2

Consent or conflict: Co-evolution of coordination and networks*

2.1 Introduction

This chapter takes the first steps of the approach outlined in Chapter 1—specifying a model and deriving implications from this model using mathematical analysis and simulation. Substantively, this chapter revolves around the theme of coordination problems and social conflict and follows the following setup. First, I introduce coordination problems as a topic for social research, its relation to social conflicts, and discuss why it is important to model coordination problems as embedded in dynamic networks. Next, I introduce a formal game-theoretical model for such problems and derive some analytic results on stable states of this model. As is characteristic for this type of models, however, the analytic results leave many open questions. Therefore, the final step in this chapter is to apply computer simulation to further study the behavior of the model under different circumstances.

A long research tradition in sociology and social psychology has shown that social networks play an important mediating role in the diffusion of behaviors and opinions through a society. In many different contexts, people are influenced by those with whom they interact (Erickson 1988; Marsden and Friedkin 1993; Merton 1968). Empirical examples of such processes include peer pressure among adolescents (Davies and Kandel 1981), diffusion of innovations (Coleman et al. 1957; Valente 2005), rebellion, and collective action (Gould 1991, 1993; Opp and Gern 1993). These findings are relevant for the study of polarization, described as the social or ideological separation of a society into two or more groups (Esteban and Scheider 2008), because the adaptation process might increase agreement within groups, while it deepens disagreement between groups. The extent to which a society will polarize into possibly opposing camps is likely to be influenced by the patterns of social relations through which members of the society influence each other and through which opinions, behaviors, and ideologies diffuse.

It is important to realize that social networks are not always static but can be altered by actors consciously selecting their relations. At least in part, this selection processes is based on behavioral traits of others; sociological research shows that people tend to choose their friends among those who behave and think as themselves (a process known as “homophily”; see Lazarsfeld and Merton 1954; McPherson et al. 2001; Zeggelink et al. 1996). The combination of influence via networks and selective network formation processes suggests that polarization can occur through two different processes—on the one hand, polarization may occur because behavior or opinions cluster locally within networks and, on the other, a society may segregate socially because people with different behaviors or opinions tend to avoid each other.

Our study of polarization takes into account that social and ideological or behavioral alienation between groups develop interdependently. In other words, the degree to which polarization on a behavioral trait occurs depends on patterns of social ties, but this social structure itself is also influenced by behavioral choices. We aim at a theoretical understanding of the interplay between polarization of behavior and social structure. We develop a model in which actors are involved in interactions with others with whom they have social relations, while this social network itself is subject to change by the actors. This model predicts how the likelihood of polarization of behavioral outcomes depends on the social structure of a society at the time actors have to decide on a certain form of collective action or have to develop an opinion on some issue that becomes salient. In addition to problems in which persistent disagreement constitutes a clear potential for conflict, the model captures other types of processes from which conventions emerge, such as lifestyle choices of pupils in school classes. In such networks, persistent disagreement is not problematic per se. In Chapter 5, we will apply our model to such a situation. In the current chapter, however, I will stick to the theme of polarization in societies.

2.1.1 Polarization, conflict, and coordination

Group mobilization and group formation have previously been modeled in different ways, for example, as social influence processes (Axelrod 1997),1 as multi-person Prisoner's Dilemmas (Takács 2001), or as collective action problems (Gould 1999). Identification with a group can also be considered a multi-person coordination problem in the sense that belonging to a group and making the same choice as others is more important than what choice is actually made (Hardin 1995). In group identification, one prefers to join a group if others do the same because benefits can be expected from group membership itself. There is little sense in speaking English if everyone else speaks French; similarly, it may not be beneficial to identify as Serbian if everyone else identifies as Yugoslavian. However, if enough people start to call themselves Serbs, it becomes attractive to join this group.

It can be argued that not only identification with a group, but also group action is often mainly a matter of coordination. Usually, group mobilization and other forms of collective action are seen as free-rider problems. According to the “logic of collective action” (Olson 1971), individual group members should not be expected to contribute to collective efforts unless they have individual (selective) incentives that compensate their efforts. Hence, collective action should not occur in most cases because every individual has reasons to free ride on the other group members. This prediction seems at odds with the real-world observation that group action does occur in many instances, from voting to mass demonstrations and collective violence. This led some scholars to argue that coordination rather than cooperation is the basic strategic interaction that underlies group action. According to Hardin (1995), the power resulting from mass action can diminish the costs of joining to a level that is sufficiently low to reduce the free-rider problem to a problem of coordination (see Heckathorn 1996; Macy 1991; Marwell and Oliver 1993). Others (Chwe 2001; Gould 1995; Klandermans 1988; Lohmann 1994) have also pointed at the importance of coordination in collective phenomena such as rebellion, uprisings, and union participation. Therefore, by modeling collective action as a coordination problem we abstract from free-riding problems and focus on settings in which actors have an incentive to join if enough others do so as well.

Modeling group identification and group mobilization as a coordination problem is not to say that actors are indifferent between behavioral alternatives as long as they coordinate with others. The coordination game that is the backbone of our model assumes that behavioral alternatives can be ranked, even if coordination with others still has priority. Consider the simple coordination game as displayed in Figure 2.1, with c = 8. The actors have two choices, L(EFT) or R(IGHT). The game has two Nash equilibria, (R,R) and (L,L), in each of which actors do not wish to deviate as long as the other actor does not deviate. However, (L,L) yields higher payoffs for both actors and is termed efficient or payoff dominant. The other equilibrium (R,R) is attractive in the sense that it is less risky—if an actor assumes that the other actor chooses R and L with equal probabilities, the expected payoff of choosing R is higher than of choosing L. Therefore, (R,R) is risk dominant (Harsanyi and Selten 1988). For our applications, choosing L may represent joining an uprising in order to accomplish a regime change, while choosing R is to stick to the status quo. Joining the uprising is risky if you are not sure that others will also do so.

Figure 2.1 The two-person version of the multi-person coordination game with payoffs as in the simulation b < c < a < d; (a − b) > (d − c); c = 4 or c = 8.

A consequence of choosing a multi-person version of the coordination game as described above implies that we can not only provide predictions on the likelihood and extent of polarization, but also on the extent to which actors coordinate on the efficient equilibrium. However, the model does not predict how the likelihood of the emergence of violent conflict depends on the polarization that might arise in a population. Rather, the theory assumes that polarization into separated but internally coordinated groups provides potential for violent group conflict, while the model specifies the conditions under which a polarized situation is more or less likely to occur.

As a measure of polarization, we use the two-group version of the index for qualitative variation (IQV; Mueller and Schuessler 1961, pp. 177–179; see also Agresti and Agresti 1978), which is defined as 4p(L)(1 − p(L)), in which p(L) is the proportion of actors in the population choosing L. The measure implies that polarization is 0 if p(L) = 0 or p(L) = 1, while it reaches its maximal value 1 for p(L) = . This measure is the standardized version of a much older version of a diversity measure that dates back to Gini (see Lieberson 1969 for an overview). We focus on polarization as defined above because with only two groups we cannot distinguish it from, for instance, fractionalization (see Montalvo and Reynal-Querol 2005; Reynal-Querol 2002). The maximum of our polarization measure is reached if both groups are of equal size, which corresponds with the maximum for conflict potential according to Esteban and Ray (1999) for the general polarization measure (see also Esteban and Scheider 2008).

2.1.2 Coordination and social networks

Earlier theoretical studies of coordination in large groups consider models in which actors interact with all other actors in the population without assuming any social structure (e.g., Kandori et al. 1993; Young 1993). However, this assumption seems highly unrealistic for many applications. Especially in large groups, actors can often only observe the behavior of those with whom they interact directly; they only observe their own personal network. This is especially true for cases in which no public information is available about the distribution of behavior in the larger population, and therefore, actors really have to rely on their close surroundings for information. They may even use the behavior of the members of their personal network as an approximation of the behavior in the larger population.2 Opp and Gern (1993) and Lohmann (1994), for example, discuss the importance of personal networks for the “Monday Demonstrations” in Leipzig, 1989, which is a typical case in which public information was highly restricted. Gould (1991, 1993, 1995), Scott (1990), Hardin (1995), and Chwe (2001) similarly emphasize the role of networks in instances of (violent) collective action.

Theoretical models dealing with local interaction have been formulated by (among others) Ellison (1993), Young (1998), and Berninghaus et al. (2002) although they do not consider possible heterogeneity of the network structure. Buskens and Snijders (2008) explicitly deal with coordination in heterogeneous social network structures, where heterogeneity refers to actors having different positions in the network, that is, they do not necessarily have the same number of relations. Considering static networks only, Buskens and Snijders (2008) show that denser and less segmented networks reach consensus more easily than less dense and more segmented networks. In addition, segmentation leads to fewer actors choosing the risky option, while density encourages more actors to choose the risky option. Thus, segmentation leads to less and density to more efficiency in the emerging behavior.

However, as I have argued above, networks are themselves also the consequences of choices of individual actors and change over time (Flap 2004). Therefore, we consider a model in which actors are organized in a dynamic social network and obtain “high” payoffs for relations with actors who behave the same and obtain “low” payoffs for relations with actors who behave differently. Assuming that maintaining social relations is costly, relations have to be chosen cautiously, which implies that relations with actors who behave differently may be terminated.

To see the merit of this latter assumption, it is worthwhile to consider the implications of assuming that maintaining relations with others who behave differently is not in some way costly. In that case, there would be no restriction on the number of relations an actor can maintain, and the network would eventually evolve into a situation in which every actor is connected to every other actor (the complete network), which would bring us back to the model of global interaction studied in earlier literature. Therefore, if one is interested in studying the effects of local, heterogeneously structured interaction (i.e., social networks) on coordination, it makes a lot of sense to assume that relations are somehow costly.

Recently, the study of dynamic networks with strategic decision making of actors in the network has developed quickly (for overviews, see Dutta and Jackson 2003; Goyal 2007; Jackson 2008; Newman et al. 2006). Specific models for coordination games played on dynamic networks are studied by Skyrms and Pemantle (2000), Jackson and Watts (2002), Goyal and Vega-Redondo (2005), and Berninghaus and Vogt (2006). These models focus on which networks are stable and on how behavior is distributed in stable networks. Typically, many configurations are possible. Therefore, we take a different approach and focus on how preconditions determine the emergence of a stable network and the related distribution of behavior. Hence, we study the extent to which polarization in segregated groups emerges and inefficient behavior persists as a result of the initial network and the initial distribution of behavior. More specifically, we aim to answer the following research question—how do polarization and efficiency of the emerging distribution of behavior depend on the initial distribution of behavior, the initial network, the tie costs, and the payoffs in coordination games?

In the next section, we present the dynamic model and analytic results on stable states. Subsequently, we describe simulations and derive predictions on the effects of initial conditions on polarization and efficiency. In the final section, we conclude and interpret the results in terms of the likelihood that conflicts may emerge.

2.2 The model

Actors are organized in a network of n actors with the n × n adjacency matrix N = (nij), that is, nij = 1 if two actors are connected and nij = 0 otherwise. We assume that relationships are undirected, so nij = nji. Relations have both benefits and costs. Actors have to choose between two types of behavior (or opinions, attitudes) and their benefits or payoffs depend on their own behavior and the behaviors of the actors with whom they have relations. They cannot differentiate their behavior depending on specific relations. In every existing relation, the payoff is related to the actor's own behavior and the behavior of the other person in correspondence with the coordination game depicted in Figure 2.1. From actors with whom actor i does not have a relation, i obtains a payoff 0. In our multi-person coordination game, this implies that the payoff of an actor i choosing R equals , in which j(z) runs over other actors who choose z, z = R, L. If actor i would choose L, the payoff would be .

We study two variants. Variant one assumes that (a − b) < (d − c), so that choosing L not only can lead to the efficient payoff d, but choosing L is also less risky because the expected payoff from playing L is relatively high if you do not know what others will do.3 The second variant assumes (a − b) > (d − c) so that choosing L is the risky choice because the expected payoff from playing L is relatively high if you do not know what others will do. The actual payoffs that we use in the simulation are indicated in Figure 2.1 as well. Buskens and Snijders (2008) demonstrate that

(2.1)

in combination with the behaviors of the actors with whom a focal actor is connected, determines whether or not this focal actor wants to change behavior. In addition, Buskens and Snijders (2008) show that network effects are small for static networks if risk values are far from 0.5, but that there are major differences in network effects depending on whether risk is just above or just below 0.5. Therefore, in the simulation risk varies between 0.467 and 0.538.

In addition, we assume that ties are costly. We abstract from separate costs for the creation or the deletion of ties. Thus, the costs of existing ties have to be paid in every period of interaction, but ties can be deleted without any cost. We assume increasing marginal tie costs, and so the more ties one has, the more “effort” is required for another tie. Increasing marginal tie costs can also be interpreted as diminishing marginal returns of relationships; an equivalent model assumes constant tie costs and benefits that decrease in an actor's number of interactions. The total costs for an actor of having t ties are

(2.2)

in which α > 0 and β ≥ 0. If β = 0, the tie costs are linear in the number of ties, and there are no increasing marginal costs of having more ties. Otherwise, there is a maximum number of network ties one can maintain given the payoffs one can obtain related to relationships. In one period of interactions the total payoff of an actor equals , where pij is the payoff i receives as a result of his or her own and j's behavior and is the number of ties of actor i.4 Finally, we define how the network is changed. Actors can add and sever ties. Because of the undirected nature of the ties, ties can only be created with the consent of both partners but can be removed unilaterally. In other words, we assume a two-sided link formation process (Jackson and Wolinsky 1996).

We also assume that actors have full information on the behavior of all actors in the network. This might seem to contradict our earlier argumentation about complete information models being unrealistic for large populations, but I stress that we still assume that actors only consider their direct neighbors’ behavior for updating their own behavior. Information on the behavior of other actors is only relevant when creating ties, and in effect, we here assume that actors can observe the behavior of potential neighbors when they consider creating ties. In Chapter 4, we will relax this assumption further and look at a model in which actors also use only local information to update their network.

2.3 Stable states

After specifying the model, the next step is to derive implications from the model about the dynamics of polarization and networks. A well-established strategy in the literature (Berninghaus and Vogt 2006; Goyal and Vega-Redondo 2005; Jackson and Watts 2002), is to first look for stable networks, that is, networks in which

• No actor wants to change behavior,
• No actor wants to sever a tie, and
• No pair of actors wants to add a tie given the behaviors in the network.

This definition corresponds to the notion of pairwise stability (Jackson and Wolinsky 1996) in networks in which only ties can be changed. The condition on behavior is added to guarantee stability in terms of the behavior chosen by the actors. In order to characterize stable networks, two definitions are helpful.

Definition 2.1 A (sub)network is t-full if and only if none of the actors have more than t ties and either (a) the addition of a tie causes at least one actor to have more than t ties, or (b) no ties can be added to the (sub)network.

Definition 2.2 For z = R, L, is the maximum number of ties an actor wants to have if he or she chooses z, and all actors with whom he or she has a relation choose z as well.

These definitions are useful because the number of ties an actor wants to maintain is determined by the tie costs and the payoffs an actor can obtain. More specifically, a new relation with another actor with whom an actor i can earn pij is only initiated if the number of ties actor i has will be less than or equal to after adding this new relation. Otherwise, the marginal costs of adding this new tie are larger than its benefits. This implies that

(2.3)

(2.4)

Theorem 2.1 If tie costs are equal to (α > b and β > 0), networks are stable if and only if one of these conditions holds:

1. All actors choose the same behavior z, where Z = R, L and the network is -full.

2. The network consists of two subgroups of actors choosing R and L and these subgroups are -full and -full, respectively, and there are no ties between the two subgroups.

Proof. As tie costs are always larger than b, a tie between actors with different behaviors is never sustainable because one of the actors wants to sever the tie or change behavior. It is easily checked that the definition of -full implies that no pair of actors wants to add a tie and no actor wants to sever a tie. Because actors are not connected with actors who behave differently, they do not want to change behavior. All other networks are unstable because in (sub)networks that are neither -full nor -full, some actors want to remove or add ties.

The theorem is a reformulation of the corresponding theorem in Berninghaus and Vogt (2006) for costs that are nonlinear in the number of ties and for the conditions we want to consider. We extend the results of Berninghaus and Vogt (2006) not only by allowing for nonlinear costs, but also by studying the likelihood of the emergence of different structures depending on initial conditions, as we will see in the section about simulation later in this chapter. Jackson and Watts (2002) study coordination and endogenous formation of networks. They analyze which networks are stochastically stable in a specific dynamic context, showing analytically that homogeneous networks emerge in which all the actors coordinate on one behavior (cf. Young 1998). Which behavior is chosen depends on the tie costs. In their analysis, conflict situations are excluded as possible long-term outcomes because they are less stable than nonconflict situations. In contrast with their study, we do not include “trembles” but analyze how the likelihood of emerging structures in a deterministic dynamic environment depends on initial conditions. In a deterministic environment networks with polarized behavior can be stable. This allows us to address the likelihood of conflict. Goyal and Vega-Redondo (2005) analyze a similar model but assume one-sided link-formation cost. More importantly, they characterize only stable states without analyzing the likelihood that certain stable networks emerge.

The condition in the theorem that (sub)networks should be -full or -full suggests that the payoff and cost structure does not allow much variation in network structure. This is true in the sense that the density (proportion of ties present in the network) of the emerging network hardly varies with the cost function, the payoffs, and the distribution of behavior in the emerging network. Some variation is still possible. Let us consider as an example a nine-actor network in which everybody wants to have two ties. If behavior is homogeneous, both three closed triads and one circle of nine actors are stable networks. Another type of variation is related to the possibility that one actor may still have fewer ties than he or she wants to have but all other actors in his or her group have their maximum number of ties (see Jackson and Watts 2002 for more details on variations in network structures). The simulation results presented below indeed confirm that the density of the emerging network is almost perfectly determined by the emerging distribution of behavior, the payoffs in the game, and the tie costs.

However, within the limits set by Theorem 2.1, many different situations are possible stable states of the co-evolution process. We may end up with all actors playing the efficient action L, all actors playing the inefficient action R, or with two separated groups of varying sizes, each group playing a different action, that is, some degree of polarization. Thus, we here see an example of the problem of equilibrium selection in coordination games sketched in Chapter 1. Given that we started out assuming that polarization in this model is related to the likelihood of social conflict, the analytical results are in this sense somewhat dissatisfying. In order to gain a better understanding of which of these situations is likely to emerge depending on the initial circumstances, we use computer simulation.

The basic logic of the simulation analysis is as follows. We take the model formulated above and apply it to a broad variation of parameters and starting conditions. Starting conditions, in this case, consist of initial network structure and initial distributions of action choices. For each combination of parameters and starting conditions, we let the co-evolution run until a stable state is reached, and we repeat this multiple times for each combination. Naturally, every stable state resulting from this process must conform to the characterization given by Theorem 2.1, but within these general bounds, variations as sketched above will occur depending on the parameters and initial conditions. We can subsequently use statistical methods to study more precisely how the features of stable states depend on parameters and initial conditions, thereby (at least partly) solving the equilibrium selection problem. In the next section, we first discuss the technical details of the simulation design.

2.4 Simulation design

To analyze the effects of the model parameters on the emergence of stable networks by means of computer simulation, we systematically vary the initial conditions of the dynamic process and relate the outcomes of the process to these conditions. The conditions include the initial network, the initial distribution of behavior, the payoff structure of the coordination game, the tie costs, and the adaptation rules in the dynamic process. Network size ranges from 2 to 50 actors. For networks of 2–8 actors, we include all 13,597 possible non-isomorphic networks. For networks of size 9–50, the number of possible networks becomes extremely large, and we take a sample stratified on the size (number of actors) and density of the network. In other words, we draw a set of random networks while density and size are fixed such that for each size there are about the same number of networks. Also for each density within each size a similar number of networks are drawn. Extreme densities for which the number of non-isomorphic networks is small are excluded. This results in a set of 95,729 networks. The probability for each actor to initially choose L equals 0.25, 0.50, or 0.75. This results in a wide range of distributions of actual initial behavior.

Using this procedure we obtain satisfactory amounts of variation in the variables that we want to use as independent variables to explain polarization and efficiency. Selectivity on the independent variables should in principle not effect the estimations of the regression models we use below. Nevertheless, we did some checks to ensure that the sampling procedure does not affect the substantive conclusions and this does not seem to be the case.

For reasons mentioned above, we vary only payoff c (see Figure 2.1) such that risk takes the values 0.467 (c = 4) and 0.538 (c = 8), fixing the other payoffs at b = 0, a = 14, and d = 20. With regard to the tie costs, we vary both the linear component and the quadratic component such that all are possible. Linear cost α is chosen as an integer number from 1 through 16 (excluding c and 14 to avoid equalities with payoffs), each with an equal probability. For α < 14, β is chosen such that all values of () are equally likely by choosing , where . If α > 14, we have . In these situations, β is chosen such that all values of () are equally likely by choosing and again .

The dynamic model assumes discrete time. All actors simultaneously choose behavior in each period. We distinguish three methods for how actors change behavior and ties between periods. All methods assume actors to be myopic, that is, optimizing under the restriction that the behavior and network of the previous period persists. It is also assumed that all actors know the behavior of all other actors. The three methods are different in the relative adjustment rates of behavior and ties. Since these adjustment rates can be expected to affect outcomes (Skyrms and Pemantle 2000), we compare the three approaches:

In any of the three methods, actors are informed about previous changes before they have to decide. If actors are indifferent between a change and the existing situation, we assume that actors do not change. We assume that actors do not change ties if both actors are indifferent between having and not having a tie. One additional assumption is needed here to handle actors who are not connected to anyone at all. These actors cannot adapt their behavior to any connected actor; moreover, in some cases, they are not able to connect to anyone else, regardless of the behavior they choose, because the other actors already have the optimal number of ties. In such cases, the unconnected actor changes behavior only if that might create opportunities to become connected in the next period. Otherwise, the actor does not change behavior. Clearly, other rules for changing behavior and relations can be conceived. We have chosen some of the most straightforward options on who might change what and when. However, further investigation of how the dynamics depend on these options is only called for if outcomes differ dramatically between them. As we will see below, this is not the case.5

For each of the 13, 597 + 95, 729 = 109, 326 networks and each of the three versions of the dynamics, we varied the initial conditions in the following way. The simulation was done four times with different values of the quadratic cost component for each version of the dynamic process and for 13,597 networks of size up to 8. We distinguished only two levels of the quadratic cost component for the 95, 729 larger networks. One random choice was made to select the other conditions. Then, each of the initial conditions was repeated four times. These repetitions with the same set of initial conditions enable us to distinguish between stochastic variation in the outcomes that is related to variations in initial conditions and randomness caused by the dynamic process. At each repetition, we let the process continue until convergence. This leads to 13, 597 × 48 + 95, 729 × 24 = 2, 950, 152 simulation runs.

To analyze the effects of the initial network and to evaluate the emerging networks, we need to define and compute some key network characteristics:

• Size represents the number of actors.
• Density is the number of existing ties divided by size × (size − 1)/2, the possible number of ties in the network (Wasserman and Faust 1994, p. 101).
• Degree is the number of ties of an actor divided by (size − 1) This “relative” definition differs slightly from the more common “absolute” definition (e.g., Wasserman and Faust 1994, p. 100) but facilitates comparison across network sizes.
• Centralization is measured through the standard deviation of the degrees as defined above. This measure is derived directly from the variance in degrees as proposed by Snijders (1981); see also Wasserman and Faust (1994, p. 101). Other centralization measures (Freeman 1979) lead to similar results in the analyses that we present below.
• Distance, a dyadic measure, is the minimum path length between two actors (Wasserman and Faust 1994, p. 110).
• Segmentation is the proportion of dyads at distance larger or equal to 3 of all dyads at distance larger or equal to 2 (Baerveldt and Snijders 1994). In accordance with Baerveldt and Snijders, we count disconnected dyads as having a distance larger than 3. In the complete network, there is clearly no segmentation, which implies that this measure is equal to 0.
• Segregation measures the extent to which ties are limited to actors with the same behavior rather than between actors with different behaviors. It is defined as the expected number of between-group ties minus the observed number of between-group ties, divided by the expected number of between-group ties, assuming random matching (Freeman 1978). This is the only measure that combines behavior and network structure. Because there are no ties between actors with the same behavior in the emerging networks, this measure always equals 1 for the emerging network.
• Number of components is the number of connected subgraphs (including isolates) that are not connected to the rest of the network. If the number of ties actors can have is low due to the high tie costs, groups of actors that behave in the same way might still fall apart into different components.

Size, density, and centralization are included because these represent the most basic network measures: the number of actors, the number of ties in the network, and the variation of ties between actors. The measures for segregation, segmentation, and components are included because they are particularly relevant for group formation processes in general and polarization in particular. While polarization indicates the extent to which actors are divided in subgroups due to their behavior, segmentation and components indicate the extent to which actors are divided in subgroups due to the network structure. Segregation is a measure for the extent to which behavioral and structural groupings coincide. The other measures are described only to define the network measures that we need in the analyses below. We also record the distribution of behavior in the initial and the emerging networks, the number of tie changes until convergence, and the number of behavior changes until convergence.

Some parameter values among the initial conditions are less interesting to analyze and are difficult to compare with the large set of cases. These are the following subsets:

• Cases for which a > 14. If a > 14, actors who choose R cannot maintain any ties. As a result, in over 80% of the cases all actors choose L. Actors will only stick to R if no one chooses L or all L-choosing actors do not want to add ties.
• Cases that start in a situation in which everyone chooses R or everyone chooses L. Behavior does not change in these cases and only the network ties are adapted until the network is -full or -full. Excluding these cases, 1, 544, 100 cases remain for the analyses. Summary statistics for the initial conditions are shown in Table 2.1.

Table 2.1 Summary statistics of initial conditions in the simulation (N = 1, 544, 100).

2.5 Simulation results

In this section, we explain properties of the stable networks in terms of polarization and efficiency by the model parameters using statistical regression analysis. Table 2.2 presents summary statistics for the stable states. One result is that the relative number of behavioral changes (number of changes per actor) is considerably lower than the relative number of network changes (number of changes per dyad). This can be understood by recognizing that changing one's behavior has much more impact than changing one tie—in case of a behavioral change, the payoffs resulting from all interactions are affected, while changing a tie affects only one relation. Therefore, an actor mostly does not change behavior more than once. Usually actors need to adapt multiple relationships before they cannot improve their situations anymore.

Table 2.2 Summary statistics of stable states (N = 1, 544, 100).

The type of network dynamics does not have a large influence on the stable networks. In both “alternating” and “fast network” dynamics, there are somewhat more changes in behavior and ties than in “actor-based” dynamics. Surprisingly, the number of tie changes is only marginally larger in “fast network” dynamics as in “alternating” dynamics although there are much more opportunities to change ties in the first one. Because of the limited differences between the types of dynamics, we only provide joint summary statistics in Table 2.2.

Polarization is our first important dependent variable. Recall that polarization is defined as , in which p(L) is the proportion of actors choosing L. In more than 60% of the stable states behavior is homogeneous and, thus, without polarization. Efficiency, as expressed by the proportion of actors choosing the L-behavior, is a little above 50%, which is only slightly higher than the average efficiency in initial networks (Table 2.1). The standard deviation of efficiency in stable states is rather large, reflecting the large proportion of homogeneous stable states. These percentages are not well interpretable because they depend to a large extent on our choice of initial conditions. Therefore, we focus below on how polarization and efficiency change with initial conditions.

2.5.1 Predicting stable states I: Polarization

To examine how polarization depends on the parameters of the simulation, we use linear regression models with polarization as the dependent variable and the initial conditions (the network structure, the initial distribution of behavior, the tie costs, and the dynamics version) as independent variables. It is important to realize that standard errors reflect uncertainties due to the randomness of the dynamic process and misspecification of the model, not “sampling.” In addition, with more than a million cases, even very small effects are significant. Therefore, we restrict ourselves to comparisons of relative sizes of standard errors between variables to provide information on the relative stability of the effects and we do not report significance levels. Standard errors are adapted for clustering within initial conditions (Rogers 1993).

Polarization in stable states has an extremely skewed distribution, with over 60% of the stable networks showing homogeneous behavior where the value of polarization is zero. Our statistical model has to take this unusual distribution into account. We decide to model polarization with two separate analyses. First, we estimate a model predicting whether stable states are heterogeneous or not using logistic regression. Then, we apply a linear regression model to predict the extent of polarization in the cases with polarization. The tie costs are included in the analysis as the difference in the number of ties an actor can have while choosing L as compared with the number of ties he or she can have while choosing R divided by size. The effect of tie costs is small and only the quadratic cost component turns out to be relevant for predicting behavior in the analysis. This component can be adequately summarized by . We add dummies for the different types of dynamics.

Because the outcomes of the analyses strongly depend on initial polarization, we ran separate analyses for different categories of initial polarization. These analyses suggested adding interactions of initial polarization with density and segregation to our model. To facilitate the interpretation of the main effects, the density is centered at the mean before taking the interaction. Initial polarization and segregation already have a mean equal to 0.

In Table 2.3, we report the results separately for high and low risk although the models do not differ substantially between risk values, to facilitate comparison with the results on efficiency reported below. The initial distribution of behavior, operationalized as the polarization at the start of the process, has a large positive effect on polarization in stable networks. There is a slightly higher probability of persisting polarization for the “alternating” version of the dynamics compared with the “actor-based,” which is the reference category. The probability of persisting polarization is highest for the “fast network” version, which is understandable since many network changes result in a network falling apart in groups with different behaviors before actors have an opportunity to adapt behavior. Substantively, this leads to the plausible interpretation that in societies in which relations are more volatile while people tend to stick more strongly to their behaviors or opinions, it is more likely that persistent disagreement arises that may induce conflicts. The difference between how many ties one wants to have choosing L compared with choosing R has a negative effect on the probability of any polarization as well as the extent of polarization. The probability of persistent polarization and the extent of polarization decreases for larger networks.

Table 2.3 Logistic and linear regression of behavioral polarization on simulation parameters with standard errors adapted for clustering within initial condition.

Considering the network effects, density has a large negative effect on the probability of any polarization as well as on the extent of polarization (given positive polarization). Segregation has a small positive effect on the probability of any emerging polarization for average initial polarization and a small negative effect on the extent of emerging polarization. Segmentation increases the likelihood of persistent polarization as well as the extent of polarization. Centralization promotes the probability of homogeneity although given that there is some polarization, polarization will be larger if initial centralization is higher.

Due to the interaction of polarization and density, the effect of density is stronger for higher initial polarization. This implies that density really helps to solve the polarization problem, and if some disagreement persists, at most a small minority will stick to the “deviant” behavior and will be excluded by the majority. Although the main effect of segregation is small, the interaction effect with initial polarization is substantial. It shows that segregation enhances polarization if the initial network is rather polarized. Thus, if differences in behavior are aligned with initial group boundaries, the situation likely becomes only worse due to network dynamics.

2.5.2 Predicting stable states II: Efficiency

Efficiency has a somewhat peculiar distribution. In more than 60% of the stable states behavior is homogeneous (efficiency is 0 or 1), while the remaining cases are more or less evenly distributed between the two extremes, such that the total distribution is U-shaped. An appropriate way to analyze such a dependent variable is logistic regression for grouped data, which predicts the number of successes (i.e., the number of actors choosing L), given network size. As the size of some network effects depend strongly on risk, we conduct the analyses separately for low risk and high risk (Table 2.4).

Table 2.4 Logistic regression for grouped data on the proportion of actors choosing L with standard errors adapted for clustering within initial condition.

The coefficients refer to the log-odds of the predicted proportion. Clearly, the initial proportion L determines to a large extent the emerging distribution of behavior. If the group starts with a majority of actors choosing L, there is a large probability that an even larger majority or the whole group will ultimately choose L. The other effects have to be interpreted in terms of the extent to which they affect this baseline dynamic. The “alternating” and “fast network” dynamics have higher rates of efficiency as compared with “actor-based” dynamics for low risk, while this is the other way around for high risk. This can be interpreted in the following way. Changes in behavior mostly have the largest impact. Thus, if actors behave differently from most actors they interact with, these actors start changing behavior and thereafter they optimize their ties. As we know from models with static networks, adaptation of behavior leads to attraction to the risk-dominant equilibrium. In the “alternating” and “fast network” versions of the dynamics, changing ties often has to be done before behavioral changes. This apparently decreases the attraction to the risk-dominant equilibrium. Except for the situation in which no ties were possible between actors choosing R, which are excluded from the analysis, tie costs have only small effects.

The network effects vary not only greatly with risk, but some also with the initial distribution of behavior. Therefore, we consider again interaction effects between the initial distribution of behavior and other variables. Especially the density and centralization effects depend considerably on initial behavior. In Table 2.4, interactions of the centered variables were included. In both models, strong and positive interaction effects exist between density and the initial distribution of behavior, such that the effect of density is negative for low initial proportions and positive for high initial proportions. For centralization and initial behavior, the interaction effect points in the other direction. The effects of density and centralization at their means are in different directions and they also switch if one compares high and low risk. At the means, density promotes efficiency for low risk but hampers efficiency for high risk. In contrast, centralization hampers efficiency under low risk but promotes efficiency under high risk. It is important to realize that the effects at the mean are relatively small compared with the interaction effects. Segregation has a negative effect on efficiency under low risk and a positive effect under high risk. This result indicates that segregation is to some extent able to stabilize the more risky equilibrium. Segmentation has a negative effect for low and high risk. Network size has only a small negative effect on efficiency.

To sum up, the strongest effect of the initial network structure on efficiency is that density galvanizes initial behavioral tendencies. If we start at a rather high level of efficiency, higher density leads to the emergence of even more efficient networks. However, if initial behavior is inefficient, the situation is likely to worsen with dense networks. Other network effects have a modest size. In the low-risk situation, segregation prevents the efficient behavior for diffusion through the whole network and therefore has a negative effect on total efficiency. However, segregation helps to maintain efficient behavior in parts of the network in the high-risk situation. On average, density decreases efficiency in the high-risk situation, which is surprising because density increased efficiency in high-risk situations for static networks (Buskens and Snijders 2008). Centralization, in contrast to density, favors minority behavior in high-risk situations. An interpretation for this would be that if this minority has a central position in the network, they can induce the majority to change their behavior.

2.6 Conclusions and discussion

In this chapter, we developed a model explaining opinion formation or mass mobilization processes as, for example, occurred during the 2012 “Arab Spring” or the “Monday Demonstrations” in Leipzig 1989–1991. The model formalizes the co-evolution of coordination and networks to study under what conditions it is more or less likely that the emergence of stable states leads to inefficient situations or situations with considerable conflict potential. We assume that actors are organized in a specific network in which coordination problems emerge. Initially, all actors behave in a certain way (or have a certain opinion or attitude toward a given issue). Depending on their own behavior and the behavior of the actors with whom they have relations, actors change their behavior and their relationships. After the network has developed into a stable situation, we consider behavioral polarization and efficiency in the emerged stable situation.

We derived some analytical results from our model, and then applied simulation techniques to derive further implications. It turns out that the initial network structure affects the emerging distribution of behavior. The most important result with respect to polarization is that dense networks lead to more homogeneous behavior, while more segmented and segregated networks have the opposite effect. The latter effect becomes especially important if the initial behavior is already polarized. If polarized societies are indeed more prone to end up in conflicts, conflicts are less likely in dense cohesive societies, while conflicts are more likely in segregated and segmented societies, especially if the initial attitudes in sensitive issues are correlated with initial groups in social networks. The effect of centralization is multifold—centralization of the initial network decreases the probability that the network polarizes, but if there is polarization, centralization slightly promotes the extent of polarization.

The most salient finding on efficiency is that network density amplifies the effect of the initial distribution of behavior. The higher the density, the larger the effect of initial inefficient behavior on the inefficiency of the emerging network. In addition, if the initial behavior is equally distributed, density still increases the likelihood that the emerging behavior will be inefficient if the efficient behavior is risky. A similar effect is found for centralization, although smaller and in the opposite direction—centralization has a positive effect on efficiency if initial efficiency is low and a negative effect if initial efficiency is high. In addition, in larger, more segmented, and more segregated networks, behavior tends to become less efficient. These results are consistent with the fact that dictatorial states often survive for quite some time without having to cope with major mass demonstrations (cf. Hardin 1995). As soon as a status quo with no major opposition is reached, it is difficult to turn this situation around. The centralization result shows that the best opportunity to escape from such an inefficient situation should come from central people who can mobilize others to start a revolt.

Although this study provides an extension to existing models on network formation in coordination problems and provides more insight into the relation between initial conditions and the emerging networks, there are still a number of limitations. First, our model assumes extreme opinion formation problems in which you can basically have only two opinions and if you do not agree, a relationship would be very unattractive. In some of the examples mentioned such as the ones where you have to choose between standing up against the regime or remaining quiet, these assumptions are clearly more realistic than in less extreme situations. Second, while coordination problems represent the evolution of norms or opinions, many social interactions might lead to conflicts of a different character. Such situations can also be related to, say, trust, cooperation, or distribution problems (Heckathorn 1996). Then, it becomes likely that actors differentiate behavior between their partners. People might trust some people and distrust others. Therefore, the study of the evolution of networks and the possible emergence of related social problems can be extended to other types of interactions in settings where behavior co-evolves with networks. In addition, the theory developed has to be tested in experimental and real-world settings to corroborate the hypotheses developed in this chapter. We will do this in Chapters 4 and 5.

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* This chapter is based on an article written in collaboration with Vincent Buskens and Jeroen Weesie. The original version appeared in the Journal of Peace Research (Buskens, Corten and Weesie 2008).

1. See also Stokman and Zeggelink (1996) for a model in which opinions and networks are updated simultaneously.

2. We will look at a more concrete version of such a mechanism in Chapter 4.

3. In terms of the two-person game, this would be the condition that (L,L) is not only the payoff-dominant equilibrium, but also the risk-dominant equilibrium (in the sense of Harsanyi and Selten 1988).

4. Although our cost function is provided in a very explicit form, the only crucial aspect is the number of ties an actor wants to have depending on whether he or she obtains mostly a or d in his or her relations. The possible number of ties is varied in the simulation over the complete relevant range.

5. Additional simulations with simultaneous updating of behavior (not reported here) also lead to similar results.