The micro dynamics of collective violence Jeroen Bruggeman∗ June 6, 2017

Abstract Collective violence in direct confrontations between two opposing groups happens in short bursts wherein small subgroups briefly attack small numbers of opponents, while the majority of participants forms a supportive audience. The mechanism is fighters’ mutual alignment of rhythmic movements during preliminary interactions, by which they overcome their fear. Taking Randall Collins’ observations of clashes as stylized facts, these bursts and subgroups’ small sizes are explained by a formal synchronization model.

From hunter-gatherers to modern citizens, people coalesce into groups that sometimes confront each other antagonistically. If this leads to a physical confrontation (without long range weapons), what will happen? In Hollywood movies, collective violence looks like a chaotic mess of everybody fighting simultaneously. Randall Collins’ (2008) extensive study of many photographs and surveillance video’s of violent conflicts showed up a very different pattern, though: initially, tension builds up, followed by short bursts of violence wherein small subgroups briefly attack small numbers of opponents, in particular stumbling, isolated or otherwise vulnerable individuals. Meanwhile, the majority forms an audience that does not commit violence (Weenink 2014), and sometimes encourages and at other times intervenes. This dynamic pattern holds for both protesters and police (Nassauer 2016), as well as groups engaging in ordinary street fights (Levine, Taylor, and Best ∗

Department of Sociology, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, the Netherlands. Email: [email protected]. For insightful comments, thank you to Don Weenink and Frans van Winden, as well as discussants at the Violence Workshop (Amsterdam, April 2017) and the Complexity of Covert Networks workshop at the Institute of Advanced Study (Amsterdam, May 2017).

1

2011). For most people it is difficult to overcome their fear of, and inhibitions towards, violence (Collins 2008), even for police with professional training and leadership (Klinger 2004). Overcoming fear is a main explanatory challenge. People with a preference for violent action tend to select each other assortatively (McPherson, Smith-Lovin, and Cook 2001) well before a fight, based on reputations and subtle cues. At a confrontation, they form small groups of typically 3 to maximally 6 individuals that include a leader, and their interactions involve rhythmic movements (Collins 2008). Because they are in close physical co-presence with a mutual focus of attention, their interactions strengthen their bonds (Sebanz et al. 2006; McNeill 1995; Collins 2004). However, strong bonds are not enough to conquer fear, though. At some point, however, their rhythmic movements synchronize, for example, they step back and forth together. They synchronize through mutual alignment based on bodily information sensed at close range. Synchronization yields collective efficacy (Sampson, Raudenbush, and Earls 1997; Bandura 2000), called emotional energy by Collins (2008) and collective effervescence by Durkheim (1912), and makes subgroup members feel one (Swann et al. 2012) by which they overcome their fear. Acting violently is very exhausting, though, and lasts much shorter than in Hollywood movies, where after the attackers dissolve into their larger group.1 For obvious ethical reasons there is no experimental replication of such attacks, but there is evidence of synchronization’s motivational effect in nonviolent experiments. When comparing treatments with asynchronous to synchronous movements, contributions to public goods are significantly higher in the latter (Reddish et al. 2013; Fischer et al. 2013), and synchronized movements enhance feelings of strength vis-`a-vis opponents (Fessler and Holbrook 2014). Current theory provides rich descriptions but leaves a number of questions unanswered. (1) How does synchronization happen? (2) Why does hand-to-hand violence occur in bursts rather than in a more gradual temporal pattern? (3) Why do some groups get emotional energy whereas in other groups in similar circumstances, emotional energy is drained (Collins 2004)? Does network structure have something to do with this difference? (4) Why are violent subgroups small whereas larger groups stand a better chance to gain the upper hand? (5) Are leaders important for mutual alignment? Be1

Highly experienced fighters do not have much fear and do not need preliminary interaction rituals to get to action. Moreover, using long range weapons or attacking unsuspecting opponents from ambush or during nightly raids is less scary than hand-to-hand combat with eye-to-eye contact, and feature different dynamics. Premediated strategies, fire arms and seasoned fighters are therefore excluded from this paper.

2

cause, for example, military officers determine the choreography of marches (McNeill 1995), one might think they are. To answer these questions, I use Kuramoto’s synchronization model, which is a system of coupled differential equations that shows how mutual alignment at the dyadic level results in synchronization at the group level (Strogatz and Stewart 1993; Nadis 2003; Kuramoto 1975; Strogatz 2000).

Synchronization Following Collins, I focus on small groups (2 ≤ n ≤ 6) embedded in larger groups (audiences) in antagonistic situations described above. In these small groups, individuals are in close co-presence where they can clearly see and hear one another. Although they pay attention, and may also speak, to each other, their ties are not necessarily symmetric. As in models of social influence (Friedkin and Johnsen 2011), a weighted tie aij (also) indicates the extent to which i is receptive to j’s influence. Accordingly, a leader is the recipient of strong ties, which he reciprocates by weak ties. This asymmetry of obedience versus authority (Homans 1974) is a novelty in the synchronization model.2 In line with empirical findings (Onnela et al. 2007), the tie strengths in the model are power law distributed. From this family of distributions (Adamic 2000), Zipf’s is used here, with x for the rank order position of a given tie, aij = αx−β , and β in the range 0 ≤ β ≤ 1. Because the strongest ties are to the leader and the weakest ties from him to his followers, β = 1 indicates strong leadership, whereas β = 0 renders all tie strengths equal and characterizes egalitarian groups. When the members focus on one another, their ties to more distant people in the larger group temporarily weaken, and are left out of the model. In models of social influence, the cells in the adjacency matrix are diP vided by their row sums, wij = aij / nj=1 aij (Friedkin and Johnsen 2011). Here this constraint is too tight; the followers have stronger outgoing (i.e. receptive) ties than the leader, and tie strengths increase during preliminary interactions. The strong constraint implies a weaker constraint, however, P wij = aij n/ i6=j aij , which is used at the outset. Subsequently, the α parameter from the Zipf distribution is used to express increasing tie strengths,3 which should stay relatively low to comply with cognitive limitations. Along 2

In the technical literature on synchronization (Arenas et al. 2008), leaders are modeled as fast moving individuals, or pacemakers, which rarely matches social dynamics, though. 3 If not all but only some ties strengthen, their rank order changes, where after the resulting distribution is captured by the two parameters.

3

with ties to specific individuals, the members are committed to their group as a whole or to its current goal, indicated by Ki . The first reading coincides with solidarity, identification and group bonding, whereas the second reading reflects motivation and expected benefits. In general, the two will correlate but can vary independently as a consequence of the group’s history, for example earlier confrontations with the same opponent. Because groups’ history goes beyond the scope of this paper, all member are modeled with the same level of commitment.4 People can be in all kinds of rhythms, but for the model the simplest functional form is used—a wave. Initially, each individual is in his own rhythm, reflecting his mood or temperament, modeled by a distribution of eigenfrequencies g(Ω) that is symmetric around its mean. The model shows how their interactions pull them towards one another’s rhythms (both frequency and phase). If the individuals get into sync, the phases θi (t) of their repetitive movements are the same, whereas fluctuating or large differences indicate a lack of synchronization (Jadbabaie, Motee, and Barahona 2004). The degree to which they are synchronized is indicated by an order parameter, where r = 1 indicates perfect group synchronization and r = 0 a complete absence thereof.5 These considerations come together in Kuramoto’s model of phase shift through mutual alignment, where time indices are left out for clarity: n X dθi wij sin(θj − θi ). = Ωi + K dt j=1

(1)

The model shows that without interactions, group member i will stay in his own rhythm. Previous studies showed that social interaction strengthens ties (Sebanz et al. 2006), in particular if there is a common opponent (Moore 1979; Szell et al. 2010; De Dreu et al. 2010). This empirical finding is expressed by increasing α in the power law distribution, which corresponds to increasing w’s in the model. As a consequence, movements initially remain incoherent; r(t) behaves chaotically, and, following the definition, fluctuating r(t) is set to zero. Figure 1 shows that at a critical threshold αc , there is a phase transition towards nearly maximal synchronization, which is a robust result of Kuramoto’s model (Arenas et al. 2008) that also holds true for 4

Heterogeneous (or temporally changing) commitments can be easily incorporated in the model. If, for example, one individual i has a lower commitment than the rest, ¯ the model shows that the remainder group needs higher commitments, 0 ≤ Ki < K, stronger ties or both to synchronize. In the vast literature on this model (Rodrigues et al. 2016; D¨ orfler and Bullo 2014; Arenas et al. 2008), varying tie strengths are extremely rare, but see Guti´errez et al. (2011), and the model’s dynamics is studied in response to increasing K, with the same value for all nodes. 5 To be precise, the order parameter is a complex function, r(t)eihθ(t)i (Strogatz 2000).

4

Figure 1: When tie strengths (α) increase, synchronization (r) jump starts at a tipping point (αc ), whereas in time (not shown) it increases gradually.

our small groups. Because K and α are multiplicative, it actually does not matter for this outcome if during interactions, tie strengths, commitments, or both increase, which increases the model’s applicability. Comparing different network topologies and tie strength distributions can be done by the algebraic connectivity of the networks, which is the second smallest eigenvalue of the network Laplacian6 , denoted λ2 . If λ2 = 0, the network is disconnected and won’t synchronize, whereas it is maximal in maximally connected networks, which synchronize most easily (at low α or low K): in a dyad, λ2 = 2, and the maximum decreases asymptotically to 1 with increasing network size. In networks with large distances, highly skewed degree distributions or topological bottlenecks, algebraic connectivity is low, and for these networks to synchronize, very high tie strengths (or commitments) are required to compensate for the lack of connectivity (Jadbabaie, Motee, and Barahona 2004; D¨orfler and Bullo 2014). For small fully connected networks (2 ≤ n ≤ 6), the model is also solved 6

The Laplacian L of an asymmetric matrix W with weighted ties (Chung 2005) is 1/2 −1/2 +Φ−1/2 WΦ1/2 L = I − Φ WΦ , where I is the identity matrix, and Φ the Perron vector 2 of W written as a diagonal matrix. For cliques, each cell in the Perron vector equals 1/n; in general these cells can be estimated with PageRank (Prystowsky and Gill 2005).

5

numerically, by randomly drawing sets of eigenfrequencies and initial values 500 times, and implementing them in both the leadership (β = 1) and egalitarian versions (β = 0). The distributions of tipping points are shown in Figure 2. The mean values of the tipping points and the algebraic connectivities of the pertaining networks are highly anti-correlated (-0.944), as expected. Two factors impede synchronization of larger groups. First, in larger groups, members can not simultaneously pay attention to everybody else. Hence the network is sparser and has smaller algebraic connectivity (De Abreu 2007). Second, the heterogeneity of individuals’ eigenfrequencies increases, which shows up in the variation of tipping points in Figure 2. The last boxplot shows, here for the triad, that the tipping points occur at substantially higher tie strengths when the range of eigenfrequencies broadens, here from [1, 3] to [1, 5]. In general, the heterogeneity of eigenfrequencies is measured by the Euclidean norm7 (D¨orfler and Bullo 2014). For a given distribution of eigenfrequencies, the Euclidean norm increases quadratically with n, which makes it progressively more difficult for larger groups to synchronize. Connectivity and heterogeneity imply that for large social networks to synchronize, non-feasibly strong ties or commitments are required. This explains why bursts of violence are committed by fully connected small groups, even though larger groups would have a better chance to beat their opponents. Figure 2 also shows that with increasing n, leaders are more of a hindrance to, than facilitators of, synchronization—a counter-intuitive result that is entirely consistent with the theoretical predictions based on algebraic connectivity. Leaders’ role is to take the initiative, but their groups synchronize neither faster nor at weaker tie strengths than egalitarian groups. When the focal group is synchronized it becomes critical. This means that the group becomes exceptionally sensitive to minor events (Daniels, Krakauer, and Flack 2017), in particular provocations or ephemeral vulnerabilities of the opponent(s), which then trigger the group into violent action. When the group is still asynchronous, and at a distance from criticality, such events may heat up emotions but do not trigger a collective attack. Finally, when in the course of action the fighters get exhausted, their attack unravels.

Discussion Kuramoto’s model yields a fivefold contribution to the discursive theory of violence, and increases its explanatory power through enhanced precision. 7

The Euclidean norm is ||Ω||2 =

qP

i6=j (Ωi

6

− Ωj )2 .

Figure 2: Distributions of tipping points.

7

First, increasing tie strengths and mutual alignment through interactions in close co-presence result in a phase transition towards synchronization. This phase transition is a well-established fact that is used in violence studies for the first time. Second, synchronization renders the focal group critical, such that minor provocations or the leader’s initiative result in a burst of violence, consistent with empirical observations (Collins 2008). Criticality is a particular trait of violent collective actions and team sports, but seems to be rare when there is no opponent. In the sociological literature, critical mass theory has it that criticality with respect to the number of participants is a general feature of collective actions (Marwell and Oliver 1993), which is explained by means of rational choice theory (Centola 2013). This might hold true in relatively simple situations, but in the turmoil of violent collective actions, it is unlikely that people can estimate the marginal utility of their contributions. Synchronization theory, in contrast, has no strong assumptions on rationality, and assumes that people mostly respond to their proximate social environment, like groups of animals on the move (Vicsek and Zafeiris 2012). A next step in its development is empirical testing, which can be done on video recorded violence. Third, the model explicates that tie strengths together with network topology—summarized by algebraic connectivity—have to compensate for individuals’ heterogeneity (D¨orfler and Bullo 2014). In other words, a network with low algebraic connectivity keeps collective efficacy low, no matter how hard individuals try their best. This explains Collins’ (2004) observation that in some interaction rituals, emotional energy is drained, and fourth, why violent groups are typically small rather than large, as Collins (2008) observed in a wide range of visual data. This seems to contradict the fact that at large open spaces, for example Tahrir square in Cairo, visual cues make it possible for large numbers of protesters to align simple gestures, like holding up shoes. But do they all fight? Because of the distances on the square, most of their visual ties are weak, and their synchronization is highly inaccurate. In physical confrontations with the police, only small groups engage, in line with the model. When modeling the effect of increasing K in large networks (Arenas et al. 2008), it turns out that synchronization starts in relatively well-connected clusters that are much smaller than the overall network, consistent with the argument made here. Fifth, the model shows that without sheer luck, leaders can not advance synchronization through mutual alignment, which also seems counterintuitive. An alternative mechanism of synchronization that leaders may use is choreography, popular for military marches throughout the world and capable of synchronizing large numbers of individuals. When soldiers march, 8

however, they have no eye contact and their ties are weak. They do not overcome their fear of combat without additional training. Choreography has a weaker psychological effect than mutual alignment (McNeill 1995), and soldiers’ preparations for violence require more effort than marching. The model sharpens our understanding of leaders’ role in violence by limiting it to their main contribution, namely to take the initiative in preliminary organization and attack (Glowacki and von Rueden 2015). This result could only be obtained by means of a model that disentangles the effects of leaders and their ties, which is not possible in field studies. The model treats motivations by commitments, but there is more to it. Most people are conditional cooperators to public goods (Chaudhuri 2011), willing to contribute if many others contribute as well. When it comes to violence, however, it is remarkable that small numbers of people are willing to take disproportional high risks for the larger group’s victory. From other studies we can glean that combatants may expect reputation benefits in terms of attributed status or prestige (Henrich and Gil-White 2001; Homans 1974). In this vein, the very presence of an audience can increase violence if combatants perceive it as a form of moralistic punishment (Kurzban, DeScioli, and O’Brien 2007). They may also experience an enhanced sense of belonging (Whitehouse et al. 2014; Baumeister and Leary 1995), but this is of little explanatory value because they may satisfy this need in different ways by, for example, dancing synchronously (McNeill 1995) and thereby preclude injury, whereas others who do commit to violence may have different motives altogether. For sure, investigating long term fitness trade-offs (Glowacki and Wrangham 2015) is more conductive to explanation than short term motives. This issue might be addressed by expanding the model with a group’s broader network embedding and history (Papachristos 2009; Glowacki et al. 2016), which seems also important to predict the severity of violence.

References Adamic, L. A. (2000). Zipf, power-laws, and Pareto–a ranking tutorial. Technical report, Xerox Palo Alto Research Center, Palo Alto, CA. http://ginger. hpl.hp.com/shl/papers/ranking/ranking.html. Arenas, A., A. D´ıaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou (2008). Synchronization in complex networks. Physics Reports 469, 93–153. Bandura, A. (2000). Exercise of human agency through collective efficacy. Current Directions in Psychological Science 9, 75–78. Baumeister, R. F. and M. R. Leary (1995). The need to belong: Desire for interpersonal attachments as a fundamental human motivation. Psychological

9

Bulletin 117, 497–529. Centola, D. (2013). Homophily, networks, and critical mass: Solving the start-up problem in large group collective action. Rationality and Society 25, 3–40. Chaudhuri, A. (2011). Sustaining cooperation in laboratory public goods experiments: a selective survey of the literature. Experimental Economics 14, 47–83. Chung, F. (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics 9, 1–19. Collins, R. (2004). Interaction Ritual Chains. Princeton, NJ: Princeton University Press. Collins, R. (2008). Violence: A Micro-Sociological Theory. Princeton: Princeton University Press. Daniels, B. C., D. C. Krakauer, and J. C. Flack (2017). Control of finite critical behaviour in a small-scale social system. Nature Communications 8, 14301. De Abreu, N. M. M. (2007). Old and new results on algebraic connectivity of graphs. Linear Algebra and its Applications 423, 53–73. De Dreu, C. K., L. L. Greer, M. J. Handgraaf, S. Shalvi, G. A. Van Kleef, M. Baas, F. S. Ten Velden, E. Van Dijk, and S. W. Feith (2010). The neuropeptide oxytocin regulates parochial altruism in intergroup conflict among humans. Science 328, 1408–1411. D¨orfler, F. and F. Bullo (2014). Synchronization in complex networks of phase oscillators: A survey. Automatica 50, 1539–1564. Durkheim, E. (1912). Les formes ´el´ementaires de la vie religieuse. Paris: Presses Universitaires de France. Fessler, D. M. and C. Holbrook (2014). Marching into battle: synchronized walking diminishes the conceptualized formidability of an antagonist in men. Biology Letters 10, 20140592. Fischer, R., R. Callander, P. Reddish, and J. Bulbulia (2013). How do rituals affect cooperation? Human Nature 24, 115–125. Friedkin, N. E. and E. C. Johnsen (2011). Social Influence Network Theory: A Sociological Examination of Small Group Dynamics. Cambridge, MA: Cambridge University Press. Glowacki, L., A. Isakov, R. W. Wrangham, R. McDermott, J. H. Fowler, and N. A. Christakis (2016). Formation of raiding parties for intergroup violence is mediated by social network structure. Proceedings of the National Academy of Sciences 113, 12114–12119. Glowacki, L. and C. von Rueden (2015). Leadership solves collective action problems in small-scale societies. Philosophical Transactions Royal Society B 370, 20150010.

10

Glowacki, L. and R. Wrangham (2015). Warfare and reproductive success in a tribal population. Proceedings of the National Academy of Sciences 112, 348–353. Guti´errez, R., A. Amman, S. Assenza, J. G´omez-Garde˜ nes, V. Latora, and S. Boccaletti (2011). Emerging meso- and macroscales from synchronization of adaptive networks. Physical Review Letters 107, 234103. Henrich, J. and F. J. Gil-White (2001). The evolution of prestige: Freely conferred deference as a mechanism for enhancing the benefits of cultural transmission. Evolution and Human Behavior 22, 165–196. Homans, G. C. (1974). Social behavior: Its elementary forms (2nd ed.). New York: Harcourt Brace Jovanovich. Jadbabaie, A., N. Motee, and M. Barahona (2004). On the stability of the Kuramoto model of coupled nonlinear oscillators. In Proceedings of the American Control Conference, pp. 4296–4301. Klinger, D. (2004). Into the Kill Zone: A Cop’s Eye View of Deadly Force. San Francisco: Jossey-Bass. Kuramoto, Y. (1975). Self-entrainment of a population of coupled non-linear oscillators. In International Symposium on Mathematical Problems in Theoretical Physics, Volume 39, pp. 420–422. Kurzban, R., P. DeScioli, and E. O’Brien (2007). Audience effects on moralistic punishment. Evolution and Human Behavior 28, 75–84. Levine, M., P. J. Taylor, and R. Best (2011). Third parties, violence, and conflict resolution. Psychological Science 22, 406–412. Marwell, G. and P. Oliver (1993). The Critical Mass in Collective Action. Cambridge, MA: Cambridge University Press. McNeill, W. H. (1995). Keeping Together in Time: Dance and Drill in Human History. Cambridge, Mass: Harvard University Press. McPherson, J. M., L. Smith-Lovin, and J. M. Cook (2001). Birds of a feather: Homophily in social networks. Annual Review of Sociology 27, 415–444. Moore, M. (1979). Structural balance and international relations. European Journal of Social Psychology 9, 323–326. Nadis, S. (2003). All together now. Nature 421, 780–782. Nassauer, A. (2016). From peaceful marches to violent clashes: A microsituational analysis. Social Movement Studies 15, 515–530. Onnela, J. P., J. Saram¨ aki, J. Hyv¨onen, G. Szab´o, D. Lazer, K. Kaski, J. Kert´esz, and A. L. Barab´ asi (2007). Structure and tie strengths in mobile communication networks. Proceedings of the National Academy of Sciences 104, 7332–7336.

11

Papachristos, A. V. (2009). Murder by structure: dominance relations and the social structure of gang homicide. American Journal of Sociology 115, 74– 128. Prystowsky, J. M. and L. Gill (2005). Calculating web page authority using the PageRank algorithm. https://pdfs.semanticscholar.org, Accessed: 2017-0323. Reddish, P., R. Fischer, and J. Bulbulia (2013). Lets dance together: Synchrony, shared intentionality and cooperation. PLoS ONE 8, e71182. Rodrigues, F. A., T. K. Peron, P. Ji, and J. Kurths (2016). The Kuramoto model in complex networks. Physics Reports 610, 1–98. Sampson, R. J., S. W. Raudenbush, and F. Earls (1997). Neighborhoods and violent crime: A multilevel study of collective efficacy. Science 277, 918–924. Sebanz, N., H. Bekkering, and G. Knoblich (2006). Joint action: Bodies and minds moving together. Trends in Cognitive Sciences 10, 70–76. Strogatz, S. H. (2000). From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20. Strogatz, S. H. and I. Stewart (1993). Coupled oscillators and biological synchronization. Scientific American 269 (6), 102–109. ´ G´ Swann, W. B., J. Jetten, A. omez, H. Whitehouse, and B. Brock (2012). When group membership gets personal: A theory of identity fusion. Psychological Review 119, 441–456. Szell, M., R. Lambiotte, and S. Thurner (2010). Multirelational organization of large-scale social networks in an online world. Proceedings of the National Academy of Sciences 107, 13636–13641. Vicsek, T. and A. Zafeiris (2012). Collective motion. Physics Reports 517, 71– 140. Weenink, D. (2014). Frenzied attacks: A micro-sociological analysis of the emotional dynamics of extreme youth violence. The British Journal of Sociology 65, 411–433. Whitehouse, H., B. McQuinn, M. Buhrmester, and W. B. Swann (2014). Brothers in arms: Libyan revolutionaries bond like family. Proceedings of the National Academy of Sciences 111, 17783–17785.

12