Tempus Volat, Hora Fugit - A Survey ofDynamic Network Models in Discrete and

Continuous Time

Cornelius Fritz∗, Michael Lebacher and Goran KauermannDepartment of Statistics, Ludwig-Maximilians-Universitat Munchen

October 24, 2019

Abstract

Given the growing number of available tools for modeling dynamic networks, thechoice of a suitable model becomes central. It is often difficult to compare the differentmodels with respect to their applicability and interpretation. The goal of this survey isto provide an overview of popular dynamic network models. The survey is focused onintroducing binary network models with their corresponding assumptions, advantages,and shortfalls. The models are divided according to generating processes, operatingin discrete and continuous time, respectively. First, we introduce the Temporal Ex-ponential Random Graph Model (TERGM) and its extension, the Separable TERGM(STERGM), both being time-discrete models. These models are then contrasted withcontinuous process models, focusing on the Relational Event Model (REM). We ad-ditionally show how the REM can handle time-clustered observations, i.e., continuoustime data observed at discrete time points. Besides the discussion of theoretical proper-ties and fitting procedures, we specifically focus on the application of the models usinga network that represents international arms transfers. The data allow to demonstratethe applicability and interpretation of the network models.Keywords — Continuous-Time, Discrete-Time, Event Modeling, ERGM, RandomGraphs, REM, STERGM, TERGM

∗cornelius.fritz@stat.uni-muenchen.de

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1 Introduction

The conceptualization of systems within a network framework has become popular withinthe last decades, see Kolaczyk (2009) for a broad overview. This is mostly because networkmodels provide useful tools for describing complex dependence structures and are applicableto a wide variety of research fields. In the network approach, the mathematical structure ofa graph is utilized to model network data. A graph is defined as a set of nodes and relationalinformation (ties) between them. Within this concept, nodes can represent individuals,countries or general entities, while ties are connections between those nodes. Dependent onthe 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 justa few. Given a suitable data structure for the system of interest, the conceptualization asa network enables analyzing dependencies between ties. A central statistical model thatallows this is the Exponential Random Graph Model (ERGM, Robins and Pattison, 2001).This model permits the inclusion of monadic, dyadic and hyperdyadic features within aregression-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 temporaldimension is added, which is relevant, as most systems commonly described as networksevolve dynamically over time. It can even be argued that most static networks are de factonot static but snapshots of a dynamic process. A friendship network, e.g., typically evolvesover time and influences like reciprocity often can be found to follow a natural chronologicalorder.

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 authorslaid the foundation for further articles and postulated a soft division of statistical networkmodels into latent space (Hoff et al., 2002) and p1 models (Holland and Leinhardt, 1981),all originating in the Eds-Rnyi-Gilbert random graph models (Erdos and Renyi, 1959). Kimet al. (2018) give a contemporary update on the field of dynamic models building on latentvariables. Snijders (2005) discusses continuous time models and reframes the independenceand reciprocity model as a Stochastic Actor oriented Model (SAOM, Snijders, 1996). Blocket al. (2018) provide an in-depth comparison of the Temporal Exponential Random GraphModel (TERGM, Hanneke et al., 2010) and the SAOM with special focus on the treatment oftime. Further, the ERGM and SAOM for networks which are observed at single time pointsare contrasted by Block et al. (2019), deriving theoretical guidelines for model selection basedon the differing mechanics implied by each model.

In the context of this compendium of articles, the scope is to give an update on the

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Erdös-Rényi-Gilbdert(Gilbert1959,Erdös-

Rényi1959)

LatenSpaceModels

StaticVariant(Hoff,Raftery,Handcock

2002)

DynamicVariant(SarkarandMoore2005)

p1models(HollandandLeinhardt

1981)

StaticVariant DynamicVariant

ERGM(FrankandStrauss1986)

TERGM(HannekeandXing2006)

STERGM(KrivitskiandHandcock

2012)

MarkovChain-BasedDiscreteTime

MarkovProcess-BasedContinuousTime

SAOM(Snijders1996)

DyNAM(Stadtfeldetal.2017)

REM(Butts2008)

LongitudinalData TimeStampedData

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

dynamic variant of the second strand of models relating to p1 models. We therefore extendthe summarizing diagram of Goldenberg et al. (2010) as depicted in Figure 1. Generally, wedivide temporal models into two sections, by differentiating between discrete and continuoustime network models.

Statistical models for time discrete data rely on a Markov chain assumption and conditionthe state of the network at time point t on previous states. This includes the TERGM andthe Separable TERGM (STERGM, Krivitsky and Handcock, 2014). There exists a widerange of recent applications of the TERGM. White et al. (2018) use a TERGM for modelingepidemic disease outcomes and Blank et al. (2017) investigate interstate conflicts. In Heet al. (2019) Chinese patent trade networks are inspected and Benton and You (2017) usea TERGM for analyzing shareholder activism. Applications of STERGMs are given forexample by Stansfield et al. (2019) that model sexual relationships and Broekel and Bednarz(2019) that study the network of research and development cooperation between Germanfirms.

In case of time-continuous data, the model regards the network as a continuously evolvingsystem. Although this evolution is not necessarily observed in continuous time, the processis taken to be latent and explicitly models the evolution from the state of the network attime point t− 1 to t (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-

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Descpription Year

Time t 2016 2017Number of countries included n 148 148Number of possible ties n(n− 1) 21 756 21 756Density 0.020 0.019Transitivity 0.202 0.207Reciprocity 0.085 0.087

Table 1: Descriptive Statistics for the years 2016 and 2017.

discrete observations of networks. That is, we observe the time-continuous developments ofthe network at discrete observation times only. Applications of the REM for non-clusteredobservations are manifold and range from explaining the dynamics of health behavior sen-timents via Twitter (Salathe 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) tostructures of project teams (Quintane et al., 2013).

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

2 Definitions and Data Description

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 Interna-tional Peace Research Institute (SIPRI, 2018) and includes data on the exchange of majorconventional weapons (MCW) together with the volume of each transfer. Since this articleonly regards binary network models, the trade network is discretized with a threshold of zero.This means that a tie from actor i to actor j indicates that the sender country i traded witha receiver country j in the respective year. This information can then be represented in anadjacency matrix Yt = (Yij,t)i,j=1,..,n ∈ Y , where Y = {Y : Y ∈ {0, 1}n×n} represents the setof all possible networks with n nodes, in our example countries. The entry (i, j) of Yt is ”1”if country i sold MCW to country j in year t and ”0” otherwise. Further, the discrete timepoints of the observations of Yt are denoted as t = 1, . . . , T . For demonstration purposes, werestrict our analysis to two time points only and consider the years 2016 and 2017. Hence welook at annual changes of the network structure and set T = 2. In many networks including

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our running example self loops are meaningless. We therefore fix Yii,t ≡ 0 ∀ i ∈ {1, . . . , n}throughout the article. Further, all sub-scripted indices (Yt) are assumed to be discrete andand all indices in brackets (Y (t)) continuous. The temporal indicator t denotes the obser-vation times of the network and to notationally differ this from time-continuous model wewrite t 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 possibleties stays the same as well. The density of a network is the proportion of realized edges outof all possible edges and is similar in both years, indicating the sparsity of the modelednetwork. Clustering can be expressed by the transitivity measure, providing the percentageof connected triplets out of all possible triplets. The reciprocity of a graph is the ratio ofreciprocated ties in a graph and is similar in both years, see Annex A for a description ofthe degree distribution (Csardi and Nepusz, 2006).

Additionally, information on different kinds of exogenous covariates may be controlled forin statistical network models. In the given example we use the logarithmic Gross DomesticProduct (GDP) (World Bank, 2017) as monadic covariates in respect to the sender andreceiver 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 (noideological distance), as a dyadic exemplary covariate. These covariates are assumed to benon-stochastic and we denote them by xt.

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 mod-els for the analysis of static network data. Holland and Leinhardt (1981) introduced themodel class, which was subsequently extended with respect to fitting algorithms and net-work statistics (see Lusher et al., 2012, Robins et al., 2007). Spurred by the popularity ofERGMs, dynamic extensions of this model class emerged, pioneered by Robins and Pattison(2001) who developed time-discrete models for temporally evolving social networks. Beforewe start with a description of the model, we want to highlight that the TERGM as wellas the STERGM are most appropriate for equidistant time points. That is, we observethe networks Yt at discrete and equidistant time points t = 1, ..., T . Only in this settingthe parameters allow for a meaningful interpretation. See Block et al. (2018) for a deeperdiscussion.

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

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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.

point t − 1 to time point t can be explained by exogenous covariates as well as structuralcomponents of preceding networks. We assume a first order Markov dependence structurethat applies to probability distributions Pθ with parameter vector θ ∈ Rp. Conditioning onthe first network, the resulting dependence structure of the model can be factorized into

Pθ(YT , ..., Y2|Y1, x1, ..., xT ) = Pθ(YT |YT−1, xT ) · · ·Pθ(Y3|Y2, x3)Pθ(Y2|Y1, x2). (1)

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

Pθ(Yt = yt|Yt−1 = yt−1, xt) =exp{θT s(yt, yt−1, xt)}

κ(θ, yt−1, xt). (2)

Generally, s(yt, yt−1, xt) specifies a p-dimensional function of sufficient network statisticswhich may depend on the previous network as well as on covariates. These network statis-

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tics 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 s(yt, yt−1, xt) explicitly allows for temporal interactions, e.g. de-layed reciprocity

sreciprocity(yt, yt−1) ∝∑i 6=j

(yji,tyij,t−1) . (3)

This statistic governs the tendency whether a tie (i, j) in t − 1 will be reciprocated in t.Another important temporal statistic is stability

sstability(yt, yt−1) ∝∑i 6=j

(yij,tyij,t−1 + (1− yij,t)(1− yij,t−1)) . (4)

In this case, the first product in the sum measures whether existing ties in t − 1 persistin t and the second term is one if non-existent ties in t − 1 remain non-existent in t. Theproportionality sign is used since in many cases the network statistics are scaled into a specificinterval (e.g. [0, n] or [0, 1]). Such a standardization is especially sensible for networks wherethe actor set changes with time. Additionally, exogenous covariates can be included, e.g.,time-varying dyadic covariates xij,t

sdyadic(yt, xt) =∑i 6=j

yij,txij,t. (5)

There exists an abundance of possibilities for defining interactions between ties in t− 1 andt. From this discussion and equation (2) it also becomes obvious, that in a situation wherethe interest lies in the transition between two periods t ∈ {1, 2}, a TERGM can be modelledsimply as an ERGM, including lagged network statistics (for example by incorporating yij,t−1as explanatory variable).

Concerning the estimation of the model, maximum likelihood appears to be a naturalcandidate due to the simple exponential family form (2). However, the normalization con-stant in the denominator of model (2) often poses an inhibiting obstacle when estimating(T)ERGMs. This can be seen by inspecting the normalization constant κ(θ, yt−1, xt) =∑

y∈Y exp{θT s(yt, yt−1, xt)}, that requires summation over all possible networks y ∈ Y . Thistask is virtually infeasible, except for very small networks. Therefore, Markov Chain MonteCarlo (MCMC) methods have been proposed in order to approximate the logarithmic like-lihood function (see Geyer and Thompson (1992) for Monte Carlo maximum likelihood andHummel et al. (2012) for its adaption to ERGMs). A notable special case arises if the

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

network statistics are restricted such that they decompose to

s(yt, yt−1, xt) =∑i 6=j

yij,tsij(yt−1, xt), (6)

with sij being a function that is evaluated only at the lagged network yt−1 and covariates xtfor tie (i, j). With this restriction, we impose that the ties in t are independent, conditionalon the network structures in t − 1. This greatly simplifies the estimation procedure andallows 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 andHandcock (2014). This model can be motivated by the fact that the stability term leads toan ambiguous interpretation of its corresponding parameter. Given that we include (4) ina TERGM and obtain a positive coefficient after fitting the model it is not clear whetherthe network can be regarded as ”stable” because existing ties are not dissolved (i.e. yij,t =yij,t−1 = 1) or because no new ties are formed (i.e. yij,t = yij,t−1 = 0). To disentangle this,the authors propose a model that allows for the separation of formation and dissolution.

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Krivitsky and Handcock (2014) define the formation network as Y + = Yt ∪ Yt−1, beingthe network that consists of the initial network Yt−1 together with all ties that are newlyadded in t. The dissolution network is given by Y − = Yt ∩ Yt−1 and contains exclusivelyties that are present in t and t− 1. Given the network in t− 1 together with the formationand the dissolution network we can then uniquely reconstruct the network in t, since Yt =Y +\(Yt−1\Y −) = Y − ∪ (Y +\Yt−1). Define θ = (θ+, θ−) as the joint parameter vector thatcontains 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 theparameter space of θ is the product of the parameter spaces of θ+ and θ− together withconditional independence of formation and dissolution given the network in t− 1:

Pθ(Yt = yt|Yt−1 = yt−1, xt) =

Pθ+(Y + = y+|Yt−1 = yt−1, xt)︸ ︷︷ ︸Formation Model

Pθ−(Y + = y−|Yt−1 = yt−1, xt)︸ ︷︷ ︸Dissolution Model

. (7)

The structure of the model is visualized in Figure 3. On the left hand side the state of thenetwork Yt−1 is given, consisting of two ties (i, h) and (h, j). In the formation network (topin the middle plot) all ties that could possibly be formed are shown in dashed and the actualformation in this example (i, j) is shown in solid. On the bottom, the two ties that couldpossibly be dissolved are shown and in this example (h, j) persists while (i, j) is dissolved.On the right hand side of Figure 3 the resulting network at time point t is displayed.

Given this structure and the separability assumption (7), it is assumed that a TERGMstructure (2) is appropriate for both, the formation and the dissolution process. For prac-tical reasons it is important to understand that the term ”dissolution” model is somewhatmisleading 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 im-plementation in software packages, but can simply be changed by switching the signs of theparameters in the dissolution model.

3.3 Software and Application

When it comes to software, there exist essentially two main R packages that are designedfor 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 beinterpreted as TERGMs when including lagged network statistics). The library contains thepackage tergm with implemented methods for fitting STERGMs using conditional maximumlikelihood. However, currently the package tergm (version 3.5.2) does not allow for fittingSTERGMs with time-varying dyadic covariates for more than two time periods jointly. The

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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 ?) or MCMCmaximum likelihood estimation routines.

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

TERGM STERGMFormation Dissolution

Lagged Network 3.590∗∗∗ - -(0.143) - -

Edges −14.502∗∗∗ −15.144∗∗∗ −16.382∗∗∗

(1.818) (2.307) (3.705)Reciprocity −0.164 −0.490 −0.071

(0.362) (0.443) (0.624)Out-Degree Sender (Geometrically weighted) −3.149∗∗∗ −4.117∗∗∗ −0.378

(0.271) (0.418) (0.498)In-Degree Receiver (Geometrically weighted) −1.961∗∗∗ −2.480∗∗∗ −0.407

(0.294) (0.382) (0.466)Edge-wise Shared Partners (Geometrically weighted) 0.020 0.038 0.137

(0.059) (0.058) (0.110)log(GDP) Sender 0.299∗∗∗ 0.371∗∗∗ 0.308∗∗∗

(0.046) (0.057) (0.088)log(GDP) Receiver 0.150∗∗∗ 0.095 0.321∗∗∗

(0.043) (0.057) (0.090)Polity Score (Absolute Difference) −0.028∗ −0.029∗ −0.016

(0.011) (0.014) (0.017)

Log Likelihood -945.282 -663.287 -258.293AIC 1908.564 1342.574 532.585∑

AIC 1908.564 1875.159

Table 2: Comparison of parameters obtained from the TERGM (first column) and theSTERGM (Formation in the second column, Dissolution in the third column). Standarderrors in brackets and stars according to p-values smaller than 0.001 (∗∗∗), 0.05 (∗∗) and 0.1(∗). Decay parameter of the geometrically weighted statistics is set to 1.

The results obtained for the arms trading data discussed in the previous section are dis-

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played in Table 2. For a detailed interpretation of effects focusing on political, social, andeconomic aspects we refer to the relevant literature (see e.g. Thurner et al., 2018). Here wewant 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 statis-tics are evaluated either on the formation or the dissolution network and both are formedwith the networks in t and t − 1. In contrast to that, in the TERGM (first column), ex-cept the lagged stability term all network statistics are evaluated on the network in t. Notefurther, that the TERGM coefficients try to explain the network structure in t based ont − 1, while the STERGM coefficients provide information either on the formation or thedissolution.

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

Similarly, we find that sending countries with high out-degrees are rare in the network, seethe significant and low values for the geometrically weighted out-degree. However, although itis 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 dissolutionmodel, 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, thecoefficient of the logarithmic GDP of the importing country. The TERGM assigns a higherprobability to observing in-going ties to countries with a high GDP. However, disentanglingthe model towards formation and dissolution we see highly significant coefficients in thedissolution model while the effect for the formation model is insignificant.

Overall we observe that the STERGM allows to decompose the dynamics, which canalso be quantified by the AIC as a model selection criterion. Based on the independenceassumption in (7) we can sum up the two AIC values and see that the AIC value of theSTERGM is smaller than of the TERGM.

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4 Relational Event Model

4.1 Time-Continuous Event Processes

The second type of dynamic network models results by comprehending network changes asa continuously evolving process (see Girardin and Limnios, 2018 as a basic reference forstochastic 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 acontinuously evolving process, where only one tie can occur at a time. This framework wasextended by Butts (2008) to model behavior, which is understood as a directed event (i, j)at a specific time, that potentially depends on the past. For instance, country i sendingweapons to j at a given time point is a behavior, hereinafter called event. The overall aimis 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-eventanalysis, or survival analysis respectively (see, e.g., Kalbfleisch and Prentice, 2002 for anoverview of time-to-event models). The central concept of this framework can be motivatedby the introduction of a multivariate time-continuous Poisson counting process

N(t) = (Nij(t) | i, j ∈ {1, . . . , n}), (8)

where Nij(t) counts how often actors i and j interacted in [0, t). Note that we indicatecontinuous time t with a tilde to distinguish from the discrete time setting with t = 1, 2, . . . , Tassumed in the previous section. Process (8) is characterized by an intensity function λij(t)for i 6= j, which is defined as:

λij(t) = limdt↓0

P(Nij(t+ dt) = Nij(t) + 1)

dt.

This is the instantaneous probability of observing a jump of size ”1” in Nij(t), which indicatesobserving the event (i, j) at time t. Since we assume that there are no self-loops λii(t) ≡0 ∀ i = 1, . . . , n holds.

4.2 Time-Continuous Observations

Butts (2008) introduced the Relational Event Model (REM) to analyze the intensity λij(t)when time-stamped data on the events are available. He assumed that the intensity isconstant over time but depends on time-varying relational information of past events andexogenous covariates. Vu et al. (2011) extended the model by postulating a semi-parametric

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intensity similar to Cox (1972):

λij(t | N(t), x(t), θ) = λ0(t)exp{θT sij

(N(t), x(t)

)}, (9)

where λ0(t) is an arbitrary baseline intensity, θ ∈ Rp the parameter vector and sij(N(t), x(t)

)a statistic that depends on the (possibly time-continuous) covariate process x(t) and thecounting process just prior to t. Examples for sij

(N(t), x(t)

)are the out- and in-degree of

countries i and j.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 thedyadic level and become global by aggregation of multiple similar dyadic effects (Stadtfeld,2018). This differing level of modeling necessitates defining the statistics sij

(N(t), x(t)

)on a dyadic level. To give an example, the dyadic version of reciprocity for the event(i, j) now regards, whether already having observed the event (j, i) prior to t has an effecton λij(t | N(t), x(t), θ), in comparison to the network level version (3) that counted thenumber of reciprocated ties between yt and yt−1. Therefore, the mathematical formulationis straightforward:

sij,reciprocity(N(t), x(t)

)= 1

(Nji(t)) > 0

),

where 1 is the indicator function. Since the effect of a past event at time δ, say, on a presentevent at time t may vary according to the elapsed time t − δ, Stadtfeld and Block (2017)introduced windowed effects, which only regard events that occurred in a pre-specified timewindow, 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 θ withouthaving to specify a parametric form of the baseline hazard λ0(t) nor a distribution on thetimes between events. In the same way, Λ0(t) =

∫ t0λ0(u)du 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 strat-ified Cox model in (9) and allowed multi-cast-events, which are events that are possiblydirected at multiple receivers. Stadtfeld et al. (2017) adopted the Stochastic Actor orientedModel (SAOM) to events. DuBois and Smyth (2010) and DuBois et al. (2013) extendedthe Stochastic Block Model (SBM) for time-stamped relational events. Further, DuBoiset al. (2013) adopted a Bayesian hierarchical model to event data when information is onlyavailable in smaller groups.

13

4.3 Time-Clustered Observations

Generally, the approach discussed above requires time-stamped network data, meaning thatwe observe the precise time points of all events. For the running example this means thatwe need the exact time point t of an arms trade between country i and j. Often, suchexact time-stamped data are not available and, in fact, trading between states can hardlybe stamped with a single time point t. Indeed, we often only observe the time-continuousnetwork process at discrete time points t = 1, . . . , T . In such setting, we may assume somekind of Markov structure in that we do not look at the entire history of the process N(t)but just model the intensity (9) in the time frame between t − 1 and t. Let therefore N(t)be adapted to Y (t) := N(t)−N(t− 1) and x(t) for t ∈ [t− 1, t]. We then reframe (9) as:

λij(t | Y (t), x(t), θ) = λ0(t)exp{θT sij

(Y (t), x(t)

)}. (10)

In other words, we assume that the intensity of events between t− 1 and t does not dependon states of the multivariate counting process prior to t − 1. The history of the countingprocess is reset after each time interval. This is a reasonable assumption, if one is primarilyinterested in short-term dependencies between the individual counting processes.

If we observe the continuous process at discrete time points it is inevitable that we observetime 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 tradingabove. Under the term tied observations this phenomenon is well known in time-to-eventanalysis and treated with several approximations. We make use of the so called Breslowapproximation (see Peto, 1972; Breslow, 1974). Let therefore

Ot = {(i, j) | Nij(t)−Nij(t− 1) > 0}

where element (i, j) is replicated Nij(t)−Nij(t− 1) times in Ot, that is if an event betweeni and j occurred multiple times in the interval from t− 1 to t then (i, j) appears respectivetimes in Ot. Given that we have not observed the exact time point of an event we alsoget no information on the baseline intensity λ0(t) in (9) for t ∈ [t − 1, t] so that the modelsimplifies to a discrete choice model structure (see, e.g., Train, 2009) which resembles thepartial likelihood by Cox (1972). Let therefore Ut denote the set of all possible ties betweencountries that may be observed at time point t, so that the partial likelihood is defined as:

PL(θ) =T∏t=1

∏(i,j)∈Ot

exp{θT sij(Y (t), x(t)

)}(∑

(i,j)∈Utexp{θT sij

(Y (t), x(t)

)})nt

, (11)

where nt = |Ot|.

14

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

4.4 Software and Application

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

As mentioned before, we do not have time stamps for the arms trades. While this is anslight misuse of the time-continuous model, we apply this analysis here for demonstrationpurposes and to allow for a comparison with the results in the previous chapter. In otherwords, we either observe an arms trade (i.e. Yij(t) = 1) or no trade (Yij(t) = 0).

The estimates are shown in Table 3, the first column represents the estimates of theBreslow, whereas the second column regards the estimation via the Kalbfleisch-Prenticeapproximation with 100 random orders. Regarding the significant terms, the estimates leadto similar conclusions. Only the estimates concerning transitivity, are slightly singnificantin 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 compar-ison to the global interpretation of the effects in section 3.3. Therefore, e.g., the positiveeffect of the out-degree of the sender translates to a higher intensity of observing (i, j) if ihad a higher out-degree.

Similar to the application of Section 3.3 reciprocated ties are not more likely to occur thannon-reciprocated ones judged by their significance. The degree-related covariates concern therole of centrality in respect to the intensity of an event. Apparently, both a high out-degree ofthe 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 andcountries with a high in-degree to receive weapons. It is notable that this interpretationis 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 networkexhibits a general tendency of having rather low out- and in-degrees. Both estimates indicatean asymmetric degree structure, yet the estimates from Section 3.3 are to be understood onthe global level and translate to less countries with high in- and out-degree than expectedunder under a completely random graph. In the REM, on the other hand, the estimates

15

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 −3581.731AIC 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 accordingto p-values smaller than 0.001 (∗∗∗), 0.05 (∗∗) and 0.1 (∗).

indicate that already having been highly involved in the network makes future trade activitymore 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 notclearly be detected. The respective estimate indicates, that having common trade partners isnot a catalyzing factor in trading among countries. Additionally, this effect was not found onthe global level of the analysis in Section 3.3, where the analog statistic is called geometricallyweighted edge-wise shared partners.

Further, we find additional confirmation on the influence of the logarithmic GDP of thesender 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 receivingweapons.

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

16

5 Discussion

5.1 Further models

Snijders (1996) formulated a two-stage process model operating in a continuous time frame-work. 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 outgo-ing ties. Consecutively, the actor needs to evaluate the probability of changing the presentconfiguration with each possible receiver, which entails each actors knowledge of the com-plete graph whenever he has the possibility to toggle one of his ties. Lastly, the decision israndomly drawn relative to the probabilities of all possible actions. In general, the SAOMis a well established model for the analysis of social networks, that was successfully appliedto 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 con-tinuous 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 networkevolves in micro-steps as a continuous time Markov process with an ERGM being its lim-iting distribution. Similar as in the SAOM, the model builds on randomly assigning theopportunity 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 andthe REM. Comparisons between these models can be drawn on the level at which each impliedgenerating mechanism works, how time perceived, and to what extend within-network andbetween-network dependence can be analyzed.

The overall aim in the TERGM is to find an adequate distribution of the adjacencymatrix Yt including information on previous realizations of the network. In the separableextension the aim remains unchanged, only splitting Yt into two smaller sub-networks thatinclude all possible ties that were and were not present in Yt−1 separately. Contrasting tothis aim, the REM tackles the intensity on a dyadic level. Therefore, models from Section 3take a global and models from Section 4 a local point-of-view, which results in substantiallydifferent 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. Wherethe TERGM can be framed as a Markov chain model in discrete time, the REM is operatingin continuous time, although it is discretized due to the sampling scheme of the international

17

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 dependenciesare 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 s(yt, yt−1, xt) that are onlyconcerned with yt, and between-network dependence is controlled for by the terms that onlydepend on yt−1. Due to the separability assumption, whereby all these statistics partiallydepend on yt and yt−1, this clear cut is not any more possible, as already noted in Section3.3. Lastly, the model framework in continuous time does not allow this distinction, becausethe model is solely concerned with the effect of covariates on the intensity of observing theevent (i, j).

Acknowledgement

The project was supported by the European Cooperation in Science and Technology [COSTAction CA15109 (COSTNET)]. We also gratefully acknowledge funding provided by theGerman Research Foundation (DFG) for the project KA 1188/10-1: International Trade ofArms: A Network Approach. Furthermore we like to thank the Munich Center for MachineLearning (MCML) for funding.

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A Annex: Additional Descriptives

Figure 4 depicts the distribution of in- and out-degrees in the network. This is the numberof in- and outwards directed ties each country had in a specific year. A strongly asymmetricrelation 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. Thedistribution of the in-degree is not that extreme but still we have roughly one third of allcountries not importing at all. These measure were calculated with the igraph package inR (Csardi and Nepusz, 2006).

0.0

0.2

0.4

0.6

0 5 10 15

Degree

Pro

port

ion

In−Degree

0.0

0.2

0.4

0.6

0 20 40 60

Degree

Pro

port

ion

Out−Degree

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

24