Exploring dependence structures in the international arms trade network: A network autocorrelation approach

Abstract

In this article, we analyse dependence structures among international trade flows of major conventional weapons from 1952 to 2016. We employ a Network Disturbance Model commonly used in inferential network analysis and spatial econometrics. The dependence structure is represented by pre-defined weight matrices that allow for correlating flows from the network of international arms exchange. Three dependence structures are proposed, representing sender-, receiver- and sender–receiver-related dependencies. The appropriateness of the presumed structures is comparatively assessed using the Akaike Information Criterion (AIC). It turns out that the dependence structure among the arms trade flows is complex and can be represented best by a specification that relates each arms trade flow to all exports and imports of the sending and the receiving state. Controlling for exogenous variables, we find that the trade volume increases with the GDP of the sending and the receiving state while the impact of geographical distance, regime dissimilarity and formal alliance membership is rather small.

Keywords

Arms trade dependence structures model selection network disturbance model spatial econometrics

1 Introduction

In this article, we investigate international trade flows of Major Conventional Weapons (MCW) using data provided by the Stockholm International Peace Research Institute (SIPRI). MCW include armoured vehicles, aircraft, naval vessels etc., and SIPRI has compiled all international arms transfers from 1950 to 2016 in a comprehensive database. It includes the sending and the receiving country as well as the volume of the trades in a certain year. The volume is measured as so-called TIV, i.e. trend-indicator value(s), and represents the military value and the production costs of the transferred products. The data can be regarded as a year-wise sequence of weighted networks where the countries are the nodes and the arms trade flows among them are the valued edges.

Several scholars have started to investigate arms trade using a network framework, but the available studies were restricted to binary relations, that is, trade or no trade, see for example Akerman and Seim (2014), Kinne (2016), Lebacher et al. (2018) and Thurner et al. (2018). The workhorse in inferential binary network analysis is the Exponential Random Graph Model (ERGM), as introduced by Holland and Leinhardt (1981) (see also Frank and Strauss, 1986; Kolaczyk, 2009; Lusher et al., 2012). Recent proposals for modelling valued networks within the ERGM class have been provided by Krivitsky (2012), or by Desmarais and Cranmer (2012) having proposed the Generalized Exponential Random Graph Model (GERGM). The first model is actually designed for count data while the GERGM approach allows for continuous edge weights. Both approaches are not suitable for modelling arms trade data because the TIVs are continuous and heavily zero-inflated (i.e., TIV $= 0$ means no trade). We discuss this issue in depth in the article.

The analysis of the binary event of an arms transfer is important since trading of arms implies governmental agreement and therefore indicates direct and indirect trust relations (see, e.g., Jackson, 2010), regardless of the amount of trading. In this article, however, we focus on the amount of trading. This objective raises the question of how to handle the high share of structural zeros (approximately 98%) because this poses substantial problems in conventional edge-based network models. Following models for insurance claims, where the amount of the claim is modelled conditional that there is a claim (Frees and Valdez, 2008), we opt for excluding the decision to trade arms and to focus only on the amount of trading, given that there is trading (i.e., TIV $> 0$ ). Of particular interest will be the dependence structure of trading amounts, that is how is the trading amount of a country (to a country) correlated with other trading amounts. In order to do so, we change the viewpoint of the network and go from a representation with the nodes as countries and the edges being arms trade flows to a network where the nodes are the trade flows between two countries and the edges represent the dependencies among them. A procedure that is sometimes referred to as second-order network (see Xu et al. (2016) for higher-order networks in general). This yields a model that at least conceptually resembles models from spatial econometrics (see, e.g., LeSage and Pace, 2009; Kauermann et al., 2012). The model class was developed for capturing spatial effects and spillovers from nearby sites and can be adjusted to represent the dependence of similar arms trade flows. Approaches from spatial econometrics even received their own labels in the social network methodology and are commonly called Network Autocorrelation Models (see, e.g., Leenders, 2002; Hays et al., 2010).

The article is organized as follows. In Section 2, we give a short description of the data, while in Section 3, we explain the shortfalls of recently proposed models for valued networks in view of our dataset. In Section 4, we introduce our model and propose different specifications and explanations for possible dependence structures. In Section 5, we present the results of our analysis. Section 6 concludes the article. Data, R codes as well as further descriptions and extensions are available in the online supplemental materials at www.statmod.org/smij/archive.html.

2 Data description

Figure 1:

Development of the exported TIV for 1950–2016 for the United States, Soviet Union/Russia and all other countries

The Stockholm International Peace Research Institute (SIPRI) is a unique data source for the international transfers of major conventional weapons. It covers more than 200 countries from 1950 to 2016, and includes the types and TIV volumes of MCW that are traded. Table 2 in the Appendix gives an overview of weapon systems included. The advantage of the TIV measure is consistency over time and comparability of different arms systems. (For a detailed description of the data and methodology, see SIPRI (2017b) or Holtom et al. (2012). The database can be accessed online at SIPRI (2017a))

Figure 2:

Log-log plot of aggregated TIV export versus outdegree (left) and aggregated TIV import versus indegree (right). The solid line in the middle represents the mean of exports or imports for a given in- or outdegree. Lower and upper bounds indicate minimum and maximum values

In Figure 1, the evolution of the aggregated TIV is shown. The contribution of the two most important exporters, namely the United States and the Soviet Union (Russia since 1992), is separately coloured. Figure 2 relates the outdegree (i.e., number of out ward directed binary relations) and indegree (i.e., number of inward directed binary relations) in a log-log plot to the aggregated export and import volumes, respectively. In terms of the slope and range (i.e. the distance between der lowest and the highest value), it can be seen that the connection between aggregated export volume and the outdegree is somewhat stronger than the relationship between import volume and indegree. Nevertheless, both volumes increase on average in the respective degree.

3 Valued network models and zero-inflated data

The modelling approach of Krivitsky (2012) allows for count data and is therefore not directly suitable for arms trade data since the TIV are continuous. The GERGM (Desmarais and Cranmer, 2012), however, is applicable to networks with continuous edge weights but suffers from two major restrictions when being applied to our dataset. The estimation, even for networks of moderate size, can be computationally very demanding, or even proved to be infeasible as also noted, for example, by Simpson et al. (2013), Ward et al. (2013), Simpson and Laurienti (2015) and Boivin and D'Elia (2017). In our application, we have 65 yearly networks with up to 160 nodes in the most recent periods, rendering estimation impossible in an acceptable amount of time. But even if we were able to handle the numerical problems, an unsurmountable obstacle constitutes the extreme number of zero-valued trades that arises if we create a weighted adjacency matrix and code all non-existing transfers as zero. Figure 3 shows the density of the networks for the whole observational period. Even in the years with the highest density, we have roughly 98% zeros in the data, excluding the possibility to use a GERGM. As indicated also in Schoeneman et al. (2017), the GERGM model cannot handle zero-inflated data.

Figure 3:

Network density (share of realized trade relations as compared to all possible trade relations) from 1950 to 2016

An alternative approach for valued networks are latent space models (see, e.g., Hoff et al., 2002; Handcock et al., 2007). This model class assumes that the endogenous network dependencies can be accounted for by representations of the nodes in a latent space. Obviously, the approach is tempting because it allows for an interpretation as in generalized linear models conditional on the latent space. In addition, latent space models were already applied in trade research (see Ward and Hoff, 2007; Ward et al., 2013; Berlusconi et al., 2017). The approach comes, however, with some disadvantages. First, the zero-inflation problem poses again a major obstacle. We applied the routines in the R-packages amen (Hoff et al., 2015) and latentnet (Krivitsky and Handcock, 2015) to our data, which led to unreliable coefficients and godness-of-fit diagnostics (we can deliver our extensive results on request). Additionally, the latent space representation proved to be very unstable with respect to the time dimension and showed no persistent interpretable pattern.

Provided we focus exclusively on observed trade, we can apply models from the family of spatial econometric models in order to detect related transfers. Here, two canonical models can be distinguished, the spatial autoregressive (SAR) model and the spatial error model (SEM). The main difference between the SAR model and the SEM is related to the assumptions on how the network structure influences the outcome. In the SEM, we have the same interpretation of the coefficients as in the linear model, whereas in the SAR model, changes in the covariates affect all related transfers (LeSage and Pace, 2009; Kauermann et al., 2012). This implies, for example, that if the logarithmic GDP of the sender has a positive coefficient, and two transfers $Y_{ab}$ (country $a$ transfers to country $b$ ) and $Y_{bc}$ (country $b$ transfers to country $c$ ) are positively related, an increase in the GDP of country $a$ would imply that the exports of country $b$ to country $c$ increase. This kind of spillover effects might be desirable in a spatial context but are not appropriate in the given network context. We therefore will work henceforth with the SEM. Refer the Supplementary Material for a more extensive discussion on that.

4 Model

4.1 Definitions and basic formulation

In $T$ time periods, we observe $n_{t}$ flows $Y_{t} = {Y_{t, ij} : Y_{t, ij} > 0, i, j = 1, . . . ., N_{t}}$ where $Y_{t, ij}$ is the valued arms trade flow from country $i$ to country $j$ among the $N_{t}$ countries that exist in $t = 1, . . ., T$ . In the following, we suppress the time index $t$ for ease of the notation. Let $V = {(i, j) : Y_{ij} > 0}$ be the (directed) index set for all country pairs $(i, j)$ where valued trade flows exist so that $Y = {Y_{v} : v \in V}$ . Let $Y$ be of dimension $n = | V |$ and numerate the elements in $V$ so that $V = {v_{1}, . . ., v_{n}}$ .

As a tool to analyse the dependence structure among $Y$ , we are using a graph $G$ consisting of nodes and edges, where we take $V$ as node set and define $E$ as symmetric edge set: $G = (V, E)$ . We define the neighbourhood in the model through

N (v_{a}) : = {v_{b} : {v_{a}, v_{b}} \in E}, a = 1, . . ., n .

The information given in the neighbourhood can be summarized in a $(n \times n)$ matrix W. Typically, the rows of the matrix are normalized such that they sum up to one by defining $W_{ab} = | N (v_{a}) |^{- 1}$ if $v_{b} \in N (v_{a})$ and zero otherwise. This is also the definition we are using. As we are interested in the dependencies among arms trade flows, we will henceforth attach the name armstradecorrelation to the concept of related (neighbouring) flows. The intention of the subsequent analysis is to find a reasonable weight matrix W, which defines an appropriate dependence structure.

4.2 Regression model

Arms trade flows are non-negative, leading us to consider a multiplicative model of the form

\begin{matrix} log (Y) & = x^{T} β + u \\ (u_{1}, . . ., u_{n})^{T} = u & \sim N_{n} (0, Σ) . \end{matrix}

(4.1)

Here, $x$ represents a vector of some covariates, $β$ gives the parameter vector and $u$ is the error term. The residuals $u$ are assumed to be correlated and $Σ$ provides the variance-covariance matrix encoding the relation of different arms trade flows. Suitable candidates for estimation of (4.1) are models from spatial econometrics or spatial statistics, respectively. We will employ the former. Spatial econometric models are common in network analysis, refer for instance Leenders (2002), Franzese Jr and Hays (2007), Hays et al. (2010), Metz and Ingold (2017), Silk et al. (2017), Ba (2017) as well as the package tnam by Leifeld et al. (2016). We follow a SEM (see LeSage and Pace, 2009) which is given by defining the error term u as

\begin{matrix} u & = ρ W u + ε, \\ ε & \sim N_{n} (0, σ^{2} I_{n}), \end{matrix}

(4.2)

where $I_{n}$ is the identity matrix of dimension $n$ and $ρ$ provides a measure for the dependency among the related error terms $u$ . The error terms $ε$ are assumed to be unstructured so that Equation (4.2) can be reformulated to

u = (I_{n} - ρ W)^{- 1} ε .

(4.3)

This implies the covariance matrix $Σ = σ^{2} (I_{n} - ρ W)^{- 1} (I_{n} - ρ W^{T})^{- 1}$ . Hence, matrix W expresses the dependence structure, and the explicit choice of W is the central model selection task that will be discussed in Section 4.3. The modelling approach is also known as Network Disturbance Model (NDM) in the network literature, delegating the dependencies among trade flows into the error term. Hence, the approach does not aim to model a direct dependence between the trade flows but implies that the exogenous covariates determine the ‘correct’ value for an arms trade flow, but one can find clusters of trade flows where the residuals are unusually high or low (see Leenders, 2002; Freeman, 2017 for further discussion). (We sketch some details on estimation and software in the Appendix).

4.3 Specifications of the weight matrix

We describe now the central modelling task, i.e. the appropriate specification of matrix W and hence the definition of the neighbourhood $N$ . We focus on three simple and typical structures that represent sender, receiver and sender-receiver dependence in a similar fashion as for example LeSage and Pace (2008) in a spatial context. Given that in the network context, the weight matrices are rather abstract, we illustrate our approach with the arms trade network of the year 1952, being the simplest network under study. The network is shown in Figure 4. Note that all subsequent illustrations of weight matrices W relate to the network shown in Figure 4 and are therefore not representative for other years. However, they allow us to visually illustrate the dependence structures.

Figure 4:

Directed valued network of arms trade in 1952. Edges (nodes) are scaled proportional to the logarithmic exports (logarithmic aggregated exports) measured in TIV

Figure 5:

Note: For references to colour, please see the online version of the article.

Armstradecorrelation $N_{sender}$ (corresponding to the Sending Country)

The first dependence structure relates an arms trade flow from country $i$ to country $j$ to all other exports originating from country $i$ and to a potential reciprocal trade flow $(j, i)$ . This structure can be motivated as follows. Countries have different productive capacities due to different home market sizes. As a consequence of higher competitiveness, some countries are able to export larger volumes as compared to other countries. This would imply a positive correlation among transfers originating from the same exporter. Furthermore, the possibility that exports to other highly competitive countries have a tendency to be reciprocal should be accounted for. Therefore, we allow that a given transfer is correlated with its reciprocal counterpart if it exists. Notationally, the structure is defined by

N_{sender} ((i, j)) = {(p, q) : p = i (p, q) = (j, i), (p, q) \in V} .

The corresponding graph representation for the year 1952 is shown in Figure 5. The trade flows are grouped within clusters that are related to specific exporting countries, most notably they consist of the trade clusters of the big exporters: United States (USA), United Kingdom (GBR), Canada (CAN) and the Soviet Union (SUN). Through the dependence on reciprocal flows, the clusters are connected among each other if there is between-cluster trade. This explains the connection between the cluster of the United States (USA), Great Britain (GBR) and Canada (CAN). The trade activity of the Soviet Union (SUN), however, is disconnected.

Figure 6:

Note: For references to colour, please see the online version of the article.

Armstradecorrelation $N_{receiver}$ (corresponding to the receiving country)

The aforementioned concept can be amended such that the armstradecorrelation of an arms trade is related to the trades of the importing country and not to those of the exporting one. This would imply the hypothesis that there exist countries that are strong importers and countries that import small volumes. Additionally, we take into account reciprocity with the same arguments as above. This means

N_{receiver} ((i, j)) = {(p, q) : (p, q) = (j, i) q = j, (p, q) \in V}

and entails that the arms trade amount from $i$ to $j$ is correlated with other imports of $j$ as well as with the potential reciprocal trade flow $(j, i)$ . This is shown in Figure 6. The resulting graph representation is very different from the previous one. There is no tendency of strong clustering as most importing countries do only have one or two suppliers and no exports at all which excludes connections via reciprocal flows. The largest clusters consist of arms imports of Israel (ISR) and imports of Switzerland (CHE). Great Britain (GBR), United States (USA) and Canada (CAN) reveal their strong connections by exhibiting their own import-related cluster. Note that this structure disregards an important characteristic of the arms trade network: receiving countries are often restricted to be dependent on one single exporter. Most importantly, the trade flows of the Soviet Union (SUN) are missing.

Figure 7:

Note: For references to colour, please see the online version of the article.

Armstradecorrelation $N_{sender - receiver}$ (corresponding to the trade activity of sender and receiver)

The structures of the previous tradecorrelations can be combined by the inclusion of the whole trade activity of $i$ as well as of $j$ . This presumed structure is the most complex as it corresponds to the hypothesis that a given arms transfer is related not only to the export and/or import capacity of a country but also to the transfers of its trading partners. Notationally, this can be represented as follows:

N_{sender - receiver} ((i, j)) = {(p, q) : p \in {i, j} q \in {i, j}, (p, q) \in V} .

In this case, an arms trade from country $i$ to country $j$ is dependent on the whole trade activity of countries $i$ and $j$ , regardless of whether or not they are reciprocal. This results in a very dense graphical representation, as shown in Figure 7. Here, the clusters are again very clearly visible, but the interconnection within and between clusters is much stronger. Striking are the trade flows United States (USA)’Great Britain (GBR) and Great Britain (GBR)’United States (USA). They link the two countries with the highest trade activity. The flows related to Switzerland (CHE) serve as a bridge between the clusters of the Western producers and the Soviet Union (SUN). With this specification, the graph is connected, and, in principle, each trade flow is allowed to be directly and/or indirectly correlated with an arbitrary other arms trade flow in the graph.

4.4 Exogenous covariates

We control for the influence of economic and political quantities and of geographical distance by including additional covariates in model (4.1). Since there is an average time lag of roughly two years between ordering and delivering of weapons and some covariates are only available from 1950 onwards, the first network under study is the one of 1952, and covariates are lagged by two years.

Economic variables: The standard measure for economic size is the gross domestic product (GDP). We consider this as a measure of market size and include it in logarithmic form for the sender as well as the receiver. Real GDP is measured in thousands of USD. Data until 2010 are taken from Gleditsch (2013a). Data for later years are drawn from the World Bank real GDP dataset (see The World Bank, 2017).

Distance: Gleditsch (2013b) provides data on the distances between capital cities measured in kilometres. We include this measure for the physical distance between countries $i$ and $j$ in logarithmic form.

Political variables: We use a dummy variable being one if the countries $i$ and $j$ have a formal alliance and zero otherwise. Given the restriction that the data is available only until 2012 (Correlates of War Project (2017), we assume that the alliances did not change between 2012 and 2014. In order to control for regime dissimilarity, we include the absolute difference of the so-called polity score, ranging from the spectrum −10 (hereditary monarchy) to +10 (consolidated democracy). The data can be downloaded as annual time-series. (See Marshall and Monty (2017) for the data and basic reference).

This choice of covariates is in line with recent empirical (Akerman and Seim, 2014; Martinez-Zarzoso and Johannsen, 2017; Bove et al., 2018; Thurner et al., 2018) and theoretical (Garcia-Alonso and Levine, 2007) arms trade research. It builds on a modelling approach similar to the gravity equation, a very common parsimonious structure for the analysis of international trade flows (Disdier and Head, 2008; Ward et al., 2013; Head and Mayer, 2014; Egger and Staub, 2016).

5 Results

5.1 Model selection

Our ultimate goal is to select a suitable dependence model from Section 4.3 As recommended by Leenders (2002) for this model class with several candidate weight matrices, we compare the proposed models using the Akaike Information Criterion (AIC) (see, e.g., Claeskens and Hjort, 2008). A simulation study in Section 5.4 shows that this is a valid strategy for identifying dependence structures.

For comparison, we also include a simple linear model with unstructured error term (OLS, $ρ = 0$ ). Estimation is done year-wise, and the AIC is calculated for all estimated models from 1952 to 2016, all in all covering $18 964$ observations. In Table 1, the results for the aggregated AIC, rescaled by the lowest one ( $Δ_{i} =$ AIC $_{i} -$ AIC $_{\min}$ ) are presented. AIC $_{\min}$ corresponds to armstradecorrelation structure $N_{sender - receiver}$ . In order to investigate the different models in more detail, we also focus on annual AIC values. To do so, we compute Akaike weights, which can be interpreted as the probability of model $i$ being the best (i.e., minimizing the Kullback-Leibler divergence given the data) among the proposed ones. The weights are given by $w_{i} = exp (- Δ_{i} / 2) / (\sum_{r = 1}^{R} exp (- Δ_{r} / 2))$ (see Burnham and Anderson, 2004 for a further discussion) and shown in Figure 8 for all four specifications.

Table 1:

Difference between aggregated AIC values and the minimum aggregated AIC value for different armstradecorrelation structures

	$N_{sender}$	$N_{receiver}$	$N_{sender - receiver}$	$ρ = 0$
$Δ_{i}$	293	958	0	$1 061$

We see that the specification where ρ is restricted to zero (OLS) is among those models that have a very low probability of being the best one. Only in the years 1990–2000, the probability for this model rises substantially as a consequence of the collapse of the Eastern Bloc. This mirrors the fact that this major event broke up long established network structures. Overall, the low probability for the restricted model provides evidence for our initial hypothesis that the trade flows are indeed correlated. For the armstradecorrelation structure that is related to the receiver ( $N_{receiver}$ ), we find a similar behaviour as for the restricted model (OLS). This reveals a major structural property of the valued arms trade network. Most importers are restricted to have only a small number of suppliers, and the arms imports from them are not strongly correlated.

Figure 8:

Annual Akaike weights for different specifications (raw values in grey, smoothed in black)

The two specifications that compete to be the best representation of dependencies are tradecorrelations $N_{sender}$ and $N_{sender - receiver}$ . They alternate in being the best model in most time points. In most cases, when the probability of the model with armstradecorrelation $N_{sender}$ decreases, the one with armstradecorrelation $N_{sender - receiver}$ increases and vice versa. But from a global perspective, it is armstradecorrelation $N_{sender - receiver}$ that provides the best model, which leads to the conclusion that overall, a single arms trade flow is related to the whole trade activity of the sender as well as the receiver.

We take this as a clear evidence that the exchange of international arms leads to a rather complex dependence structure. This corroborates existing binary network analyses (see Thurner et al., 2018) and shows that, the assumption of independent observations, given the covariates, is a strong simplification of the real process. Based on these promising results, we continue to analyse the model with armstradecorrelation structure $N_{sender - receiver}$ .

5.2 Coefficients

Figure 9:

Note: For references to colour, please see the online version of the article.

Economic variables: The coefficient on the logarithmic GDP of the sender is, with the exception of the year 1952, consistently positive and significant, showing a slight tendency to decrease with time. This can be seen from Figure 9, left panel, top plot and reflects the fact that countries with greater economic size have a higher tendency of possessing a highly developed arms industry and are able to produce and export expensive military equipment. When it comes to the GDP of the receiver (right panel, top plot), we also see a slight downward trend of the coefficients. As a rule, the coefficient is always positive and significant and comparable in size to the GDP of the exporter, that is, the economic size of the importer is also an important determinant of its import volume.

Distance: In the left panel at the middle plot of Figure 9, it can be seen that the coefficient related to the distance is mostly insignificant and even positive when significant.

Political variables: Before 1990, having an alliance (left panel, bottom plot) in tendency increased the amount of arms trading, but at least since 1990, this effect is almost zero. The coefficient on the absolute differences of the polity score (right panel, middle plot) shows a time-related trend. In the beginning (1953–1973), the coefficient is rather less than zero, giving the intuitive result that differences in political regimes reduce the amount of arms traded. From 1976 to 1991, the coefficient can be regarded as zero, which mirrors the fact that at the height of the Cold War, the belonging to a political bloc was more important than distances in the polity scores. After the end of the Soviet Union, the coefficient remains close to zero until 2011 where the coefficients become significantly positive for six consecutive years, providing the surprising result that differences in the polity score do not lead to a reduction of the traded amount of arms contrary to recent results of a study on the binary decision to trade (see Thurner et al., 2018).

Armstradecorrelation: The bottom right panel of Figure 9 shows the trade correlation coefficient using the structure $N_{sender - receiver}$ . The coefficient is consistently positive but exhibits some variation. The strongest downward movement can be found in the years 1990–1996, showing how massively the end of the Cold War altered the structure of the international trade of arms. From 1996 onwards, the coefficient increases again. In the most recent years, the armstradecorrelation fluctuates around $0.6$ , close to the values before 1990.

5.3 Residual analysis

Figure 10:

QQ-plot (left) and histogram (right) of standardized residuals

We give a graphical summary in Figure 10 for the pooled unstructured estimated residuals $\hat{ε}$ . The plot on the left gives a QQ-plot with the quantiles of the standardized residuals on the vertical axes against the quantiles from a standard normal distribution on the horizontal axis. The right panel shows a histogram of the residuals together with a standard normal. Besides some irregularities in the tails, the results look quite acceptable.

Figure 11:

Kernel density estimates of the armstradecorrelation residuals for selected countries in the time periods 1952–1991 (left) and 1992–2016 (right)

As shown in the previous section, the coefficient on the armstradecorrelation structure $N_{sender - receiver}$ is always positive, and it is therefore natural to ask whether it is possible to identify trade clusters with higher or lower than expected trade flows. In order to do so, we attribute to each country those elements of the vector of estimated ‘armstradecorrelation-residuals’ ( $\hat{ρ} W \hat{u}$ ) where the respective country is the sender. This is illustrated in Figure 11. For the periods 1952–1991 (left) and 1992–2016 (right), kernel density estimates of the estimated ‘armstradecorrelation-residuals’ for the most important countries are given. A strong difference in the distributions of the residuals can be found. Strikingly the residuals are mostly negative for the United States (USA), Great Britain (GBR), Germany (DEU), France (FRA) and Canada (CAN) while the distribution for the Soviet Union (SUN) is shifted to the right and the one of China (CHN) is somewhat in-between. In the period 1992–2016, the distributions are shifted closer together, but the remarkable divide between the big Western exporters and Russia (RUS) is still visible. This leads to the insight that the positive coefficient on armstradecorrelation is partly driven by the differences of the tradeclusters of the Western world and the one around the Soviet Union/Russia.

5.4 Simulation study: Model selection using the AIC

In the following simulation study, we show that our model selection approach is suitable for the identification of appropriate dependence structures.

We take the binary arms trade networks from 1952 to 2016 as given and specify four different data generating processes (DGPs) for the arms trading amount in order to make the simulation study as comparable to the real analysis as possible. The DGPs are as follows:

\begin{matrix} Y & = 5 + x_{1} + x_{2} + x_{3} + u \\ (u_{1}, . . ., u_{n})^{T} = u & = ρ_{k} W_{k} u + ε, k = 1, 2, 3, 4 \\ ε & \sim N_{n} (0, I_{n}), \end{matrix}

where the covariates $x_{1}$ , $x_{2}$ and $x_{3}$ are simulated independently from a normal distribution with mean zero and variance equal to one. The weighting matrices $W_{k}$ are constructed as in Section 4.3 and $W_{1}$ corresponds to $N_{sender}$ , $W_{2}$ to $N_{receiver}$ and $W_{3}$ to $N_{sender - receiver}$ . We set $ρ_{1} = ρ_{2} = ρ_{3} = 0.5$ and include an independence model with $ρ_{4} = 0$ .

Table 2

Comparison of AIC values for different DGPs in the rows (DGP W₁: N_sender, DGP W₂: N_receiver, DGP W₃: N_{sender−receiver} and DGP W₄: ρ = 0) estimated with different models in the columns (Model 1: N_sender , Model 2: N_receiver , model 3: N_{sender−receiver} and model 4: ρ = 0). For each DGP-model combination, the mean and median AIC value are given together with the share of instances where the respective model has the smallest AIC value

		Model 1 $N_{sender}$	Model 2 $N_{sender}$	Model 3 $N_{sender - receiver}$	Model 4 ρ = 0
DGP W₁	share min(AIC)	0.954	0.005	0.027	0.014
	mean(AIC)	843.589	874.236	860.855	873.261
	median(AIC)	904.849	940.157	925.573	939.792
DGP W₂	share min(AIC	0	1	0	0
	mean(AIC)	947.322	864.561	939.660	946.358
	median(AIC)	1 013.132	925.942	1 004.702	1 012.253
DGP W₃	share min(AIC	0.124	0.062	0.678	0.136
	mean(AIC)	840.238	843.637	836.495	843.345
	median(AIC)	901.777	905.435	896.311	905.221
DGP W₄	share min(AIC	0.117	0.136	0.131	0.616
	mean(AIC)	833.962	833.967	833.899	832.975
	median(AIC)	894.109	894.079	894.082	892.974

In a first step, we simulate all four DGPs 100 times for each of the 65 real binary networks. This provides us $26 000$ simulated datasets that are fitted in a second step for all four models that correspond to the DGPs specified earlier, yielding $104 000$ fitted models. The results are shown in Table 2 with four different DGPs in the rows and different models in the columns. For each DGP-model combination, we provide the share of instances where the respective model was judged as the best model by means of the AIC (i.e., has minimal AIC) as well as the mean and median of the AIC values. The correct DGP-Model combinations are on the diagonal, and it is clearly visible that the correct DGP-model combination gives the lowest mean and median AIC values, and is therefore chosen most often as the best model. We also see that it is somehow harder to decide for Model $N_{sender - receiver}$ in the DGP $W_{3}$ because the structures of the weighting matrices $W_{1}$ and $W_{2}$ are part of $W_{3}$ . Nevertheless, the AIC leads to the adequate decision on average. Another notable result is that, given that no correlation structure exists (DGP $W_{4}$ , $ρ = 0$ ), it is harder to reject the models that impose some structure, but again, the relative and absolute majority of decision goes to the correct model.

6 Conclusion

In this article, we analysed the volume of international arms trade flows by employing a Network Disturbance Model. Three different dependence structures are proposed, and the best one is selected based on the minimization of the AIC. Using this approach, we find that a specification that relates the error term of an arms trade flow to the whole trade activity of the sending and receiving state performs well in most years. This indicates that the network of international arms trade exhibits a very complex dependence structure among the arms trade flows. The analysis of the armstradecorrelation coefficient shows that the correlation pattern among the flows is rather stable and breaks only for the transition period after the end of the Cold War. The development of the coefficients of the covariates discloses how the influence of economic and political factors has changed with time.

From a political economy perspective, the evolution of the coefficients on the covariates corroborates earlier results with respect to GDP. It does, however, yield surprising results on the distance measure and on political variables. Whereas the low influence of geographic distance confirms recent results from the analysis of the binary arms trade network (see Thurner et al., 2018), the low impact of regime dissimilarity (absolute difference of polity scores) and formal alliance membership stands in contrast to the effects on the formation of binary network ties and provides the new insight that regime dissimilarity and joint alliance membership are seemingly more important for the tie formation than for the volume of arms transfer given they trade.

Acknowledgements

We would like to thank SIPRI, and especially Siemon Wezeman for providing data and detailed information on the underlying data collection approach. This article has profited from the COST Action CA15109 European Cooperation for Statistics of Network Data Science (COSTNET), supported by COST (European Cooperation in Science and Technology). We thank two anonymous reviewers for their careful reading and useful suggestions.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors acknowledge the financial support by the German Research Foundation (DFG) under grants TH 697/9-1 and KA 1188/10-1 ‘International Arms Trade: A Network Approach’.

A1
Types of weapons included in the SIPRI arms trade database

Type Explanation

Aircraft All fixed-wing aircraft and helicopters, including unmanned aircraft with a minimum loaded weight of 20 kg. Exceptions are microlight aircraft, powered and unpowered gliders and target drones.

Air-defence systems (a) All land-based surface-to-air missile systems and (b) all anti-aircraft guns with a calibre of more than 40 mm or with multiple barrels with a combined calibre of at least 70 mm. This includes self-propelled systems on armoured or unarmoured chassis.

Anti-submarine warfare weapons Rocket launchers, multiple rocket launchers and mortars for use against submarines, with a calibre equal to or above 100 mm.

Armoured vehicles All vehicles with integral armour protection, including all types of tank, tank destroyer, armoured car, armoured personnel carrier, armoured support vehicle and infantry fighting vehicle. Vehicles with very light armour protection (such as trucks with an integral but lightly armoured cabin) are excluded.

Artillery Naval, fixed, self-propelled and towed guns, howitzers, multiple rocket launchers and mortars, with a calibre equal to or above 100 mm.

Engines (a) Engines for military aircraft, for example, combat-capable aircraft, larger military transport and support aircraft, including large helicopters; (b) Engines for combat ships—fast attack craft, corvettes, frigates, destroyers, cruisers, aircraft carriers and submarines; (c) Engines for most armoured vehicles—generally engines of more than 200 horsepower output. $^{}$

Missiles (a) All powered, guided missiles and torpedoes and (b) all unpowered but guided bombs and shells. This includes manportable air defence systems and portable guided anti-tank missiles. Unguided rockets, free-fall aerial munitions, anti-submarine rockets and target drones are excluded.

Sensors (a) All land-, aircraft- and ship-based active (radar) and passive (e.g., electro-optical) surveillance systems with a range of at least 25 km, with the exception of navigation and weather radars, (b) all fire-control radars, with the exception of range-only radars and (c) anti-submarine warfare and anti-ship sonar systems for ships and helicopters. $^{}$

Satellites All reconnaissance and communications satellites.

Ships (a) All ships with a standard tonnage of 100 tonnes or more, and (b) all ships armed with artillery of 100-mm calibre or more, torpedoes or guided missiles and (c) all ships below 100 tonnes where the maximum speed (in kmh) multiplied with the full tonnage equals 3500 or more. Exceptions are most survey ships, tugs and some transport ships

Other (a) All turrets for armoured vehicles fitted with a gun of at least 12.7 mm calibre or with guided anti-tank missiles, (b) all turrets for ships fitted with a gun of at least 57-mm calibre and (c) all turrets for ships fitted with multiple guns with a combined calibre of at least 57 mm and (d) air refuelling systems as used on tanker aircraft. $^{*}$

Source: SIPRI (2017b).

Notes: In cases where the system is fitted on a platform (vehicle, aircraft or ship), the database only includes those systems that come from a different supplier from the supplier of the platform.

The Arms Transfers Database does not cover other military equipment such as small arms and light weapons (SALW) other than portable guided missiles such as man-portable air defence systems and guided anti-tank missiles. Trucks, artillery under 100-mm calibre, ammunition, support equipment and components (other than those mentioned earlier), repair and support services or technology transfers are also not included in the database.

Type	Explanation
Aircraft	All fixed-wing aircraft and helicopters, including unmanned aircraft with a minimum loaded weight of 20 kg. Exceptions are microlight aircraft, powered and unpowered gliders and target drones.
Air-defence systems	(a) All land-based surface-to-air missile systems and (b) all anti-aircraft guns with a calibre of more than 40 mm or with multiple barrels with a combined calibre of at least 70 mm. This includes self-propelled systems on armoured or unarmoured chassis.
Anti-submarine warfare weapons	Rocket launchers, multiple rocket launchers and mortars for use against submarines, with a calibre equal to or above 100 mm.
Armoured vehicles	All vehicles with integral armour protection, including all types of tank, tank destroyer, armoured car, armoured personnel carrier, armoured support vehicle and infantry fighting vehicle. Vehicles with very light armour protection (such as trucks with an integral but lightly armoured cabin) are excluded.
Artillery	Naval, fixed, self-propelled and towed guns, howitzers, multiple rocket launchers and mortars, with a calibre equal to or above 100 mm.
Engines	(a) Engines for military aircraft, for example, combat-capable aircraft, larger military transport and support aircraft, including large helicopters; (b) Engines for combat ships—fast attack craft, corvettes, frigates, destroyers, cruisers, aircraft carriers and submarines; (c) Engines for most armoured vehicles—generally engines of more than 200 horsepower output. $^{*}$
Missiles	(a) All powered, guided missiles and torpedoes and (b) all unpowered but guided bombs and shells. This includes manportable air defence systems and portable guided anti-tank missiles. Unguided rockets, free-fall aerial munitions, anti-submarine rockets and target drones are excluded.
Sensors	(a) All land-, aircraft- and ship-based active (radar) and passive (e.g., electro-optical) surveillance systems with a range of at least 25 km, with the exception of navigation and weather radars, (b) all fire-control radars, with the exception of range-only radars and (c) anti-submarine warfare and anti-ship sonar systems for ships and helicopters. $^{*}$
Satellites	All reconnaissance and communications satellites.
Ships	(a) All ships with a standard tonnage of 100 tonnes or more, and (b) all ships armed with artillery of 100-mm calibre or more, torpedoes or guided missiles and (c) all ships below 100 tonnes where the maximum speed (in kmh) multiplied with the full tonnage equals 3500 or more. Exceptions are most survey ships, tugs and some transport ships
Other	(a) All turrets for armoured vehicles fitted with a gun of at least 12.7 mm calibre or with guided anti-tank missiles, (b) all turrets for ships fitted with a gun of at least 57-mm calibre and (c) all turrets for ships fitted with multiple guns with a combined calibre of at least 57 mm and (d) air refuelling systems as used on tanker aircraft. $^{*}$
Source: SIPRI (2017b).
Notes: In cases where the system is fitted on a platform (vehicle, aircraft or ship), the database only includes those systems that come from a different supplier from the supplier of the platform.
The Arms Transfers Database does not cover other military equipment such as small arms and light weapons (SALW) other than portable guided missiles such as man-portable air defence systems and guided anti-tank missiles. Trucks, artillery under 100-mm calibre, ammunition, support equipment and components (other than those mentioned earlier), repair and support services or technology transfers are also not included in the database.

Countries included in the analysis with three-digit country codes and time-period of inclusion in the model

Country	Code	Included	Country	Code	Included	Country	Code	Included
Afghanistan	AFG	1950–2016	German Dem. Rep.	GDR	1950–1991	Pakistan	PAK	1950–2016
Albania	ALB	1950–2016	Germany	DEU	1950–2016	Panama	PAN	1950–2016
Algeria	DZA	1962–2016	Ghana	GHA	1957–2016	Papua New Guin.	PNG	1975–2016
Angola	AGO	1975–2016	Greece	GRC	1950–2016	Paraguay	PRY	1950–2016
Argentina	ARG	1950–2016	Guatemala	GTM	1950–2016	Peru	PER	1950–2016
Armenia	ARM	1991–2016	Guinea	GIN	1958–2016	Philippines	PHL	1950–2016
Australia	AUS	1950–2016	Guinea-Bissau	GNB	1973–2016	Poland	POL	1950–2016
Austria	AUT	1950–2016	Guyana	GUY	1966–2016	Portugal	PRT	1950–2016
Azerbaijan	AZE	1991–2016	Haiti	HTI	1950–2016	Qatar	QAT	1971–2016
Bahrain	BHR	1971–2016	Honduras	HND	1950–2016	Romania	ROM	1950–2016
Bangladesh	BGD	1971–2016	Hungary	HUN	1950–2016	Russia	RUS	1992–2016
Belarus	BLR	1991–2016	India	IND	1950–2016	Rwanda	RWA	1962–2016
Belgium	BEL	1950–2016	Indonesia	IDN	1950–2016	Saudi Arabia	SAU	1950–2016
Benin	BEN	1961–2016	Iran	IRN	1950–2016	Senegal	SEN	1960–2016
Bhutan	BTN	1950–2016	Iraq	IRQ	1950–2016	Serbia	SRB	1992–2016
Bolivia	BOL	1950–2016	Ireland	IRL	1950–2016	Sierra Leone	SLE	1961–2016
Bosnia Herzegov.	BIH	1992–2016	Israel	ISR	1950–2016	Singapore	SGP	1965–2016
Botswana	BWA	1966–2016	Italy	ITA	1950–2016	Slovakia	SVK	1993–2016
Brazil	BRA	1950–2016	Jamaica	JAM	1962–2016	Slovenia	SVN	1991–2016
Bulgaria	BGR	1950–2016	Japan	JPN	1950–2016	Solomon Islands	SLB	1978–2016
Burkina Faso	BFA	1960–2016	Jordan	JOR	1950–2016	Somalia	SOM	1960–2016
Burundi	BDI	1962–2016	Kazakhstan	KAZ	1991–2016	South Africa	ZAF	1950–2016
Cambodia	KHM	1953–2016	Kenya	KEN	1963–2016	Soviet Union	SUN	1950–1991
Cameroon	CMR	1960–2016	North Korea	PRK	1950–2016	Spain	ESP	1950–2016
Canada	CAN	1950–2016	South Korea	KOR	1950–2016	Sri Lanka	LKA	1950–2016
Cape Verde	CPV	1975–2016	Kuwait	KWT	1961–2016	Sudan	SDN	1956–2016
Central Afr. Rep.	CAF	1960–2016	Kyrgyzstan	KGZ	1991–2016	Suriname	SUR	1975–2016
Chad	TCD	1960–2016	Laos	LAO	1950–2016	Swaziland	SWZ	1968–2016
Chile	CHL	1950–2016	Latvia	LVA	1991–2016	Sweden	SWE	1950–2016
China	CHN	1950–2016	Lebanon	LBN	1950–2016	Switzerland	CHE	1950–2016
Colombia	COL	1950–2016	Lesotho	LSO	1966–2016	Syria	SYR	1950–2016
Comoros	COM	1975–2016	Liberia	LBR	1950–2016	Taiwan	TWN	1950–2016
DR Congo	ZAR	1960–2016	Libya	LBY	1951–2016	Tajikistan	TJK	1991–2016
Congo	COG	1960–2016	Lithuania	LTU	1990–2016	Tanzania	TZA	1961–2016
Costa Rica	CRI	1950–2016	Luxembourg	LUX	1950–2016	Thailand	THA	1950–2016
Cote dIvoire	CIV	1960–2016	Macedonia	MKD	1991–2016	Timor-Leste	TMP	2002–2016
Croatia	HRV	1991–2016	Madagascar	MDG	1960–2016	Togo	TGO	1960–2016
Cuba	CUB	1950–2016	Malawi	MWI	1964–2016	Trinidad Tobago	TTO	1962–2016
Cyprus	CYP	1960–2016	Malaysia	MYS	1957–2016	Tunisia	TUN	1956–2016
Czech Republic	CZR	1993–2016	Mali	MLI	1960–2016	Turkey	TUR	1950–2016
Czechoslovakia	CZE	1950–1991	Mauritania	MRT	1960–2016	Turkmenistan	TKM	1991–2016
Denmark	DNK	1950–2016	Mauritius	MUS	1968–2016	Uganda	UGA	1962–2016
Djibouti	DJI	1977–2016	Mexico	MEX	1950–2016	Ukraine	UKR	1991–2016
Dominican Rep.	DOM	1950–2016	Moldova	MDA	1991–2016	Un. Arab Emirates	ARE	1971–2016
Ecuador	ECU	1950–2016	Mongolia	MNG	1950–2016	United Kingdom	GBR	1950–2016
Egypt	EGY	1950–2016	Morocco	MAR	1956–2016	United States	USA	1950–2016
El Salvador	SLV	1950–2016	Mozambique	MOZ	1975–2016	Uruguay	URY	1950–2016
Equatorial Guin.	GNQ	1968–2016	Myanmar	MMR	1950–2016	Uzbekistan	UZB	1991–2016
Eritrea	ERI	1993–2016	Namibia	NAM	1990–2016	Venezuela	VEN	1950–2016
Estonia	EST	1991–2016	Nepal	NPL	1950–2016	Vietnam	VNM	1976–2016
Ethiopia	ETH	1950–2016	Netherlands	NLD	1950–2016	South Vietnam	SVM	1950–1975
Fiji	FJI	1970–2016	New Zealand	NZL	1950–2016	Yemen	YEM	1991–2016
Finland	FIN	1950–2016	Nicaragua	NIC	1950–2016	North Yemen	NYE	1950–1991
France	FRA	1950–2016	Niger	NER	1960–2016	South Yemen	SYE	1950–1991
Gabon	GAB	1960–2016	Nigeria	NGA	1960–2016	Yugoslavia	YUG	1950–1992
Gambia	GMB	1965–2016	Norway	NOR	1950–2016	Zambia	ZMB	1964–2016
Georgia	GEO	1991–2016	Oman	OMN	1950–2016	Zimbabwe	ZWE	1950–2016

Appendix

A.1 Descriptives

Table 2 gives the categories of arms that are included in the analysis. All types with explanations are taken from SIPRI (2017b). The 171 countries that are included in our analysis can be found in Table 3, together with the three-digit country codes that are used to abbreviate countries in the article. In addition, the time periods, for which we coded the countries as existent are included. Note that the SIPRI dataset contains more than 171 arms trading entities, but we excluded non-states and countries with no (reliable) covariates available. In the covariates, some missings are present in the data. No time series of covariates for the selected countries is completely missing (those countries are excluded from the analysis), and the major share of them is complete, but there are series with some missing values. This is sometimes the case in the year 1990 and/or 1991 where the former socialist countries split up or had some transition time. In other cases, values at the beginning or at the end of the series are missing. We have decided to impute the missing values via linear interpolation, using the R package imputeTS by Moritz (2016).

1.2 Technical details on the estimation

The estimation of the models is done in R (see R Development Core Team, 2014). For models with one weight matrix, there is an implementation in the package spdep (version 0.7-7) by Bivand et al. (2013) (see also Bivand and Piras, 2015), based on a two-step method. First, the profiled likelihood is maximized with respect to $ρ$ , by using the simplex method of Nelder and Mead (1965). In the second step, the other parameters are found by weighted least squares. Given those estimates, the concentrated likelihood is updated for $ρ$ . The major computational burden comes with the calculation of the log-determinant of $(I_{n} - ρ W)$ . The problem is solved by using the eigenvalues of W, that is, $log (| I_{n} - ρ W |) = \sum_{i = 1}^{n} log (1 - ρ λ_{i})$ , with $λ_{i}$ being the eigenvalues of W.

Further important packages used in this article are the igraph package by Csardi and Nepusz (2006) as well as the statnet suite by Handcock et al. (2008) for the visualization of networks. For the figures and tables, the packages ggplot2 by Wickham (2016) and stargazer by Hlavac (2013) are used.

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