A theory of targeted and indiscriminate state violence in networks

Abstract

Theories explaining why states choose to use targeted or indiscriminate violence against civilians hinge on the state’s capacity to gain information about who to target and its ability to do enough damage to prevent defection to the rebel’s side. In contrast to these theories, I show that the choice of strategy depends on the characteristics of the community experiencing the violence, not the state employing it. This article argues that even when states can target certain civilians, they may choose to employ indiscriminate violence owing to the characteristics of the civilians’ social network structure. The state’s optimal strategy of violence is driven by two factors: the degree distribution of civilians’ social networks and the correlation between citizens’ motivation to leave a network and citizens’ value to other nodes in the network. When the degree distribution is uniform, and motivation and value are positively correlated, indiscriminate violence is more often preferred.

Keywords

repression social networks state violence

Introduction

The use of indiscriminate violence by states is surprising because it can cause mobilization in favor of insurgents, but states employ it anyway (Mason & Krane, 1989; Goodwin, 2001; Kalyvas, 2006). Figure 1 shows that states continuously use violence against civilians, and that a large proportion of this violence is indirect in the form of air strikes, shelling and chemical weapons. The use of indiscriminate violence varies by time and location, provoking the question of under what conditions states prefer this strategy. Existing arguments focus on the state’s capacity to effectively use violence, the role of limited information (Kalyvas, 2006), and the inability of a rebel group to protect civilians (Zhukov, 2014), but predict that indiscriminate violence should be used rarely or only when conflicts end quickly, and generally view indiscriminate violence as a second-best option for the state. I offer a theory suggesting that indiscriminate violence can be the state’s preferred strategy even when it can engage in targeted violence, conditional on the visible characteristics of the structure of civilians’ social networks.

I study a formal social network model of violence and displacement in order to understand the choice of the state to use targeted or indiscriminate violence against a community of civilians in order to remove them from a territory. I focus on the effects of network structure because social networks play an important role in determining the effects of repression and violence strategies on behavior (Siegel, 2011). Violence necessarily has impacts beyond its direct victims because it reshapes social structures through the removal of network members. I analyze the effects of the change in network structure in response to violence directly and examine how they determine the strategic behavior of the state.

The theory captures a setting in which a state has chosen a particular village or city to forcibly remove the

population from and can either strike the village in an indiscriminate fashion or remove particular members within it.¹ This territory contains a social network of civilians, which determines how they will respond to the violence. Civilians weigh their own incentives to leave the territory against the value of their ties in the network. By using violence, the state can manipulate these ties. The state’s choice is a gamble: it can target a certain influential member of the community and hope that other network members respond to this removal, or remove network members at random and potentially cause influential civilians and other civilians to leave in response. To directly compare the two strategies, I consider the case when both eliminate one node in expectation.

I show that the characteristics of civilian social networks shape the optimal strategy for the state. The state’s choice is determined by both the network structure and the distribution of individual attributes of civilians in the network. The results of the model demonstrate that indiscriminate violence can be preferred to targeted violence based on characteristics at two levels. At the network level, as the distribution of degree among nodes becomes more uniform, indiscriminate violence becomes more valuable. At the individual level, as value for others in the network and motivation to leave the network are more positively correlated throughout the community (for example, when the civilians with the greatest network ties also have the largest benefits from leaving), the unraveling effect of indiscriminate violence is greater. Therefore, I argue that, along with Kalyvas’s (2006) list of conditions favoring indiscriminate violence (imbalances of power, low resources or low information), under certain conditions and holding all else equal, social network characteristics make indiscriminate violence a strategic choice.

Substantively, the analysis indicates that strategies of violence can have heterogeneous effects depending on the composition of the community experiencing violence. This differs from past theories that view the state’s choice of violence as a function of the state’s levels of information and technology. I show that the characteristics of those receiving the violence dictate the optimal strategy as well. At the group level, more uniform networks will provoke more indiscriminate violence. Within groups, leaders that are more mobile or have greater ties to insurgency and thus greater motivation to leave for other opportunities or to mobilize are more likely to provoke indiscriminate violence. At the state level, the theory indicates that the state’s optimal strategy can vary within time periods and across locations.

I then extend the model in two ways. First, I incorporate potential mobilization against the state as a result of driving away nodes from the network. The effect that this extension has on the model results depends on the distribution of motivation to leave the network, but in general makes indiscriminate violence more attractive. Second, I extend the result on degree distribution to larger networks and show through simulations that a similar result holds in a more abstract network environment.

The study builds upon Siegel (2011), in which simulations are used to demonstrate how mobilization responds to targeted or random removal of nodes from a network. I study a similar network setup, but employ simpler network structures in order to obtain an analytical solution. I also focus on understanding the state’s strategic choice between targeted and indiscriminate violence, whereas Siegel (2011) focuses on the collective response of the victims.

Outside of the literature on violence repression, this article makes contributions to the general study of social networks in political science. I consider networks in which edges dissuade instead of motivate, bridging a gap between literature in psychology and sociology on affectual ties and formal network analysis (Slater, 1963; Freud, 1975; Goodwin, 1997). By directly considering the spillover effects of node removal (the effects of a node being removed on the behavior of other nodes) and asymmetries of influence in networks, the analysis is also related to work on collective behavior, thresholds and optimal seeding (Granovetter, 1978; Ballester, Calvó-Armengol & Zenou, 2006; Banerjee et al., 2014; Larson, Lewis & Rodriguez, 2017; Akbarpour, Malladi & Saberi, 2018).

Next, I introduce the theory and the findings in more depth before formalizing it. I then introduce the extensions, analyzing the effect that incorporating mobilization for rebel groups in the state’s expected utility has on the findings. Then I extend the results on degree distribution to n-node networks and use simulations to establish a more general result. Subsequently, I offer a translation of the formal findings that is of use to empirical researchers interested in studying the relationship between strategies of violence and social network structure. Following this, I briefly conclude.

An overview of the theory

In the model a state seeks to remove civilians from a network through violence. This assumption of the goal of violence follows from the empirical finding that states use violence against civilians in order to manipulate their locations and social structures (Greenhill, 2008; Steele, 2011; Valentino, 2014; Lichtenheld, 2020). Displacement is used as a tool by armed groups to take control of territory (Balcells & Steele, 2016). In the model, removing civilians from a network can be seen as either forcing them to leave the geographic area where the network is located or causing them to change their behavior by no longer engaging in some activity that is damaging to the state.² Such a strategy may be likely when a territory is valuable economically or strategically, or owing to the dynamics of the conflict (Lichtenheld, 2020). Regardless, the high level of forced migration in civil wars provides an empirical basis for this assumption.

The network in the model is a representation of civilians (also called network members, agents, or nodes) in a territory that is valuable to the state during conflict. Before violence occurs all members of the network are in this territory and can be targeted by the state. The state first chooses a strategy of violence, either targeted or indiscriminate, which eliminates nodes in the network. If the state chooses targeted violence it pays a generic cost. This cost can be thought of as an ex ante cost the state must pay in order to use the strategy, such as gathering information on the network.³

After the choice of strategy civilians who are not eliminated can remain in or leave the territory. Network edges between civilians serve to keep nodes grounded in a place, and breaking edges by removing civilians makes flight, or leaving the network, more likely. When making the choice to leave, network members weigh the net benefit to leaving against the value of their ties in the network. The goal of the state is to choose the strategy that has the greatest impact on the network by removing the greatest number of civilians in expectation, through either killing them or removing network members that keep them grounded in place and causing them to leave the network. Leaving the network is any action that leads civilians to no longer inhibit the state’s plans in the territory, such as fleeing, joining a group in another location, joining a group to fight back against the state or submitting to the state’s cause.

I model the choice of a network member to stay or leave as a comparison between the value that other network members have for the civilian and one’s motivation to leave the network. Network ties can only affect a civilian’s choice through their value. In general, this captures the value that the social network has for an individual, in terms of affectual ties (Goodwin, 1997). In other words, people who form ties often like being around each other, and leaving a network and these ties can be costly.

Motivation to leave the network can capture several incentives: civilians may consider joining a rebel group, which would require them to leave the territory. They also may have incentives to move through social contacts outside of the territory or labor market opportunities. The generic net benefit here can capture any force that may lead a civilian to move from their home or change their behavior toward actions less damaging to the state. In this model, both of these parameters are taken as exogenous and the only piece that varies in response to violence is network structure through the elimination of nodes. Because the purpose of this model is to generate predictions on the strategies of violence used by states, I do not consider more complex civilian strategies. Instead, the stochastic actions of civilians reflect reasonable levels of information for the state about what civilians may do in response to violence.

I restrict the state’s ability to target civilians by assuming that it can only target the network member with the highest level of value for other civilians in the network. I make this restriction because it assumes a reasonable level of information for the state: it can observe some characteristics of individuals in the community that proxy for value, such as status or profession, but cannot observe individual-level preferences. Moreover, it is easier for the state to identify high-status members of communities like community or religious leaders, whereas less influential members are less identifiable.

In this setting, I provide two key results: first, as the distribution of ties in a network changes, so does the optimal strategy for the state. Targeted violence is more efficient when the targeted node has a higher degree than other nodes in the network. This is due to the heightened ability of central nodes to cause spillovers when they are eliminated: when the targeted node has a high degree, removing it removes the largest possible number of edges from the network. Moreover, it is harder to cause this node to leave the network in response to the probabilistic removal of any other node. Because indiscriminate violence eliminates this node with a probability less than one, it becomes less valuable relative to targeted violence, even when costly.

Second, I show that as value in the network and motivation to leave the network are negatively correlated, targeted violence becomes more valuable relative to indiscriminate violence (and vice versa). The logic of this result comes from the potential spillovers that can occur from the elimination of a certain agent. When value and motivation are negatively correlated, targeted violence is more effective because eliminating an influential but unmotivated agent removes it from the network when it otherwise wouldn’t leave while also leading the motivated agents to leave the network. In this case, the state will be willing to pay the costs of targeting because the expected spillovers from targeting an influential node are much larger than those of eliminating an agent that is already motivated to leave and doesn’t greatly impact the behavior of other network members.

Conversely, when an agent is highly motivated to leave a network and also is valuable for others in the network, that agent will be eliminated under costly targeted violence. However, that agent is also likely to leave the network in response to the elimination of any other node because of its high level of motivation. Thus, the state can employ indiscriminate violence, not pay a cost, and still have this node leave the network in expectation. On the other hand, when a network member is both not influential and not motivated to leave, it is unlikely to leave in response to the targeting of the most influential node. However, because it has a chance to be eliminated directly under indiscriminate violence, the probability of it leaving the network is higher if the state chooses this strategy. Thus, indiscriminate violence has greater potential for spillover effects as the correlation between the two attributes increases.

This demonstrates that, while targeted violence is more effective for the state under a variety of conditions, in some cases indiscriminate violence can generate the same level of destruction as targeted violence without needing to pay the cost of gathering information for targeting. This suggests that in situations where the state knows that the leader has ties outside of the village that it can easily access or is highly connected to a rebel group that could offer shelter, indiscriminate violence can be as effective as targeted violence. Later, I discuss other possible operationalizations of the abstract variables considered in the theory.

In general, these results demonstrate the value of considering the effects of violence in a network setting: The effect of the elimination of an agent on the network as a whole is a function of that agent’s attributes, in terms of its value to another agent in the network, and the attributes of other agents, in terms of their willingness to leave the network. When eliminating one agent causes more agents to leave the network relative to the elimination of a different agent, targeting this agent is more valuable to the state. Conversely, when the elimination of agents has a similar expected impact on the network after violence, randomly eliminating agents can be as or more effective.

In the baseline model, I do not explicitly consider the role of rebel groups in this interaction. This is not because the actions of rebel groups are not important, but because the model takes the incentives of the state to clear the network of civilians as a given, treating the interaction between the state and the rebel group as exogenous. This allows focus on the interaction between states and the structure of civilian social networks. However, this is not to say that the theory does not speak to situations in which the strategy of the state is affected by rebel groups: there are clear cases where state violence is used indiscriminately against civilians in order to defeat rebel groups (Downes, 2008). However, there are also cases of states engaging in clearance operations against civilians to acquire resources or to establish greater legitimacy in the territory (Azam, 2002; Lichtenheld, 2020). The state may have incentives to clear civilians from territories for a multitude of reasons. The theory offered here helps to understand the type of violence employed in these situations. However, rebel groups do have an influence on patterns of state violence (see, for example, Valentino, 2014; Azam & Hoeffler, 2002; Schutte, 2017), so I consider an extension in which nodes that leave the network can mobilize in favor of the rebel group. How this addition changes the strategy of the state depends on the distribution of motivation to leave the network for the nodes in the network, but in general, as mobilization becomes more damaging to the state, the relative value of indiscriminate violence increases.

I proceed by distinguishing between targeted and indiscriminate violence in simple networks. I consider variations of a three-agent network, determining the optimal strategy of violence by the state.⁴ I then incorporate rebel groups and then extend the model into an n-node setting.

Comparing indiscriminate and targeted violence in networks

Setup

I consider a simple network model of a state’s choice between targeted and indiscriminate violence. In the model, the state is confronted with a three-node network. The goal of the state is to remove as many nodes as possible from the network.⁵ The state can choose between targeted violence, in which a single node is eliminated with certainty, and pay cost $κ > 0$ , or indiscriminate violence, where it is expected that one node will be eliminated, and all nodes can be eliminated with equal probability.⁶ This operationalization of indiscriminate violence captures the ex ante decision that the state faces, instead of an ex post realization of which node was removed. This is closer to a realistic depiction of the trade-off that the state faces. For every node that leaves the network the state gains utility normalized to 1.

Formally, the network is represented with a set of three individuals, {a, b, c} with a $3 \times 3$ adjacency matrix g where, for generic nodes i and j, $g_{i j} = 1$ is an edge between $i \neq j$ in the network. I assume that all edges are undirected, meaning $g_{i j} = 1 g_{j i} = 1$ . Edges are weighted by $α_{i j} > 0$ , which is the value to i of maintaining its tie to j. Intuitively, this represents how valuable it is for i to have j in its network, in terms of the affection i has for j or the economic value that j provides for i. I_i is defined to be the value of ties to other actors in the network for i, such that $I_{i} = \sum_{i \neq 1}^{g} α_{i j}$ . $α_{i j}$ need not equal $α_{j i}$ . Civilians decide to leave the network by comparing the cost or benefit from leaving, $θ_{i}$ ⁷, with the strength of their ties in the network, I_i . $Θ_{i}$ is drawn from a uniform distribution with support $[- 1, σ_{i}]$ . An increase in $σ_{i}$ increases both the mean and variance of the motivation level for i. Thus, a civilian remains in the network if $I_{i} \geq θ_{i}$ . In this interaction, the state is fully informed about the network structure and the value of ties between nodes, α but only knows the distribution of the individual level costs or benefits to leaving, $Θ$ .

For simplicity, I restrict the basis on which the state can choose the node it eliminates when using targeted violence. Formally, I define T_i as the total value of node i to all other nodes in the network, where $T_{i} = {\sum_{i \neq j}^{g} α}_{j i}$ I assume that the state targets the node with the highest total value in the network. In other words, this is the node that has the highest sum of weighted ties in the network.

I consider three-node networks, and assume without loss of generality that nodes can be ordered as $T_{a} \geq T_{b} \geq T_{c}$ . This limits the cases of networks to those where a is always connected. I vary the number of edges in the networks, analyzing the effects of violence on networks with three, two, and one edges. So that networks are equal except for their structure, the following constraints must be imposed: $σ_{c} \leq 0$ , so that c remains in the network when the network only has an edge between a and b, $σ_{b} \leq α_{b a}$ so that b remains in the network when its only edge is with a, and $σ_{a} \leq α_{a b}$ for the same reason. I proceed by outlining the conditions under which the state would prefer to target a single node to eliminate one node in expectation in these network settings. I generate results in terms of the constraint on the cost of targeted violence, $κ$ , for the state to prefer targeted violence. I show the relationship between network structure and constraints on this cost in Table I and describe the results below. The full derivation of these results are available in the Appendix.

Table I.

The results of the analysis for different levels of network density. The bottom row displays the constraints of $κ$ for targeted violence to be preferred

Fully connected	Network with two edges	Network with one edge
$\begin{array}{l} κ \leq \frac{23}{27} (1 - \frac{1 + α_{b c}}{1 + σ_{b}}) - \frac{4}{27} (1 - \frac{1 + α_{a c}}{1 + σ_{a}}) - \frac{2}{27} ((1 - \frac{1}{1 + σ_{b}}) + (1 - \frac{1}{1 + σ_{a}})) \end{array}$	$\begin{array}{l} κ \leq \frac{21}{27} (1 - \frac{1}{1 + σ_{b}}) - \frac{2}{27} (1 - \frac{1}{1 + σ_{a}}) - \frac{4}{27} (1 - \frac{1 + α_{a c}}{1 + σ_{a}}) \end{array}$	$\begin{array}{l} κ \leq \frac{21}{27} (1 - \frac{1}{1 + σ_{b}}) - \frac{6}{27} (1 - \frac{1}{1 + σ_{a}}) \end{array}$

Analysis

Fully connected network

In a fully connected network, all nodes have edges to each other and the elimination of any node can affect whether the other nodes remain. The structure of this network is shown in the first panel of Table I. Using targeted violence the state eliminates node a and pays cost $κ$ . b leaves the network if the edge between a and b is essential for b, or $θ_{b} > α_{b c}$ , which occurs with probability

1 - \frac{1 + α_{b c}}{1 + σ_{b}}

So the payoff to targeted violence in this case is:

1 + (1 - \frac{1 + α_{b c}}{1 + σ_{b}}) - κ

Under indiscriminate violence, the state eliminates one agent in expectation, such that each agent has a 1/3 probability of being eliminated. Note that these probabilities are independent: more than one agent can be eliminated under indiscriminate violence. However, it is also possible that the state will fail to eliminate a single agent. The tension between the two strategies comes from the fact that with some probability indiscriminate violence can eliminate more nodes than targeted violence, but also may not eliminate any nodes or may eliminate nodes that have a comparatively smaller effect on others’ behavior. In this network, eliminating a has the potential to cause b to leave the network, and vice versa. Moreover, indiscriminate violence has some probability of eliminating c, whereas targeted violence does not. The full presentation of these results is shown in the Appendix. The payoff to indiscriminate violence in this case is:

1 + \frac{4}{27} ((1 - \frac{1 + α_{b c}}{1 + σ_{b}}) + (1 - \frac{1 + α_{a c}}{1 + σ_{a}})) + \frac{2}{27} ((1 - \frac{1}{1 + σ_{b}}) + (1 - \frac{1}{1 + σ_{a}}))

Network with two edges

When the edge between b and c is removed, both strategies become more efficient for the state. These results are shown in the Appendix. Under targeted violence, the state eliminates a and b leaves the network if $θ_{b} > 0$ , which is a lower constraint than in the network with three edges, as c no longer has value for b. The payoff to targeted violence is

1 + (1 - \frac{1}{1 + σ_{b}}) - κ

which is strictly higher than the payoff in the full network.

Indiscriminate violence also becomes more efficient relative to the three-edge network, through the same mechanism as it also has a positive probability of eliminating a. The payoff is

1 + \frac{6}{27} (1 - \frac{1}{1 + σ_{b}}) + \frac{2}{27} (1 - \frac{1}{1 + σ_{a}}) + \frac{4}{27} (1 - \frac{1 + α_{a c}}{1 + σ_{a}})

However, compared with targeted violence, indiscriminate violence is a worse strategy than in the three edge network because the edge between c and a still influences a in the case where b is eliminated by chance and so the relative payoff to targeted violence is higher.

Network with one edge

Finally, when the edge between a and c is removed, the payoff to targeted violence does not change. However, indiscriminate violence becomes more efficient because the elimination of b is more likely to cause a to leave the network, as c no longer has value for a. This is shown formally in the Appendix. The payoff to indiscriminate violence in this case is:

1 + \frac{6}{27} (1 - \frac{1}{1 + σ_{b}}) + \frac{6}{27} (1 - \frac{1}{1 + σ_{a}})

Results

Overall, this analysis has a fundamental and simple finding: if targeted violence is more costly than indiscriminate violence, there are conditions under which indiscriminate violence is a best response to the structure of the network and the relationship between the value in a network and motivation. How low this cost needs to be for targeted violence to be preferred is dictated by several factors.

First, indiscriminate violence becomes more efficient as value in the network, T_i and motivation to leave the network, $Θ_{i}$ are more positively correlated. In other words, when the most influential agents are also the most highly motivated to leave the network, the state more often prefers to employ indiscriminate violence. This statement follows from the fact that the right-hand sides of the constraints on $κ$ are decreasing in $σ_{a}$ . This means there is a stricter constraint on the costs of targeted violence when a is motivated: the cost of targeted violence must be lower. The intuition for this result is straightforward: as a becomes more motivated, she is more likely to leave the network in response to the elimination of any agent. Thus, the state can remove a from the network without paying a cost by using indiscriminate violence. Conversely, when $σ_{b}$ is large, targeted violence is more efficient because b is more likely to leave when a is targeted.

Second, the distribution of degree in the network affects whether indiscriminate or targeted violence is the preferred strategy. Targeted violence has its highest value relative to indiscriminate violence when the node with the highest value has a higher degree than other nodes. Specifically, the difference in expected payoffs between targeted and indiscriminate violence is highest in the two-edge network, and second highest in the one-edge network except when $α_{a c}$ is much larger than $α_{b c}$ or $σ_{a}$ is much smaller than $σ_{b}$ , representing the case when a stays in the network if b is eliminated, which makes targeted violence more valuable. The intuition for this result is as follows: when each node has the same degree, the removal of any node has the potential to influence whether or not another node leaves the network directly, and all nodes have value for each other, making it more likely for each node to stay in the network. However, when the node with the highest level of value has more edges than other nodes, then its elimination has a greater potential to directly cause other nodes to leave. Moreover, when this node has a higher degree, it is harder to make it leave the network in response to the elimination of other nodes because they have value for the node. Consider the network with two edges. In this network, if a is eliminated, b leaves the network if its level of motivation is positive. However, if under indiscriminate violence c is eliminated, b’s behavior is only influenced if a’s motivation to leave is larger than the value of the edge between a and b, which is never true by assumption. Because a has shorter paths to other nodes in the network, the conditions for targeted violence to have spillover effects are less stringent than those for indiscriminate violence in such a setting. However, in the network with three edges this is not the case. In this network, eliminating a only causes b to leave if its motivation to leave is larger than the value of the edge between b and c, and so the bar is higher for targeted violence to have spillover effects. Because of this, the payoff to targeted violence is always weakly largest in the two edge network. I explore the relationship between the distribution of degree in a network and the optimal strategy of violence further in simulations.

The role of rebel groups

A key assumption in the above setting is that a node being eliminated is just as valuable as a node choosing to leave the network. This assumption may be tenable in settings where a state seeks to access a valuable territory and is not concerned about the downstream effects of villagers being forced away from their land. It is also applicable in situations where villagers can aid rebels by funneling them key resources only accessible in their place of residence. In these cases, the state benefits from driving villagers away. However, in conflicts where a rebel group poses a credible threat to the state, forcing villagers away rather than directly eliminating them may be more costly, as these victims of violence may begin to aid rebel forces, either by joining them directly or providing them with information. This suggests that driving villagers away in some cases may be costly to the state. I consider the effect of that assumption here. I show that the results of the baseline model hold when incorporating rebel groups into the analysis.

Specifically, I adapt the above model to incorporate a cost to driving a node away from a network rather than directly eliminating them. I assume that a node that is driven away from the network chooses to conspire with a rebel group against the state in the larger conflict. To model this, I amend the payoff that the state receives from driving away a node to be 1 – 1/P, where $P > 1$ is the population of possible rebel group members and this term represents the marginal impact of a node joining the rebel group. The assumption is that every node that is not eliminated but leaves the network increases the chance that the state loses the conflict. As P becomes smaller, causing a node to leave the network is less valuable for the state and makes directly eliminating nodes more valuable. I determine the moderating effect of this externality on the state’s preferred strategy.

I show the constraints on $κ$ necessary for targeted violence to be preferred in the Online Appendix. The effect of incorporating antistate mobilization into the model depends on the distributions of $Θ_{a}$ and $Θ_{b}$ . As P decreases, meaning that the marginal impact of a node joining the rebel group is greater, targeting becomes more effective when $σ_{a}$ is large. This is because a leaving the network owing to b being eliminated becomes less valuable, and this can only take place under indiscriminate violence. Conversely, when $σ_{b}$ increases and $α_{b c}$ decreases, a decrease in P makes indiscriminate violence more valuable, as b is more likely to leave in response to targeted violence, but the value of them leaving the network is less.

In general, increasing the effect of mobilization in the form of decreasing P increases the relative value of indiscriminate violence more often than it increases the relative value of targeted violence. For a decrease in P to increase the value of targeted violence in the fully connected network, it must be true that

σ_{a} > \frac{23 α_{b c} - 4 α_{a c} - 21 σ_{b} - 4 α_{a c} σ_{b}}{15 σ_{b} - 23 α_{b c} - 6}

and

σ_{b} < \frac{1}{15} (6 + 23 α_{b c})

Consider the case where $α_{a c} = 0.6, α_{b c} = 0.5$ , and $σ_{b} = 1$ . Then $σ_{a}$ must be greater than $5.72$ for a decrease in P to increase the value of targeted violence. This can be understood in relation to the findings in the initial model setup: what made targeting valuable for the state was the ability to selectively eliminate the node that would have the greatest spillover effects on the rest of the network. However, when these spillovers become less valuable to the state’s goals, indiscriminate violence becomes relatively more effective because it causes spillovers less often. Thus, it is more likely that states will resort to indiscriminate violence against civilians when the backlash among connected nodes is damaging to the state.

n-Node networks

Thus far, the analysis has focused on small networks with the same number of nodes and variation in the number of edges. However, as networks become larger they can take on a variety of forms, and two networks with the same level of mean density can have different effects on the behavior of nodes. Here, I focus on establishing a result for n-node networks. I concentrate on networks characterized by connected components of essential edges, where the removal of a node within a component leads the entire component to exit the network.⁸ I hold the number of nodes and components constant but allow for variation in the number of nodes within components.

Connecting this variation to real-world networks, a network with a particularly large component would represent a tight-knit, homogeneous setting where most network members are acquainted and sensitive to each other’s behavior. For example, small villages with large kinship networks would probably display a large component of essential ties. However, when considering essential ties, many networks may be splintered into several small components. For example, networks in larger villages and cities, or networks in which nodes have heterogeneous characteristics, are likely to have many components.

Specifically, I consider fragile networks with n nodes, and j components, where all edges in the network are essential, and all nodes in each component are connected, such that the elimination of an edge removes an entire component. I focus on variation in the distribution of nodes in components. I assume that targeted violence eliminates the node with the largest degree, which by assumption is any node in the largest component. Let the size of component i be c_i and denote the largest component as c_b . Then, the payoff to the state of targeted violence is $c_{b} - k$ , where k is once again the cost of targeting.

Again, assume that indiscriminate violence eliminates any node with probability 1/n. The elimination of any node also leads its component to exit the network, so the payoff to indiscriminate violence will be written in terms of component size. Specifically, the probability that a member of component i is eliminated is

1 - {(\frac{n - 1}{n})}^{c_{i}}

and so the payoff to indiscriminate violence can be written as

\sum_{i = 1}^{j} (1 - {(\frac{n - 1}{n})}^{c_{i}}) c_{i}

The state therefore prefers indiscriminate violence when

k > c_{b} - \sum_{i = 1}^{j} (1 - {(\frac{n - 1}{n})}^{c_{i}}) c_{i}

I generate results through simulations. Specifically, I generate networks with n = 10,000 nodes and j = 50 components, and randomly vary the sizes of the components over 10,000 simulations. With these constructed networks, I generate the expected payoff to the state of either strategy, and compare the difference in these expected payoffs, or the level that k would need to be for indiscriminate violence to be preferred.

First, I note that the payoff for either strategy increases as the size of the largest component increases. Figure 2 shows the relationship between the size of the largest component (which is also the payoff for targeted violence, $- k$ ) and the expected payoff for indiscriminate violence. This result is obvious for targeted violence, but not for indiscriminate violence. The logic is as follows: the probability that a component is eliminated through indiscriminate violence increases along with its size. As the largest component grows, it is both more likely to be eliminated through indiscriminate violence, and its elimination is more valuable.

However, I also show that targeted violence becomes more valuable relative to indiscriminate violence as the size of the largest component increases. The relationship between the size of the largest component and the difference in expected payoffs between targeted and indiscriminate violence is shown in Figure 3. While increasing the size of the largest component does increase the expected payoff of indiscriminate violence, it does so at a rate that is slower than for targeted violence, which gets the payoff from eliminating the largest component with certainty.

Third, as the standard deviation of component sizes decreases, the value of indiscriminate violence increases relative to targeted violence. This is shown in Figure 4. This is related to a second finding: as the standard deviation of the distribution increases, there are more components in the tails of the distribution. If the largest components are much larger than the mean, then there is less probability that indiscriminate violence will eliminate a component or components similar to the size of the largest component. Similarly, if some components are much smaller than the mean, there is still a positive probability that indiscriminate violence will eliminate these components and not others. However, as components are closer to the mean size, then it is more likely that indiscriminate violence will eliminate a component similar to the size of the largest component.

The variation in networks examined here reflects real world networks. In some locations, networks are extremely tight, such that there might be only one large component, meaning that a path exists between nearly every node. For example, in homogeneous, small villages, nearly every villager may be in a large component of essential ties with a few isolated clusters. In such a setting, this analysis suggests that targeted violence will be more effective, as it will eliminate the largest component despite being more costly. Conversely, in networks that are more heterogeneous and larger, such as villages with a multitude of ethnic groups or cities, there may be a multitude of components representing different groups. In this setting, indiscriminate violence will be expected to accomplish nearly the same outcome as targeted violence and be less costly.

Moreover, the assumptions made about the information available to states in such a setup are realistic. It is likely that states would be able to identify separate components in the network, so long as components are defined by some observable feature. For example, social network ties are more likely as geographical distance decreases, suggesting that neighborhoods or clusters of dwellings could proxy for components in networks. Researchers should also be able to identify the distribution of components in clusters by collecting detailed social network data and identifying variables that predict edges or by using community detection algorithms. Thus, a fruitful research agenda would be to empirically determine the relationship between clustering in networks and strategies of violence.

Translating the results

The theory presented here suggests several comparative statics that can be tested empirically using observable proxies. In this section, I discuss how the results of the theory can be translated into empirical research. I focus on two variables and discuss how they can be operationalized. The two variables are the motivation of the most valuable node and the degree distribution of a network. Note that the following discussion does not cover the full universe of possible proxies, and, while in an ideal research design these proxies would not be prone to measurement error, researchers should consider whether these proxies can be measured without error in their research setting.

All else equal, the theory shows that as the motivation to leave the network of the most valuable node increases, so does the value of using indiscriminate violence. This presents two measurement problems for the state and for researchers: identifying who the most valuable node is, and identifying their level of motivation. I consider reasonable proxies for each in turn. First, the state and researchers must be able to identify who the most valuable node is. In the theory, I defined the targeted node to be one that provides the most value to other nodes in the network. A reasonable approach to identifying this node in a real-world network would be to begin with the degree, as the total value increases with the degree if the value is always positive, as it is defined in the model. Thus, without other contextual factors, the most valuable node would be that with the greatest number of ties. However, with access to more information, this node could be better identified by researchers. For instance, in networks characterized by high levels of religiosity, religious leaders may be the most valuable node, even if other types of leaders have similar degrees. In non-religious networks, a local leader may be the most valuable. However, a researcher must take it upon themselves to define the value and degree within the context they are studying.

Along with identifying the most valuable node, researchers must also be able to proxy for their motivation. Again, a suitable proxy for motivation to leave the network depends on the context in the study location. However, several factors may play a role in this motivation in many contexts. The first is mobility. If the location of the violence is largely isolated and the transportation infrastructure is underdeveloped, it may be less likely that members of the network can or would wish to leave the network. Similarly, if other locations close to where the violence is occurring are also dangerous, then leaving the network may be more costly. Conversely, these conditions may affect the distribution of motivation throughout the network: those who can afford to travel or find safety in a new location may have a higher motivation to leave. Thus, valuable nodes who are also better off than the rest of the network may have higher motivation than others in some situations. Along with this, external social ties may make relocation less costly. Those who can rely on social support in other locations may be more likely to leave. Thus, those who have relocated in the past or who have already been displaced previously may have higher motivation to leave the network. Similarly, those who can find others who share their same political or ethnic identity may be more able to leave (Balcells, 2018).

The theory also shows that as the degree distribution of the network is less uniform, targeted violence becomes more effective. As argued earlier, there are several proxies for degree distribution. One is the distribution of geographical distance from the center of the location where the network exists. When all of the nodes are close together geographically, it is more likely that they will have similar numbers of network ties. However, as the geographical expanse of the network expands, it is likely that degree will vary widely and clusters of ties may exist. Along the same lines, when the network is homogeneous along ethnic and religious lines, the distribution of degree may be more uniform. When the network is less homogeneous, there may be more clusters of ties and a less uniform distribution.

Thus, while these two variables are likely to be context dependent and suitable proxies may vary across locations, they are also likely to be observable to both states and researchers, given access to basic data about the makeup of the population of the network. This suggests that (1) the level of information necessary for the state to effectively weigh the trade-offs to the different strategies of violence in the model is realistic and (2) the researchers can use these findings to identify locations where certain strategies of violence are likely to be used.

Conclusion

The central finding of this analysis is that the structure of networks plays a key role in determining the most effective strategy of violence for a state. The structure of the social network and the characteristics of the individuals within it determine whether a state chooses to employ targeted and indiscriminate violence. As value to other nodes in a network and motivation to leave a network are more positively correlated, indiscriminate violence becomes more valuable. As degree distribution becomes less uniform, violence in general becomes more effective, but targeted violence is more often preferred.

The findings have implications for the study of civil wars and violence against civilians. Primarily, they demonstrate that beyond factors like information and operational discipline, network conditions matter in states’ choice of violence strategy. The theory presented in this article can allow insight into settings outside violence as well. For instance, when states seek to relocate citizens in order to build infrastructure, there may be conditions when offering key community leaders large incentives to leave can cause others to leave with them. Alternatively, uniform incentives may be preferred when community leaders are more willing to leave the community, owing to pull factors. The theory may also speak to phenomena as diverse as migration for temporary work and disaster preparedness, in that both require individuals to make decisions to leave networks for some outside benefit.

However, it is important to note that the theory presented in this article is not a prescription for strategies of violence, but instead a description of the trade-offs states face when choosing to use violence. States already use indiscriminate violence often, as shown in Figure 1, and the intention of this theory is to contribute to the understanding of why this violence takes place. Moreover, while states have access to the same data that researchers can use in predicting and understanding patterns of violence, shedding light on how states may use this data to commit acts of violence can allow outside observers to preempt such attacks. Greater understanding of the deployment of different strategies of violence by states can aid efforts to prevent these atrocities by enabling external actors to predict where indiscriminate violence is likely to take place and act against it.

Footnotes

Replication data

The code for the simulation analysis in this article, along with the Online Appendix, are available at . All of the analyses were conducted using R.

Acknowledgements

For helpful comments and suggestions, I am grateful to Carlo Prato, Kimuli Kasara, Mike Ting, John Huber, Timothy Frye, John Marshall, Jennifer Larson, Tamar Mitts, Giovanna Invernizzi, Dylan Groves, Salif Jaiteh, Jenny Jun, Linan Yao, Heewon Yoon, and participants in the Columbia University Political Economy Breakfast, the 2018 ICSID workshop and the 2019 MPSA conference, along with several anonymous reviewers.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Daniel R Thomas

Notes

Appendix

References

Akbarpour

Mohammad

Malladi

Suraj

Saberi

Amin

(2018) Just a few seeds more: Value of network information for diffusion. Working Paper. https://economics.sas.upenn.edu/system/files/2018-10/NetworkSeeding.pdf.

Azam

Jean-Paul

(2002) Looting and conflict between ethnoregional groups: Lessons for state formation in Africa. Journal of Conflict Resolution 46(1): 131–153.

Azam

Jean-Paul

Hoeffler

Anke

(2002) Violence against civilians in civil wars: Looting or terror? Journal of Peace Research 39(4): 461–485.

Balcells

Laia

(2018) Dynamics of internal resettlement during civil war: Evidence from Catalonia (1936–39). Journal of Peace Research 55(2): 236–251.

Balcells

Laia

Steele

Abbey

(2016) Warfare, political identities, and displacement in Spain and Colombia. Political Geography 51(March): 15–29.

Ballester

Coralio

Calvó-Armengol

Antoni

Zenou

Yves

(2006) Who’s who in networks. wanted: The key player. Econometrica 74(5): 1403–1417.

Banerjee

Abhijit

Chandrasekhar

Arun G

Duflo

Esther

Jackson

Matthew O

(2014) Gossip: Identifying central individuals in a social network. https://www.nber.org/system/files/working_papers/w20422/w20422.pdf

Donnay

Karsten

Dunford

Eric T

McGrath

Erin C

Backer

David

Cunningham

David E

(2019) Integrating conflict event data. Journal of Conflict Resolution 63(5): 1337–1364.

Downes

Alexander B

(2008) Targeting Civilians in War. Ithaca, NY: Cornell University Press.

10.

Freud

Sigmund

(1975) Group Psychology and the Analysis of the Ego. New York: WW Norton & Company.

11.

Goodwin

Jeff

(1997) The libidinal constitution of a high-risk social movement: Affectual ties and solidarity in the huk rebellion, 1946 to 1954. American Sociological Review: 62(1): 53–69.

12.

Goodwin

Jeff

(2001) No Other Way Out: States and Revolutionary Movements, 1945-1991. Cambridge: Cambridge University Press.

13.

Granovetter

Mark

(1978) Threshold models of collective behavior. American Journal of Sociology 83(6): 1420–1443.

14.

Greenhill

Kelly M

(2008) Strategic engineered migration as a weapon of war. Civil Wars 10(1): 6–21.

15.

Kalyvas

Stathis N

(2006) The Logic of Violence in Civil War. Cambridge: Cambridge University Press.

16.

Kalyvas

Stathis N

(2008) Ethnic defection in civil war. Comparative Political Studies 41(8): 1043–1068.

17.

Larson

Jennifer M

Lewis

Janet I

Rodriguez

Pedro L

(2017) From chatter to action: how social networks inform and motivate in rural Uganda. British Journal of Political Science 52(4): 1769–1789.

18.

Lichtenheld

Adam G

(2020) Explaining population displacement strategies in civil wars: A cross-national analysis. International Organization 74(2): 253–294.

19.

Mason

T David

Krane

Dale A

(1989) The political economy of death squads: Toward a theory of the impact of state-sanctioned terror. International Studies Quarterly 33(2): 175–198.

20.

Patty

John W

Penn

Elizabeth Maggie

(2014) Sequential decision making and information aggregation in small networks. Political Science Research and Methods 2(2): 243–271.

21.

Schutte

Sebastian

(2017) Geographic determinants of indiscriminate violence in civil wars. Conflict Management and Peace Science 34(4): 380–405.

22.

Siegel

David A

(2011) When does repression work? Collective action in social networks. Journal of Politics 73(4): 993–1010.

23.

Slater

Philip E

(1963) On social regression. American Sociological Review: 339–364.

24.

Staniland

Paul

(2012) Between a rock and a hard place: Insurgent fratricide, ethnic defection, and the rise of pro-state paramilitaries. Journal of Conflict Resolution 56(1): 16–40.

25.

Steele

Abbey

(2011) Electing displacement: Political cleansing in Apartadó, Colombia. Journal of Conflict Resolution 55(3): 423–445.

26.

Valentino

Benjamin A

(2014) Why we kill: The political science of political violence against civilians. Annual Review of Political Science 17: 89–103.

27.

Zhukov

Yuri M

(2014) Theory of indiscriminate violence. Unpublished PhD dissertation, Harvard University.

28.

Zhukov

Yuri M

Davenport

Christian

Kostyuk

Nadiya

(2019) Introducing xsub: A new portal for cross-national data on subnational violence. Journal of Peace Research: 56(4): 604–614.