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

TL;DR

This survey reviews tie-oriented dynamic network models, comparing discrete and continuous time approaches, highlighting their assumptions, advantages, limitations, and applications to real-world network data.

Contribution

It provides a comprehensive overview of binary dynamic network models, including TERGM, STERGM, and REM, with insights into their theoretical properties, fitting procedures, and practical applications.

Findings

Models effectively capture different temporal dynamics.

Application to real networks demonstrates interpretability.

Continuous and discrete models have complementary strengths.

Abstract

Given the growing number of available tools for modeling dynamic networks, the choice of a suitable model becomes central. The goal of this survey is to provide an overview of tie-oriented 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. First, we introduce the Temporal Exponential Random Graph Model (TERGM) and 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 on two networks that represent international arms transfers and email exchange. The data allow to demonstrate the applicability and interpretation of the network models.

Tables

Table 1. Table 1 : Descriptive statistics for the international arms trade network (left) and the European research institutions email correspondence (right).

		Arms Trade Network		Email Network
Time	$t$	2016	2017	Period $1$	Period $2$
Number of events		$-$	$-$	$\mathrm{4\hspace{0.17em}957}$	$\mathrm{2\hspace{0.17em}537}$
Number of nodes	$n$	180	180	88	88
Number of possible ties	$n\left(n-1\right)$	32 220	32 220	7 656	7 656
Density		0.021	0.020	0.123	0.087
Transitivity		0.195	0.202	0.407	0.345
Reciprocity		0.081	0.083	0.7	0.687
Repetition		$-$	0.641	$-$	0.574

Table 2. Table 2: Arms trade network: Comparison of parameters obtained from the TERGM (first column), STERGM (Formation in the second column, Dissolution in the third column) and REM (fourth column). Standard errors in brackets and stars according to p 𝑝 p -values smaller than 0.001 0.001 0.001 ( ∗∗∗ ), 0.05 0.05 0.05 ( ∗∗ ) and 0.1 0.1 0.1 ( ∗ ). Decay parameter of the geometrically weighted statistics is set to log ⁡ ( 2 ) 2 \log(2) and the Kalbfleisch-Prentice approximation was used with 100 random orderings of the events to find the estimates of the REM.

	TERGM	STERGM		REM
		Formation	Dissolution
Repetition	${3.671}^{\ast \ast \ast }$	$-$	$-$	${2.661}^{\ast \ast \ast }$
	$\left(0.132\right)$	$-$	$-$	$\left(0.143\right)$
Edges	$-{15.632}^{\ast \ast \ast }$	$-{17.186}^{\ast \ast \ast }$	$-{16.987}^{\ast \ast \ast }$	$-$
	$\left(1.809\right)$	$\left(2.168\right)$	$\left(3.587\right)$	$-$
Reciprocity	$-0.258$	$-0.620$	$-0.058$	$-0.109$
	$\left(0.306\right)$	$\left(0.436\right)$	$\left(0.619\right)$	$\left(0.181\right)$
In-Degree (GWID)	$-{1.823}^{\ast \ast \ast }$	$-{2.106}^{\ast \ast \ast }$	$-0.412$	${0.060}^{\ast \ast }$	In-Degree Receiver
	$\left(0.278\right)$	$\left(0.379\right)$	$\left(0.442\right)$	$\left(0.015\right)$
Out-Degree (GWOD)	$-{3.220}^{\ast \ast \ast }$	$-{4.126}^{\ast \ast \ast }$	$-0.326$	${0.010}^{\ast \ast }$	Out-Degree Sender
	$\left(0.304\right)$	$\left(0.462\right)$	$\left(0.533\right)$	$\left(0.004\right)$
GWESP	$0.050$	$0.076$	$0.150$	$0.010$	Transitivity
	$\left(0.066\right)$	$\left(0.071\right)$	$\left(0.126\right)$	$\left(0.029\right)$
Polity Score	$-{0.024}^{\ast }$	$-{0.028}^{\ast }$	$-0.016$	$-0.016$
	$\left(0.010\right)$	$\left(0.014\right)$	$\left(0.017\right)$	$\left(0.009\right)$
log(GDP) Sender	${0.313}^{\ast \ast \ast }$	${0.394}^{\ast \ast \ast }$	${0.323}^{\ast \ast \ast }$	${0.395}^{\ast \ast \ast }$
	$\left(0.048\right)$	$\left(0.054\right)$	$\left(0.088\right)$	$\left(0.039\right)$
log(GDP) Receiver	${0.165}^{\ast \ast \ast }$	${0.135}^{\ast }$	${0.327}^{\ast \ast \ast }$	${0.192}^{\ast \ast \ast }$
	$\left(0.043\right)$	$\left(0.054\right)$	$\left(0.087\right)$	$\left(0.032\right)$
Log Likelihood	$-$949.833	$-$675.327	$-$258.425
AIC	1917.666	1366.654	532.849
$\sum$ AIC	1917.666	1899.503

Table 3. Table 3: Email exchange network: Comparison of parameters obtained from the TERGM (first column), STERGM (Formation in the second column, Dissolution in the third column) and REM (fourth column). Standard errors in brackets and stars according to p 𝑝 p -values smaller than 0.001 0.001 0.001 ( ∗∗∗ ), 0.05 0.05 0.05 ( ∗∗ ) and 0.1 0.1 0.1 ( ∗ ). Decay parameter of the geometrically weighted statistics is set to log ⁡ ( 2 ) 2 \log(2) .

	TERGM	STERGM		REM
		Formation	Dissolution
Repetition	${1.367}^{\ast \ast \ast }$	$-$	$-$	${2.27}^{\ast \ast \ast }$
	$\left(0.107\right)$	$-$	$-$	$\left(0.084\right)$
Edges	$-{5.755}^{\ast \ast \ast }$	$-{4.853}^{\ast \ast \ast }$	$-{2.237}^{\ast \ast \ast }$	$-$
	$\left(0.237\right)$	$\left(0.247\right)$	$\left(0.224\right)$	$-$
Reciprocity	${0.398}^{\ast \ast \ast }$	${2.498}^{\ast \ast \ast }$	${2.586}^{\ast \ast \ast }$	${1.655}^{\ast \ast \ast }$
	$\left(0.112\right)$	$\left(0.157\right)$	$\left(0.226\right)$	$\left(0.075\right)$
In-degree (GWID)	${1.060}^{\ast \ast }$	${1.349}^{\ast }$	$0.709$	$-0.004$	In-Degree Receiver
	$\left(0.333\right)$	$\left(0.648\right)$	$\left(0.415\right)$	$\left(0.003\right)$
Out-degree (GWOD)	$0.031$	$-0.411$	$-0.369$	$-0.0001$	Out-Degree Sender
	$\left(0.312\right)$	$\left(0.431\right)$	$\left(0.397\right)$	$\left(0.003\right)$
GWESP	${1.560}^{\ast \ast \ast }$	${0.655}^{\ast \ast \ast }$	${0.429}^{\ast \ast \ast }$	${0.070}^{\ast \ast \ast }$	Transitivity
	$\left(0.110\right)$	$\left(0.111\right)$	$\left(0.086\right)$	$\left(0.008\right)$
Log Likelihood	$-$1723.732	$-$1000.506	$-$505.431
AIC	3459.464	2011.012	1020.862
$\sum$ AIC	3459.464	3031.874

Table 4. Table 4 : Countries included in the analysis of the international trade network with the ISO3 codes, that are used in the graphical representations of the network.

Country Name	ISO3	Country Name	ISO3	Country Name	ISO3
Afghanistan	AFG	Germany	DEU	Niger	NER
Albania	ALB	Ghana	GHA	Nigeria	NGA
Algeria	DZA	Greece	GRC	Norway	NOR
Andorra	AND	Grenada	GRD	Oman	OMN
Angola	AGO	Guatemala	GTM	Pakistan	PAK
Antigua and Barbuda	ATG	Guinea	GIN	Palau	PLW
Argentina	ARG	Guinea-Bissau	GNB	Panama	PAN
Armenia	ARM	Guyana	GUY	Papua New Guinea	PNG
Australia	AUS	Haiti	HTI	Paraguay	PRY
Austria	AUT	Honduras	HND	Peru	PER
Azerbaijan	AZE	Hungary	HUN	Philippines	PHL
Bahamas	BHS	Iceland	ISL	Poland	POL
Bahrain	BHR	India	IND	Portugal	PRT
Bangladesh	BGD	Indonesia	IDN	Qatar	QAT
Barbados	BRB	Iran	IRN	Romania	ROM
Belarus	BLR	Iraq	IRQ	Russia	RUS
Belgium	BEL	Ireland	IRL	Rwanda	RWA
Belize	BLZ	Israel	ISR	Saint Kitts and Nevis	KNA
Benin	BEN	Italy	ITA	Saint Lucia	LCA
Bhutan	BTN	Jamaica	JAM	Saint Vincent and the Grenadines	VCT
Bolivia	BOL	Japan	JPN	Samoa	WSM
Botswana	BWA	Jordan	JOR	San Marino	SMR
Brazil	BRA	Kazakhstan	KAZ	Sao Tome and Principe	STP
Brunei Darussalam	BRN	Kenya	KEN	Saudi Arabia	SAU
Bulgaria	BGR	South Korea	KOR	Senegal	SEN
Burkina Faso	BFA	Kosovo	KOS	Serbia	YUG
Burundi	BDI	Kuwait	KWT	Seychelles	SYC
Cambodia	KHM	Kyrgyzstan	KGZ	Sierra Leone	SLE
Cameroon	CMR	Laos	LAO	Singapore	SGP
Canada	CAN	Latvia	LVA	Slovakia	SVK
Cape Verde	CPV	Lebanon	LBN	Slovenia	SVN
Central African Republic	CAF	Lesotho	LSO	Solomon Islands	SLB
Chad	TCD	Liberia	LBR	South Africa	ZAF
Chile	CHL	Libya	LBY	Spain	ESP
China	CHN	Lithuania	LTU	Sri Lanka	LKA
Colombia	COL	Luxembourg	LUX	Sudan	SDN
Comoros	COM	Macedonia (FYROM)	MKD	Suriname	SUR
DR Congo	ZAR	Madagascar	MDG	Swaziland	SWZ
Congo	COG	Malawi	MWI	Sweden	SWE
Costa Rica	CRI	Malaysia	MYS	Switzerland	CHE
Cote dIvoire	CIV	Maldives	MDV	Tajikistan	TJK
Croatia	HRV	Mali	MLI	Tanzania	TZA
Cuba	CUB	Malta	MLT	Thailand	THA
Cyprus	CYP	Marshall Islands	MHL	Timor-Leste	TMP
Czech Republic	CZE	Mauritania	MRT	Togo	TGO
Denmark	DNK	Mauritius	MUS	Trinidad and Tobago	TTO
Dominica	DMA	Mexico	MEX	Tunisia	TUN
Dominican Republic	DOM	Micronesia	FSM	Turkey	TUR
Ecuador	ECU	Moldova	MDA	Turkmenistan	TKM
Egypt	EGY	Mongolia	MNG	Uganda	UGA
El Salvador	SLV	Montenegro	YUG	Ukraine	UKR
Equatorial Guinea	GNQ	Morocco	MAR	United Arab Emirates	ARE
Estonia	EST	Mozambique	MOZ	United Kingdom	GBR
Ethiopia	ETH	Myanmar	MYM	United States	USA
Fiji	FJI	Namibia	NAM	Uruguay	URY
Finland	FIN	Nauru	NRU	Uzbekistan	UZB
France	FRA	Nepal	NPL	Vanuatu	VUT
Gabon	GAB	Netherlands	NLD	Viet Nam	VNM
Gambia	GMB	New Zealand	NZL	Zambia	ZMB
Georgia	GEO	Nicaragua	NIC	Zimbabwe	ZWE

Table 5. Table 5 : Email exchange network: Comparison of parameters obtained from the TERGM (first column) and STERGM (Formation in the second column, Dissolution in the third column). Standard errors in brackets and stars according to p 𝑝 p -values smaller than 0.001 0.001 0.001 ( ∗∗∗ ), 0.05 0.05 0.05 ( ∗∗ ) and 0.1 0.1 0.1 ( ∗ ). Decay parameter of the geometrically weighted statistics is set to log ⁡ ( 2 ) 2 \log(2) .

	TERGM	STERGM		REM
		Formation	Dissolution
Repetition	${1.986}^{\ast \ast \ast }$	$-$	$-$	${2.354}^{\ast \ast \ast }$
	$\left(0.058\right)$	$-$	$-$	$\left(0.048\right)$	$-$
Edges	$-{3.435}^{\ast \ast \ast }$	$-{4.485}^{\ast \ast \ast }$	$-{0.769}^{\ast \ast \ast }$	$-$
	$\left(0.059\right)$	$\left(0.073\right)$	$\left(0.107\right)$	$-$
Reciprocity	${1.090}^{\ast \ast \ast }$	${2.755}^{\ast \ast \ast }$	${1.761}^{\ast \ast \ast }$	${1.699}^{\ast \ast \ast }$
	$\left(0.069\right)$	$\left(0.079\right)$	$\left(0.113\right)$	$\left(0.043\right)$
In-Degree (GWID)	$-{1.239}^{\ast \ast \ast }$	$-{0.870}^{\ast \ast \ast }$	$-{0.307}^{\ast }$	${0.007}^{\ast \ast \ast }$	In-Degree Receiver
	$\left(0.099\right)$	$\left(0.156\right)$	$\left(0.152\right)$	$\left(0.002\right)$
Out-Degree (GWOD)	$-{1.574}^{\ast \ast \ast }$	$-{1.757}^{\ast \ast \ast }$	$-0.056$	${0.009}^{\ast \ast \ast }$	Out-Degree Sender
	$\left(0.097\right)$	$\left(0.155\right)$	$\left(0.159\right)$	$\left(0.001\right)$
GWESP	${0.497}^{\ast \ast \ast }$	${0.508}^{\ast \ast \ast }$	${0.124}^{\ast \ast }$	${0.199}^{\ast \ast \ast }$	Transitivity
	$\left(0.028\right)$	$\left(0.029\right)$	$\left(0.043\right)$	$\left(0.011\right)$

Equations

$${\mathbb{P}}_{\theta }(Y_T,...,Y_2|Y_1,x_1,...,x_T)={\mathbb{P}}_{\theta }(Y_T|Y_{T-1},x_T)\cdots {\mathbb{P}}_{\theta }(Y_3|Y_2,x_3){\mathbb{P}}_{\theta }(Y_2|Y_1,x_2).$$

$${\mathbb{P}}_{\theta }(Y_t=y_t|Y_{t-1}=y_{t-1},x_t)=\frac{\exp \{{\theta }^{T}s(y_t,y_{t-1},x_t)\}}{\kappa (\theta ,y_{t-1},x_t)}.$$

$$s_{delrecip}(y_t,y_{t-1})\propto \sum _{i\ne j}{y}_{ji,t}{y}_{ij,t-1}.$$

$$s_{stability}(y_t,y_{t-1})\propto \sum _{i\ne j}\left(y_{ij,t}{y}_{ij,t-1}+(1-y_{ij,t})(1-y_{ij,t-1})\right).$$

$$s_{dyadic}(y_t,x_t)=\sum _{i\ne j}{y}_{ij,t}{x}_{ij,t}.$$

$$s(y_t,y_{t-1},x_t)=\sum _{i\ne j}{y}_{ij,t}{\tilde{s}}_{ij}(y_{t-1},x_t),$$

$$s_{GWOD}(y_{t})=\exp\{\alpha_{O}\}\sum_{k=1}^{n-1}\big{(}1-(1-\exp\{-\alpha_{O}\})^{k}\big{)}O_{k}(y_{t}),$$

$$s_{GWESP}(y_{t})=\exp\{\alpha_{S}\}\sum_{k=1}^{n-2}\big{(}1-(1-\exp\{-\alpha_{S}\})^{k}\big{)}S_{k}(y_{t}),$$

$${\mathbb{P}}_{\theta }(Y_t=y_t|Y_{t-1}=y_{t-1},x_t)=$$

$$\underset{\text{Formation Model}}{\underbrace{{\mathbb{P}}_{{\theta }^{+}}(Y^{+}=y^{+}|Y_{t-1}=y_{t-1},x_t)}}\underset{\text{Dissolution Model}}{\underbrace{{\mathbb{P}}_{{\theta }^{-}}(Y^{-}=y^{-}|Y_{t-1}=y_{t-1},x_t)}}.$$

$${\mathbb{P}}_{{\theta }^{+}}(Y^{+}=y^{+}|Y_{t-1}=y_{t-1},x_t)=\frac{\exp \{({\theta }^{+}{)}^{T}s(y^{+},y_{t-1},x_t)\}}{\kappa ({\theta }^{+},y_{t-1},x_t)},$$

$${\mathbb{P}}_{\theta }(Y_t=y_t|Y_{t-1}=y_{t-1},x_t)$$

$$=\frac{\exp \{{\theta }^{T}s(y_t,y_{t-1},x_t)\}}{\kappa (\theta ,y_{t-1},x_t)}$$

$$s_{recip}(y^{+},y_{t-1})=s_{recip}(y^{+})\propto \sum _{i\ne j}{y}_{ji}^{+}{y}_{ij}^{+}$$

$$N(\tilde{t})=(N_{ij}(\tilde{t})\mid i,j\in \{1,\dots ,n\}),$$

$${\lambda }_{ij}(\tilde{t})=\underset{dt\downarrow 0}{\lim }\frac{\mathbb{P}(N_{ij}(\tilde{t}+dt)=N_{ij}(\tilde{t})+1)}{dt}.$$

$$\displaystyle\lambda_{ij}(\tilde{t}\mid N(\tilde{t}),x(\tilde{t}),\theta)=\lambda_{0}(\tilde{t})\text{exp}\big{\{}\theta^{T}s_{ij}\big{(}N(\tilde{t}),x(\tilde{t})\big{)}\big{\}},$$

$$\displaystyle s_{ij,reciprocity}\big{(}N(\tilde{t}),x(\tilde{t})\big{)}=\mathbb{I}\big{(}N_{ji}(\tilde{t})>0\big{)},$$

$$\displaystyle s_{ij,SOD}\big{(}N(\tilde{t}),x(\tilde{t})\big{)}=\sum_{h=1}^{n}\mathbb{I}\big{(}N_{ih}(\tilde{t})>0\big{)}.$$

$$\displaystyle s_{ij,TRA}\big{(}N(\tilde{t}),x(\tilde{t})\big{)}=\sum_{h=1}^{n}\mathbb{I}\big{(}N_{ih}(\tilde{t})>0\big{)}\mathbb{I}\big{(}N_{hj}(\tilde{t})>0\big{)}.$$

$$\displaystyle\mathcal{L}(\theta)=\prod_{(i,j,\tilde{t})\in\Omega}\lambda_{ij}(\tilde{t}\mid N(\tilde{t}),x(\tilde{t}),\theta)\text{exp}\Big{\{}-\underset{\mathcal{T}}{\int}\sum_{k,h=1}^{n}\lambda_{kh}(u\mid N(u),x(u),\theta)du\Big{\}}.$$

$${\mathcal{P}\mathcal{L}}_{cont}(\theta )=\prod _{(i,j,\tilde{t})\in \Omega }\frac{{\lambda }_{ij}(\tilde{t}\mid N(\tilde{t}),x(\tilde{t}),\theta )}{{\sum }_{(k,h)\in U_{\tilde{t}}}{\lambda }_{kh}(\tilde{t}\mid N(\tilde{t}),x(\tilde{t}),\theta )}.$$

$$\displaystyle\lambda_{ij}(\tilde{t}\mid\tilde{Y}(\tilde{t}),x(\tilde{t}),\theta)=\lambda_{0}(\tilde{t})\text{exp}\big{\{}\theta^{T}s_{ij}\big{(}\tilde{Y}(\tilde{t}),x(\tilde{t})\big{)}\big{\}}.$$

$$O_t=\{(i,j)\mid N_{ij}(t)-N_{ij}(t-1)>0\}$$

$$\displaystyle\mathcal{PL}_{clust}(\theta)=\prod_{t=1}^{T}\frac{\prod_{(i,j)\in\mathit{O}_{t}}\text{exp}\{\theta^{T}s_{ij}\big{(}\tilde{Y}(t),x(t)\big{)}\}}{\big{(}\sum_{(k,h)\in\mathit{U_{t}}}\text{exp}\{\theta^{T}s_{kh}\big{(}\tilde{Y}(t),x(t)\big{)}\}\big{)}^{n_{t}}},$$

Full text

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 on two networks that represent international arms transfers and email exchange. The data allow to demonstrate the applicability and interpretation of the network models.

Keywords — Continuous-Time, Discrete-Time, Event Modeling, ERGM, Random Graphs, REM, STERGM, TERGM

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 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 $p_{1}$ models (Holland and Leinhardt, 1981), all originating in the Edös-Rényi-Gilbert random graph models (Erdös and Rényi, 1959; Gilbert, 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.

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 $p_{1}$ 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. In this endeavor, we emphasize reviewing tie-oriented models. Tie-oriented models are concerned with formulating a stochastic model for the existence of a tie contrasting the actor-oriented approach by Snijders (2002), which specifies the model from the actors point of view (Block et al., 2018). The Dynamical Actor-Oriented model (DyNAM, Stadtfeld and Block, 2017) adopts this actor-oriented paradigm to event data. This type of model was formulated with a focus on social networks (Snijders, 1996). Contrasting this, tie-oriented models can be viewed as more general, since they are also applicable to non-social networks.

Statistical models for time discrete data rely on an autoregressive structure and condition the state of the network at time point $t$ on previous states. This includes the TERGM and the Separable TERGM (STERGM, Krivitsky and Handcock, 2014). There exists a variety 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 $t-1$ to $t$ (Block et al., 2018). In this paper we discuss the relational event model (REM, Butts, 2008) for the analysis of event data. Applications of the REM 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). 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. Henceforth, we use the term time-clustered for this special data structure.

In reviewing dynamic network models, we assume a temporal first-order Markov dependency. To be more specific, this implies that the network at time point $t$ only depends on the previous observation of the network. This characteristic is widely used in the analysis of longitudinal networks (Hanneke et al., 2010; Krivitsky and Handcock, 2014) and the resulting conditional independence among states of the network facilitates the estimation with an arbitrary number of time points. In that respect, it suffices to only include two observational moments for illustrative purposes, since the interpretation and estimation with a longer series of networks is unchanged. Lastly, the comparison of the methods at hand in a clear-cut manner is hence enabled.

The paper is structured as follows. In Section 2 we give basic definitions that are used throughout the paper and present the two data examples that will be analyzed as illustrative examples. After that, Section 3 introduces time-discrete and Section 4 time-continuous network models. They are applied in Section 5 on two data sets and Section 6 concludes. Additional results relating to the applications can be found in the Supplementary Material.

2 Definitions and Data Description

2.1 Definitions

This article regards directed binary networks, with ties representing directed relations between two nodes at a time point. The respective information can be represented in an adjacency matrix $Y_{t}=(Y_{ij,t})_{i,j=1,..,n}\in\mathcal{Y}$, where $\mathcal{Y}=\{Y:Y\in\{0,1\}^{n\times n}\}$ represents the set of all possible networks with $n$ nodes. The entry $(i,j)$ of $Y_{t}$ is ”1” if a tie is outgoing from node $i$ to $j$ in year $t$ and ”0” otherwise. Further, the discrete time points of the observations of $Y_{t}$ are denoted as $t=1,\ldots,T$. We restrict our analysis to two time points in both exemplary networks, which suffices for comparison. Hence, we set $T=2$. In many networks, including our running examples self-loops are meaningless. We therefore fix $Y_{ii,t}\equiv 0~{}\forall~{}i\in\{1,\ldots,n\}$ throughout the article. Further, all sub-scripted temporal indices ($Y_{t}$) are assumed to take discrete and all indices in brackets ($Y(t)$) continuous values. The temporal indicator $t$ denotes the observation times of the network and to notationally differ this from time-continuous model we write $\tilde{t}$ for continuous time.

To sufficiently compare different models, we use two application cases. The first one represents the international trade of major weapons, which is given by discrete snapshots of networks that are yearly aggregated over time-continuous trade instances, i.e., the time-stamped information is not observed. Whereas the second application, a network of email traffic, comes in time-stamped format, that can be aggregated to discrete-time observations.

2.2 Data Set 1: International Arms Trade

The data on international arms trading for the years 2016 and 2017 are provided by the Stockholm International Peace Research Institute (SIPRI, 2019). To be more specific, information on the exchange of major conventional weapons (MCW) together with the volume of each transfer is included. In order to have a binary network representation, we discretize the data and set edges $Y_{ij,t}$ to ”1” if country $i$ sent arms to country $j$ in $t$.

The left side of Table 1 gives some descriptive measures (Csardi and Nepusz, 2006) and Figure 2 visualizes the arms trade network using the software Gephi (Bastian et al., 2009). 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 triangles out of all connected triplets. Reciprocity in a graph is the ratio of reciprocated ties and is similar in both years. As expressed by the high percentage of repeated ties, most countries seem to continue trading with the same partners.

Additionally, 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 concerning 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 zero (no ideological distance) to 20 (highest ideological distance), as a dyadic exemplary covariate. These covariates are assumed to be non-stochastic and we denote them by $x_{t}$. See the Supplementary Material for a list of all included countries and their ISO3 code.

2.3 Data Set 2: European Research Institution Email Correspondence

The second network under study represents anonymized email exchange data between institution members of a department in a European research institution (see Paranjape et al., 2017, Email EU Core, 2019). In this data set, we observe events $\omega=(i,j,\tilde{t})$ that represent emails sent from department member $i$ to department member $j$ at a specific time point $\tilde{t}$.

The data contains $n=89$ persons and is recorded over $802$ days. For this paper, we select the first two years and split them again into two years, labeled Period 1 and Period 2. Within the first period, $8\,068$ events are recorded and $4\,031$ in the second period. We only regard one-to-one email correspondences, therefore we exclude all group mails from the analysis. In the right column of Table 1 the descriptive measures for the two aggregated networks are given and in Figure 3 they are visualized. All descriptive statistics are higher in the email exchange network as compared to the arms trade network. In comparison to the arms trade network, the aggregated network is more dense with more than 10% of all possible ties being realized. In both years the transitivity measure is relatively higher in both time periods. The high share of reciprocated ties is intuitive given that the network represents email exchange between institution members that may collaborate. No covariates are available for this network. See Annex A for the visualization of the degree distributions of both applications.

3 Dynamic Exponential Random Graph Models

3.1 Temporal Exponential Random Graph Model

The Exponential Random Graph Model (ERGM) is 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 $Y_{t}$ at discrete and equidistant time points $t=1,...,T$. 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 $t-1$ to time point $t$ can be explained by exogenous covariates as well as structural components of the present and preceding networks. Using a first-order Markov dependence structure and conditioning on the first network, the resulting dependence structure of the model can be factorized into

[TABLE]

In the formulation above, it is assumed that the joint distribution can be decomposed into yearly transitions from $Y_{t-1}$ to $Y_{t}$. Further, it is assumed that the same parameter vector $\theta$ governs all transitions. Often, this is an unrealistic assumption for networks observed at many time-points because the generative process may change other time. Therefore, it can be useful to allow for different parameter vectors for each transition probability (i.e. $\theta_{T}$, $\theta_{T-1},...$). In such a setting, the parameters for each transition can either be estimated sequentially (e.g. Thurner et al., 2018) or by using smooth time-varying effects (e.g. Lebacher et al., 2019).

Given the dependence structure (1), the TERGM assumes that the transition from $Y_{t-1}$ to $Y_{t}$ is generated according to an exponential random graph distribution with the parameter $\theta$:

[TABLE]

Generally, $s(y_{t},y_{t-1},x_{t})$ specifies a $p$-dimensional function of sufficient network statistics which may depend on the present and previous network as well as on covariates. These network statistics can include static components, designed for cross-sectional dependence structures (see Morris et al., 2008 for more examples). However, the statistics $s(y_{t},y_{t-1},x_{t})$ explicitly allow temporal interactions, e.g. delayed reciprocity

[TABLE]

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

[TABLE]

In this case, the first product in the sum measures whether existing ties in $t-1$ persist in $t$ and the second term is one if non-existent ties in $t-1$ remain non-existent in $t$. The proportionality sign is used since in many cases the network statistics are scaled into a specific interval (e.g. $[0,n]$ or $[0,1]$). 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 covariates $x_{ij,t}$

[TABLE]

There exists an abundance of possibilities for defining interactions between ties in $t-1$ and $t$. From this discussion and equation (2) it also becomes evident, that in a situation where the interest lies in the transition between two periods, a TERGM can be modeled simply as an ERGM, including lagged network statistics. This can be done for example by incorporating $y_{ij,t-1}$ as explanatory variable in (5) which is mathematically equivalent to the stability statistic (4). In the application we call this statistic repetition (Block et al., 2018).

Concerning the estimation of the model, maximum likelihood estimation 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 $\kappa(\theta,y_{t-1},x_{t})=\sum_{\tilde{y}\in\mathcal{Y}}\exp\{\theta^{T}s(\tilde{y}_{t},y_{t-1},x_{t})\}$, that requires summation over all possible networks $\tilde{y}\in\mathcal{Y}$. 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). The article by Caimo and Friel (2011) provides an alternative algorithm that uses MCMC-based inference in a Bayesian model framework. Another approach is to employ maximum pseudolikelihood estimation (MPLE, Strauss and Ikeda, 1990) that can be viewed as a local alternative to the likelihood (van Duijn et al., 2009) but is often regarded as unreliable and poorly understood in the literature (Hunter et al., 2008, Handcock et al., 2003). However, the MPLE is asymptotically consistent (Desmarais and Cranmer, 2012) and the often suspect standard errors can be corrected via bootstrap (Leifeld et al., 2018). A notable special case arises if the network statistics are restricted such that they decompose to

[TABLE]

with $\tilde{s}_{ij}$ being a function that is evaluated only at the lagged network $y_{t-1}$ and covariates $x_{t}$ for tie $(i,j)$. With this restriction, we impose that the ties in $t$ are independent, conditional on the network structures in $t-1$. This greatly simplifies the estimation procedure and allows to fit the model as a logistic regression model (see for example Almquist and Butts, 2014) without the issues related to the MPLE.

A problem, that is very often encountered when fitting (T)ERGMs with endogenous network statistics is called degeneracy (Schweinberger, 2011) and occurs if most of the probability mass is attributed to network realizations that provide either full or empty networks. One way to circumvent this problems is the inclusion of modified statistics, called geometrically weighted statistics (Snijders et al., 2006). Using the definitions of Hunter (2007), the geometrically weighted out-degree distribution (GWOD) controls for the out-degree distribution with one statistic, via

[TABLE]

with $O_{k}(y_{t})$ being the number of nodes with out-degree $k$ in $t$ and $\alpha_{O}$ as the weighting parameter. Correspondingly, the in-degree distribution is captured by the geometrically weighted in-degree distribution (GWID) statistic by exchanging $O_{k}(y_{t})$ with $I_{k}(y_{t})$, which counts the number of nodes with in-degree $k$, and $\alpha_{O}$ with $\alpha_{I}$. While on the one hand, the weighting often effectively counteracts the problem of degeneracy, the statistics become more complicated to interpret. Negative values of the associated parameter typically indicate a centralized network structure.

Regarding statistics capturing clustering, the most common geometrically weighted triangular structure is called geometrically weighted edge-wise shared partners (GWESP) and builds on the number of two-paths that indirectly connect two nodes $i$ and $j$ given the presence of an edge $(i,j)$:

[TABLE]

where $\alpha_{S}$ is a weighting parameter. The number of edges with $k$ shared partners ($S_{k}(y_{t})$) is uniquely defined in undirected networks. If the edges are directed it must be decided, which combination should form a triangle, see Lusher et al. (2012) for a discussion. As a default, the number of directed two-paths is chosen (Goodreau et al., 2009). Generally, a positive coefficient for GWESP indicates that triadic closure increases the probability of edge occurrence and globally a positive value for the associated parameter means more triadic closure as compared to a regime with a negative value (Morris et al., 2008).

3.2 Separable Temporal Exponential Random Graph Model

An useful improvement of the TERGM (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. $y_{ij,t}=y_{ij,t-1}=1$) or because no new ties are formed (i.e. $y_{ij,t}=y_{ij,t-1}=0$). To disentangle this, the authors propose a model that allows for the separation of formation and dissolution.

Krivitsky and Handcock (2014) define the formation network as $Y^{+}=Y_{t}\cup Y_{t-1}$, being the network that consists of the initial network $Y_{t-1}$ together with all ties that are newly added in $t$. The dissolution network is given by $Y^{-}=Y_{t}\cap Y_{t-1}$ and contains exclusively ties that are present in $t$ and $t-1$. Given the network in $t-1$ together with the formation and the dissolution network we can then uniquely reconstruct the network in $t$, since $Y_{t}=Y^{+}\backslash(Y_{t-1}\backslash Y^{-})=Y^{-}\cup(Y^{+}\backslash Y_{t-1})$. Define $\theta=(\theta^{+},\theta^{-})$ 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 $\theta$ is the product of the parameter spaces of $\theta^{+}$ and $\theta^{-}$ together with conditional independence of formation and dissolution given the network in $t-1$:

[TABLE]

The structure of the model is visualized in Figure 4. On the left hand side the state of the network $Y_{t-1}$ is given, consisting of two ties $(i,h)$ and $(h,j)$. In the top network all ties that could possibly be formed are shown in dashed and the actual formation in this example $(i,j)$ is shown in solid. On the bottom, the two ties that could possibly be dissolved are shown and in this example ($h,j$) persists while $(i,j)$ is dissolved. On the right hand side of Figure 4 the resulting network at time point $t$ is displayed. Given this structure and the separability assumption (3.2), it is assumed that the formation model is given by

[TABLE]

with $\kappa(\theta^{+},y_{t-1},x_{t})$ being the normalization constant. Accordingly, the dissolution model can be defined. Inserting the separable models in (3.2) makes it apparent, that the STERGM is a subclass of the TERGM since

[TABLE]

with $\theta=(\theta^{+},\theta^{-})^{T}$, $s(y_{t},y_{t-1},x_{t})=(s(y^{+},y_{t-1},x_{t}),s(y^{-},y_{t-1},x_{t}))^{T}$ and the normalization constant set accordingly.

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.

The network statistics are used similarly as in a cross-sectional ERGM. In Krivitsky and Handcock (2014) they are called implicitly dynamic because they are evaluated either at the formation network $y^{+}$ or the dissolution network $y^{-}$, which are both formed from $y_{t-1}$ and $y_{t}$. For example, the number of edges is separately computed now for the formation and the dissolution network, giving either the number of edges that newly formed or the number of edges that persisted. For example, reciprocity in the formation network is defined as

[TABLE]

and in case of the dissolution model, $y^{+}$ is simply exchanged with $y^{-}$. Similarly, and edge covariates or the geometrically weighted statistics shown in equations (5), (7) and (8) are now functions of $y^{+}$ or $y^{-}$ not $y_{t}$.

3.3 Model Assessment

In analogy to binary regressions models, the (S)TERGM can be evaluated in terms of their receiver-operator-

characteristic (ROC) curve or precision-recall (PR), where the latter puts more emphasis on finding true positives (e.g. Grau et al., 2015). A comparison between different models is possible using, for example, the Akaike Information Criterion (AIC, Claeskens and Hjort, 2008). Here we want to highlight, that the AIC fundamentally builds on the log likelihood, which in most realistic applications is only available as an approximation, see Hunter et al. (2008) for further discussion.

However, in statistical network analysis it is often argued that suitable network models should not exclusively provide good predictions for individual edges, but also be able to represent topologies of the observed network. The dominant approach to asses the goodness-of-fit of (S)TERGMs is based on sampling networks from their distribution under the estimated parameters and then comparing network characteristics of these sampled networks with the same ones from the observed network (Hunter et al., 2008). For this approach, it is recommendable to utilize network characteristics that are not used for specifying the model. For instance, models that include the GWOD statistic (7) may not be compared to the simulated values of it but against the out-degree distribution.

Hanneke et al. (2010) point out that for networks with more than one transition from $t-1$ to $t$ available it is possible to employ a ”cross-validation-type” assessment of the fit. The parameters can be fit repetitively to all observed transitions except one hold-out transition. It is then checked, how well the network statistics from the hold-out transition period are represented by the ones sampled from the coefficients obtained from all other transitions.

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, changes in the network are not occurring at discrete time points but as a continuously evolving process, where only one tie can be toggled 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. Correspondingly, the observations in this section are behaviors which are given as triplets $\omega=(i,j,\tilde{t})$ and encode the sender $i$, receiver $j$, and exact time point $\tilde{t}$. This fine-grained temporal information is often called time-stamped or time-continuous, we adopt the latter name. Furthermore, we only regard dyadic events in this article, i.e., a behavior only includes one sender and receiver.

The concept of behavior, hereinafter called event, generalizes the classical concept of binary relationships based on graph theory as promoted by Wasserman and Faust (1994). This event framework does not intrinsically assume that ties are enduring over a specific time frame (Butts, 2009; Butts and Marcum, 2017). For example in an email exchange network, sending one email at a specific time point is merely a brief event, which does not convey the same information as a durable relationship. Therefore, the time-stamped information cannot adequately be represented in a binary adjacency matrix without having to aggregate the relational data at the cost of information loss (Stadtfeld, 2012). Nevertheless, a friendship between actor $i$ and $j$ at a given time point can still be viewed as an event that has an one-to-one analogy to a tie in the classical framework.

The overall aim of Relational Event Models (REM, Butts, 2008) is to understand the dynamic structure of events conditional on the history of events (Lerner et al., 2013). This dynamic structure, in turn, controls how past interactions shape the propensity of future events. To make this model feasible, 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). The central concept of the REM can be motivated by the introduction of a multivariate time-continuous Poisson counting process

[TABLE]

where $N_{ij}(\tilde{t})$ counts how often actors $i$ and $j$ interacted in $[0,\tilde{t})$. Note that we indicate continuous time $\tilde{t}$ with a tilde to distinguish from the discrete time setting with $t=1,2,\ldots,T$ assumed in the previous section. Process (12) is characterized by an intensity function $\lambda_{ij}(\tilde{t})$ for $i\neq j$, which is defined as:

[TABLE]

This is the instantaneous probability of observing a jump of size ”1” in $N_{ij}(\tilde{t})$, which indicates observing the event $(i,j,\tilde{t})$. Since we assume that there are no self-loops $\lambda_{ii}(\tilde{t})\equiv 0~{}\forall~{}i=1,\ldots,n$ holds.

4.2 Time-Continuous Observations

Butts (2008) introduced the REM to analyze the intensity $\lambda_{ij}(\tilde{t})$ of process (12) when time-continuous 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):

[TABLE]

where $\lambda_{0}(\tilde{t})$ is an arbitrary baseline intensity, $\theta\in\mathbb{R}^{p}$ the parameter vector and s_{ij}\big{(}N(\tilde{t}),x(\tilde{t})\big{)} a statistic that depends on the (possibly time-continuous) covariate process $x(\tilde{t})$ and the counting process just prior to $\tilde{t}$.

Generally, similar statistics as already introduced in Section 3 can be included in

s_{ij}\big{(}N(\tilde{t}),x(\tilde{t})\big{)}. Solely the differing level of the model needs to be accounted for, since model (13) takes a local time-continuous point of view to understand the relational nature of the observed events. This necessitates defining the statistics s_{ij}\big{(}N(\tilde{t}),x(\tilde{t})\big{)} from the position of specific ties, in contrast to the globally defined statistics $s(y_{t},y_{t-1},x_{t})$ in (2). To give an example, the tie-level version of reciprocity for the event $(i,j)$ is defined as

[TABLE]

where $\mathbb{I}(\cdot)$ is the indicator function. It only regards, whether already having observed the event $(j,i)$ prior to $\tilde{t}$ has an effect on $\lambda_{ij}(\tilde{t}\mid N(\tilde{t}),x(\tilde{t}),\theta)$, in comparison to the network level version (3) of delayed reciprocity that counted all reciprocated ties between the networks $y_{t}$ and $y_{t-1}$.

Degree statistics can be specified as either sender- or receiver-specific. If we, e.g., want to control for the out-degree of the sender the corresponding tie-oriented statistic is:

[TABLE]

The in-degree of the receiver can be formulated accordingly.

Clustering in event sequences may be captured by different types of nested two-path configurations. For instance, the tie-oriented version of directed two-paths, henceforth called transitivity, is given by:

[TABLE]

The inclusion of monadic and dyadic exogenous covariates becomes straightforward by setting s_{ij,dyadic}\big{(}N(\tilde{t}),x(\tilde{t})\big{)} equal to the covariate values of interest. Since the effect of a past event at time $\delta$, say, on a present event at time $\tilde{t}$ may vary according to the elapsed time $\tilde{t}-\delta$, 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.

If time-continuous observations are available each dimension of the observed counting process is conditional on the past independent. This, in turn, enables the construction of a likelihood, which can subsequently be maximized. Assuming that $\Omega$ is the set of all observed events and $\mathcal{T}$ the interval of observation, the likelihood can be written as:

[TABLE]

This likelihood is straightforward to maximize in the case of a parametric baseline intensity $\lambda_{0}(t)$, for example Butts (2008) assumes $\lambda_{0}(t)=\gamma_{0}$. Alternatively, Butts (2008) analyzed events with ordinal temporal information. In this setting, the likelihood is equal to the partial likelihood introduced by Cox (1972) for estimating parameters of semi-parametric intensities as in (13). Letting $\mathit{U}_{t}$ denote the set of all possible events that could have occurred at time point $t$ but did not, the partial likelihood for continuous event data is defined as:

[TABLE]

Consecutively, $\Lambda_{0}(t)=\int_{0}^{t}\lambda_{0}(u)du$ can be estimated with a Nelson Aalen estimator (see Kalbfleisch and Prentice, 2002 for further details on the estimation).

When dealing with large amounts of event data the main obstacle is evaluating the sum over the intensities of all possible ties in (15) (Butts, 2008). One exact option is to trade a longer running time for a slimmer memory footprint by means of a coaching data structure. Vu et al. (2011) exploit this by saving prior values of the sum and subsequently changing it event-wise by elements of $\mathit{U}_{\tilde{t}}$ whose covariates changed. Alternatively, Vu et al. (2015) proposes approximate routines that utilize case-control sampling and stratification for the Cox model (Langholz and Borgan, 1995). More precisely, the sum is only calculated over a sampled subset of possible events in addition to stratification. Lerner and Lomi (2019) go one step further and sample events out of $\Omega$ for the calculation of $\mathcal{PL}(\theta)$ in (15).

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 (13). 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-continuous network data, meaning that we observe the precise time points of all events. To give an instance, in the first data example, this means that we need the exact time point $\tilde{t}$ of an arms trade between country $i$ and $j$. Often, such exact time-stamped data are not available and, in fact, trading between states can hardly be stamped with a single time point $\tilde{t}$. Indeed, we often only observe the time-continuous network process at discrete time points $t=1,\ldots,T$. In such setting, we may assume a Markov structure in that we do not look at the entire history of the process $N(\tilde{t})$ but just condition the intensity (13) on the history of events from the previous observation $t-1$ to $\tilde{t}$. Technically this means that $N(t)$ is adapted to $\tilde{Y}(\tilde{t}):=N(\tilde{t})-N(t-1)$ and $x(\tilde{t})$ for $\tilde{t}\in[t-1,t]$. We then reframe (13) as:

[TABLE]

In other words, we assume that the intensity of events between $t-1$ and $t$ does not depend on states of the multivariate counting process (12) prior to $t-1$. For this reason, all endogenous statistics introduced in Section 4.2 are now evaluated on $\tilde{Y}(\tilde{t})$ instead of $N(\tilde{t})$. This is a reasonable assumption, if one is primarily interested in short-term dependencies between the individual counting processes. It enables a meaningful comparison to the models from Section 3 that assume an analog discrete Markov property. However, we want to emphasize that this dependence structure is not vital to inferential results.

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. Under the term tied observations this phenomenon is well known in time-to-event analysis and treated with several approximations. One option is the so-called Breslow approximation (see Peto, 1972; Breslow, 1974). Let therefore

[TABLE]

where element $(i,j)$ is replicated $N_{ij}(t)-N_{ij}(t-1)$ times in $\mathit{O}_{t}$, that is if an event between $i$ and $j$ occurred multiple times in the interval from $t-1$ to $t$ then $(i,j)$ appears respective times in $\mathit{O}_{t}$. Given that we have not observed the exact time point of an event we also get no information on the baseline intensity $\lambda_{0}(\tilde{t})$ in (13) for $\tilde{t}\in[t-1,t]$ so that the model simplifies to a discrete choice model structure (see, e.g., Train, 2009) which resembles the partial likelihood (15) and is defined as:

[TABLE]

where $n_{t}=|\mathit{O}_{t}|$. Alternatively, one can replace the denominator in (17) by considering all possible orders of the unobserved events in $\mathit{O}_{t}$ giving the average likelihood as introduced by Kalbfleisch and Prentice (2002). Since this can be a combinatorial and hence numerical challenge, random sampling of time point orders among the time-clustered observations can be used with subsequent averaging, which we call Kalbfleisch-Prentice approximation (see Kalbfleisch and Prentice, 2002). Further techniques to deal with unknown time ordering are augmenting the clustered events into possible paths of ordered events and adapting the maximum likelihood estimation proposed for the SAOM by Snijders et al. (2010) or using random sampling of the ordering. This can be legitimized in cases where we may assume independence among events happening in one year, since the events take a long time to materialize (Snijders, 2017).

4.4 Model Assessment

In comparison to the assessment for models operating in discrete time, widely accepted methods dealing with relational event data are scarce. The proposals either stem from time-to-event analysis or regard link prediction, which is the task of predicting the most likely next event given the history of past events (Liben-Nowell and Kleinberg, 2007). One example of the former option is the usage of Schönfeld residuals by Vu et al. (2017) to check the assumption of proportional intensities, which is central to semi-parametric models as the one proposed by Cox (1972). For the latter approach, we need to define a predictive measure that quantifies how well the next event is predicted. Vu et al. (2011) proposed the recall measure that estimates the percentage of test events which are in the list of $K$ most likely next events according to a given model. Evaluating this percentage for different values of $K$ permits a visualization of the predictive capabilities of the model. The strength of the predicted intensity allows the ordering of events according to the probability of being observed next. If we model the propensity of time-clustered events that represent binary adjacency matrices one can alternatively adopt the analysis of the ROC and PR curve introduced in Section 3.3.

5 Application

When it comes to software, there exist essentially three 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. The library contains the package tergm with implemented methods for fitting STERGMs using MCMC approximations of the 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 using either maximum pseudo-likelihood or MCMC maximum likelihood estimation routines. In order to obtain Bayesian Inference in ERGMs, the package bergm by Caimo and Friel (2014) can be used. Besides implementations in R, the stand-alone program PNet (Wang et al., 2006) allows for simulating, fitting and evaluating (T)ERGMs. In order to ensure comparable estimates we estimate the TERGM, as well as the STERGM, with the statnet library, using MCMC-based likelihood estimation techniques. We use the package ergm and include delayed reciprocity and the repetition of previous ties as dyadic covariates. The STERGM is fitted using the tergm package.

Marcum and Butts (2015) implemented the R package $\mathtt{relevent}$ (version 1.0-4) to estimate the REM for time-stamped data. It was followed by the package $\mathtt{goldfish}$ (version 1.2) by Stadtfeld and Hollway (2018) for modeling event data with precise and ordinal temporal information with an actor- and tie-oriented variant of the REM. Furthermore, it is highly customizable in terms of endogenous and exogenous user terms and will be used in the following applications.

We want to remark that the STERGM coefficients are implicitly dynamic, while in the TERGM all network statistics except the lagged network and delayed reciprocity terms are evaluated on the network in $t$. All covariates of the REM are continuously updated and the intensity at time point $\tilde{t}\in[t-1,t]$ only depends on events observed in $[t-1,\tilde{t})$. Like the building period proposed by Vu et al. (2011), the events in $t-1$ are only used for building up the covariates and not directly modeled. Due to no compositional changes, we did not scale any statistics. Moreover, we refer to the Supplementary Material for the model assessment.

5.1 Data Set 1: International Arms Trade

The results obtained for the arms trading data 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 (e.g. Thurner et al., 2018). Here we want to comment on a few aspects only. While we do not have timestamps for the arms trades, the longitudinal networks can still be viewed as time-clustered observations enabling the techniques from Section 4.3.

Both, the TERGM (column 1) and the REM (column 4) identify the repetition of previous ties as a driving force in the dynamic structure of the network. Degree-related covariates, which are GWID and GWOD in the (S)TERGM and in- and out-degree in the REM, capture centrality in the network. The coefficients of the GWID and GWOD are negative and have low $p$-values in the TERGM. This stands in contrast to the STERGM, where these effects are only pronounced in the formation model (column 2), while they are insignificant effect in the dissolution model (column 3). Hence, these effects suggest a centralized pattern in the formation network, which is also captured by the TERGM. In the REM an analogous pattern can be detected, since a higher in-degree of the receiver increases the respective intensity, thus spurs trade relations. Similar interpretations hold for the out-degree of the sender. Overall, countries that have a high out-degree are more likely to send weapons and countries with a high in-degree to receive weapons, which again results in a centralized network structure as indicated by the estimates in the TERGM and STERGM.

Lastly, consistent effects among the models were also found for the exogenous covariates. Consider, for instance, the coefficient of the logarithmic GDP of the importing country. The TERGM assigns a significantly higher probability to observe in-going ties to countries with a high GDP just like the REM. However, disentangling the model towards formation and dissolution we see strongly significant coefficients in the dissolution model while the effect for the formation model is weakly significant.

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.

5.2 Data Set 2: European Research Institution Email Correspondence

As already indicated by the descriptive statistics in Table 1, the email network seems to be driven by three major structural influences: repetition, reciprocity, and transitive clustering. The estimates from Table 3 demonstrate, that all models were able to identify these forces.

According to the REM (column 4), the event network of email traffic in the research institution is not centralized and primarily based on collaboration between coworkers. We can draw those conclusions from insignificant estimates of degree-related statistics and highly significant estimates regarding reciprocity and repetition. In the TERGM (column 1) we find a positive and significant effect of GWID, while no effect can be found in the STERGM (columns 2 and 3). The estimates of repetition and reciprocity in the REM and TERGM are very pronounced. For instance, the estimates of the REM imply that a reciprocated event is 19.6 times more likely than an event with the same covariates only not being reciprocated. Interestingly, the STERGM detects a lower effect of GWESP in the formation and dissolution than the TERGM. The effect of the delayed reciprocity in the TERGM is less relevant than reciprocity in the formation and dissolution model. This strongly differing effect size results from the mathematical formulation of the statistics given in equations (3) and (11).

Contrasting the AIC values of the TERGM and STERGM shows that the dynamic structure of the email network is again better explained by the STERGM. In the Supplementary Material we fit the TERGM and STERGM to multiple time points.

6 Conclusion

6.1 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). Estimation of this model variant is predominantly carried out with the R package $\mathtt{RSiena}$ (Ripley et al., 2013).

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 to the SAOM, the model builds on randomly assigning the opportunity to change, followed by a function that governs the probability of a tie change. This model is still tie-oriented, meaning that dyadic ties instead of actors are chosen and then have the option to change the current network.

6.2 Summary

In this article, we put emphasis on tie-oriented dynamic network models. Comparisons between these models can be drawn on the level at which each implied generating mechanism works and how time is perceived. The overall aim in the TERGM is to find an adequate distribution of the adjacency matrix $Y_{t}$ conditioning on information of previous realizations of the network. In the separable extension, the aim remains unchanged, only splitting $Y_{t}$ into two smaller sub-networks that include all possible ties that were and were not present in $Y_{t-1}$ separately. While the (S)TERGM proceeds in discrete time, the REM tackles modeling the intensity on the tie level in continuous time conditional on past events. Therefore, the TERGM and STERGM take a global and REM a local point-of-view, which results in substantially different interpretations of the estimates.

Furthermore, we analyzed two data sets that represent two types of network data that are traditionally either modeled by the TERGM and REM. By extending the REM to time-clustered observations and aggregating events to binary adjacency matrices a meaningful comparison between the STERGM, TERGM, and REM is enabled.

Acknowledgement

We thank the anonymous reviewers for their careful reading and constructive comments. 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.

Appendix A Annex: Additional Descriptives

Figures 5 and 6 depict the distributions of in- and out-degrees in the two networks. Building on in- and out-degree of all nodes, these distributions represent the relative frequency of all possible in- and out-degrees in the observed networks, which is calculated with the $\mathtt{igraph}$ package in R (Csardi and Nepusz, 2006).

In the arms trade network, 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.

The email exchange network shows a different structure. Here, many medium-sized in-degrees can be found and only roughly $10\%$ of all nodes have receiver no emails. For the out-degree, this number doubles (roughly 20% have not sent emails). Further, the distribution of the out-degree is more skewed then the one for the in-degree.

Appendix B Supplementary Material

B.1 Countries in the International Arms Trade Network

B.2 Simulation-based Goodness-of-fit in (S)TERGMs

In Figures 7 and 10 we show simulation-based godness-of-fit (GOF) diagnostics for the the TERGM model and in Figures 8, 9, 11 and 12 for the STERGM in the formation and dissolution model, respectively. The figures are created by the R package ergm (version 3.10.4) and follow the approach of Hunter et al. (2008). In all three models, the fitted model is used in order to simulate 100 new networks. Based on these, different network characteristics are computed and visualized in boxplots.

The standard characteristics used are the complete distributions of the in-degree, out-degree, edge-wise shared partners and minimum geodesic distance (i.e. number of node pairs with shortest path of length $k$ between them). The solid black line indicates the measurements of these characteristic in the observed network. These statistics show whether measures like GWID, GWOD and GWESP are sufficient to reproduce global network patterns. Because many shares are rather small, we visualize the simulated and observed measures on a log-odds scale.

On the bottom of the figures it is shown how well the actual network statistics are reproduced. Note, that both models compare different things as the TERGM is evaluated at $y_{t}$ while the STERGM regards $y^{+}$ and $y^{-}$. Overall, all plots indicate a satisfying fit of the respective models.

B.2.1 Data Set 1: International Arms Trade

B.2.2 Data Set 2: European Research Institution Email Correspondence

B.3 ROC-based Goodness-of-fit

B.3.1 Data Set 1: International Arms Trade

As already stated in Section 3.3 of the main paper techniques for assessing the fit of a probabilistic classification can be used when working with binary network data. In the case of observations at discrete time points this allows an informal comparison of the models proposed in Section 3 and 4.3.

In the case of the TERGM and STERGM the application of the ROC- and PR-curve follows from the conditional probability of observing a specific tie (see equation (2.5) of Hunter and Handcock, 2006). For the REM we predict the intensities of all possible events given the information of $t-1$ and use this value as a score in the calculation of the ROC curve. While the latter approach is non-standard and can only be applied to REMs that regard durable ties, it enables a direct comparison between the models as shown in Figure 13. The results of the ROC curve indicate a generally good fit of all models. In the STERGM more parameters are estimated, which seems to lead to a slightly bigger area under curve (AUC) values as compared to the REM and TERGM. Similar to the conclusions from the ROC curve, the PR curve favors the TERGM and STERGM over the REM.

B.3.2 Data Set 2: European Research Institution Email Correspondence

The second data set regards reoccurring events in the REM, which are aggregated for the analysis of the TERGM and STERGM. Therefore, the ROC and PR curve are only available for the TERGM and STERGM. The results are depicted in Figure 14. For this data set, the ROC curves favor the TERGM. Yet, when emphasis is put on finding the true positives, the PR curve detects a better model fit of the STERGM.

As explained in Section 4.4 one option to asses the goodness-of-fit of REMs is the recall measure as proposed by Vu et al. (2011). We apply the measure in three different situations that may be of interest when measuring the predictive performance of relational event models: predict the next tie, next sender, and next receiver. The worst case scenario in terms of predictions of a model would be random guessing of the next sender, receiver, or event, the resulting recall rates are indicated by the dotted lines. The results in Figure 15 exhibit a good predictive performance of the REM, i.e. in about 75$\%$ of the events the right sender and receiver is among the 25 most likely senders and receivers.

B.4 Application with multiple time points

In the main article, we fitted a TERGM as well as a STERGM to two time points, called period 1 and period 2. However, it is possible to fit these models to multiple transitions. In order to do so, we took the first two years of the email exchange network and aggregated the deciles into 10 binary networks. Using again a first order Markov assumption and conditioning on the first network, this allows to fit a TERGM as well as a STERGM to the remaining nine networks. For comparison we additionally fit a REM to the data set. The estimation is done using the function mtergm from the package btergm (version 3.6.1) (Leifeld et al., 2018) that implements MCMC-based maximum likelihood. The STERGM is fitted again using the package tergm (version 3.5.2).

The corresponding results can be found in Table 5 in column 1 to 3. Note that the parameter estimates still refer to the transition from $t-1$ to $t$ and can interpreted in the same way as in the main article. In that regard note, that it is now assumed that the coefficients stay constant with time. Possible approaches to relax this assumption were given in the main article. The estimates of the REM (column 4) are consistent with the (S)TERGM but slightly differ to the main article, since now we condition only on the first out of 10 periods and hence model more events.