Tempus Volat, Hora Fugit -- A Survey of Dynamic Network Models in Discrete and Continuous Time

05/23/2019 ∙ by Cornelius Fritz, et al. ∙ Universität München 0

Given the growing number of available tools for modeling dynamic networks, the choice of a suitable model becomes central. It is often difficult to compare the different models with respect to their applicability and interpretation. The goal of this survey is to provide an overview of popular dynamic network models. The survey is focused on introducing binary network models with their corresponding assumptions, advantages, and shortfalls. The models are divided according to generating processes, operating in discrete and continuous time, respectively. First, we introduce the Temporal Exponential Random Graph Model (TERGM) and its extension, the Separable TERGM (STERGM), both being time-discrete models. These models are then contrasted with continuous process models, focusing on the Relational Event Model (REM). We additionally show how the REM can handle time-clustered observations, i.e., continuous time data observed at discrete time points. Besides the discussion of theoretical properties and fitting procedures, we specifically focus on the application of the models using a network that represents international arms transfers. The data allow to demonstrate the applicability and interpretation of the network models.

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1 Introduction

The conceptualization of systems within a network framework has become popular within the last decades, see Kolaczyk (2009) for a broad overview. This is mostly because network models provide useful tools for describing complex dependence structures and are applicable to a wide variety of research fields. In the network approach, the mathematical structure of a graph is utilized to model network data. A graph is defined as a set of nodes and relational information (ties) between them. Within this concept, nodes can represent individuals, countries or general entities, while ties are connections between those nodes. Dependent on the context, these connections can represent friendships in a school (Raabe et al., 2019), transfers of goods between countries (Ward et al., 2013), sexual relations between people (Bearman et al., 2004) or hyperlinks between websites (Leskovec et al., 2009) to name just a few. Given a suitable data structure for the system of interest, the conceptualization as a network enables analyzing dependencies between ties. A central statistical model that allows this is the Exponential Random Graph Model (ERGM, Robins and Pattison, 2001). This model permits the inclusion of monadic, dyadic and hyperdyadic features within a regression-like framework.

Although the model allows for an insightful investigation of within-network dependencies, most real-world systems are typically more complex. This is especially true if a temporal dimension is added, which is relevant, as most systems commonly described as networks evolve dynamically over time. It can even be argued that most static networks are de facto not static but snapshots of a dynamic process. A friendship network, e.g., typically evolves over time and influences like reciprocity often can be found to follow a natural chronological order.

Of course, this is not the first paper concerned with reviewing temporal network models. Goldenberg et al. (2010) wrote a general survey covering a wide range of models. The authors laid the foundation for further articles and postulated a soft division of statistical network models into latent space (Hoff et al., 2002) and models (Holland and Leinhardt, 1981), all originating in the Edös-Rényi-Gilbert random graph models (Erdös and Rényi, 1959). Kim et al. (2018) give a contemporary update on the field of dynamic models building on latent variables. Snijders (2005) discusses continuous time models and reframes the independence and reciprocity model as a Stochastic Actor oriented Model (SAOM, Snijders, 1996). Block et al. (2018) provide an in-depth comparison of the Temporal Exponential Random Graph Model (TERGM, Hanneke et al., 2010) and the SAOM with special focus on the treatment of time. Further, the ERGM and SAOM for networks which are observed at single time points are contrasted by Block et al. (2019), deriving theoretical guidelines for model selection based on the differing mechanics implied by each model.

Figure 1: Tree diagram summarizing the dependencies between models originating in the Erdös-Rényi-Gilbert graph model, models situated in a box with a grey background are discussed in this article. This graph is an update of Figure 6.1 in Goldenberg et al. (2010).

In the context of this compendium of articles, the scope is to give an update on the dynamic variant of the second strand of models relating to models. We therefore extend the summarizing diagram of Goldenberg et al. (2010) as depicted in Figure 1. Generally, we divide temporal models into two sections, by differentiating between discrete and continuous time network models.

Statistical models for time discrete data rely on a Markov chain assumption and condition the state of the network at time point

on previous states. This includes the TERGM and the Separable TERGM (STERGM, Krivitsky and Handcock, 2014). There exists a wide range of recent applications of the TERGM. White et al. (2018) use a TERGM for modeling epidemic disease outcomes and Blank et al. (2017) investigate interstate conflicts. In He et al. (2019) Chinese patent trade networks are inspected and Benton and You (2017) use a TERGM for analyzing shareholder activism. Applications of STERGMs are given for example by Stansfield et al. (2019) that model sexual relationships and Broekel and Bednarz (2019) that study the network of research and development cooperation between German firms.

In case of time-continuous data, the model regards the network as a continuously evolving system. Although this evolution is not necessarily observed in continuous time, the process is taken to be latent and explicitly models the evolution from the state of the network at time point to (Block et al., 2018). In this paper we discuss the relational event model (REM, Butts, 2008) for the analysis of event data. Eventually, the REM is adapted to time-discrete observations of networks. That is, we observe the time-continuous developments of the network at discrete observation times only. Applications of the REM for non-clustered observations are manifold and range from explaining the dynamics of health behavior sentiments via Twitter (Salathé et al., 2013), inter-hospital patient transfers (Vu et al., 2017), online learning platforms (Vu et al., 2015), and animal behavior (Tranmer et al., 2015) to structures of project teams (Quintane et al., 2013).

The paper is structured as follows. In Section 2 we present the international arms trade network of major conventional weapons (MCW) that will be analyzed as an illustrative example and give basic definitions that are used throughout the paper. After that, Section 3 introduces time-discrete and Section 4 time-continuous network models. In Section 5 further models are shortly discussed and differences between the proposed models are exhibited.

2 Definitions and Data Description

Descpription Year
Time 2016 2017
Number of countries included 148 148
Number of possible ties 21 756 21 756
Density 0.020 0.019
Transitivity 0.202 0.207
Reciprocity 0.085 0.087
Table 1: Descriptive Statistics for the years 2016 and 2017.

As a running example throughout this paper, we use data on international arms trading. The arms trading data are provided in a comprehensive database by the Stockholm International Peace Research Institute (SIPRI, 2018) and includes data on the exchange of major conventional weapons (MCW) together with the volume of each transfer. Since this article only regards binary network models, the trade network is discretized with a threshold of zero. This means that a tie from actor to actor indicates that the sender country traded with a receiver country in the respective year. This information can then be represented in an adjacency matrix , where represents the set of all possible networks with nodes, in our example countries. The entry of is ”1” if country sold MCW to country in year and ”0” otherwise. Further, the discrete time points of the observations of are denoted as . For demonstration purposes, we restrict our analysis to two time points only and consider the years 2016 and 2017. Hence we look at annual changes of the network structure and set . In many networks including our running example self loops are meaningless. We therefore fix throughout the article. Further, all sub-scripted indices () are assumed to be discrete and and all indices in brackets () continuous. The temporal indicator denotes the observation times of the network and to notationally differ this from time-continuous model we write for continuous time.

Table 1 gives some descriptive measures and Figure 2 visualizes the arms trade network. There are no compositional changes of the involved countries, whereby the number of possible ties stays the same as well. The density of a network is the proportion of realized edges out of all possible edges and is similar in both years, indicating the sparsity of the modeled network. Clustering can be expressed by the transitivity measure, providing the percentage of connected triplets out of all possible triplets. The reciprocity of a graph is the ratio of reciprocated ties in a graph and is similar in both years, see Annex A for a description of the degree distribution (Csardi and Nepusz, 2006).

Additionally, information on different kinds of exogenous covariates may be controlled for in statistical network models. In the given example we use the logarithmic Gross Domestic Product (GDP) (World Bank, 2017) as monadic covariates in respect to the sender and receiver of weapons. We also include the absolute difference of the so called polity IV index (Center for systemic Peace, 2017), ranging from 20 (highest ideological distance) to zero (no ideological distance), as a dyadic exemplary covariate. These covariates are assumed to be non-stochastic and we denote them by .

Figure 2: The International Arms Trade as a binary network in 2016 (left) and 2017 (right). Isolated countries are not depicted for clarity and the node size relates to its total degree.

3 Dynamic Exponential Random Graph Models

3.1 Temporal Exponential Random Graph Model

The Exponential Random Graph Model (ERGM) is certainly among the most popular models for the analysis of static network data. Holland and Leinhardt (1981) introduced the model class, which was subsequently extended with respect to fitting algorithms and network statistics (see Lusher et al., 2012, Robins et al., 2007). Spurred by the popularity of ERGMs, dynamic extensions of this model class emerged, pioneered by Robins and Pattison (2001) who developed time-discrete models for temporally evolving social networks. Before we start with a description of the model, we want to highlight that the TERGM as well as the STERGM are most appropriate for equidistant time points. That is, we observe the networks at discrete and equidistant time points . Only in this setting the parameters allow for a meaningful interpretation. See Block et al. (2018) for a deeper discussion.

Hanneke et al. (2010) is the main reference for the TERGM, a model class that utilizes the Markov structure and, thereby, assumes that the transition of a network from time point to time point

can be explained by exogenous covariates as well as structural components of preceding networks. We assume a first order Markov dependence structure that applies to probability distributions

with parameter vector

. Conditioning on the first network, the resulting dependence structure of the model can be factorized into

(1)

Depending on the phenomenon of interest, it is also possible to allow for different parameter vectors for each transition probability (i.e. , ). Given the dependence structure (1), the TERGM assumes that the transition from to is generated according to an exponential random graph distribution with the parameter :

(2)

Generally, specifies a -dimensional function of sufficient network statistics which may depend on the previous network as well as on covariates. These network statistics can include static components, designed for cross-sectional dependence structures, e.g., out-degree, in-degree, reciprocity or transitivity (see Morris et al., 2008 for more examples). However, the formulation explicitly allows for temporal interactions, e.g. delayed reciprocity

(3)

This statistic governs the tendency whether a tie in will be reciprocated in . Another important temporal statistic is stability

(4)

In this case, the first product in the sum measures whether existing ties in persist in and the second term is one if non-existent ties in remain non-existent in . The proportionality sign is used since in many cases the network statistics are scaled into a specific interval (e.g.  or ). Such a standardization is especially sensible for networks where the actor set changes with time. Additionally, exogenous covariates can be included, e.g., time-varying dyadic covariates

(5)

There exists an abundance of possibilities for defining interactions between ties in and . From this discussion and equation (2) it also becomes obvious, that in a situation where the interest lies in the transition between two periods , a TERGM can be modelled simply as an ERGM, including lagged network statistics (for example by incorporating as explanatory variable).

Concerning the estimation of the model, maximum likelihood appears to be a natural candidate due to the simple exponential family form (

2). However, the normalization constant in the denominator of model (2) often poses an inhibiting obstacle when estimating (T)ERGMs. This can be seen by inspecting the normalization constant , that requires summation over all possible networks . This task is virtually infeasible, except for very small networks. Therefore, Markov Chain Monte Carlo (MCMC) methods have been proposed in order to approximate the logarithmic likelihood function (see Geyer and Thompson (1992) for Monte Carlo maximum likelihood and Hummel et al. (2012) for its adaption to ERGMs). A notable special case arises if the network statistics are restricted such that they decompose to

(6)

with being a function that is evaluated only at the lagged network and covariates for tie . With this restriction, we impose that the ties in are independent, conditional on the network structures in

. This greatly simplifies the estimation procedure and allows to fit the model as a logistic regression model (see for example

Almquist and Butts, 2014).

3.2 Separable Temporal Exponential Random Graph Model

A useful extension of the TERGM model (2) is the STERGM proposed by Krivitsky and Handcock (2014). This model can be motivated by the fact that the stability term leads to an ambiguous interpretation of its corresponding parameter. Given that we include (4) in a TERGM and obtain a positive coefficient after fitting the model it is not clear whether the network can be regarded as ”stable” because existing ties are not dissolved (i.e. ) or because no new ties are formed (i.e. ). To disentangle this, the authors propose a model that allows for the separation of formation and dissolution.

Figure 3: Conceptual representation, illustrating formation and dissolution in the STERGM.

Krivitsky and Handcock (2014) define the formation network as , being the network that consists of the initial network together with all ties that are newly added in . The dissolution network is given by and contains exclusively ties that are present in and . Given the network in together with the formation and the dissolution network we can then uniquely reconstruct the network in , since . Define as the joint parameter vector that contains the parameters of the formation and the dissolution model. Building on that, Krivitsky and Handcock (2014) define their model to be separable in the sense that the parameter space of is the product of the parameter spaces of and together with conditional independence of formation and dissolution given the network in :

(7)

The structure of the model is visualized in Figure 3. On the left hand side the state of the network is given, consisting of two ties and . In the formation network (top in the middle plot) all ties that could possibly be formed are shown in dashed and the actual formation in this example is shown in solid. On the bottom, the two ties that could possibly be dissolved are shown and in this example () persists while is dissolved. On the right hand side of Figure 3 the resulting network at time point is displayed.
Given this structure and the separability assumption (3.2), it is assumed that a TERGM structure (2) is appropriate for both, the formation and the dissolution process. For practical reasons it is important to understand that the term ”dissolution” model is somewhat misleading since a positive coefficient in the dissolution model implies that nodes (or dyads) with high values for this statistic are less likely to dissolve. This is also the standard implementation in software packages, but can simply be changed by switching the signs of the parameters in the dissolution model.

3.3 Software and Application

When it comes to software, there exist essentially two main R packages that are designed for fitting TERGMs and STERGMs. Most important is the extensive statnet library (Goodreau et al., 2008) that allows for simulation-based fitting of ERGMs (which can be interpreted as TERGMs when including lagged network statistics). The library contains the package tergm with implemented methods for fitting STERGMs using conditional maximum likelihood. However, currently the package tergm (version 3.5.2) does not allow for fitting STERGMs with time-varying dyadic covariates for more than two time periods jointly. The package btergm (Leifeld et al., 2018) is designed for fitting TERGMs as shown in equations (2) using either maximum pseudo-likelihood (often regarded as unreliable, see vanduijn2009) or MCMC maximum likelihood estimation routines.

In order to ensure comparable estimates we estimate the TERGM as well as the STERGM with the statnet library, using MCMC based likelihood inference techniques. We use the package ergm and include the lagged previous network as a dyadic covariate, which is in fact equivalent to the stability term (4) after some reformulation (see Block et al., 2018). The STERGM is fitted using the tergm package.

TERGM STERGM
Formation Dissolution

Lagged Network
- -
- -
Edges
Reciprocity
Out-Degree Sender (Geometrically weighted)
In-Degree Receiver (Geometrically weighted)
Edge-wise Shared Partners (Geometrically weighted)
log(GDP) Sender
log(GDP) Receiver
Polity Score (Absolute Difference)

Log Likelihood
-945.282 -663.287 -258.293
AIC 1908.564 1342.574 532.585
AIC 1908.564 1875.159
Table 2:

Comparison of parameters obtained from the TERGM (first column) and the STERGM (Formation in the second column, Dissolution in the third column). Standard errors in brackets and stars according to

-values smaller than (), () and (). Decay parameter of the geometrically weighted statistics is set to .

The results obtained for the arms trading data discussed in the previous section are displayed in Table 2. For a detailed interpretation of effects focusing on political, social, and economic aspects we refer to the relevant literature (see e.g. Thurner et al., 2018). Here we want to comment on a few aspects only. First of all, concerning the general interpretation, note that the STERGM coefficients are implicitly dynamic because the corresponding statistics are evaluated either on the formation or the dissolution network and both are formed with the networks in and . In contrast to that, in the TERGM (first column), except the lagged stability term all network statistics are evaluated on the network in . Note further, that the TERGM coefficients try to explain the network structure in based on , while the STERGM coefficients provide information either on the formation or the dissolution.

Given that, it is not surprising that the coefficients can substantially differ in terms of significance and sign of the coefficients. For example, the statistic geometrically weighted In-degree of the receiver has a coefficient that is high in absolute terms and a low -value in the TERGM. However, the effect is mainly driven by the formation, which can be seen by a weak and insignificant effect in the dissolution model but an even stronger and significant effect in the formation model. Hence, the TERGM suggests that nodes with a relatively high in-degree are overall rather unlikely. However, this does not apply for the persistence of ties, where a high in-degree of the receiver only slightly weakens the probability of retaining the import relationship.

Similarly, we find that sending countries with high out-degrees are rare in the network, see the significant and low values for the geometrically weighted out-degree. However, although it is unlikely that a country with a high out-degree adds a new edge in the formation process, the effect is insignificant in the dissolution model. This indicates that in the dissolution model, the persistence of ties is not strongly driven by the receivers’ out-degree.

Comparable effects can be found for the exogenous covariates.Consider, for instance, the coefficient of the logarithmic GDP of the importing country. The TERGM assigns a higher probability to observing in-going ties to countries with a high GDP. However, disentangling the model towards formation and dissolution we see highly significant coefficients in the dissolution model while the effect for the formation model is insignificant.

Overall we observe that the STERGM allows to decompose the dynamics, which can also be quantified by the AIC as a model selection criterion. Based on the independence assumption in (3.2) we can sum up the two AIC values and see that the AIC value of the STERGM is smaller than of the TERGM.

4 Relational Event Model

4.1 Time-Continuous Event Processes

The second type of dynamic network models results by comprehending network changes as a continuously evolving process (see Girardin and Limnios, 2018 as a basic reference for stochastic processes). The idea was originally introduced by Holland and Leinhardt (1977). According to their view, tie changes are not occurring at discrete time points but as a continuously evolving process, where only one tie can occur at a time. This framework was extended by Butts (2008) to model behavior, which is understood as a directed event at a specific time, that potentially depends on the past. For instance, country sending weapons to at a given time point is a behavior, hereinafter called event. The overall aim is to understand the dynamic structure of events conditional on the information of the past (Lerner et al., 2013).

To model the event based approach, we leverage results from the field of time-to-event analysis, or survival analysis respectively (see, e.g., Kalbfleisch and Prentice, 2002 for an overview of time-to-event models). The central concept of this framework can be motivated by the introduction of a multivariate time-continuous Poisson counting process

(8)

where counts how often actors and interacted in . Note that we indicate continuous time with a tilde to distinguish from the discrete time setting with assumed in the previous section. Process (8) is characterized by an intensity function for , which is defined as:

This is the instantaneous probability of observing a jump of size ”1” in , which indicates observing the event at time . Since we assume that there are no self-loops holds.

4.2 Time-Continuous Observations

Butts (2008) introduced the Relational Event Model (REM) to analyze the intensity when time-stamped data on the events are available. He assumed that the intensity is constant over time but depends on time-varying relational information of past events and exogenous covariates. Vu et al. (2011) extended the model by postulating a semi-parametric intensity similar to Cox (1972):

(9)

where is an arbitrary baseline intensity, the parameter vector and a statistic that depends on the (possibly time-continuous) covariate process and the counting process just prior to . Examples for are the out- and in-degree of countries and .

To understand the relational nature of the observed events, model (9) takes a local time-continuous point of view, whereby all global structural effects are assumed to originate on the dyadic level and become global by aggregation of multiple similar dyadic effects (Stadtfeld, 2018). This differing level of modeling necessitates defining the statistics on a dyadic level. To give an example, the dyadic version of reciprocity for the event now regards, whether already having observed the event prior to has an effect on , in comparison to the network level version (3) that counted the number of reciprocated ties between and . Therefore, the mathematical formulation is straightforward:

where is the indicator function. Since the effect of a past event at time , say, on a present event at time may vary according to the elapsed time , Stadtfeld and Block (2017) introduced windowed effects, which only regard events that occurred in a pre-specified time window, e.g. a year. We will come back to this point in the next section.

In case of survival data, Cox (1972) introduced the partial likelihood to estimate without having to specify a parametric form of the baseline hazard nor a distribution on the times between events. In the same way, can be estimated with a Nelson Aalen estimator (see Kalbfleisch and Prentice, 2002 for further details on the estimation).

Extensions of this model building on already well established methods in social network and time-to-event analysis were numerously proposed. Perry and Wolfe (2013) used a stratified Cox model in (9) and allowed multi-cast-events, which are events that are possibly directed at multiple receivers. Stadtfeld et al. (2017) adopted the Stochastic Actor oriented Model (SAOM) to events. DuBois and Smyth (2010) and DuBois et al. (2013) extended the Stochastic Block Model (SBM) for time-stamped relational events. Further, DuBois et al. (2013) adopted a Bayesian hierarchical model to event data when information is only available in smaller groups.

4.3 Time-Clustered Observations

Generally, the approach discussed above requires time-stamped network data, meaning that we observe the precise time points of all events. For the running example this means that we need the exact time point of an arms trade between country and . Often, such exact time-stamped data are not available and, in fact, trading between states can hardly be stamped with a single time point . Indeed, we often only observe the time-continuous network process at discrete time points . In such setting, we may assume some kind of Markov structure in that we do not look at the entire history of the process but just model the intensity (9) in the time frame between and . Let therefore be adapted to and for . We then reframe (9) as:

(10)

In other words, we assume that the intensity of events between and does not depend on states of the multivariate counting process prior to . The history of the counting process is reset after each time interval. This is a reasonable assumption, if one is primarily interested in short-term dependencies between the individual counting processes.

If we observe the continuous process at discrete time points it is inevitable that we observe time clustered observations, meaning that two or more events happen at the same time point. This is a to some extend inherent problem, as motivated on the basis of the arms trading above. Under the term tied observations this phenomenon is well known in time-to-event analysis and treated with several approximations. We make use of the so called Breslow approximation (see Peto, 1972; Breslow, 1974). Let therefore

where element is replicated times in , that is if an event between and occurred multiple times in the interval from to then appears respective times in . Given that we have not observed the exact time point of an event we also get no information on the baseline intensity in (9) for so that the model simplifies to a discrete choice model structure (see, e.g., Train, 2009) which resembles the partial likelihood by Cox (1972). Let therefore denote the set of all possible ties between countries that may be observed at time point , so that the partial likelihood is defined as:

(11)

where .

Alternatively, one can replace the denominator in (11) by considering all possible orders of the unobserved events in . Since this can be a combinatorial and hence numerical challenge, some random sampling of time point orders among observations, that are time clustered, can be used as well with subsequent averaging, which we call Kalbfleisch-Prentice approximation (see Kalbfleisch and Prentice, 2002).

4.4 Software and Application

Marcum and Butts (2015) implemented the R package to estimate the REM for time-stamped data. It was followed by the package by Stadtfeld and Hollway (2018) for generally modeling time-stamped data. The latter package is highly customizable in terms of endogenous user terms and will be used in the following application to the arms trade network.

As mentioned before, we do not have time stamps for the arms trades. While this is an slight misuse of the time-continuous model, we apply this analysis here for demonstration purposes and to allow for a comparison with the results in the previous chapter. In other words, we either observe an arms trade (i.e. ) or no trade ().

The estimates are shown in Table 3, the first column represents the estimates of the Breslow, whereas the second column regards the estimation via the Kalbfleisch-Prentice approximation with random orders. Regarding the significant terms, the estimates lead to similar conclusions. Only the estimates concerning transitivity, are slightly singnificant in the Kalbfleisch-Prentice approximation but not in the Breslow approximation.

It should be noted that the general interpretation is now on the dyadic level, in comparison to the global interpretation of the effects in section 3.3. Therefore, e.g., the positive effect of the out-degree of the sender translates to a higher intensity of observing if had a higher out-degree.

Breslow Kalbfleisch-Prentice
Reciprocity 0.029 0.154
(0.189) (0.176)
Out-degree Sender 0.037 0.032
(0.004) (0.004)
In-degree Receiver 0.153 0.142
(0.188) (0.0159 )
Transitivity 0.033 0.062
(0.031) (0.031)
log(GDP) Sender 0.479 0.441
(0.037) (0.038)
log(GDP) Receiver 0.221 0.184
(0.03) (0.031)
Polity Score (Absolute Difference) 0.033 0.030
(0.009) (0.008)
Log Likelihood 3621.419 3581.731
AIC 7256.84 7177.46
Table 3: Parameters obtained for the REM using the Breslow (first column) and Kalbfleisch-Prentice (second column) approximation. Standard errors in brackets and stars according to -values smaller than (), () and ().

Similar to the application of Section 3.3 reciprocated ties are not more likely to occur than non-reciprocated ones judged by their significance. The degree-related covariates concern the role of centrality in respect to the intensity of an event. Apparently, both a high out-degree of the sender and in-degree of the receiver result in a higher intensity, thus spur trade relations. Consequentially, countries that have a high out-degree are more likely to send weapons and countries with a high in-degree to receive weapons. It is notable that this interpretation is different but not inconsistent with the findings regarding the geometrically weighted in- and out-degree in the TERGM, both having negative coefficients indicating that the network exhibits a general tendency of having rather low out- and in-degrees. Both estimates indicate an asymmetric degree structure, yet the estimates from Section 3.3 are to be understood on the global level and translate to less countries with high in- and out-degree than expected under under a completely random graph. In the REM, on the other hand, the estimates indicate that already having been highly involved in the network makes future trade activity more probable. In contrast, a country that was never active is less likely to send weapons, which again results in the asymmetric degree structure mentioned above.

Local clustering as indicated by the significantly positive parameter of transitivity can not clearly be detected. The respective estimate indicates, that having common trade partners is not a catalyzing factor in trading among countries. Additionally, this effect was not found on the global level of the analysis in Section 3.3, where the analog statistic is called geometrically weighted edge-wise shared partners.

Further, we find additional confirmation on the influence of the logarithmic GDP of the sender and receiver on the intensity of a trade, which is in line with Section 3.3. For instance, the economic power of the exporter country has a strong effect on the intensity of receiving weapons.

Lastly, it should be mentioned that the indicated AIC values cannot be compared to the models in Section 3.3.

5 Discussion

5.1 Further models

Snijders (1996) formulated a two-stage process model operating in a continuous time framework. The dynamics are considered to evolve according to unobserved micro-steps. At first, a sender out of all eligible actors gets the opportunity to change the state of all his outgoing ties. Consecutively, the actor needs to evaluate the probability of changing the present configuration with each possible receiver, which entails each actors knowledge of the complete graph whenever he has the possibility to toggle one of his ties. Lastly, the decision is randomly drawn relative to the probabilities of all possible actions. In general, the SAOM is a well established model for the analysis of social networks, that was successfully applied to a wide array of network data, e.g., in Sociology (Agneessens and Wittek, 2012; de Nooy, 2002), Political Science (Kinne, 2016; Bichler and Franquez, 2014), Economics (Castro et al., 2014), and Psychology (Jason et al., 2014).

Another notable model that can be regarded as a bridge between the ERGM and continuous time models is the Longitudinal ERGM (LERGM, Snijders and Koskinen, 2013; Koskinen et al., 2015). In contrast to the TERGM, the LERGM assumes that the network evolves in micro-steps as a continuous time Markov process with an ERGM being its limiting distribution. Similar as in the SAOM, the model builds on randomly assigning the opportunity to change, followed by a function that governs the probability of a tie change.

5.2 Resume

In this article, we put emphasis on two popular dynamic network models, the TERGM and the REM. Comparisons between these models can be drawn on the level at which each implied generating mechanism works, how time perceived, and to what extend within-network and between-network dependence can be analyzed.

The overall aim in the TERGM is to find an adequate distribution of the adjacency matrix including information on previous realizations of the network. In the separable extension the aim remains unchanged, only splitting into two smaller sub-networks that include all possible ties that were and were not present in separately. Contrasting to this aim, the REM tackles the intensity on a dyadic level. Therefore, models from Section 3 take a global and models from Section 4 a local point-of-view, which results in substantially different interpretations of the estimates as seen in Sections 3.3 and 4.4.

The most apparent difference is the perception of time in the respective models. Where the TERGM can be framed as a Markov chain model in discrete time, the REM is operating in continuous time, although it is discretized due to the sampling scheme of the international arms trade network.

As a result from viewing the network as either evolving in continuous or discrete time, the possibilities to differentiate between within-network and between-network dependencies are affected. The only model that can clearly isolate these two dependencies is the TERGM, where the within-network dependence is captured by all terms of that are only concerned with , and between-network dependence is controlled for by the terms that only depend on . Due to the separability assumption, whereby all these statistics partially depend on and , this clear cut is not any more possible, as already noted in Section 3.3. Lastly, the model framework in continuous time does not allow this distinction, because the model is solely concerned with the effect of covariates on the intensity of observing the event .

Acknowledgement

The project was supported by the European Cooperation in Science and Technology [COST Action CA15109 (COSTNET)]. We also gratefully acknowledge funding provided by the German Research Foundation (DFG) for the project KA 1188/10-1: International Trade of Arms: A Network Approach

. Furthermore we like to thank the Munich Center for Machine Learning (MCML) for funding.

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Appendix A Annex: Additional Descriptives

Figure 4 depicts the distribution of in- and out-degrees in the network. This is the number of in- and outwards directed ties each country had in a specific year. A strongly asymmetric relation is revealed, indicating that about 70 of the countries do not export any weapons, while a small percentage of countries accounts for the major share of trade relations. The distribution of the in-degree is not that extreme but still we have roughly one third of all countries not importing at all. These measure were calculated with the package in R (Csardi and Nepusz, 2006).

Figure 4: Barplots indicating the distribution of the in- and out-degrees in the years 2016 and 2017, black bars indicate the values of year 2016 and grey bars of 2017.