the availability of large scale data on social interactions , .
As a relevant example of conflicts in social cooperation we select Wikipedia (WP), an intriguing example of value production in an online collaborative environment . WP is a free web-based encyclopedia project, where volunteering individuals collaboratively write and edit articles about any topic they choose. The availability of virtually all data concerning the visiting and editing of article pages provides a solid empirical basis for investigating topics such as online content popularity ,  and the role of opinion-formation processes in a peer-production environment .
The editing process in WP is usually peaceful and constructive, but some controversial topics might give rise to extreme cases of disagreement over the contents of the articles, with the editors repeatedly overriding each other’s contributions and making it harder to reach consensus. These ‘edit wars’ (as they are commonly called) result from complex online social dynamics, and recent studies  have shown how to detect and classify them, as well as how they are related to burstiness and what are their circadian patterns in editing activity .
Although collaborative behavior might appear without direct interactions between individuals, communication tends to have a positive effect on cooperation and trust . Indeed, more immediate forms of communication (voice as opposed to text, for example) have been seen to increase the level of cooperation in online environments . In WP, direct communication is implemented via ‘talk pages’, open forums where editors may discuss improvements over the contents of articles and exchange their related opinions . Discussions among editors are not mandatory , but there is a significant correlation between talk page length and the likelihood of an edit war, indicating that many debates happen in articles and talk pages, simultaneously , .
agent-agent dynamics: Individuals share their views and opinions about changes in the article using an open channel accessible to all editors (the talk page or some other means of communication), thus effectively participating in an opinion-formation process through information sharing.
agent-medium dynamics: Individuals edit the article if it does not properly summarize their views on the subject, thus controlling the temporal evolution of the article and coupling it to the opinion-formation mechanism.
We describe the opinion-formation process taking place in the talk page by means of the well-known bounded confidence mechanism first introduced by Deffuant et al., where real discussions take place only if the opinions of the people involved are sufficiently close to each other. Conversely, we model article editing by an ‘inverse’ bounded confidence process, where individuals change the current state of the article only if it differs too much from their own opinions. Particularly, we focus our attention on how the coupling between agent-agent and agent-medium interactions determines the nature of the temporal evolution of an article. This we consider as a further step towards the theoretical characterization of conflict in social cooperative systems such as WP .
The text is organized as follows: In Section 2 we introduce and discuss the model in detail. In Section 3 we describe our results separately for the cases of a fixed editor pool and a pool with editor renewal, and finally make a comparison with empirical observations on WP conflicts. In Section 4 we present concluding remarks.
Let us first assume that there is a system of N agents as potential editors for a collective medium. The state of an individual i at time t is defined by its scalar, continuous opinion , while the medium is characterized by a certain value in that same interval. The variable x represents the view and/or inclination of an agent concerning the topic described by the common medium, while A is the particular position actually represented by the medium.
Although it may seem too reductive to describe people’s perceptions by a scalar variable x, many topics can actually be projected to a one-dimensional struggle between two extreme, opposite options. In the Liancourt Rocks territorial dispute between South Korea and Japan , for example, the values represent the extreme position of favoring sovereignty of the islets for a particular country. Other topics are of course multifaceted, generating discussions that depend on the global affinity of individuals and multiple cultural factors . While this complexity could be tackled by the use of vectorial opinions , , our intention here is not to describe extremism as realistically as possible, but to study the rise of collaborative conflict even in the simplest case of binary extremism.
In the case of WP, the scalar variable A represents the opinion expressed by the written contents of an article, which carries the assumption that all agents perceive the medium in the same way. Real scenarios of public opinion might be more complex, given the tendency of individuals to attribute their own views to others and thus perceive false consensus , usually out of a social need to belong . Even so, we consider A to be a sensible description of a WP article, one that could initially be measured by human judgment in the form of expert opinions, or in an automated way by quantifying lexical features and the use of certain language constructs. We note, however, that the actual value of A is not the main concern of our study. Instead, we are interested in how opinion differences in collaborative groups may eventually lead to conflict, specifically when such opinion differences are perceived with respect to a common medium that all individuals modify collectively.
2.1 Agent-agent dynamics
The parameter is usually referred to as the confidence or tolerance for pairwise interactions, while is a convergence parameter. AAD is then a symmetric compromise between similarly-minded individuals: people with very different opinions simply do not pay attention to each other, but similar agents debate and converge their views by the relative amount .
The dynamics set by Eq. (1) has received a lot of attention in the past , starting from the mean-field description of two-body inelastic collisions in granular gases , . Its final, steady state is comprised by isolated opinion groups that arise due to the instability of the initial opinion distribution near the boundaries. Furthermore, increases as in a series of bifurcations . In the limit corresponding to a ‘stubborn’ society, the asymptotically final value of also depends on , . The bounded-confidence mechanism has been extended in many ways over the years, considering interactions between more than two agents , vectorial opinions , –, and coupling with a constant external field .
2.2 Agent-medium dynamics
meaning that individuals edit the medium when it differs too much from their opinions, but adopt the medium’s view when they already think similarly. Observe that the maximum meaningful value of is 1/2 (i.e. convergence to the average of opinions), while the maximum implies changing the medium (opinion) so that it completely reflects the agent’s (medium’s) point of view.
The previous rules comprise our model for the dynamics of conflicts in WP given a fixed agent pool, that is, without agents entering or leaving the editing process of the common medium. In a numerical implementation of the model, every time step t consists of N updates of AAD given by Eq. (1) and of AMD following Eqs. (2) and (3), so that time is effectively measured in number of edits and the broad inter-event time distribution between successive edits (observed in empirical studies ) does not have to be considered directly. Given a fixed agent pool, AAD favors opinion homogenization in intervals of length and can thus create several opinion groups for low tolerance, while AMD makes the medium value follow the majority group. Then, for a finite system there is a nonzero probability that any agent outside the majority group will be drawn by the medium to it, and the system will always reach consensus after a transient regime characterized by fluctuations in the medium value .
However, in real WP articles the pool of editors tends to change frequently. Some editors leave (due to boredom, lack of interest or fading media coverage on the subject, or are banned from editing by editors at a higher hierarchical level) and newly arrived agents do not necessarily share the opinions of their predecessors. Such feature of agent renewal during the process or writing an article may destroy consensus and lead to a steady state of alternating conflict and consensus phases, which we take into account by introducing thermal noise in the model. Along with any update of AAD/AMD, one editor might leave the pool with probability and be substituted by a new agent with opinion chosen uniformly at random from the interval . The quantity then formally acts as the inverse temperature of the system, signaling a dynamical phase transition between different regimes of conflict .
3.1 Fixed agent pool
In the presence of a fixed agent pool () with finite size N, the dynamics always reaches a peaceful state where all agents’ opinions lie within the tolerance of the medium. To show this, let us calculate the probability that an unsatisfied editor i changes the medium A for n consecutive times, such that afterwards and the agent can finally stop its stream of edits. For fixed and following Eq. (2), the final distance between editor and medium is , so the inequality is satisfied if . The probability of agent i not participating in AAD for n time steps is , while the probability of choosing it for AMD is . Then the total probability of this stream of edits is , which for large N and might be very small, but always finite. After editor i gets into the tolerance interval of the medium, it will not perform additional edits and will eventually adopt the majority opinion close to the medium value. Similar events with other unsatisfied agents will finally result in full consensus and put an end to the dynamics.
The existence of a finite relaxation time τ to consensus (for finite systems) contrasts drastically with the behavior of the bounded confidence mechanism alone, where consensus is never attained for . In other words, the presence of agent-medium interactions promotes an agreement of opinions that would otherwise not exist in the agent-agent dynamics, even though it may happen after a very long time (i.e. ). If we think of the medium as an additional agent with maximum tolerance (in the sense that it always interacts with the rest no matter what) and against which agents have a different tolerance (as opposed to ), this result is reminiscent of previous observations for a bounded-confidence model with heterogeneous thresholds , . There, even a small fraction of ‘open-minded’ agents with relatively high tolerance may bridge the opinion difference between the rest of the agents and lead to consensus.
In order to analyze all possible typical behaviors of the fixed agent pool dynamics, we perform extensive numerical simulations in systems of size ranging from to 104, letting the dynamics evolve for a maximum time . We then characterize the temporal evolution of medium and agent opinions as a function of , and , while keeping for all results in this section. Finally, since the value of has no major effect other than regulating the convergence time of AAD , , from now on we fix it to the maximum value in order to speed up the simulations as much as possible.
In regime II identified with intermediate values of (Figure 1(B) and (E)), the fixed pool dynamics produces quasi-periodic oscillations in the medium value A, which appear after an initial stage of opinion group formation and end up very quickly in total consensus. Quite surprisingly, the final consensual opinion is not (as in regime I) or that of the initial mainstream group, but some intermediate value closer to the extremist groups at the boundaries. This is indicative of a symmetry-breaking transition: as increases, a symmetric stationary state at is replaced by a final state close to 0 or 1. The oscillations in regime II can initially be understood as a struggle over medium dominance among the different opinion groups created by AAD. The AMD mechanism couples the medium dynamics with these groups, exchanging agents between them and thus modifying their positions, until the majority group wins over the rest and consensus is achieved. For small oscillations are more well-defined and last for longer, while extremist groups tend to diffuse towards the mainstream.
In regime III for large (Figure 1(C) and (F)), extremist agents directly converge to a mainstream group and an article at the center. Since in this case is so large, after any jump of the article extremist agents can enter its tolerance interval and start drifting inwards. The limiting condition for this behavior is , a line separating regime III from the rest. A smaller value produces a more erratic medium evolution, with occasional jumps up and down.
This symmetry-breaking mechanism may be understood analytically via a rate equation formalism . The resulting rate equation can be solved numerically assuming three editor groups: a mainstream at and two extremists with opinions close to the boundaries. The solution shows stability for the medium at the mainstream opinion when is small, but becomes unstable and oscillating for . The bifurcation transition is very sensitive on the position of the extremists, depending not only on but also on the initial conditions. This is in part the cause of the ‘noisy’ landscape of regime II in Figure 3(A), which appears regardless of the measure used to draw the phase diagram.
3.2 Agent renewal
In real systems the pool of collaborators is usually not fixed: Editors come and go and very often the number of editors fluctuates in time as external events may incite more or less attention. To keep things simple we only focus on systems with a fixed number of editors (N agents), but we allow agent replacement with probability . In our numerical simulations this happens prior to editing, and new agents have initially random opinions coming from a uniform distribution.
If there is always an opinion range outside the article tolerance region and new agents may enter such range and edit the article. From WP data we know that even peaceful articles have few disputes now and then so such a scenario is realistic. This is thus in contrast with the case of a fixed opinion pool, where consensus is theoretically always achieved.
then consensus is always reached after a finite number of steps, but if there are realizations that do not reach consensus ever. We show here an example: if the medium value is , then for an editor at will disagree with the article and change it by , so the new medium value would be . Afterwards an agent at can restore the article to its previous state and avoid consensus.
The lack of full consensus does not mean that the system is always in a conflict state. There are periods when A remains unchanged and these peaceful times are ended by conflicts in which the opinion of the article is continuously disputed between agent groups of different opinion. If the dispute is settled (i.e. all agents are satisfied by the article) a new peaceful period may start. The ratio of these peaceful and conflicting periods changes with the parameters and may be considered as a good candidate for an order parameter. Thus we define the order parameter P as the relative length of the peaceful periods.
The above transition is the result of a competition between two timescales. New agents arrive outside of the article’s tolerance interval with an ‘insertion’ timescale . In order to have the conflicts must be resolved before a new extremist agent arrives. Let us note that the convergence is very fast if there is only one extremist group. The problem is solved by displacing the article opinion by the required amount, which can be done in few (N independent) steps. This is what happens in the left panel of Figure 7. On the other hand, if we have two extremist groups on both sides of the opinion interval the relaxation is much slower and this is manifested in a much longer relaxation time. Thus, at the transition the insertion timescale is equal to the relaxation time of the case with two extremist groups, which is analogous to the fixed agent pool version of the model.
where , , n denotes the integer part of (which is actually the number of steps the medium can make in one direction) and c is a constant depending on .
The above approach works well for and (regime III of the fixed pool case). If the mainstream group gets dissatisfied either by the large jump ( is too large) or by the small tolerance ( too small) of the article, the reasoning presented in  breaks down and new effect comes into play, namely the relaxation times of the fixed pool system becomes be enormous (regime I).
This is why, starting from a conflict initial condition, the lower phase diagram in Figure 5 shows for . On the other hand, in order to initiate such a conflict one needs to have a situation where two extremists appear on both ends of the opinion space outside of the article tolerance interval. If we have a single extremist then the consensus will be reached within a few time steps, independently of N. So the probability that we create a long-lasting conflict state decreases proportionally to the agent replacement probability. This is why we see only peace on the finite-time realizations leading to the upper phase diagram in Figure 5. Had we waited long enough, a conflict would have been formed for and would have persisted further on.
Eternal peace (): The system reaches consensus very fast and remains there for ever.
Peace ( and above the phase transition line): The system is mainly in a consensual state and only interrupted by short disputes.
War ( and below the phase transition line): The system is mainly in a state of disagreement.
Perpetual war (): In this regime and in the thermodynamic limit no consensus may exist.
3.3 The case of Wikipedia
Although the model described and analyzed above is simplified enough to be extendable to various cases of collaboration, we specially intend to use it to explain some of the empirical observations regarding edit wars in WP.
Wikipedia is huge, not only in its number of articles and users but in the number of times articles are edited. In most cases articles are not written in a collaborative way, i.e., they have single authors or a few authors who have written and edited different parts of the article without any significant interaction . In contrast, a few cases show significant constructive and/or destructive interactions between editors. The latter situation has been named ‘edit war’ by the WP community and defined as follows: “An edit war occurs when editors who disagree about the content of a page repeatedly override each other’s contributions, rather than trying to resolve the disagreement by discussion” .
To start an empirical analysis of such opinion clashes and the way they are entangled with collaboration, we need to be able to locate and quantify edit wars.
3.3.1 Controversy measure
An algorithm to quantify edit wars and measure the amount of social clashes for WP articles has been introduced and validated before , and then used to study extensively the dynamical aspects of WP conflicts . An independent study  has also shown that this measure is among the most reliable in capturing very controversial articles.
We quantify the ‘controversiality’ of an article based on its edit history by focusing on ‘reverts’ (i.e. when an editor completely undoes another editor’s edit and brings the article back to the state just before the last version). Reverts are detected by comparing all pairs of revisions of an article throughout its history, namely by comparing the MD5 hash code  of the revisions. Specifically, a revert is detected when two versions in the history line are exactly the same. In this case the latest edit (leading to the second identical revision) is marked as a revert, and a pair of editors, referred to as reverting and reverted editors, are recognized.
where , are the number of edits for the article committed by the reverting/reverted editor. This measure can be easily calculated for each article, irrespective of the language, size, and length of its history.
List of the most controversial articles in different language WPs according to M .
George W. Bush
Unidentified flying object
9/11 conspiracy theories
List of WWE personnel
Andrés Manuel López Obrador
Grêmio Foot-Ball Porto Alegrense
September 11 attacks
Newell’s Old Boys
Sport Club Corinthians Paulista
Muhammad al-Durrah incident
Race and intelligence
God in Christianity
Luiz Inácio Lula da Silva
Nuclear power debate
Guns N’ Roses
FC Universitatea Craiova
Hungarian radical right
Disney Channel (Romania)
Legionnaires’ rebellion & Bucharest pogrom
Hungarian Guard Movement
Ferenc Gyurcsány’s speech in May 2006
The Mortimer case
Sexual orientation change efforts
Romanian Orthodox Church
Koreans in Japan
Ali bin Talal al Jahani
Korea origin theory
List of upcoming TVB series
2006 Lebanon War
People’s Mujahedin of Iran
Criticism of the Quran
Jewish settlement in Hebron
Kamen Rider Series
Beitar Jerusalem F.C.
Second Sino-Japanese War
GoGo Sentai Boukenger
Tiananmen Square protests of 1989
3.3.2 Dynamics of conflict and war scenarios
Single war to consensus: In most cases controversial articles can be included in this category. A single edit war emerges and reaches consensus after a while, stabilizing quickly. If the topic of the article is not particularly dynamic, the reached consensus holds for a long period of time (top left in Figure 8).
Multiple war-peace cycles: In cases where the topic of the article is dynamic but the rate of new events (or production of new information) is not higher than the pace to reach consensus, multiple cycles of war and peace may appear (top center in Figure 8).
Never-ending wars: Finally, when the topic of the article is greatly contested in the real world and there is a constant stream of new events associated with the subject, the article tends not to reach a consensus and M increases monotonically and without interruption (top right in Figure 8).
The empirical war scenarios described previously are in qualitative agreement with the theoretical regimes of our model in the case of agent renewal, as seen from both the sample time series in Figure 7 and the regimes of war and peace in the phase diagrams of Figure 5 and Figure 6. Unfortunately, the theoretical order parameter P is quite difficult to measure in real systems as editor opinions are not known. What we know instead is the controversy measure M of Eq. (7). As mentioned before, M counts conflict events (i.e. mutual reverts) and weights them by the maturity of the editor. This process can actually be repeated for the model: The editor maturity is then defined as the number of time steps an agent has been in the pool of editors (a quantity constantly reset by agent replacement), and a conflict event is considered as the time an editor modifies the article, since this implies the agent is not satisfied with the state of the medium.
Percentage of banned users to the total number of editors at three different classes.
Num. editors w.>1,000edits
Ban. editors w.>1,000edits
% ban. w.>1,000edits
Num. editors w.>100edits
Ban. editors w.>100edits
% ban. w.>100 edits
Num. editors w.>1edits
Ban. editors w.>1edits
% ban. w.>1edits
Here we have studied through modeling the emergence, persistence and resolution of conflicts in a collaborative environment of humans such as WP. The value production process takes place through interaction between peers (editors for WP) and through direct modification of the product or medium (an article). While in most cases this process is constructive and peaceful, from time to time severe conflicts emerge. We modeled the dynamics of conflicts during collaboration by coupling opinion formation with article editing in a generalized bounded-confidence dynamics. The simple addition of a common value-production process leads to the replacement of frozen opinion groups (typical of the bounded-confidence dynamics) with a global consensus and a tunable relaxation time. The model with a fixed pool shows a rich phase diagram with several characteristic behaviors: (a) an extremely long relaxation time, (b) fast relaxation preceded by oscillating behavior of the medium opinion, and (c) an even faster relaxation with an erratic medium. We have observed a symmetry-breaking, bifurcation transition between regimes (a) and (b), as well as divergence of the relaxation time in the transition between regimes (b) and (c).
If the pool is not fixed and editors are exchanged with new ones at a given rate, we obtain two different phases: conflict and peace. A conflict measure can be defined for the modeled system and be directly compared to its empirical counterpart in real WP data. It is then possible to follow the temporal evolution of this measure of controversy and obtain a good qualitative agreement with the empirical observations. These results lead us to plausible explanations for the spontaneous emergence of current WP policies, introduced to moderate or resolve conflicts.
Two remarks are at place here. In this study we have used a particular collaboration environment and compared our results with WP. The main reason behind is that for the free encyclopedia we have a full documentation of actions; however, we should emphasize that as web-based collaborative environments are abundant, we believe that our approach and results are much more general. Second, we are aware of the fact that the model contains a number of stringent simplifications: There are cultural differences between the WPs (e.g., in the usage of the talk page), and as in all human-related features there are large inhomogeneities in the opinions, in the tolerance level and in the activity of editors. Some of these aspects are under current study and will be taken into account for future research.
The authors acknowledge support from the ICTeCollective EU FP7 project. JK thanks FiDiPro (TEKES) and the DATASIM EU FP7 project for support. JT thanks the support of European Union and the European Social Fund through project FuturICT.hu (grant no.: TAMOP-4.2.2.C-11/1/KONV-2012-0013).
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