Rejoinder: DyNAMs and the Grounds for Actor-oriented Network Event Models

Abstract

We thank the editorial board for the opportunity to engage in a public dialogue about our work and the commentators for providing interesting and challenging comments. We are especially pleased that Tom Snijders and Carter Butts took the time to comment on our article, because the dynamic network actor model (DyNAM) builds upon the work of both authors. Moreover, their contrasting comments highlight important distinctions between actor-oriented and tie-based network models as well as in methodological thinking more generally. We take this opportunity to clarify some misconceptions and further develop a lively discussion within our research community.

First, we identify what we perceive as different ideas about the aim of statistical network modeling. Second, we outline our conclusions about the fundamental difference between actor- and tie-oriented models for relational events. Third, we respond to specific comments presented by Carter Butts (this volume, pp. 47–56) and Tom Snijders (this volume, pp. 41–47) in separate sections. Finally, we discuss and emphasize the role of actor-oriented network models, like the DyNAM, in the portfolio of statistical network models.

1. The Aim of Statistical Network Models

We believe that the comments of Butts and Snijders reflect different views of the purposes of statistical (network) modeling that are difficult to reconcile. In this section, we illustrate these two positions and emphasize our belief that it can be highly beneficial to consider the nature of social processes when developing statistical models; we have done this when we developed DyNAMs for coordination networks.

On one hand, Butts states that “general modeling frameworks often can, and should, support multiple phenomenological interpretations [italics in original]” (this volume, p. 53) and that “we should then mobilize whatever representation [of a modeling framework] provides the greatest advantage or insight” (this volume, pp. 53–54). This suggests that statistical models should be generic, theory-free tools that support a very broad range of applications by researchers. He argues in his commentary that the relational event model (REM) is such a framework because it can, but need not, include an interpretation of agency. In this view, the job of a researcher would be to use these generic modeling frameworks and concentrate theory into how estimated parameters are interpreted.

On the other hand, the development by Snijders of the stochastic actor-oriented model (SAOM) sought the “integration of theoretical and statistical model[s]” to facilitate “more stringent theory development because it requires a completely explicit theory and provides a much more direct test of the theory” (Snijders 1996:149). Snijders echoes this alignment of statistical and theoretical models with the following comment: “Determining which model best represents the coordination between the two actors in creating a tie will depend on the social situation that is defined by the network dynamics, and this choice of model will influence the parameter interpretation” (this volume, p. 43). In this view, a researcher should choose (or develop) a statistical model that closely reflects the assumed underlying process, with parameter interpretation tracking more closely the model formulation.

We agree with the former position inasmuch as it is hardly useful to develop a new statistical model for every theoretical or empirical case. A contribution to statistical network methodology ought to be applicable to a broad range of issues. The DyNAM model for coordination ties we have proposed here does this, and we provide broad examples from different social science disciplines in which networks are the outcomes of an agreement process between actors. But we do not agree with the notion that broadly applicable tools need to be generic in how they reflect individual behavior and social processes. Our understanding is in line with the latter position that statistical models become more useful for studying particular contexts when they more closely approximate those real-world processes. It is for this reason that we have developed an actor-oriented model for time-stamped coordination data. Where researchers are interested in agency—often the case in social networks and, indeed, social science—actor-oriented models often bring the statistical model closer to social theory.

2. Differences between Actor- and Tie-oriented Event Models

Another distinction discussed in the comments was the one that occurs between actor- and tie-oriented models. It is a common misconception that a tie-oriented network model can provide exactly the same insights as an actor-oriented network model and that parameters can be interpreted the same way. Here, we argue that the distinction between actor- and tie-oriented network models is not mainly a distinction on the level of model interpretation but first and foremost a mathematical one.¹

The distinction lies in how each model explains the observation of events. Whereas tie-oriented network models express competing propensities of relational events between pairs of actors as a one-step process, actor-oriented network models explain events as a two-step process of (1) sender activity and (2) receiver choice.²Figure 1 (taken from Stadtfeld and Block 2017) shows this conceptually.

Figure 1.

The difference between tie- and actor-oriented models on the micro level.

We show these differences mathematically using similar notation. Both models can be understood as a Markovian process in which a rate $λ$ determines the probabilities of events between actors $i$ and $j$ from actor set $A$ to emerge through time, given the process state $x$ (consisting of the state of networks, past events, and attributes), statistic functions $s (\cdot)$ , $t (\cdot)$ , $u (\cdot)$ (representing, for example, whether an event occurs within transitive structures), and weighting parameter vectors $θ$ , $β$ , $γ$ ; details on the notation are provided in Butts (2008), Stadtfeld (2012), Stadtfeld and Block (2017), and Stadtfeld, Hollway, and Block (this volume, pp. 1–40).

λ_{ij}^{Tie} (x, γ, u, A) = \exp (γ^{T} u (x, i, j))

\begin{matrix} λ_{ij}^{Actor} (x, θ, β, s, t, A) = \underset{step 1 : actor rate}{\underset{︸}{\exp (θ^{T} s (x, i))}} \frac{\exp (β^{T} t (x, i, j))}{\underset{step 2 : receiver choice}{\underset{︸}{\sum_{k \in A} \exp (β^{T} t (x, i, k))}}} \end{matrix}

Note that although the tie-oriented model in equation (1) consists of just one step, determining the general propensity of an event to be observed, the actor-oriented model in equation (2) consists of two steps: the first determines the activity rate of actor i and the second the conditional probability of i choosing j as the event receiver. This separates actor-oriented models into those parameters that govern an actor’s propensity to send events (modeled in the actor rate) and those by which they choose event receivers (modeled in the receiver choice).

This separation of rate and evaluation distinguishes actor-oriented models and means that it is not trivially a special case of the tie-oriented model. The models are equivalent if and only if the actor rate of step 1 is the same as the multinomial choice denominator of step 2 in equation (2) (Snijders 2001:388).

The rate of an actor in such a model would thus be proportional to the attractiveness of the available receiver choice options in the denominator—a strange assumption.

Expanding on the above equations further reveals this false equivalence between the models. Butts presents the probability of an actor i to form a tie with j rather than any other actor k in the tie-oriented model conditioning on i being the next actor to form a tie as

P_{i \to j}^{Tie} (x, γ, A | i active) = \frac{\exp (γ^{T} u (x, i, j))}{\sum_{k \in A \ {i}} \exp (γ^{T} u (x, i, k))}

He correctly points out that this is the same as the conditional probability of a specific event receiver choice j in the actor-oriented model (the probability of step 2) when $γ$ , $β$ , u and t are chosen equivalently:

P_{i \to j}^{Actor} (x, β, A | i active) = \frac{\exp (β^{T} t (x, i, j))}{\sum_{k \in A \ {i}} \exp (β^{T} t (x, i, k))}

However, although the choice of j conditioning on i’s activity is equivalent, the probability P_i under which actor i is the next to send an event is fundamentally different in the two models. In the actor-oriented DyNAM model, the probability directly follows from comparing the actor rates (step 1); in the tie-oriented REM, it follows from the aggregation and comparison of all tie rates in the network:

P_{i}^{Actor} (x, θ, s, A) = \frac{\exp (θ^{T} s (x, i,))}{\sum_{k \in A} \exp (θ^{T} s (x, k))}

P_{i}^{Tie} (x, γ, u, A) = \frac{\sum_{j \in A} \exp (γ^{T} u (x, i, j))}{\sum_{k, l \in A} \exp (γ^{T} u (x, k, l))}

Note that the tie-oriented actor activity probability $P_{i}^{Tie}$ depends on exactly the same parameters $γ$ and statistic functions $u (\cdot)$ as the multinomial choice probabilities $P_{i \to j}^{Tie}$ in equation (3). Rates in the actor-oriented model depend on a different parameter vector $θ$ and statistic functions $t (\cdot)$ —instead of $β$ and $t (\cdot)$ in equation (4)—and can thus be specified differently.

This flexibility may enable different parameters to be included in the rate (step 1) and choice function (step 2) and thus the development and testing of more refined social theory.³ Even where this flexibility is not exploited and symmetric specifications are chosen for the rate and evaluation functions of actor-oriented models, parameter estimates may emerge as quite different, allowing researchers to recognize, say, that what makes actors “hurry” or “foot-drag” in general is not necessarily the same as what makes actors decide to whom to send an event given their available choice options.

The argument offered by Butts that both tie- and actor-oriented network event models can be reduced to multinomial models (McFadden 1974) ignores these important differences. Actor-oriented models identify “choice” as the second in a two-step process. And although the statement by Butts that “standard REMs can be interpreted in terms of decision processes” (this volume, p. 51) is not incorrect, it misses the point. REM parameters ultimately do more than just guide receiver choice. The actor-choice interpretation of the tie-based REM follows only with some elaborated conditioning and is always confounded with the simultaneous interpretation of parameters relating to actor activity. Thus the meaningful mathematical and conceptual difference between the two classes that defines the tie-oriented model as a one-step process and the actor-oriented model as a two-step process⁴ must be reflected in parameter interpretation and is consequential for researchers.

3. Model Interpretation in Light of a Theoretical Framework

In this section we respond with clarifications to two specific comments from Butts that concern how differences between actor- and tie-oriented network models are reflected in (choice-theoretic) interpretations and provide examples where an actor-oriented perspective is more in line with assumptions about the nature of social processes.

First, Butts argues that the tie-oriented REM can be reduced to a multinomial choice model as shown in equation (3) when one conditions “on no other event occurring before i acts” (this volume, p. 50). It is technically correct that the multinomial choice transformation can be achieved by “simply [conditioning] on the (statistical) event that all competing events not involving ego occur after ego’s next action.” However, this is not such a simple assumption. As we showed in equations (5) and (6), the probabilities on which this conditioning is based are fundamentally different. Butts acknowledges this: “This useful feature of REM processes actually goes beyond multinomial choice, in that it provides a language for specifying not only actors’ relative preferences among options but also their propensity to pursue those options” (this volume, p. 51). This is a consequential admission. Importantly, this conflation of sender activity and receiver choice must always be the case with the REM and it is not possible, as it is with the DyNAM, to specify parameters that affect only receiver choices, not relative actor activities.

Consider the example of states’ bilateral fisheries agreements (Hollway and Koskinen 2016a, 2016b; Stadtfeld et al. 2017), and assume that states strive for bilateral agreements embedded in triadic structures. If a small state forms a treaty with a high-degree node (e.g., the European Union), this connection will open numerous two-paths to countries connected to the European Union, each of which provides an opportunity for triadic closure. Because the REM does not distinguish between opportunity and choice, it would expect this small state to subsequently form more treaties given that more attractive opportunities are available. However, there may be theoretical reasons to expect that although small states may profit from closing these triads, they lack the resources necessary to propose (all) these agreements. Treaty partner choice may be affected by options for transitive closure; the rates, however, might not. Snijders indeed proposes to use this feature of the actor-oriented DyNAM to develop more nuanced social models that distinguish between actors’ activities and their choices by separating out the rate and evaluation functions.

Second, Butts argues from equation (4) that one can derive that the probability of actor i to select an actor j is “a monotone declining function” of $γ^{T} u (x, i, k)$ , where k is any other actor in the choice set, implying that actors weigh recipient choices against one another. This comment was meant to contradict the argument in our article that this comparison of alternatives by actors is an important argument for actor-oriented network models. We make two observations in relation to this. First, the probability of event $i \to j$ in tie-oriented network models is a monotonically declining function of any $γ^{T} u (x, k, l)$ for an event $k \to l$ that would imply that actors weigh their options against everybody’s options and not just their own. Second, this statement again holds only when considering the conditional probability in equation (3) that looks only at i’s next receiver choice. If we use equation (1) as the relevant comparison case, we can see that the tie rate from i to j does not depend on any k.

So can we say that actor-oriented DyNAMs are choice-theoretic models but the tie-based REM is not? We agree with Butts that such a claim (which we never made) is false because it relates only to how models are interpreted. Both models are flexible in their interpretation; one can interpret REM parameters as reflecting underlying choice patterns, and actor-oriented models can be interpreted solely in terms of propensities of particular senders to send ties to particular receivers. Researchers should, however, consider that in the case of tie-oriented models, structural parameters necessarily affect timing and choices simultaneously, which is not the case for actor-oriented models like the DyNAM.

4. Reflections on the DyNAM and Actor-oriented Models

The observations by Snijders invite us to reflect on how the DyNAM might be elaborated and extended as an actor-oriented network model. For example, as Snijders observes, the actor orientation allows us to specify undirected ties as different types of agreement processes. We used a multinomial–multinomial model in Stadtfeld, Hollway, and Block (2017), but a multinomial–binomial option as proposed in Snijders and Pickup (2017) would be easy enough to incorporate where different processes are thought to govern actors’ proposition and acceptance decisions.⁵ A limited strategic perspective in the form of a “one-step ahead expected utility rule” (Snijders, this volume, p. 43) would be an intriguing addition to these options. Although strategic thinking is often thought to be inconsistent with a myopic Markov process, we see no contradiction so long as researchers specify the situations under which such strategic behavior would be expected. Relevant statistics that model the expected reaction can be measured on the process state and incorporated into the model specification. As such, parameters could be interpreted as relating to unobserved strategic decisions similar to how unobserved preferences are inferred from individual choices.

One major challenge for using the DyNAM that we readily admit is that it is demanding of data. Like most statistical network models, it relies on complete (or as near to complete as possible) network data, which can be taxing to collect in many settings. Moreover, as is the case with an event model like the REM, it also requires complete information about the precise start and, where appropriate, end dates of ties. On the one hand, this makes DyNAM a natural fit for settings that publicly record the beginning and end dates of ties, such as those often found in the public archives of political systems. On the other hand, some ties and other variables of interest to researchers may not have unique start and end dates. We faced this problem in our article with some fisheries treaties that were concluded on the same day or due to romantic relationships being grouped by week. Because there were only a few such cases, we treated uncertainty in our sequence-based inference by randomly permuting the ordering of events with the same time stamp. But we expect that similar challenges may appear in other data sets too, and Snijders helpfully suggests model-based and multiple imputation strategies for inferring most likely sequences that can then be used to weight the final results. Snijders also suggests that in some applications dependencies are likely to be realized only when there is a real sequence to the events, so that information can be given its due time to diffuse and operate on actors’ subsequent decisions. Indeed, it may not be very consequential which bilateral fisheries agreement was concluded first among those with the same start date; the conclusion of one is unlikely to instantly affect the conclusion of another when tie formation necessitates some planning time.

5. Conclusions

Our article “Dynamic Network Actor Models: Investigating Coordination Ties through Time” argues that dynamic coordination networks (in which ties are the outcome of agreement processes) can be found across many social science disciplines and proposes a class of statistical models that is tailored to their analysis from an actor-oriented perspective. The DyNAM model we introduce builds upon important prior work. It represents a recombination of elements from the stochastic actor-oriented network models of Snijders (1996) and the REMs for analyzing time-stamped network data of Butts (2008). We are thankful for the opportunity to receive and respond to their careful and thought-provoking comments on our article.

A key conceptual division represented in these comments, and in our response to them, is between tie- and actor-oriented network event models. We have argued here that actor-oriented models separate rate and evaluation, whereas tie-based models conflate the two in each parameter. This makes the models mathematically distinct and so one cannot be subsumed within the other, as Butts suggests. We also argued that the claim that the tie-oriented model by Butts can be straightforwardly understood in choice-theoretic terms is incomplete for the same reason. This claim ignores how REM parameters incorporate not only the choice of event receivers but also the timing of actions. Last, we have argued that by modeling these as separate processes, SAOMs and DyNAMs provide greater flexibility in model specification.⁶

We hope that this rejoinder adds to an increasing awareness of theoretical and mathematical differences between tie- and actor-oriented network models. Our arguments here are in line with other publications in which we demonstrate underlying mathematical and conceptual differences between statistical network models (Block et al., 2016, 2017; Stadtfeld and Block 2017; Stadtfeld et al. 2017), with the aim of informing researchers’ choice of model. We believe that the DyNAM represents an important addition to a diverse methodological toolbox providing alternatives to empirical researchers investigating specific forms of social mechanisms (such as the dynamics of coordination networks).

To facilitate the comparison of the two network event models discussed here, we have implemented both DyNAM and REM in a software framework called goldfish (as illustrated in our detailed appendix, this volume, pp. 32–36). The new effect classes that we proposed in our article are thus now also available to be included in the tie-oriented REM, as well as the more elaborated treatment of time that was introduced in Stadtfeld and Block (2017). The comments by Snijders should encourage us and others to extend this modeling approach further. We believe that the actor-oriented DyNAM holds broad promise for applications in many social scientific fields, and we look forward to observing these developments in time.

Footnotes

Notes

Author Biographies

The author biographies can be found on page 40 of this volume.

References

Block

Per

Koskinen

Johan

Hollway

James

Steglich

Christian

Stadtfeld

Christoph

. 2017. “Change We Can Believe In: Comparing Longitudinal Network Models on Consistency, Interpretability and Predictive Power.” Social Networks. Retrieved September 19, 2017 (https://doi.org/10.1016/j.socnet.2017.08.001).

Block

Per

Stadtfeld

Christoph

Snijders

Tom A. B.

2016. “Forms of Dependence: Comparing SAOMs and ERGMs from Basic Principles.” Sociological Methods and Research. Retrieved September 19, 2017 (https://journals-sagepub-com.web.bisu.edu.cn/doi/abs/10.1177/0049124116672680).

Butts

Carter T.

2008. “A Relational Event Framework for Social Action.” Pp. 155–200 in Sociological Methodology, vol. 38, edited by Xie

. Hoboken, NJ: Wiley-Blackwell.

DuBois

Christopher

Butts

Carter T.

McFarland

Daniel

Smyth

Padhraic

. 2013. “Hierarchical Models for Relational Event Sequences.” Journal of Mathematical Sociology57(6):297–309.

DuBois

Christopher

Smyth

Padhraic

. 2010. “Modeling Relational Events via Latent Classes.” In Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining.

Hollway

James

. 2015. The Evolution of Global Fisheries Governance 1960–2010. PhD dissertation, Department of Politics and International Relations, University of Oxford.

Hollway

James

Koskinen

Johan H.

2016b. “Multilevel Bilateralism and Multilateralism: States’ Bilateral and Multilateral Fisheries Treaties and Their Secretariats.” Pp. 315–32 in Multilevel Network Analysis for the Social Sciences, edited by Lazega

Emmanuel

Snijders

Tom A. B.

Cham, Switzerland: Springer International.

Hollway

James

Koskinen

Johan H.

2016a. “Multilevel Embeddedness: The Case of the Global Fisheries Governance Complex.” Social Networks44:281–94.

Marcum

Christopher Steven

Butts

Carter T.

2015. “Constructing and Modifying Sequence Statistics for relevent Using informr in R.” Journal of Statistical Software64(1):1–34.

10.

McFadden

Daniel

. 1974. “Conditional Logit Analysis of Qualitative Choice Behavior.” Pp. 105–42 in Frontiers in Econometrics, edited by Zarembka

Paul

. New York: Academic Press.

11.

Perry

Patrick O.

Wolfe

Patrick J.

2013. “Point Process Modelling for Directed Interaction Networks.” Journal of the Royal Statistical Society, Series B (Statistical Methodology)75(5):821–49.

12.

Snijders

Tom A. B.

1996. “Stochastic Actor-oriented Models for Network Change.” Journal of Mathematical Sociology21(1):149–72.

13.

Snijders

Tom A. B.

2001. “The Statistical Evaluation of Social Network Dynamics.” Pp. 361–95 in Sociological Methodology, vol. 31, edited by Sobel

Michael E.

Becker

Mark P.

Boston, MA: Blackwell.

14.

Snijders

Tom A. B.

Pickup

Mark

. 2017. “Stochastic Actor-oriented Models for Network Dynamics.” In Oxford Handbook of Political Networks, edited by Victor

Jennifer Nicoll

Lubell

Mark

Montgomery

Alexander H.

Oxford, United Kingdom: Oxford University Press.

15.

Stadtfeld

Christoph

. 2010. “Who Communicates with Whom? Measuring Communication Choices on Social Media Sites.” Pp. 564–69 in Proceedings of the 2010 IEEE Second International Conference on Social Computing (SocialCom). Minneapolis, MN.

16.

Stadtfeld

Christoph

. 2012. Events in Social Networks: A Stochastic Actor-oriented Framework for Dynamic Event Processes in Social Networks. Karlsruhe, Germany: KIT Scientific.

17.

Stadtfeld

Christoph

Block

Per

. 2017. “Interactions, Actors, and Time: Dynamic Network Actor Models for Relational Events.” Sociological Science4:318–52.

18.

Stadtfeld

Christoph

Geyer-Schulz

Andreas

. 2011. “Analyzing Event Stream Dynamics in Two Mode Networks: An Exploratory Analysis of Private Communication in a Question and Answer Community.” Social Networks33(4):258–72.

19.

Duy Q.

Asuncion

Arthur U.

Hunter

David R.

Smyth

Padhraic

. 2011. “Continuous-time Regression Models for Longitudinal Networks.” Advances in Neural Information Processing Systems24:2492–2500.