Network data capture information on relations between actors. The manifold types of links between actors in the network encompass, for instance, stable ties associated with some duration. In the field of political science, for example, military alliance agreements are active for a certain number of years Cranmer2012,Leeds2019. A different kind of link consists of bilateral instantaneous events - like hostile actions measured in real time Boschee2015. Note that instantaneous events can be viewed as the limit case of stable ties if the duration of these ties goes to zero buttts2017. While instantaneous events can happen anytime, they are not always observable in a high resolution of time. Under these circumstances, we can count the instantaneous events occurring in a given time interval, which gives a network-based counting process. We define the respective class of processes as a multivariate counting process that simultaneously guides all dyadic interaction within an event network and investigate it in this article. Comprehensive monographs and survey articles on statistical network analysis are available by kolaczyk2009,kolaczyk2017,goldenberg2010, Lusher2012. Recent overviews of dynamic network modeling can be found in Fritz2019,Kim2018.
In real-life applications, most networks exhibit some kind of dynamics: structure changes over time driven by endogenous and exogenous determinants, being covariates that capture the present or past network dependencies and additional information external to the evolution of the network, respectively. One way to conceive the generating process of networks is as a discrete Markov chain, where the realized path consists of the observed networks and the state space is the set of all observable networks. The transition probabilities defining the chain are given by a distribution over all possible networks Robins2001. For stable ties, this view results in the temporal exponential random graph model (TERGM, hanneke2010). Alternatively, we can perceive the networks as evolving over time guided by a continuous Markov process holland1977. In this case, network dynamics can be modeled by the stochastic actor-oriented model (SAOM, snijders1996) or in the case of instantaneous events with a precise time-stamp by the relational event model (REM) as proposed by butts2008. Although modern sensory technology eases the collection of such fine-grained data lazer2009, exact continuous information is meanwhile not obtainable for every observed event. In our case, for example, data on the transactions of combat aircraft trades are collected yearly, but the exact time point of each event (e.g. day of delivery) is impossible to verify SIPRI. Therefore, instead of observing instantaneous events, we only protocol counts of events during given intervals. As a consequence, the resulting event data can also be comprehended as valued networks, weighted by the count of events that happened within the given intervals. Though the body of literature on dynamic network models is steadily growing, the consideration of valued dynamic networks is less developed and mainly limited to cross-sectional analyses (see desmarais2012,krivitsky2012,Robins1999,Krivitsky2009).
In this article, we introduce a tie-oriented model for the analysis of network-based event data. Tie-oriented models assume a bilateral intensity governing the occurrence of events within a dyad, as opposed to actor-oriented models suggested by stadtfeld2012 and extended in Hoffman2019,stadtfeld2017. They partition the intensity into an egocentric sender-specific intensity and a probability selecting the receiver conditional on the sender along the lines of the discrete choice model of mcfadden1973. To represent the dynamic evolution of the network-based process, we start with a framework that operates in continuous time at the tie level. This approach is then extended in multiple ways. Most importantly, we make use of the separable decomposition of network dynamics introduced by krivitsky2014 and adopt this to time-continuous event data. Furthermore, we extend the model towards a semiparametric specification and use penalized B-Splines to obtain flexible time-varying coefficients eilers1996. To capture latent actor-specific heterogeneity, we include random effects for each actor in the network differentiating between the sender and receiver of events. As an application case, we use the strategically most crucial international deliveries of weapons, namely combat aircraft from 1950 to 2017 Forsberg1994,sipridata2017. Combat aircraft comprises all “unmanned aircraft with a minimum loaded weight of 20 KG” siprimeth. They are very costly and the number of units transferred constitute a highly valuable information for military strategists Forsberg1997. This is the reason why we propose to focus on unit sales as the important quantity.
The remainder of this article is structured as follows: the next section formally introduces the tie-oriented model based on a network-based counting process together with extensions to separable, time-varying, and random effects and an estimation procedure. Consecutively, we introduce the application case and apply our novel method. The paper concludes with Section4.
2 Network-based Counting Process
2.1 A Framework for Discrete and Continuous Time Event Data
We start by proposing the model for time-continuous event data, which are observed at discrete time points. We use the temporal indicator and mathematically define the network-valued process as a Poisson process on a valued network given by:
where is the total number of actors in the network. Process (1) counts the relational events between all actors in the network during the interval . It is characterized by the network-valued intensity rate . The th entry of this intensity is defined as the probability that we observe an instantaneous jump of size 1 in
. Heuristically, this is the probability of the occurrence of a directed event from actorto at time point . By definition we set and .
We assume that the process is observed in discrete time points leading to the time-discrete observations , which are defined as cumulated events through:
set to 0. Based on the properties of a Poisson process, these increments follow a matrix-valued Poisson-distribution:
We represent with the integrated intensity on the time interval so that . Accordingly, we define the observed values of as .
Generally, we are interested in modeling conditional on the past network topology and exogenous covariates, which are denoted by . Covariates can be node-specific (regarding either a feature of the sender or receiver), dyadic (regarding a relation between the sender and receiver), or global (regarding the complete network). Building on a first-order Markov property, we allow the intensity to depend on the past network behavior and exogenous covariates through:
This is equivalent to the assumption of dyadic independence of events to occur in each time interval given information on the past and exogenous covariates. We further specify the intensity in time-varying semiparametric form through:
where is the baseline intensity,
is a multidimensional vector consisting of network statistics and theoretically derived exogenous covariates in. We discuss different specifications of the statistics in Section 3 where we describe the application case in more detail. The coefficient vector is possibly time-varying and needs to be estimated from the data.
Note that with time possible compositional changes of the actor set can occur. To compensate this in the model, we include indicator functions similar to risk indicators in time-to-event analysis Kalbfleisch2002. To be specific, we multiply the intensity by an indicator function, that determines whether actors and are both present in the network at time :
with denoting the set of actors partaking in the network at time point . With these actor set changes the possible range of the network statistics does change as well, leading to values that are not scaled coherently for a comparison across years. To solve this issue, we divide all network statistics by their maximal value to allow for a cohesive interpretation.
2.2.1 Separability Assumption
It is reasonable to assume that interaction patterns are substantially different for already linked and still unlinked actors. To properly capture this characteristic, holland1977 proposed a process-based model for binary ties taking the values “0" or “1" by two separate intensity functions. One intensity toggles entries from “0" to “1" (formation of ties) and another one from “1" to “0" (dissolution of ties). Thereby, separate and potentially differential effects of statistics depending on previous interaction behavior are enabled. This model, henceforth called separable model, was later adopted to the SAOM by incorporating a so-called gratification function Snijders1997,Snijders2003 and to the TERGM by extending it to the separable TERGM krivitsky2014. In the following, we combine the framework of relational event models with the separability approach.
More specifically, we postulate two different conditions for the network-based process under which the effect of all covariates changes. One condition governs events between unlinked actors and is characterized by the onset intensity. The second condition only regards events among actors that already interacted with each other and is driven by the repetition intensity. In accordance with the the Markov assumption specified in (3), we define the onset intensity at time to control all events which did not occur in . Accordingly, the repetition intensity drives the events that did occur at least once in . This can be incorporated by splitting the intensity into two conditional intensities:
where and are defined along the lines of (3) and specified by the corresponding time-varying parametric effects and jointly represented by . The possibly overlapping vectors of statistics are denoted accordingly as and , respectively. Setting enables the inclusion of a time-varying intercept in the onset model, this holds similarly for the repetition model. Consecutively, the complete separable model is given by replacing (3) with
2.2.2 Spline-based Time-Varying Effects
Let the th component of statistic be defined as with the matching coefficient . We expand each component in a semiparametric way by replacing it with a B-Spline basis function (see DeBoor2001). More specifically, we place equidistant knots on a grid in , where the number of knots can chosen relatively high Kauermann2011. In principle, we could choose individual grids for each component of , but for the sake of a simple notation, we select the same one for all covariates. We now rewrite each coefficient as:
where is the B-spline basis evaluated at and denotes the corresponding coefficient vector. To obtain a smooth fit we penalize the difference of adjacent basis coefficients as proposed by eilers1996. This leads to the overall penalized log-likelihood function:
with . The penalty results from the quadratic form with penalty matrix constructed from pairwise differences of the spline coefficients and as the penalty (and hence tuning) parameter. This vector controls the smoothness of the fit and is chosen data based following a mixed model approach as described in detail in Ruppert2003a, see also wood2017.
2.2.3 Accounting for Nodal Heterogeneity
The specification of the model introduced so far implicitly assumes that the nodal heterogeneity is fully captured by the structural statistics . As already thoroughly discussed by thiemichen2016 or steffensmeier2018, this can be considered a questionable assumption. It seems, therefore, advisable to include sender- and receiver-specific random effects to account for unobserved heterogeneity. Let therefore denote a latent sender-specific effect of actor and the receiver-specific effect of actor . This leads to the heterogeneous intensity
where is specified in (9).
The vector-valued function is estimated by finding the argument maximizing the penalized likelihood resulting from (10) and considering the penalty on coefficient vector as a improper prior distribution. This leads to a generalized additive mixed model, which is extensively discussed in wood2017 and Ruppert2003a, ruppert2009. In order to utilize these techniques, we initially calculate all covariates for each actor-tuple and at each point in time. By doing that, we transform the data into a generalized version of the so-called counting-process representation, which is known from time-to-event analysis Tutz2016,friedman1982,whiteheadd1980. For each snapshot of the event network, this procedure generates a design matrix of conditionally independent observations with a target variable expressing the number of events that occurred between a specific tuple of actors and covariates given by .
For the estimation, we utilize the versatile package (wood2017, version 1.8-31). Thereby, we follow wood2017b who enhance the pseudo-quasi-likelihood (PQL) method by breslow1993 for the analysis of larger data sets. The main extensions are threefold. Firstly, the tuning parameters are not estimated until convergence in each iteration of the estimation procedure but updated by only one Newton step. Secondly, efficient methods for computing the matrix cross-products in each iteration are run in parallel Li2019. Thirdly, the covariates are discretized along a marginal grid. Hence, the design matrices for the smooth covariates take significantly less memory. wood2017b describe the method in detail as it is implemented in the function of the already mentioned package. Well-calibrated frequentist confidence bands for the estimated function are guaranteed by Bayesian large sample properties Wood2013.
So far, quantitative work on the international arms trade utilizing statistical network analysis has been mostly restricted to binarized networks. Here, the occurrence of a trade relationship between two countries in a specific year was modeled conditional on endogenous and exogenous statistics by the gravity model of trade Akerman2014, johannsen2014,lebacher2019,Thurner2019. Only lebacher2019,Thurner2019 employ TERGMs and extensions of it. Contrary, Lebacher2019 fit a a network disturbance model on the yearly aggregated trend indicator values (TIV, siprimeth) of the international arms trades, maintaining the valued character of deliveries. All these contributions rely on data provided by the Stockholm International Peace Research Institute (sipridata2017) and they consider all types of major conventional weapons indiscriminately.
In the following, we concentrate on the counts of combat aircraft deliveries as reported in the SIPRI data, where each combat aircraft delivery is perceived as an event. Our focus on the transfers of aircraft is due to the fact that these weapon systems usually incorporate the highest technological sophistication, therefore, they are are restricted to close allies. Furthermore, they are of crucial strategic importance for international deterrence but also for counterinsurgency in intrastate conflict Hoeffler2016. Lastly, their sizes and cost makes the available data highly reliable Forsberg1994,Forsberg1997. Previous research on the trade of combat aircraft was limited to the quantitative analysis of a small subset of countries or fighter programs Hoeffler2016,Vucetic2011,Vucetic. Contrasting these endeavors, we take a global point of view on the combat aircraft trade. Here, a closer look at the data reveals that countries commonly partition major deals with their stable trade partners into multiple deliveries occurring over the span of several years. For instance, the United States and Japan signed a deal in 1984 comprising 32 quantities of aircraft, which were realized between 1988 and 2016. The additional information provided by this segmentation of trade deals into isolated deliveries would be lost when only regarding binarized networks111In the Supplementary Material we provide the results regarding alternative models for the data. Overall, there is no relevant difference to the findings presented subsequently..
Two examples of the network representing aggregated events over 6 years are depicted in Figure 1. Generally, the networks exhibit a structure with hubs around the United States (USA), Russia (RUS), France (FRA), and United Kingdom (UK). Coincidentally, this set of country also shows the highest average hub-scores over time Kleinberg1999. Analog to the distribution of the in- and out-degrees in binary networks, we can examine the distribution of the yearly-aggregated in- and outgoing event counts averaged over time. This enables a better understanding of the topology of the observed networks. Figure 2 (a) suggests a strong centralization in the outward event count distribution with some countries being the sender of up to 1300 deliveries in one year and on average
of the countries not exporting at all. The inward count distribution is not as skewed as can be seen in Figure2 (b). There are few countries that receive many aircraft deliveries, although the mode is still at zero.
3.2 Model Specification
We now employ the outlined model to the international combat aircraft trade network spanning from 1950 to 2017. The event networks are observed yearly. In this context, denotes the number of observed combat aircraft units delivered in year between country and and its distribution follows from (2). Given this information, we estimate the time-continuous intensities of all country-dyads, which are per assumption governed by the repetition intensity if the respective countries traded in the previous year, and by the onset intensity otherwise as defined in (6)222In the Supplementary Material, we, additionally, compare different time frames to define which events are driven by the onset and repetition intensity, e.g. having delivered combat aircraft in the last one or two years.. All actors in the network are countries and an event represents the delivery of combat aircraft between two countries. To appropriately capture interdependencies of the observed event counts, we incorporate a wide range of endogenous statistics, whose mathematical representation is given in Table 1 and visualized in Figure 3.
|(a) In-Degree Sender|
|(b) In-Degree Receiver|
|(c) Out-Degree Sender|
|(d) Out-Degree Receiver|
|(f) Shared Supplier|
As already investigated in multiple applications barabasi1999,Snijders2003,Newman2002, the degree structure plays a crucial role in the observed event network. In the case of directed events, the in- and out-degree of a country determine its relative location in the network Wasserman1994. In our application, the degrees reflect the number of different countries with whom a specific country had at least one transaction in a specific year as an importer (in-degree) and exporter (out-degree). To reveal the impact of these measures on the intensity of observing an event, we include four degree-related statistics concerning the sender and receiver in our specification, as illustrated in Figure 3 (a) - (d). For example, a positive effect of the sender’s out-degree can be loosely understood as the tendency to trade with countries that are already sending a lot.
Besides degree-based statistics, Holland1971,Davis1970 highlight the role of triangular structures in networks. Adopted to event relations, it refers to the change in intensity of an event between countries and , if they are indirectly connected by an additional two-path, i.e. third country. Since the aircraft deliveries between countries are directed, there are multiple ways to define two-paths. We incorporate two triadic structures: transitivity, Figure 3 (e), and shared supplier, Figure 3 (f). While transitivity in an event network suggests that already having observed a delivery from country to and to affects the intensity of an event from to , the shared supplier mechanism reflects the tendency towards trading with countries that import combat aircraft from a common exporter. Likewise, we control for reciprocity, which is the tendency of countries to respond to previous events directed at them, Figure 3 (g).
Political economy models of arms trade Levine1994,Thurner2019 as well as the gravity model of arms trade guide the selection of appropriate exogenous covariates. Thurner2019,Akerman2014 included the dyadic distance in kilometers between the capitals of country and as well as the logarithmic gross domestic product (GDP in US ) of the sender and receiver countries as covariates in the model. Pamp2018,lebacher2019 emphasize the impact of military expenditures as a further proxy for the Newtonian power of attraction, which we include in logarithmic form as a sender- and receiver-specific covariate. The respective yearly data was collected by SIPRI in US
and combined by Nordhaus2012 with data from Stuckey2012. We use this combined data set, but due to remaining missing data we employ linear interpolation, if at least 60of the time series for a specific country is available. Moreover, we incorporate two dyadic variables controlling whether country and signed an alliance treaty or are similar to each other in terms of their regimes in power, following johannsen2014,Thurner2019. The alliance treaty obligations and provisions project identified military alliance agreements Leeds2019 and regime dissimilarity is operationalized by the absolute difference in the Polity IV scores of countries and marshall2017. This measure indicates year-wise regime characteristics of all countries and takes values from -10 (strongly autocratic) to 10 (strongly democratic). Thus, the absolute differences lie between 0 (strong similarity) and 20 (strong dissimilarity) for each country-dyad and year. The sources and used period of all incorporated exogenous covariates are described in more detail in the Supplementary Material.
3.3.1 Fixed Effects
In Figures 4 to 7 the entire results of the time-varying estimates are given accompanied by alternative time-constant coefficients as dotted horizontal lines. The latter are obtained by setting . All exponentially transformed estimates at a specific point in time can be interpreted (ceteris paribus) as the multiplicative change of the intensity (5) corresponding to the effect of covariates in relative risk models Kalbfleisch2002. Therefore, an effect estimated at zero does not change the relative risk of an event to happen, but positive or negative coefficients lead to a higher or lower relative risks of the event to occur, respectively. Additionally, the occurrence of an event is equivalent to the increment of one in the counts of aircraft units, since one event represents in our application case a combat aircraft delivery.
From simple inspection it can be concluded that in all cases, time-varying coefficients are carrying completely different information as compared to time-constant coefficients. This is evidence for the necessity to account for the multiple systemic changes that happened within the international aircraft market during the considered time interval. From a statistical point of view, the time-varying effects can also be underpinned by a lower cAIC value when compared to time-constant effects (see Section 3.4 for additional details on the cAIC).
Moreover, we observe different shapes of the curves of the time-varying coefficients when comparing onset and repetition conditions leading to the conclusion that the import of all covariates on these two separate conditions is different.
Time-varying effects relating to the degree structure are shown in Figure 4. Figure 4 (a) indicates a steady negative influence of the sender‘s in-degree in the onset condition from around 1965 onward. It can be concluded, that the count of dyadic events are lower if the sender‘s in-degree is high. This may be justified by the observation that only a small subset of countries are adequately equipped to be producing and exporting aircraft.This technological possibility, in turn, increases self-sufficient behavior, thus alleviates the need of additional imports. Contrary, in the repetition condition, the in-degree of the receiver exhibits a positive effect for the post-Cold War period from 1990 to 2010, Figure 4 (b). Otherwise, the effect is insignificant. Concerning the receiver, a negative effect of the in-degree can be observed from 1950 to 1980 in the onset model, Figure 4 (c). When proceeding to deliver aircraft, the effect of the receiver‘s in-degree is similar to sender‘s in-degree, Figure 4 (d). For the sender’s out-degree, the effect in the onset model is negative until around 1980 and thereupon positive. In the latter case, the effect mirrors a higher tendency of delivering combat aircraft, if the sender is already a prolific exporter country. During the complete observational period we observe that receivers are not senders themselves, thus exhibit low out-degrees, Figure 4 (g) and (h). This behavior does not depend on the condition of the dyadic intensity.
Triadic structures play a major role during the Cold War. Afterwards, the impact disappears but is again strengthened after 2000 under the onset condition, Figure 5 (a) and (c). In particular, an increasing number of indirect transitive connections between country and results in a greater count of aircraft deliveries between 1950 and 1990. Similarly, receiving combat aircraft from the same third country increases the unit sales between the receivers during the Cold War period, Figure 5 (c). A possible consequence of this process is the strengthening of a block structure. For a consecutive delivery, the triadic effects are less pronounced and in the case of shared suppliers, Figure 5 (d), constantly insignificant. The count of reciprocal events, on the other hand, raises trade from 1990 to 2005, Figure 5 (e). This may be a consequence of an opening in the international market after the fall of the Soviet Union, leading to multiple emergent countries. If the relationship is maintained, reciprocal events are encouraged throughout the period of observation, although to a smaller degree, Figure 5 (f).
While the logarithmic GDP of the receiver has a relatively weak positive influence when starting a trade relation, Figure 6 (a), its repetition is only affected after the end of the Cold War, Figure 6 (b). On the sender-side, the estimates of both models are constantly positive, Figure 6 (c) and (d). In contrast to the effect in the onset model, the logarithmic GDP of the sender has a higher effect from 1950 to 1980 in the repetition condition. Moreover, the military expenditure of the receiver is one of the main drivers in this model, Figure 6 (f). Here, a higher military spending of possible sender countries augments the count of receiving combat aircraft deliveries, specifically during the 50s. Conversely, the exogenous covariate only slowly gains attention in the onset condition after the Cold War, Figure 6 (e). While the effect of the military expenses of the sender stays overall positive when delivering aircraft for the first time, it inhibits it to be repeated in the next year, Figure 6 (g) and (h).
The findings in Figure 7 (a) and (b) indicate that similar regimes are overall more likely to start trading combat aircraft. Only at the height of the Cold War from 1970 to 1980, the effect is estimated at approximately 0, Figure 7 (a). The strength of the effect is less salient in the repetition condition than in the onset condition of the model, Figure 7 (b). Furthermore, the time-varying coefficients discover a steadily decreasing influence of beginning to transact with allies, Figure 7 (c). This finding suggests evidence of the overall deteriorating importance of international alliances in combat aircraft transactions if the countries did not trade in the previous year. We don’t observe a similar downward trend in the case of repeating an event, Figure 7 (d). Lastly, the distance between the respective capitals generally hinders events to occur, Figure 7 (e) and (f). Therefore, countries tend to trade with spatially more distant than close partners. Maybe this is due to the relative spatial isolation of the main exporters’ capitals, Moscow (Russia/USSR) and Washington, D.C. (USA).
3.3.2 Random Effects
The random effects permit an extended analysis of the unexplained heterogeneity in the model. More precisely, the random effects express country-specific deviations from an overall behavioral trend, which is captured by the time-varying effects. Additionally, they correct for repeated measurements of the countries as simultaneous senders and receivers of events in each year. The model introduced in Section 2 comprises two country-specific random effects for all countries as a sender and receiver of combat aircraft deliveries. The results are given in Figure 8 and visualized on a world map in Figure 9.
In the first quadrant of Figure 8 countries with a positive random sender and receiver effect are shown. This composition of random effects suggests that the respective countries are senders and receivers of more combat aircraft events than marginally expected. Countries in the Middle East, e.g., Israel (ISR), Libya (LBY), and Jordanian (JOR), are allocated to this group.
Negative sender but positive receiver effects are identified for countries in South-East Asia (Thailand (THA), Cambodia (KHM), Laos (LAO), Myanmar (MYR), and Sri Lanka (LKA)). In comparison to the average behavior, these countries are rather reluctant as senders and confident as receivers of combat aircraft deliveries. The latent sender effect of Mexico (MEX) is the most negative coefficient estimated. This suggests Mexico’s reliance on the import of combat aircraft, although its high economic status would imply additional participation in the event network as a sender.
The third quadrant contains all countries, which were less active than expected as a sender and receiver of events. This strand of countries is either economically strong, yet exhibiting a passive trading behavior, e.g., Luxembourg (LUX), or relatively poor and missing preconditions to send or receive weapons, e.g., Trinidad and Tobago (TTO).
Lastly, a negative random coefficient regarding receiving arms is mostly associated with European countries. The corresponding sender effect is positive. Hence, these countries are situated in the fourth quadrant of Figure 8. The East European countries Moldova (MDA), Ukraine (UKR), and Belarus (BLR) have the highest positive sender effect paired with relatively low receiver effects.
In terms of continent-wide tendencies, we locate Africa in the first three quadrants. South America is principally assigned to the first and second quadrant. Asia, Oceania, and North America are more dispersed and exhibit a less homogeneous country behavior.
3.4 Model Comparison and Assessment
We compare the estimated model to alternative specifications, which are chosen to reflect all subsequent extensions of Section 2.2 and are indicated in Table 2. Model 1 includes all effects linearly without the separable extension. This is we assume that and omit the separation of the statistics into and . This separability is added in Model 2 according to Section 2.2.1. Model 3 includes time-varying coefficients as introduced in Section 2.2.2. Lastly, Model 4 is the model whose findings were presented in Section 3.3. Hence, also random effects are taken into account, that are explained in Section 2.2.3.
|Separability||Time-Varying Effects||Random Effects||cAIC|
One way to compare these models is by means of information criteria, i.e. the Akaike Information Criterion (AIC,Akaike1974). As already discussed in the context of linear mixed models Greven2010 and generalized mixed models Saefken2014, the usage of the conditional or marginal AIC does not appropriately incorporate the uncertainty of estimating the covariance parameters of the random effects (in our applicationand ). Therefore, we utilize a corrected conditional AIC proposed by wood2016. The resulting cAIC values are given in Table 2 and indicate a superior model fit when all extensions introduced in Section 2.2 are included.
Furthermore, we asses the fit of the selected Model 4 through a graphical tool, that compares the expected and observed frequencies of combat aircraft deliveries over all years. The expected frequency of count , denoted by , can be computed through:
where is the predicted intensity under and the Poisson distribution of following from (2). This procedure is closely related to the rootogram proposed by Kleiber2016 and dates back to Tukey1977. It can detect whether the distributional form of the target variable could be adequately represented by the estimated model and over- or underdispersion is present in the data. We can infer from Figure 10 that the estimated model captures even high event counts between countries averaged over the complete time span.
We introduced a novel model for the analysis of relational event data. Originating in a counting process operating in continuous time that we only observe at specific time points, we derived a tie-level intensity, whose parameters can be estimated according to the maximum likelihood principle. Extensions to separable models, which govern the onset and repetition of events by two functions, as well as the incorporation of time-varying and random coefficients are given. Eventually, we applied the procedure to the international combat aircraft network from 1950 to 2017. In doing that, we could use the additional information given by the counts of yearly aircraft deliveries to estimate a time-continuous intensity, contrary to existing work on binarized networks. Moreover, the separability detects fundamentally different processes governing the onset and repetition of event relationships, while the time-varying effects uncover a systemic change during the Cold War period. Furthermore, we identified triangular network statistics and the economic nodal covaraites of the sender as the principal drivers of the onset condition of the proposed intensity. Here, a decaying effect of bilateral military alliances became apparent. For the repetition condition this effect remained consistently positive and a high military expenditure of the receiver was shown to be the driving force. Finally, the random effects enable a visual comparison of the unexplained heterogeneity between the modeled countries (Figure 9) and correct the estimates for repeated measurements as well as possible overdispersion.