1 Introduction
The analysis of international trade based on network data has a long tradition with Leontief (1941) as an early reference. Nevertheless, network analysis as a genuinely relational perspective is only in the beginning to become a serious substitute technique for general equilbrium models based on inputoutput matrices (Jones 2012) or for gravity models (Egger and Staub 2016; Head and Mayer 2013). Recent work based on a network perspective includes del RíoChanona et al. (2017), Zhou et al. (2016) and Fan et al. (2014) to name but a few. While general commercial trade data has several sources, including the World InputOutput dataset (Timmer et al. 2012) and the World Import and Export data (Center for International Data 2017), the information about global arms trading is much more limited. A major source in this respect is the data provided by the Stockholm International Peace Research Institute (SIPRI). It provides information on the exchange of major conventional weapons (MCW) for the period 1950  2016, see Akerman and Seim (2014). For each year, the database provides a comparable trading amount, expressed in so called trend indicator values (TIVvalues), that mirror production costs of the transferred weapons and not explicitly the value of the contract. Recording whether there is trade at all, i.e. whether the trading amount is positive or zero, leads to a yearly binary network which is visualized for the most recent consecutive networks (2015, 2016) in Figure 1 with the isolates (nonconnected countries) excluded for better clarity.
It can be seen that there is indeed some hierarchical structure in the network as there exist some highly interconnected central nodes, e.g. United States (USA), France (FR), Germany (DEU) and Russia (RUS), and many countries that are only receiving arms, around them. If one wishes to understand changes in the network, i.e. the transition from the structure in 2015 to the structure in 2016, a formal modelling approach is needed. This is the intention of our paper and we model the yearly changes in arms trading.
The central workhorse in statistical network analysis are Exponential Random Graph Models (ERGM) as introduced by Holland and Leinhardt (1981), see also Frank and Strauss (1986) and Lusher et al. (2012). A general overview and survey about the state of the art in statistical network analysis is provided by Goldenberg et al. (2010), Hunter et al. (2012), Fienberg (2012) or Kolaczyk (2009, 2017). The first generation of these models has been static and applied to cross sectional networks. The arms trade data require dynamic models to investigate how trading relations are build up or given up over time. Statistical models for dynamic or temporal networks have been developed just in the recent years, and different techniques have been proposed. Robins and Pattison (2001)
were the first to extend the ERGM class to discretetime Markov chain models, to allow for network dynamics, see also
Snijders et al. (2010). Hanneke et al. (2010) or Leifeld et al. (2017) consider network models when the networks are observed on a discrete time scale, like years, as in our example. They propose the temporal Exponential Random Graph Models (tERGM) which makes use of a Markov structure and includes previous network statistics as covariates in the model. A related approach is presented by Almquist and Butts (2014), discussing assumptions that allow for translating the often computationally intractable analysis of dynamic ERGMs into logistic regression models.
Koskinen et al. (2015) extend the model using Bayesian methods which allows the parameters in the dynamic network model to change with time. An extended view from a general perspective how to model dynamic networks is provided by Holme (2015). This also includes models for continuous time, such as stochastic actorbased models (Snijders et al. 2010) or dynamic stochastic block models (see for instance Xu 2015). A recent novel modelling strategy for networks observed at discrete time points has been proposed by Krivitsky and Handcock (2014). They do not model the state of the network itself but focus on network changes which either happen with new edges occurring (labelled formation) or dissolving (labelled as dissolution). Assuming independence between the two processes, conditional on the previous network, leads to so called separable network models. This modelling class will be employed in this paper in order to model changes of arm trading over a period of more than 65 years. Given the long time window that we consider and given the multiple political evolutions and revolutions that took place during this time, it appears necessary to allow the model parameters to change with time. We do this by incorporating smoothing techniques and making use of so called generalized additive models (GAM). The latter model class has been proposed by Hastie and Tibshirani (1990) and extended massively by Wood (2017) to allow for smooth, semiparametric modelling in a generalized regression framework (see also Ruppert et al. 2009). We combine the two models classes in this paper in order to allow for nonparametric smooth network dynamics describing changes in arm trading behaviour.Besides smooth timevarying effects we need to take heterogeneity into account. Classical network models assume homogeneity of the nodes (actors), which for our data refers to homogeneity of countries with respect to arm trading. This is apparently questionable and heterogeneity needs to be included in the model (see also Thiemichen et al. 2016, for a discussion on node heterogeneity). To do so, we allow for countryspecific random heterogeneity, both, as sender (exporter) as well as receiver (importer). Using random effects for sending and receiving, labelled as productivity and attractiveness in the model of Duijn et al. (2004) is not new, but we accommodate them by smooth, timevarying random effects following Durbán et al. (2005) which leads to smooth random sender and receiver effects. The fitted effects will be investigated separately with techniques from functional data analysis (FDA), see for instance Ramsay and Silverman (2005). This allows to exhibit countries that have changed their role in the arm trading network over the observation period.
The international transfer of weapons has been extensively investigated. Systematic historical accounts (Harkavy 1975; Krause 1995) accentuate the aspect of the technological advantages of highly developed countries. The countries’ international status in the world system relies on the own endowment of and the possibility to supply highly developed weapons systems. In this perspective, the diffusion of defense technologies is considered a power projection leading to a web of security relations. Formal politicoeconomic models of the international trade of arms (for an overview see GarciaAlonso and Levine 2007) proposed a oligopolistic supplyanddemand model of the weapons trade. The supplier’s decision criteria are per assumption twofold and potentially contradictory: they include not only welfare returns for arms sales, but exporters have to balance these economic incentives against potentially negative security externalities. Transferred weapons may harm the exporter’s security interests in the future. Currently, there are no published empirical network analyses of the international arms trade in a proper sense. Akerman and Seim (2014)
is actually a binarized gravity model of arms transfer relations.
Kinne (2016) applies a twostage model along Snijders (2017)Stochastic actororiented model (SOAM) where the first stage consists of a network model of the formation of trade agreements whereas the transfer of arms stage is conceived as transfer relation between two countries where exports and imports are summed up and estimated as a ordered dimension (lowhigh). The paper by
Thurner et al. (2017) relies on the tERGM model and is the first dynamic model of arms transfers.The aims of our paper are twofold. First, we combine different novel statistical models and techniques to analyse dynamic network models. These include in its core separable temporal network models (Krivitsky and Handcock 2014), generalized additive models (Wood 2017) and functional data analysis (Ramsay and Silverman 2005). Secondly, the combined model allows us to analyse the dynamics in arm trading using current data of the Stockholm Peace Research Institute. The paper is organized as follows. In Section 2 we provide information about the data. Section 3 explains the combination of the statistical models. Section 4 provides the results and their interpretation. Section 5 concludes the paper.
2 Data description and preprocessing
Data on the international trade of major conventional weapons (MCW) is provided by the Stockholm International Peace Research Institute (SIPRI). This institute collects information on all transfers of MCW from 1950 to 2016 in a comprehensive data base. The data on trade flows includes information on the sender country, the receiver country as well as the type of MCW that is traded, this includes for example aircraft, armoured vehicles and ships. See Table 1 in the Appendix A.1 for a comprehensive overview of the types of arms included. The volume of the trades is measured in so called TIVvalues, shorthand for trendindicator value. Those values do not represent the sales prices of arms transfers but are based on production costs and represent the value of military resources that are exported. For detailed explanations on the methodology see SIPRI (2017b). The SIPRI Arms Transfers Database can be accessed free of charge online at SIPRI (2017a) after agreeing to its terms and conditions.
In order to analyse the binary network of international arms trades, we discretize the network. This means that we disregard the information on the value of the trade flow and work only with binary variables, being one if there is a trade flow greater zero between two countries and zero else. The number of countries included and their threedigit countries codes are given in Table
2 of Appendix A.1, together with the years where they are included in the network. Note that we have excluded all nonstate organizations like the Khmer Rouge or the Lebanon Palestinian Rebels from the dataset as well as countries with no reliable covariate information available. Figure 10 in the Appendix A.1 also provides a collection of summary statistics of the network.The analysis of the degree of the network nodes is of vital interest in statistical network analysis, see Barabási and Albert (1999). It is defined as the distribution of the degree sequence and gives important insights into the basic properties of the network studied. As we have 67 networks to analyse, we compute the timeaverage degree distribution and provide information on the minimal and maximal value of the realized degree distribution.
This is plotted in a loglog version in Figure 2 for both, outdegree and indegree. It becomes very clear that most of the countries are restricted to have no exports at all with a timeaverage share of 78% of countries having outdegree zero, while the outdegree distribution has a long tail, indicating that there are a few countries, having a very high outdegree. The highest observed outdegree is 66 and can be attributed to the United States. Other countries with exceptional high outdegree for almost the whole time period are Russia (Soviet Union), France, Germany, United Kingdom, China, Italy and Canada. In the right plot the indegree distribution can be seen. Here the pattern is different. The highest value observed is 16 and corresponds to Saudia Arabia. In contrast to the outdegree distribution, the countries with a high indegree are changing with time. In the beginning of the observational period the countries with the highest indegree were Germany, Indonesia, Italy, Turkey and Australia, but in more recent times these are the United Arab Emirates, Saudi Arabia, Singapore, Thailand, Oman and the United States.
3 Model
3.1 Dynamic Formation and Dissolution Model
In this section we formalize the network model. Let be the network at time point , which consists of a set of actors, labelled as and a set of directed edges, represented through the index set . Note that this is a slight misuse of index notation since does not necessarily refer to the th element if we consider as adjacency matrix. This is because the actor set is allowed to change with time, so that and are not running indices from to , where is the number of elements in . Instead indices and represent the th and th country, respectively. We define if country exports weapons to country and since selfloops are meaningless elements are not defined.
We aim to model the network in based on the previous year network in . To do so we have to take into account that the actor sets and may differ. In particular we have to consider the case of newly formed countries. New countries of interest are those that are present in but do not provide information about their network embedding in the previous period. For exports this is not a concern as it is almost never the case that a new country starts sending arms immediately after entering the network. Notable exceptions are Russia, the Czech Republic and Slovakia. However, these countries have clear defined predecessor states (the Soviet Union and Czechoslovakia) which can be used in order to gain information about the position of these countries in the precedent network. Regarding the imports, there is a share of countries that start receiving arms immediately with entering the network. Notwithstanding, those transactions represent a share of less then 0.3% of the observed trade flows. Therefore, we regard this cases as negligible and include in the model only countries where information on the current and previous time period is available. We formalize this approach by defining as the subgraph of with actor set containing elements. Accordingly, represents the subgraph of with actor set . Note that both subgraphs share the same set of actors and if and coincide.
From a modelling perspective, we follow Hanneke et al. (2010) and assume that the network in can be modelled given preceding networks, using a firstorder Markov structure to describe transition dynamics for those actors included in the set . Striving for a high interpretability of the model, we employ an extended version of the separable temporal exponential random graph model (StERGM) proposed by Krivitsky and Handcock (2014) which is in fact an extension of the tERGM model proposed by Hanneke et al. (2010). The basic idea of this approach is to focus on the formation of potential new edges and the persistence of existing edges, rather than simply modelling the existence of edges.
Let therefore represent the formation network, that consists of edges that are either present in or in . For the persistence network, we define , being the network that consists of edges that are present in and in . Based on the actor set and given the formation and persistence network as well as the network in the network in is uniquely defined by
(1) 
Note that both, as well as depend on time as well, which we omitted in the notation for ease of readability. We assume that for each discrete time step, the processes of formation and persistence are separable. That is, the process that drives the formation of edges does not interact with the process of the persistence of the edges conditional on the previous network. Formally this is given by the conditional independence of and :
(2) 
where the lower case letters denote the realisations of the random networks and gives the parameters of the model. We will also include nonnetwork related covariates in our analysis, but we suppress this here in the notation for simplicity.
Note that it is not possible to use the lagged response as predictor, as by construction and . That is, an edge that existed in cannot be formed newly and an edge that was not existent in cannot be dissolved. It follows that the formation model exclusively focuses on the binary variables with . This assures that in no edge between actors and was present and both actors are observable at both time points. Equivalently, the model for consists of observations with , assuring that only edges that could potentially persist enter the model.
A central model class for networks are ERGMs. For the formation network this would yield the probability model
(3) 
The sum in the denominator is over all possible formation networks from the set of potential edges that can form given the network . The inner product
, relates a vector of statistics
to the parameter vector . The analogous model is assumed for the persistence of edges and not explicitly given here for the interest of space.We will subsequently work with a simplified model which is computationally much more tractable. We assume that the formation or persistence of an edge at time point does solely depend on the past state but not on the current state of the network. This is achieved by restricting the statistics such, that they decompose to
for some statistics . This assumption is extensively discussed by Almquist and Butts (2014) and can be well justified by the notion that the lagged network accounts for the major share of the dependency among the edges in the current network. It also allows for intuitive interpretations as can be seen as follows. Let represent the formation network , excluding the entry . Then, for the following logistic model holds
(4) 
Note that model (4) describes networks dynamics, but the model itself is static. Hence we model dynamics but do not allow for dynamics in the model itself. This is a very implausible restriction which we give up by allowing the model parameters to change with time , that is we replace the parameter by , representing a smooth function in time. In other words, we allow the parameters in the model to smoothly interact with time. This leads to a timevarying coefficient model in the style as proposed by Hastie and Tibshirani (1993). The focus of interest is therefore not only on the formation and persistence of edges (trade flows) but also on how these effects change in the 67 years long observation period.
3.2 Network Statistics and Explanatory Variables
The following network statistics are included in the analysis and visualised in Figure 3. With the exception of Reciprocity they are normed to a percentage range between 0 and 100.
Outdegree: The outdegree of actor in time point is defined as
In our model we include this statistics for the sender () and the receiver (). Note that the receivers outdegree is essentially the same measure as Outdegree Popularity often used in Stochastic actororiented models (Snijders
et al. 2010) showing whether countries having a high outdegree are attractive as a receiver of arms.
Reciprocity: This statistic measures whether the potential receiver (in case of the formation model) was a sender in the dyadic relationship in the previous period:
Transitivity: As measures for higherorder dependencies we include two versions of transitive triads (see for example Robins et al. 2007). The first threenode statistic included in our model is called transitivity and defined as
This statistic can be interpreted as an intermediate exporting relationship as it essentially counts the directed twopaths from to in . The second threenode statistic we operationalise is called sharedsuppliers and given by the shared number of actors that export to a given pair of countries:
By their nature, local three node statistics can sum up to at most which justifies the denominator.
Naturally, the network of international arms trade is not exclusively driven by network statistics but also influenced by variables from the realms of politics and economics. Those variables are either noderelated or dyadrelated and will be called exogenous variables in this paper. For all exogenous variables that are included in the model we use lagged versions, first in order to be consistent with the idea that the determination of the network in is based on and second, because there is a substantial time lag between the ordering and the delivery of MCW.
Colony: Trying to capture the influence of a strong political dependency, we include the binary variable . It is one, if country was the colonial power and was the respective colony and zero otherwise. The data stems from the Colonial/Dependency Contiguity table, freely available online (Correlates of
War Project 2017a).
Formal Alliances:
As a consequence of the strong relationship between the directed network of arms trade and the undirected network of dyadic formal alliances (consisting mostly of defence agreements), the binary variable is included in the model, being one if countries and had a formal alliance in the previous period. As the network of formal alliances is symmetric, we have . Given the restriction that the data is available only until 2012 (Correlates of
War Project (2017b) for the most recent version of the data and Gibler (2008) for the article of record for the data set) we assume that the alliances did not change between 2012 and 2015.
Regime Dissimilarity:
A common result in network analysis is that similar actors are subject to stronger gravity than dissimilar ones. We therefore include a measure for the similarity of political regimes. A standard measure is the so called polity score, ranging from the spectrum (hereditary monarchy) to (consolidated democracy). This data can be downloaded as annual crossnational timeseries until 2015, see Center for
systemic Peace (2017) for the data and Marshall and
Jaggers (2002) as a basic reference. In our model we operationalise the distance between political regimes by using the absolute differences between the scores: .
Economic Quantities: Regarding economic factors, the standard measure is the gross domestic product (in millions), that is included in logarithmic form for the sender () as well as the receiver (). The GDP data are taken from
Gleditsch (2013), which, to the best of our knowledge is the only dataset that also covers socialist and communist countries prior to 1990. This is merged from the year 2010 on with recent real GDP data from the World Bank real GDP dataset (World Bank 2017).
Military Personnel:
It seems plausible that the size of the military personnel of the receiving country has an impact on the formation and persistence of edges. Accordingly, the military personnel (in thousands) of the receiving countries is added in logarithmic form (). The data is available from the Correlates of
War Project (2017c) in the national material capabilities data set with Singer
et al. (1972) as the basic reference on the data. Again we have to assume that the values hold until 2015.
3.3 Modelling Heterogeneity
The proposed network model assumes homogeneity, meaning that all differences between nodes in the network are fully described by the network statistics and the explanatory variables proposed above. As noted in Section 2 already, there is a rather small number of countries that dominate the market and a large number of countries that are restricted to imports. Moreover there are some countries that change their relative position in the trade network during the course of time. This mirrors a substantial amount of heterogeneity which itself may be dynamic. This heterogeneity will be accommodated in our model by the inclusion of smooth, timevarying random effects for the sender states as well as for the receiver states. We thereby follow the modelling strategy of Durbán and AguileraMorillo (2017). To be specific, for the formation model we assume that the model includes two timedependent random coefficients and
. The effects are assumed to be a realization of a stochastic process with continuous and integrable functions. The smooth effects are based on a Bspline basis and we impose normal distributions as penalty on the corresponding spline coefficients.
3.4 Complete Model and Estimation
Putting all the above elements together, the specification of the formation model of equation (4) is given by
(5) 
Analogously we get the persistence model.
Given this specification, we are able to estimate the model by using the flexible generalized additive mixed model (GAMM) framework provided by Wood (2017) (see also Wood 2006) which is implemented in the mgcv package (version 1.812) by Wood (2011). We sketch some details on the estimation procedure in the Appendix A.2.
4 Results
4.1 Timevarying fixed effects
The results of the timevarying effects are grouped into networkrelated covariates (presented in Figure 4) and political and economic variables (presented in Figure 5
). In these Figures, the left column gives the coefficients for the formation model and the right column for the persistence model. The values for the coefficients are presented as solid lines with shaded regions, indicating two standard error bounds. The zeroline is presented as a dashed line and the estimates for timeconstant coefficients are given by the dotted horizontal line. Note that the coefficients at a given time point can be interpreted as the coefficients in a simple logit model. Additionally, for the same coefficient in the formation and persistence model the effect size can be compared directly.
NetworkEffects
Outdegree
: The degreestatistics are almost constant and close to zero for both models as they have been partly absorbed by the inclusion of the smooth random effects. An exception is given by the coefficient on the outdegree of the receiver in the formation model, that is consistently negative and shows a slight upward trend in its timevarying version which however is not significant based on the confidence intervals. Nevertheless we can interpret this as evidence that countries that have a high outdegree are unattractive as importing countries.
Reciprocity: The coefficient on reciprocity is positive and significant for the formation model, meaning that we observe more reciprocity as to be expected by a random graph. For the persistence model we find that reciprocity has a positive effect that is significant from 1980 on, indicating an increased importance of reciprocity for lasting trade relations.
Transitivity: Looking at threenode statistics it can be seen (at least since the 1975s) that the variable transitivity has a positive impact on the formation as well as on the persistence. No time variation can be captured for the persistence model. For the formation, however we see a consistent positive effect, that almost linearly increases until the end of the cold war. After that, the effect climbs steeply upwards, peaking at the end of the 20th century, followed by a decline back to the level of 1990. A possible interpretation of the pronounced swing between 1990 and 2010 is, that after the break up of the two hostile blocs a lot of longstanding armtrading partnerships ceased to exist and the exporting countries tried to tap into new markets with new customers, where they did not have previous relationships with. Therefore, they created new export relationships with those countries where already an indirect exporting relationship was present.
The coefficients related to the shared suppliers show, that the effect on the formation model is positive and constant in time, while the effect for the persistence model starts from being significantly negative and turns out to have virtually no effect from the 1980ths on. Until 1980, two countries with one or more common suppliers had an increased probability to engage in trade with each other. These relationships were unlikely to persist in the 50th up to the 80th. In the later decades, the negative effect on the persistence is not any longer significant.
Covariate Effects
Colony:
We observe that having a colonial past increases the probability of forming and maintaining a trade flow from the former colonial power to the former colony. However, no time variation can be carved out and the effect is insignificant.
Formal Alliance:
The impact of a bilateral formal alliance on the formation of an edge is positive and significant but has decreased strongly from 1960 to 2016. The longterm trend reflects the increase of bilateral formal alliances, that are not necessarily accompanied by arms trade. The exceptions from the general trend is the combination of the small downward bump starting in 1990 with the following upward movement around the year 2000 mirroring the reorganization of formal alliance agreements in that time period. While the influence on the formation is declining, the coefficient for the persistence model is constant over time and corroborates the intuitive expectation that an existing alliance increases the probability that a given edge persists.
Regime Dissimilarity:
Some timerelated variation can be found in the coefficient on the absolute difference of the polity scores. For the formation model, this coefficient is all along negative and significant. The negative effect on the formation is modest at the time period from 1950 to 1985. With the decay of the eastern bloc, the resistance to send new arms to dissimilar regimes increases strongly until 2000 potentially because the pool of democracies became larger. After that, the absolute effect of different polity scores declines again, coming back to the long term constant effect. For the persistence model, we see that a strong difference in political regimes had a negative effect on the persistence of a trade relationship in the beginning of the observational period, but the effect became almost zero after 1960.
Economic Quantities:
The coefficients on the logarithmic GDP for the sender and the receiver are positive and almost constant for both models. This strengthens our expectations that greater economic power of the sender as well as the receiver increases the probability of forming and maintaining trade relations.
Military Personnel: For the military personnel of the receiver no strong effect for the formation can be found. For the persistence, however, we finde a postive significant effect. This means that countries with a big army tend to be attractive for being the receiver in persistent trade relations.
4.2 Timevarying Smooth Random Effects
We now pay attention to the actorspecific heterogeneity. In Figure 6, the countryspecific effects for the sender, as well as the receiver countries are visualized for the formation model on the left and the persistence model on the right.
In order to interpret these effects we make use of functional data analysis. More specifically, we employ functional principal component analysis to the multivariate time series of random effects seen in Figure 6 (see also Ramsay and Silverman 2005 and the Appendix A.3). The resulting functional principal component analysis is shown in Figure 7 for the formation model and in Figure 8
for the persistence model. On the left hand side the scores of the first two principal components are plotted, where the latter are visualised on the right hand side. The share of variance explained by the respective component is provided in the brackets. The first principal component is close to be constant and represents the share of variance induced by different overall levels of the random effect curves. The dynamic of the random effects is captured by the second principal component, delivering a tendency for an upward movement if positive and downward if negative. Hence, looking on the horizontal axes, we see countries that build up many arm trade links as exporters (importers) on the right hand side while countries that are reluctant building up export (import) links are on the left hand side. Looking on the vertical axes, we see countries that decrease their role as exporter (importer) over the time on the bottom, and vice versa countries that increase the number of export (import) links over time on the top. All these effects are conditional on the remaining covariate effects shown before. Therefore, the analysis captures nonmodelled systematic patterns in the error terms.
4.2.1 Results of the Functional Component Analysis
Formation Model: Countries like Mexico (MEX), Israel (ISR) or Japan (JPN) play prominent roles in the component score plot for the sender effect of Figure 7. This visualizes that Mexico (MEX) has the lowest overall level, being the country with the least tendency to form new trade exports. On the other side is Israel (ISR), being the country with the highest tendency to establish new arms exports. There is also a lot of variation over time. A notable case is the development of Japan (JPN), starting with an already low sender effect, that is decreased even further with time. But also other patterns that are not so obvious from Figure 6 appear. For example countries like Bulgaria (BGR), Turkey (TUR), Ukraine (UKR), South Africa (ZAF), United Arabic Emirates (ARE) and Denmark (DNK) are countries that started with medium and small sized sender effects for the formation model but have increased their ability to form new export relations very strongly.
On the other hand a nontrival share of European countries (Italy (ITA), Netherlands (NLD), Great Britain (GBR), France (FRA), Sweden (SWE), Switzerland (CHE), Hungary (HUN)) together with the United States (USA) and Canada (CAN) have reduced their high sender effects.
Another global pattern becomes visible since the top left plot in Figure 7 looks like a lying mushroom. That is, those countries that started on a low level (i.e. negative component 1) show, with the exception of Japan (JPN) and Turkey (TUR), not very much upward or downward variability (i.e. low level for component 2). In contrast, those countries that have a level above zero diffuse more strongly up or down. This can be interpreted as a path dependency for those that had a high level and increase it further but also reflects the relative decrease of importance of the ”old giants” in the MCW trade like Great Britain (GBR), France (FRA), United States (USA), Egypt (EGY) and, of course, the Soviet Union (SUN) and Chechoslovakia (CZE).
The dynamics for the receiver effects of the formation model are more evenly distributed. The countries United Arab Emirates (ARE) and Oman (OMN) is easy to identify as they have a high level (large value of component 1) and a strong growth over time (high value for component 2). But also other oil exportdriven countries like Jordan (JOR), Saudi Arabia (SAU), Equatorial Guinea (GNQ), Iraq (IRQ), Chad (TCD), Qatar (QAT), Algeria (DZA) and Bahrain (BHR) are within the first quadrant. These are importing countries which even increased their (partly already high) number of import partners over time. Interesting upward mover from a low level are Afghanistan (AFG), Philippines (PHL) and Luxembourg (LUX). Not very surprisingly the southwest quadrant at the bottom of Figure 7 is populated with (former) socialist countries as Cuba (CUB), Ukraine (UKR), North Korea (PRK), Yugoslavia (YUG) and Moldova (MDA) that have lost parts of their supply channels after the end of the cold war. But also Japan (JPN) can be found in this quadrant.
Persistence Model: The patterns for the sender effects of the persistence model can be seen at the top of Figure 8. Again Japan (JPN) stands out, with a very low score for the first component it represents the country with the lowest prospect to maintain export relationships. Israel (ISR) is also again notable, together with North Korea (PRK), Moldova (MDA) and the Soviet Union (SUN) it has the highest tendency to be a sender in persistent arms trade (large value of component 1). All in all less strong time dynamics is visible, with the exception of the big former socialist exporters (Czechoslovakia (CZE), Soviet Union (SUN) and China (CHN)) as downward movers (fourth quadrant). This mirrors the fact that these countries had established lasting relationships in arms trade with strongly subordinate receivers. An interpretation that might also explain the decrease of the sender’s effect of Great Britain (GBR) and France (FRA) which lost subordinated customers with the decolonization. On the other hand other European countries like Germany (DEU), Finland (FIN), Belgium (BEL), Bulgaria (GBR), Ukraine (UKR), Switzerland (CHE), Italy (ITA), Ireland (IRL), Norway (NOR) and Sweden (SWE) have managed to increase their sender effect with time (high value for component 2). The greatest increase (high value on component 2), starting form a relatively low level, can be attributed to Canada (CAN).
On the bottom of Figure 8 the results on the receiver effects of the receiver model are given. This plot is very conclusive for detecting countries that moved upward in their tendency to establish lasting import relationships. Notably, Norway (NOR) and Japan (JPN) with a high value on the second component but also Ireland (IRL), Jamaica (JAM), Finland (FIN), China (CHN), Estonia (EST) and Azerbaijan (AZE) increased their sender effects. Again we find prominent positions for Israel (ISR) and United Arab Emirates (ARE), being countries that have a high level (high value for component one) and increased their tendency to be the receiver in persistent trade relations (high value for component two). Also the positions of Costa Rica (CRI) and Germany (DEU) are noteworthy as they represent to some extend the two sides of the extremes from a low starting level to a higher one (Costa Rica (CRI) in the second quadrant) and a high starting level to a low one (Germany (DEU) in the fourth quadrant).
4.2.2 Further Political Interpretations
For some of the results noted above, exceptional scores can be traced back to concrete poltical and economic circumstances not captured in the model. The astonishing low tendency to export of Japan (JPN), although being among the wealthy countries with a highly developed export industry, is clearly due to the highly restrictive arms export principles introduced in 1967, and tightened in 1976. This ban on exports was only lifted in 2014 (Ministry of Foreign Affairs of Japan 2014). Otherwise the results are pointing to the fact that some countries like Israel (ISR) follow in a striking way the path of developing highly internationally competitive weapons systems while for others as for example Mexico (MEX), contrary, it is not possible to enter any other market despite being among the world’s largest economies. We consider these special paths as induced by cumulative advantages and learning over time in the one case, whereas in the case of Mexico (MEX) we observe the stickiness and path inertia of a country having not been able to make its defence products sold externally. We also see that many emerging economies (Turkey (TUR), United Arab Emirates (UAE) and South Africa (ZAF)) and left overs of the collapsed Soviet Union defense industries (especially Bulgaria (BGR) and Ukraine (UKR)) rush into the global market of military products. The astonishing position of Germany (DEU) in the persistence plot reflects the development from postwar Germany, that had very restricted importing possibilities, to the modern state of Germany that has its own highperforming arms industry which relaxed the dependence on foreign arm imports. Clearly, a close look on the individual circumstances in country specific casestudies would be necessary to provide deeper and more differnetiated insights. Especially the role of politics and political regimes (e.g. North Korea (PRK) and Israel (ISR)) and oilexports (e.g. United Arabic Emirates (ARE) and Oman (OMN)) are interesting. This is, however, beyond the scope of this paper and will be subject to further research.
4.3 Model Evaluation
An important challenge in statistical network analysis is the question of how to evaluate the goodness of fit of a proposed model. While classical measures as the receiver operator characteristic (ROC) curve and the related measure the area under the curve (AUC) are insightful when it comes to weight the true positive rate (TPR) against the false positive rate (FPR), the evaluation of the fit in network models is somewhat more subtle. This is mainly because a good network model should be able to reflect the global network characteristics. To be more precise, this means it is not enough to reach an acceptable rate of correctly classified edges but also the global network structures like the mean outdegree, the share of reciprocity and observed transitivity should be mirrored sufficiently by the predictions. In order to do so, simulationbased approaches are advocated by
Hunter et al. (2008), Almquist and Butts (2014) and Hanneke et al. (2010) and implemented in the standard network packages in R (Handcock et al. (2016) for the ergm package, Krivitsky and Handcock (2016) for the tergm package and Ripley et al. (2013) for the RSiena package).We employ both approaches in order to evaluate the outofsample predictive power of our model. We first fit the formation model as well as the persistence model, based on the information in , to the data in and use the estimated coefficients for the prediction of new formation or persistence of existing ties in . As the networks are very sparse in the early years and hence also the quality of predictions is very volatile we restrict the analysis of the outofsamplepredictions to the years 19702016.
As the predictions are probabilistic by their nature, we weight the TPR against the FPR for varying threshold levels, yielding the ROC curve and the AUC for each year of prediction. The corresponding plots are given in Figure 9 with the ROC curves at the top and the AUC time on the bottom, for the formation (left) and the persistence (right). Generally speaking, the formation model has the greater prognostic power then the persistence model. While the AUC value of the formation model is very high for the whole observational period it is much lower for all years in the persistence model. The AUC for the formation model is very high in the beginning but declines steeply from 1990 on. This seems not too surprising as the end of the cold war resulted in a radical reorganization of the international arms trade. If we look on the forecasts, we see that the rules that determined the formation of new arms trade relations normalized with the beginning of the 21st century as the quality of the forecasts goes back to the pre1990 level.
Interestingly enough the same logic does not apply to the dissolution model. While it seems harder to predict whether a trade relation is dissolved or not, we do not see a longlasting strong decline in the predictive accuracy after 1990. This can be taken as evidence that the end of the cold war has not disrupted too strongly on established trade relations.
Secondly, we simulate for each year potential networks by sampling formation and persistence networks with each potential network being viewed as a realization of a Bernoulli trial, using the predicted probability of the occurrence or persistence of an edge. Then, based on equation (1), the predicted network for is constructed. From this, we evaluate six global network characteristics and compare them to the actual characteristics from the true MCW trade network in . The corresponding Figure 11 is given in the Appendix A.4. The results are reassuring and the simulated networks mirror the real network properties in an acceptable way. Hence, we conclude the model proposed in the paper is able to predict the global patterns of the network in a reasonable way.
5 Discussion
The idea of applying inferential network models to trade networks is not new but our dynamic approach including dynamic heterogeneity is novel. The arm trading network has very special properties, for example a very strong actor heterogeneity and strong changes of the network characteristics over time as the network is steadily growing. Our main contribution to the literature is however not the development of a whole new model class, but the combination of already existing techniques in order to incorporate the properties of the networks under study as good as possible.
For our study of the discretized MCW networks from 1950 to 2016 we employ a dynamic separable network model as proposed by Krivitsky and Handcock (2014) for the basic structure and add techniques proposed by Hastie and Tibshirani (1993) and Durbán et al. (2005). This enables us to study the process of formation and dissolution of trade flows separately as well as the inclusion of timevarying coefficients and smooth timevarying random effects that are further analysed by methods from functional data analysis as described in Ramsay and Silverman (2005). With this we detect new insights with regard to latent dynamics of the formation and persistence of transfer relations. The results mirror the efforts of countries to open new markets  or vice versa the loss of trade partners. This is one important dimension of success and failure of market competition. Our main results from the analysis are strong network effects, especially for the triadic network statistics transitivity and shared suppliers as well as reciprocity. We also find interesting time variation in the coefficients on formal alliances and homophily, measured by the absolute difference of the polity score. For the economic quantities we have solid positive effects on the probability that a tie forms newly or persists. A careful analysis of the random effects exhibits a high variation among the countries as well as along the time dimension. By using functional principal component analysis we decompose the functional time series of smooth random effects in order find countries that have increased or decreased their relative importance in the network. Especially, interesting patterns for the countries Japan, United Arabic Emirates, North Korea and Israel can be found. The evaluation of the fit confirms that the chosen model is able to give good outofsample predictions.
Apart from that, we have to admit that the inclusion of the random effects is potentially problematic because the individual specific effects might be correlated with the covariates, especially the outdegree measures which might lead to consistency issues. However, the inclusion of timevarying fixed effects for each individual seems to be infeasible at the moment, which also prevents us from using a Hausman test (
Hausman 1978). It therefore remains to say that the recent implementation is the best we could get given the high actorspecific and timebased variation in the data.References
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Appendix A Appendix
a.1 Descriptives
Table 1 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 2, together with the threedigit country codes that are used to abbreviate countries in the paper. In addition to that, the time periods, for which we coded the countries as existent are included. Note that the SIPRI data set contains more than 171 arm trading entities but we excluded nonstates 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 and/or where the former socialist countries splitted 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 on three general rules to fill the gaps: First, if a value for a certain country is missing in but there are values available in and , the mean of those is used. If the values are missing at the end of the observational period, the last value observed is taken. In case of missing values in the beginning, the first value observed is taken.
The number of countries included each year in the network is provided in the upper left panel of Figure 10
. It can be seen that the network is growing almost each year until 1992, with two big leaps that show the effects of the decolonization, beginning in 1960 and the end of the Soviet Union after 1991. Typical descriptive statistics for the analysis of networks are
Density, Reciprocity and Transitivity, all shown in Figure 10. The Density is defined as the number of edges divided by the number of possible edges. Reciprocity is defined as the share of trade flows being reciprocal. Transitivity is defined as the ratio of triangles and connected triples in the graph.Type  Explanation 

Aircraft  All fixedwing 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. 
Airdefence systems  (a) All landbased surfacetoair missile systems, and (b) all antiaircraft guns with a calibre of more than 40 mm or with multiple barrels with a combined caliber of at least 70 mm. This includes selfpropelled systems on armoured or unarmoured chassis. 
Antisubmarine 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, selfpropelled 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, combatcapable 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 antitank missiles. Unguided rockets, freefall aerial munitions, antisubmarine rockets and target drones are excluded. 
Sensors  (a) All land, aircraft and shipbased active (radar) and passive (e.g. electrooptical) surveillance systems with a range of at least 25 kilometres, with the exception of navigation and weather radars, (b) all firecontrol radars, with the exception of rangeonly radars, and (c) antisubmarine warfare and antiship 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 100mm 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 antitank missiles, (b) all turrets for ships fitted with a gun of at least 57mm calibre, and (c) all turrets for ships fitted with multiple guns with a combined calibre of at least 57 mm, and (d) air refueling systems as used on tanker aircraft. 
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 manportable air defence systems and guided antitank missiles. Trucks, artillery under 100mm calibre, ammunition, support equipment and components (other than those mentioned above), repair and support services or technology transfers are also not included in the database.  
Source: SIPRI (2017b) 
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  GuineaBissau  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  TimorLeste  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 
a.2 Details on the Estimation Procedure
The recent implementation of Generalised Additive Mixed Models (GAMM) in the R package mgcv allows for smooth varying coefficients as proposed by Hastie and Tibshirani (1993). These models can be represented in GAMMs by multiplying the smooths by a covariate (in the given application the smooths of time are multiplied by the covariates given in equation (5)). See Wood (2017) for more details.
The functions for the smooths are based on PSplines as proposed by Eilers and Marx (1996), giving low rank smoothers using a Bspline basis using a simple difference penalty applied to the parameters. For the smooth timevarying coefficients on the fixed effects a maximum number of 65 knots is used, combined with a secondorder Pspline basis (quadratic splines) and a firstorder difference penalty on the coefficients.
The nonlinear random smooths are estimated similar to those proposed by Durbán et al. (2005). As a basic idea, one views the individual smooths as splines with random coefficients, i.e. each country has a random effect, that is in fact a function of time that is approximated by regression splines. The parameters of the splines are assumed to be normally distributed with mean zero and the same variance for all curves, which translates into having the same smoothness parameter for all curves. This concept is implemented efficiently in the GAMM structure of the mgcv package by using the nesting of the smooth within the respective actor. In order to avoid overfitting and keeping computation tractable, a firstorder penalty with nine knots is employed. The smoothness selection is done for all smooths by the restricted maximum likelihood criterion (REML).
As the data set is rather big with more than 1.3 million observations in the formation model, the fitting procedure of the model is computationally expensive and was virtually impossible with standard implementations in R before the introduction of the bam() function in the mgcv package in 2016 that needs less memory and is much faster than other comparable packages. The estimation routine employs techniques as proposed in Wood et al. (2015). Those methods use discretization of covariate values and iterative updating schemes that require only subblocks of the model matrix to the computed at once which allows for the application of parallelization tools.
For all computations we also used the statistical programming language R (R Development Core Team 2008), version 3.3.1. Important packages used for visualization of networks and computation of network statistics are the statnet suite of network analysis packages (Handcock et al. 2008) as well as the package igraph (Csardi and Nepusz 2006). For the Tables the stargazer package from Hlavac (2013) was employed. For the model evaluation and visualization we used the pROC package of Robin et al. (2011).
a.3 Details on the PCA of the Timevarying Smooth Random Effects
For the analysis of the smooth random effects we are following the discretization approach of Ramsay and Silverman (2005, Chap. 8). As noted in Section 3.3 we assume the random effects ( and in the formation and the persistence model, respectively) to be realizations of a stochastic process , for individual countries and .
In order to summarize the information provided by these functions we are searching for a weight function that gives us the principal component scores . In order to do so, the weight function among all possible functions must be found that maximizes subject to the constraint . From our model we get individual estimated functions for all observations (countries) and can discretize the functions on a grid. We use equidistant points on the interval of length . This gives a discretized time series matrix with country specific observations in the rows and the estimated functions, evaluated at the discrete time points, in the columns:
Therefore, in fact we are searching for a solution for the discrete approximation of such that the solution that maximizes the mean square satisfies . This is now a standard problem, with the solution
being found by the eigenvector that corresponds to the largest eigenvalue of the covariance matrix of
.a.4 OutofSamplePredictions for Simulated Networks
As a standard principle in network analysis, a model should be able the reflect global network characteristics. We evaluate six of them for our outofsample forecasts. The first three characteristics are related to the number of actors that are actively engaged in the arms trade. The statistic Size is defined as the count of predicted edges in each year. This measure helps to evaluate the ability of the model to predict amount of realized arms trade in each year. As it is also of interest to measure how dense the predicted arms trade network is, we include Density, relating the size of the network to the number of edges that could have potentially realized. We define the Order of the network as number of actors that are engaged in either exporting or importing arms. The results will provide an impression whether the model has the ability not only to classify the right amount of edges (as in Size), but also their nesting within the countries.
As we have emphasized the importance of local network statistics we evaluate whether the local network statistics are able to generate the corresponding global statistics. Therefore, we include the Mean Indegree (beeing the same as Mean Outdegree), as well as the share of Reciprocity. In order to evaluate the accuracy of our predictions with respect to triangular relationships we furthermore include the measure Transitivity, that divides the number of triangles by the number of connected triples in the graph. In this statistic, the direction of the edges is ignored. The analysis of this measure gives an impression how well the two chosen transitivity measures capture the overall clustering in the network.
The results are presented in Figure 11. In each of the six panels we see the respective network statistics plotted against time. The solid red line gives the network statistics, evaluated at the real MCW network. The boxplots show the network statistics, evaluated for each year for the 1.000 simulated networks.
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