Improving and Extending STERGM Approximations Based on Cross-Sectional Data and Tie Durations

Abstract

Temporal exponential-family random graph models (TERGMs) are a flexible class of models for network ties that change over time. Separable TERGMs (STERGMs) are a subclass of TERGMs in which the dynamics of tie formation and dissolution can be separated within each discrete time step and may depend on different factors. The Carnegie et al. (2015) approximation improves estimation efficiency for a subclass of STERGMs, allowing them to be reliably estimated from inexpensive cross-sectional study designs. This approximation adapts to cross-sectional data by attempting to construct a STERGM with two specific properties: a cross-sectional equilibrium distribution defined by an exponential-family random graph model (ERGM) for the network structure, and geometric tie duration distributions defined by constant hazards for tie dissolution. In this paper we focus on approaches for improving the behavior of the Carnegie et al. approximation and increasing its scope of application. We begin with Carnegie et al.’s observation that the exact result is tractable when the ERGM is dyad-independent, and then show that taking the sparse limit of the exact result leads to a different approximation than the one they presented. We show that the new approximation outperforms theirs for sparse, dyad-independent models, and observe that the errors tend to increase with the strength of dependence for dyad-dependent models. We then develop theoretical results in the dyad-dependent case, showing that when the ERGM is allowed to have arbitrary dyad-dependent terms and some dyad-dependent constraints, both the old and new approximations are asymptotically exact as the size of the STERGM time step goes to zero. We note that the continuous-time limit of the discrete-time approximations has the desired cross-sectional equilibrium distribution and exponential tie duration distributions with the desired means. We show that our results extend to hypergraphs, and we propose an extension of the Carnegie et al. framework to dissolution hazards that depend on tie age.

1. Introduction

One of the most general methodological frameworks for the statistical modeling of networks is the class of exponential-family random graph models (ERGMs). These models specify probability distributions on networks as functions of the cross-sectional network statistics (Robins et al., 2007), and are commonly used to model static networks.

Temporal exponential-family random graph models (TERGMs) are a discrete-time generalization of ERGMs that model transitions between network states in terms of (possibly time-dependent) network statistics (Hanneke et al., 2010). Separable TERGMs (STERGMs) are a subclass of TERGMs in which the dynamics of tie formation and dissolution are independent within each time step and are allowed to depend on different factors. The within-step independence assumption does limit model flexibility, but it does not preclude using STERGMs to model many processes of interest, and its effects can often be mitigated by reducing the time step size. STERGMs have been characterized as sufficiently general that they can be used for a broad range of applications (see below for examples), while also possessing good interpretability properties (Krivitsky and Handcock, 2014). For a review of these models and other terminology used in this introduction, see Section 2.2.

While the statistical theory for TERGMs has been well developed (Hanneke et al., 2010; Krivitsky and Handcock, 2014), the practical estimation of these models remains a formidable challenge. The barriers are both data-related and computational. When complete longitudinal (panel) network data are available, the relatively efficient conditional MLE method can be used (Krivitsky and Handcock, 2014); dealing with missing data in this context, however, is not trivial (Almquist and Butts, 2018). STERGMs have been used successfully in many applied research contexts where complete network panel data are available. Applications in the last decade range from supply chains (Park et al., 2018), climate policy (Jasny and Fisher, 2019), twitter (Almquist et al., 2017) and political blogs (Almquist and Butts, 2013) to venture capital networks (Zhang et al., 2019), biotechnology R&D networks (Broekel and Bednarz, 2018), product competition (Xie et al., 2020), landscape management (Angst and Hirschi, 2017) and global weapons transfers (Lebacher et al., 2021).

In application settings where the network data must be actively collected, rather than passively scraped, the cost to obtain panel data on a census of the network can be prohibitive if the population is large. Here the more common study design in practice is to collect a single observation of a cross-sectional network state, or an egocentric sample of such a network state, along with information on the timing of edges. In principle, such cross-sectional study designs can be used for TERGM estimation by supplementing the data with an equilibrium assumption, and a gradient descent algorithm based on the equilibrium generalized method of moments estimator (EGMME) has been developed to estimate the TERGM in this context (Krivitsky, 2022). However, this EGMME algorithm has not proven to be generally successful, owing to the computational burden of simulating TERGMs and estimating the gradient with finite differences, and the lack of a good method for initializing coefficients in the general case. While STERGMs are a proper subclass of TERGMs, even they cannot be estimated reliably using the EGMME algorithm.

To improve the efficiency of estimation in the EGMME context, Carnegie et al. (2015) proposed a method for approximating the coefficients of a class of STERGMs by adjusting the coefficients of an ERGM for (possibly heterogeneous) edge durations with constant dissolution hazards. This approximation is analogous to one commonly used in biostatistics and epidemiology when representing the relationship between disease prevalence, incidence, and duration (Alho, 1992). Carnegie et al.’s method seeks to produce a STERGM with a cross-sectional equilibrium distribution that matches the given ERGM distribution, and edge duration distributions that match those defined by the constant dissolution hazards. It gains efficiency over the EGMME algorithm by leveraging the relative ease of estimating ERGMs (Hummel et al., 2012), including in the context of egocentrically sampled data (Krivitsky and Morris, 2017), and has come to be called the “edges dissolution approximation” (EDA). While these EDA STERGMs are less general than arbitrary STERGMs – in particular, the dissolution models are restricted to dyad-independent terms – they achieve a useful balance of theoretical model flexibility and practical model estimability from limited data.

Implementing the EDA as presented in Carnegie et al. (2015) involves the following steps. First, an ERGM containing the cross-sectional terms of interest is fit to data for a single cross-sectional network, possibly specified by target statistics. The STERGM formation model terms are the same as those in the ERGM, and the STERGM dissolution model terms are a dyad-independent submodel of the ERGM. Next, the dissolution coefficients are estimated directly from the target mean edge durations using a constant hazard model, equating the dissolution probability to the reciprocal of the target mean duration.¹ Finally, the formation coefficients are obtained by taking the ERGM coefficients and subtracting off the corresponding dissolution coefficients. These approximate coefficients can be used either as initial values for the EGMME algorithm, or directly as an estimate of the STERGM coefficients, with the latter being the more common practice.

The primary application for the EDA to date has been to epidemic modeling. This is not surprising, as the EDA was first developed using this application for testing proof of concept, and the R package EpiModel (Jenness et al., 2018) integrated the EDA into accessible software for epidemic simulations. As a population science, epidemiology exemplifies the application setting most likely to benefit from the EDA: the population of interest is sufficiently large that empirical research on network dynamics is not feasible without methods designed to use sampled data and efficient computation. EDA STERGMs are used in epidemic modeling as a principled, data-driven framework for simulation studies; to explore the effects of network structure and dynamics on transmission, and as a laboratory to test the potential impact of interventions. While mathematical simulation studies have been used for epidemic modeling since computers first made this possible (Bartlett, 1957), what distinguishes the EDA STERGM framework is its superior ability to capture heterogeneities in the contact network, and its statistical foundation: the STERGM is used for both estimation of the dynamic network properties and its subsequent simulation. For estimation, the model for the dynamic contact network takes advantage of the EDA STERGM’s flexibility to simultaneously capture multiple attribute heterogeneities and key forms of dyad dependence from egocentrically sampled network data. For simulation, this estimated model generates complete dynamic networks over time, leveraging the EDA STERGM’s equilibrium properties to ensure that the observed cross-sectional network structures and tie durations are reproduced stochastically in expectation. There is a growing body of work in epidemiology that relies on EDA STERGMs, with applications to HIV (Jenness et al., 2021; Goodreau et al., 2017; Mittler et al., 2019; Anderson et al., 2021; Le Guillou et al., 2021), monkeypox (Spicknall et al., 2022), sexually transmitted bacterial infections (Earnest et al., 2020), COVID-19 (Al-Khani et al., 2020; Lopez-Abente et al., 2020; Hu et al., 2021), MRSA (Goldstein et al., 2017), non-human infections (Webber et al., 2016; Robinson et al., 2018; Wilson-Aggarwal et al., 2019), and multi-infection interactions (Jenness et al., 2017; Jones et al., 2022). There is at least one application of the EDA STERGM outside of epidemiology, to cyberattacks (Amusan et al., 2020). The fact that all of these papers (and over 60 more that we know of) use EpiModel is testament both to the importance of software for making new methods accessible, and the effectiveness of the EDA as a feasible platform for population-level network research.

The EDA as presented in Carnegie et al. (2015) successfully resolves key challenges to using EDA STERGMs in practice, but it does have some limitations. First, the approximation error tends to rise as edge duration and model density fall, or as strength of dyad dependence increases. The simulation results of Carnegie et al. (2015) and Section 3.2 below show that the errors due to the approximation can be on the order of the target values themselves, and thus comparable to the sampling and measurement uncertainty in the data (Althubaiti, 2016). Carnegie et al. showed that the approximation error goes to zero as the STERGM time step size goes to zero when the ERGM is dyad-independent, but error reduction in the dyad-dependent case was not addressed theoretically, only via simulation. Second, Carnegie et al.’s developments were restricted to constant dissolution hazards and ordinary (non-hyper) graphs, limiting model flexibility and the scope of potential applications. Finally, Carnegie et al.’s presentation assumed that the dissolution model was a subset of the ERGM model, which could give the impression that choosing the ERGM restricted the options for dissolution modeling. As a result, this nesting was required in early versions of the EpiModel software.

We address these limitations in the present paper. In Section 3.1.3 we derive a new form of the EDA that is adapted to sparse models. We show in Section 3.1.4 that this new approximation outperforms the old approximation for sparse, dyad-independent models, in the sense that it reduces the error in cross-sectional edge probability. In Section 3.2, we explore the behavior of both approximations for various dyad-dependent models, noting that while the new approximation is nearly exact for sparse models near dyad independence, there are regimes where the errors are large for both approximations. In Section 3.3, we turn to theoretical results for dyad-dependent models. We prove that when the ERGM is allowed to have arbitrary dyad-dependent terms and some dyad-dependent constraints, both the old and new approximations are asymptotically exact as the size of the STERGM time step goes to zero. We also show that the continuous-time limit of the discrete-time approximations has the desired cross-sectional equilibrium distribution and exponential tie duration distributions with the desired means. We extend these results to hypergraphs in Section 3.4, and propose an extension of the EDA framework to age-dependent dissolution hazards in Section 3.5. Our definition of EDA STERGMs in Section 2.2 allows us to clarify that the dissolution model need not be a subset of the ERGM model, and all of our results apply to this more general definition of the EDA. We discuss the implications of these findings and their place in the broader literature in Section 4.

2. Materials and Methods

2.1. Software

We will use a mix of formal derivations and simulations in this paper. Simulations use R (R Core Team, 2023) and the statnet suite of packages (Pavel N. Krivitsky et al., 2003–2022). Model terms used in this paper are as implemented in the ergm package (Hunter et al., 2008) for specifying the statistics of the network.

2.2. Terminology, Notation, and Conventions

We begin by reviewing some basic network-theoretic terminology. A network consists of a set of nodes (also called vertices) and a set of edges (also called ties). It may be directed (in which case an edge is an ordered pair of nodes) or undirected (in which case an edge is an unordered pair of nodes). A dyad is a potential edge; we will also identify a dyad with an ordered pair of nodes in the directed case and with an unordered pair of nodes in the undirected case.

When the node set and directedness are understood, as will typically be the case in this paper, we can identify a network state with its edge set. We will thus permit ourselves the use of standard set-theoretic notation for network states, including cardinality $\left(\right|\cdot \left|\right)$, membership $(\in )$, and empty set $\left(\varnothing \right)$, understanding that these refer to the edge set of the network state. In particular, if $d$ is a dyad and $u$ is a network state, the notation $d\in u$ means $d$ is an edge in $u$. When two network states have the same node set and directedness, we will also use standard notation for union $(\cup )$, intersection $(\cap )$, difference $(\setminus )$, symmetric difference $\left(\mathrm{\Delta }\right)$, and containment $(\subseteq )$, again referring to the edge sets of the network states.

In general, we may impose constraints on the network state, allowing only certain edge sets with a given node set and directedness. We will call a network state valid if its edge set satisfies the constraints. We will use $x$, $y$, and $z$ to refer to valid network states; these symbols will always denote valid network states, whether or not we use the term valid in referring to them. A dyad $d$ is called free if its edge state is not (unconditionally) fixed by the constraints, i.e., if there are valid network states $x$ and $y$ with $d\in x$ and $d\notin y$. A dyad that is not free is called fixed. An edge is called free (resp. fixed) if its underlying dyad is free (resp. fixed). A choice of constraints is called dyad-independent if any free dyad can have its edge state toggled in any valid network state without violating the constraints. Constraints that are not dyad-independent are called dyad-dependent.

A mapping from the space of all network states (with a given node set and directedness) to the real numbers is called a network statistic. We will use the term network statistics to refer to a vector-valued function, each component of which is a network statistic for the same node set and directedness. (The case of a single component is allowed.) We say that network statistics $g$ are dyad-independent if for all network states $u,$ $v$ (not necessarily valid) and all dyads $d$ we have $g(u\cup \{d\left\}\right)-g(u\setminus \{d\left\}\right)=g(v\cup \{d\left\}\right)-g(v\setminus \{d\left\}\right)$. Network statistics that are not dyad-independent are called dyad-dependent.

An ERGM with a given node set and directedness is defined by a choice of constraints, network statistics $g$, and (real) canonical coefficients $\theta$. The network statistics and canonical coefficients combine to determine the cross-sectional distribution $\pi$ of the ERGM by the formula

$$\pi (x)=\frac{1}{C}\mathrm{e}\mathrm{x}\mathrm{p}(\theta \cdot g(x))$$ (1)

where $x$ is a valid network state and

$$C=\sum _y \mathrm{e}\mathrm{x}\mathrm{p}(\theta \cdot g(y)),$$

the sum being taken over all valid network states $y$. An ERGM is called dyad-independent if its constraints and network statistics are both dyad-independent; otherwise, it is called dyad-dependent.

A STERGM with a given node set and directedness is defined by a choice of constraints, formation (network) statistics $g^{+}$, dissolution (network) statistics $g^{-}$, (real) formation coefficients ${\theta }^{+}$, and (real) dissolution coefficients ${\theta }^{-}$. Throughout this paper, with the exception of Section 3.5 (which we handle there), we will assume that the constraints are cross-sectional, and that $g^{+}$ and $g^{-}$ are given by cross-sectional statistics. This means that if the network at time $t$ is $x$, then the probability that the network at time $t+1$ is $y$ is given by

$$T_{xy}=\frac{1}{C_x}\cdot \frac{\exp \left({\theta }^{+}\cdot g^{+}\left(x\cup y\right)\right)}{\exp \left({\theta }^{+}\cdot g^{+}\left(x\right)\right)}\cdot \frac{\exp \left({\theta }^{-}\cdot g^{-}\left(x\cap y\right)\right)}{\exp \left({\theta }^{-}\cdot g^{-}\left(x\right)\right)}$$ (2)

where

$$C_x=\sum _z\frac{\exp ({\theta }^{+}\cdot g^{+}(x\cup z))}{\exp ({\theta }^{+}\cdot g^{+}(x))}\cdot \frac{\exp ({\theta }^{-}\cdot g^{-}(x\cap z))}{\exp ({\theta }^{-}\cdot g^{-}(x))}.$$

These models are STERGMs in the sense of Krivitsky and Handcock (2014) except when the constraints are dyad-dependent, which breaks the separability of formation and dissolution within a time step. We will use the term STERGM to refer to these models throughout this paper (understanding that in the case of dyad-dependent constraints, we should technically be using the more general term TERGM instead). We say that a STERGM is dyad-independent if its constraints, formation statistics, and dissolution statistics are all dyad-independent; otherwise, we say that the STERGM is dyad-dependent.

An EDA STERGM is defined by a choice of ERGM, positive integer $L$, durational targets $\overrightarrow{D}\in (1,\mathrm{\infty }{)}^L$, and mapping $\varphi$ from dyads to $\{1,\dots ,L\}$. Given a network state $u$ and a value of $k\in \{1,\dots ,L\}$, we let $u_k=\{d\in u:\varphi (d)=k\}$. We define network statistics $h$ mapping into ${\mathbb{R}}^L$ such that $h_k\left(u\right)=\left|u_k\right|$ for all network states $u$ and all $k\in \{1,\dots ,L\}$. Letting $g$ denote the network statistics of the ERGM and $\theta$ the canonical coefficients of the ERGM, we define the EDA STERGM in the form of Carnegie et al. (2015) as follows: its node set, directedness, and constraints match those of the ERGM; its formation statistics are $(g,h)$; its dissolution statistics are $h$; its formation coefficients are $\left(\theta ,-\mathrm{l}\mathrm{o}\mathrm{g}\left(D_1-1\right),\dots ,-\mathrm{l}\mathrm{o}\mathrm{g}\left(D_L-1\right)\right)$; its dissolution coefficients are $\left(\mathrm{l}\mathrm{o}\mathrm{g}\left(D_1-1\right),\dots ,\mathrm{l}\mathrm{o}\mathrm{g}\left(D_L-1\right)\right).$² Letting $\pi$ denote the ERGM distribution, this means that we can write (2) as

$$T_{xy}=\frac{1}{C_x}\cdot \frac{\pi (x\cup y)}{\pi (x)}\cdot \prod _k\frac{1}{{(D_k-1)}^{\left|{(x\text{Δ}y)}_k\right|}}$$ (3)

where

$$C_x=\sum _z\frac{\pi (x\cup z)}{\pi (x)}\cdot \prod _k\frac{1}{{(D_k-1)}^{\left|{(x\text{Δ}z)}_k\right|}}$$

and $k$ ranges over $\{1,\dots ,L\}$. In general, there are other choices of statistics $h$ and coefficients for the EDA STERGM that give rise to the same transition probabilities (3); we have chosen the above for simplicity of description.³ Note that even if $x$ and $y$ are valid, the network $x\cup y$ need not be valid (if the constraints are dyad-dependent). We will assume that $\pi$ has been extended (by the formula (1)) to be defined on $x\cup y$ whenever $x$ and $y$ are valid, but that it is normalized with respect to the space of valid networks only.

In the context of EDA STERGMs, we say that a free dyad $d$ (or an edge on that dyad) is of type $k$ if $\varphi (d)=k$. The interpretation is that $D_k$ is the corresponding durational target. We say that $y$ equals $x$ plus one edge of type $k$ if $x\subseteq y$ and $|y\setminus x|=\left|(y\setminus x{)}_k\right|=1$. We say $y$ equals $x$ minus one edge of type $k$ if $x$ equals $y$ plus one edge of type $k.$

We say (informally) that a network is sparse if it has far fewer than the maximum number of edges given its directedness and the size of its node set, and that a model is sparse if it tends to produce sparse networks. For the precise results of Section 3.1, edge probabilities of 1/3 or less are sufficient.

The arguments in this paper apply both to undirected models and to directed models. Loops may be either allowed or forbidden, regarding the prohibition of loops as a particular dyad-independent constraint. Our results also apply to bipartite models, regarding the bipartite condition as a particular dyad-independent constraint. (A network is bipartite if its vertex set can be partitioned into two subsets such that each edge in the network has one endpoint in each subset, and a model is bipartite if its vertex set can be partitioned into two subsets such that the model constraints prohibit edges with both endpoints in the same subset.)

We define and discuss hypergraphs in Section 3.4; until then, we will restrict our attention to (non-hyper) graphs as defined in this Section 2.2.

3. Results

3.1. The Dyad-Independent Case

For the dyad-independent case we derive exact results for the formation coefficients following Carnegie et al. (2015), define a new approximation for sparse models, derive the errors in both the old and new approximations, and derive the regime in which the new approximation performs better than the old approximation.

3.1.1. Notation

Suppose we are given a dyad-independent ERGM with statistics $g$ and coefficients $\theta$. The linear predictor of a free dyad $d$ is defined to be ${\eta }_d=\theta ·(g(\left\{d\right\})-g(\varnothing ))$, which can be interpreted as the log odds of $d$ being an edge in a random network drawn from the ERGM distribution $\pi$. In other words, if $p_d\in (0,1)$ denotes the probability that free dyad $d$ is an edge under $\pi$, then ${\eta }_d=\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(p_d\right)$. Since the ERGM is dyad-independent, its behavior is fully specified by its linear predictors for free dyads and its edge states for fixed dyads. In particular,

$$\pi (x)=\frac{\prod _{\text{free}d\in x}\exp ({\eta }_d)}{\sum _y\prod _{\text{free}d\in y}\exp ({\eta }_d)}.$$

Likewise, the behavior of a dyad-independent STERGM of the type (2) is fully specified by its formation and dissolution linear predictors for free dyads, and its edge states for fixed dyads. The formation linear predictor for free dyad $d$ is defined to be ${\eta }_d^{+}={\theta }^{+}\cdot \left(g^{+}(\{d\})-g^{+}(\varnothing )\right)$, and the dissolution linear predictor for free dyad $d$ is defined to be ${\eta }_d^{-}={\theta }^{-}\cdot \left(g^{-}(\{d\})-g^{-}(\varnothing )\right)$. We can then write

$$T_{xy}=\frac{1}{C_x}\cdot \left(\prod _{d\in (x\cup y)\setminus x} \mathrm{e}\mathrm{x}\mathrm{p}\left({\eta }_d^{+}\right)\right)\cdot \left(\prod _{d\in x\setminus (x\cap y)} \mathrm{e}\mathrm{x}\mathrm{p}\left(-{\eta }_d^{-}\right)\right)$$

where

$$C_x=\sum _z \left[\left(\prod _{d\in (x\cup z)\setminus x} \mathrm{e}\mathrm{x}\mathrm{p}\left({\eta }_d^{+}\right)\right)\cdot \left(\prod _{d\in x\setminus (x\cap z)} \mathrm{e}\mathrm{x}\mathrm{p}\left(-{\eta }_d^{-}\right)\right)\right].$$

The formation linear predictor ${\eta }_d^{+}$ for free dyad $d$ can be interpreted as the log odds of $d$ being an edge at time $t+1$ given that it is not an edge at time $t.$ The dissolution linear predictor ${\eta }_d^{-}$ for free dyad $d$ can be interpreted as the log odds of $d$ being an edge at time $t+1$ given that it is an edge at time $t$.

We also define $q_d\in (0,1)$ to be the “formation probability” of free dyad $d$ as given by the relation ${\eta }_d^{+}=\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(q_d\right)$.

In the remainder of this section, $d$ will be fixed but arbitrary, and we will drop the subscript $d$ on ${\eta }_d$, $p_d$, ${\eta }_d^{+}$, ${\eta }_d^{-}$, and $q_d$. We denote by $D\in (1,\infty )$ the durational target for the dyad under consideration.

3.1.2. Overview of Results

As we are dealing with dyad-independent ERGMs, we will be able to focus on one free dyad at a time. Expressed in terms of linear predictors, the prescription set forth in Carnegie et al. (2015) is to take

$$\begin{array}{l}{\eta }^{+}=\eta -\log \left(D-1\right),\\ {\eta }^{-}=\log \left(D-1\right).\end{array}$$ (4)

For dyad-independent STERGMs of the type (2), specifying the dissolution linear predictor ${\eta }^{-}$ is equivalent to specifying the mean duration $D$ via the relation ${\eta }^{-}=\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(1-1/D)=\mathrm{l}\mathrm{o}\mathrm{g}(D-1)$. Given this value of ${\eta }^{-}$, the choice of value for ${\eta }^{+}$ is supposed to render equilibrium edge probability approximately equal to the edge probability in the original ERGM.

In Section 3.1.3, we show that both the target mean duration $D\in (1,\infty )$ and ERGM edge probability $p\in (0,1)$ can be matched exactly by the STERGM when $\frac{p}{(1-p)D}<1$, i.e. $D-\mathrm{e}\mathrm{x}\mathrm{p}(\eta )>0$, and that the STERGM linear predictors accomplishing this exact matching are uniquely given by

$$\begin{array}{l}{\eta }^{+}=\eta -\log \left(D-\exp \left(\eta \right)\right),\\ {\eta }^{-}=\log \left(D-1\right).\end{array}$$ (5)

We are particularly interested in the sparse limit, which corresponds to $\eta \ll 0$; since $D>1$, we can then approximate $\eta -\log \left(D-\exp \left(\eta \right)\right)$ as $\eta -\mathrm{l}\mathrm{o}\mathrm{g}(D)$, so that the prescription (5) becomes

$$\begin{array}{l}{\eta }^{+}=\eta -\log \left(D\right),\\ {\eta }^{-}=\log \left(D-1\right).\end{array}$$ (6)

We will refer to (4) as the “old” EDA, to (5) as the “exact” EDA, and to (6) as the “new” EDA. While the exact result (5) could be implemented in practice (say, via an operator term (Krivitsky et al., 2023)), for models that are very sparse across all dyad types, the difference between (5) and (6) should be negligible.

3.1.3. Derivation of the Exact Solution and the Sparse Limit

Here, we derive the general result (5) and (thus) obtain the sparse limit (6). The derivation of (5) is similar to that given in Carnegie et al. (2015). We use notation as in Section 3.1.1, for a fixed but arbitrary free dyad. We take $D\in (1,\mathrm{\infty })$ and $q\in (\mathrm{0,1})$, with ${\eta }^{-}=\mathrm{l}\mathrm{o}\mathrm{g}(D-1)$ and ${\eta }^{+}=\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(q)$, so that the finite irreducible aperiodic Markov chain on edge states of this dyad (across time steps) possesses a unique stationary distribution, which is also its equilibrium distribution. We additionally let $p^{\mathrm{*}}\in (0,1)$ denote the equilibrium (cross-sectional) edge probability of this dyad under the STERGM.

We first derive a general relationship between $p^{\mathrm{*}}$, $q$, and $D$. In equilibrium, the probability to form an edge on this dyad must equal the probability to dissolve an edge on this dyad, so as to conserve the total probability that there is an edge on this dyad. We start with an edge with probability $p^{\mathrm{*}}$, and given that we start with an edge, we dissolve it with probability $1/D$. We start with a non-edge with probability $1-p^{\mathrm{*}}$, and given that we start with a non-edge, we form an edge with probability $q$. Thus $\left(1-p^{\mathrm{*}}\right)q=p^{\mathrm{*}}/D$, or $q=\frac{p^{\mathrm{*}}}{\left(1-p^{\mathrm{*}}\right)D}$.

Note that given any two of $p^{\mathrm{*}}$, $q$, and $D$, the relationship $q=\frac{p^{\mathrm{*}}}{\left(1-p^{\mathrm{*}}\right)D}$ uniquely determines the third. Thus, if we want to have $p^{\mathrm{*}}=p$ for a given value of $D$, then we must have $q=\frac{p}{(1-p)D}$, and since $q\in (\mathrm{0,1})$, this requires $\frac{p}{(1-p)D}\in (\mathrm{0,1})$. Conversely, suppose that $\frac{p}{(1-p)D}\in (\mathrm{0,1})$, and define $q=\frac{p}{(1-p)D}$. Since $p^{\mathrm{*}}$ is uniquely determined by the relationship $q=\frac{p^{\mathrm{*}}}{\left(1-p^{\mathrm{*}}\right)D}$, it follows that $p^{\mathrm{*}}=p$.

Thus, we can achieve cross-sectional edge probability $p$ and mean edge duration $D$ under the STERGM when $\frac{p}{(1-p)D}\in (\mathrm{0,1})$, and the value of $q$ accomplishing this is uniquely given as $q=\frac{p}{(1-p)D}$. We know ${\eta }^{+}=\mathrm{logit}\left(q\right)$ and $\eta =\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(p)$; using these relations together with $q=\frac{p}{(1-p)D}$ yields after simplification that ${\eta }^{+}=\eta -\mathrm{l}\mathrm{o}\mathrm{g}(D-\mathrm{e}\mathrm{x}\mathrm{p}(\eta ))$, as in (5).

We now consider the sparse limit. We have ${\eta }^{+}=\mathrm{logit}\left(q\right)$ and $\eta =\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(p)$; in the sparse limit $p\ll 1$, we may approximate $\mathrm{logit}\left(p\right)$ by $\log \left(p\right),\frac{p}{(1-p)D}$ by $p/D$, and $\mathrm{logit}\left(q\right)=\mathrm{logit}\left(\frac{p}{\left(1-p\right)D}\right)$ by $\mathrm{l}\mathrm{o}\mathrm{g}(p/D)$; the statement $\log \left(p/D\right)=\log \left(p\right)-\log \left(D\right)$ is then the statement that ${\eta }^{+}=\eta -\log \left(D\right)$ in the sparse limit, so we have derived (6).

Note that this argument is analogous to that commonly used in biostatistics and epidemiology when representing the relationship between disease prevalence, incidence and duration (our $p$, $q$ and $D$ respectively) (Alho, 1992). For homogeneous infection and recovery processes, the general form of that relationship is $\frac{p}{1-p}=qD$, and the equivalent “sparse limit” is $p=qD$.

3.1.4. Formal Derivation of the Approximation Errors

In this section we derive and compare the errors of the old and new approximations (4) and (6), assuming that the model is dyad-independent. We focus on a single free dyad, with notation as in Section 3.1.1. By the assumption of dyad independence, the mean duration will be matched exactly, so the error will be entirely in the equilibrium edge probability. The target value is $p$, the ERGM edge probability, and we let $p_{\text{old}}$ and $p_{\text{new}}$ denote the equilibrium edge probabilities in the STERGM using the old and new approximations, respectively. We can determine $p_{\text{old}}$ and $p_{\text{new}}$ by equating the exact results (5) for $p_{\text{old}}$ and $p_{\text{new}}$ to the approximations (4) and (6) for $p$ (with the same value of $D$ throughout, since we know all of (4)–(6) yield $D$ as the mean duration), and then solving for what $p_{\text{old}}$ and $p_{\text{new}}$ must be in order for these equations to be satisfied.

In order to do the derivation only once, we let $\alpha$ be a parameter taking values in $\{0,1\}$ and define $p_{\alpha }$ by

$$\mathrm{logit}\left(p_{\alpha }\right)-\log \left(D-\exp \left(\mathrm{logit}\left(p_{\alpha }\right)\right)\right)=\mathrm{logit}\left(p\right)-\log \left(D-\alpha \right)$$

so that $p_{\text{new}}=p_0$ and $p_{\text{old}}=p_1$. Noting that

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(p_{\alpha }\right)-\mathrm{l}\mathrm{o}\mathrm{g}\left(D-\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(p_{\alpha }\right)\right)\right)=-\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{D}{\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(p_{\alpha }\right)\right)}-1\right)$$

we obtain

$$-\log \left(\frac{D}{\exp \left(\mathrm{logit}\left(p_{\alpha }\right)\right)}-1\right)=\mathrm{logit}\left(p\right)-\log \left(D-\alpha \right).$$

Solving for $p_{\alpha }$, we find

$$p_{\alpha }=p\cdot \frac{D}{D+p+\alpha (p-1)}.$$

Thus we have the relative error

$$\frac{p_{\alpha }-p}{p}=\frac{D}{D+p+\alpha \left(p-1\right)}-1=\frac{-p-\alpha \left(p-1\right)}{D+p+\alpha \left(p-1\right)}.$$

This means that

$$\frac{p_{\text{old}}-p}{p}=\frac{-2p+1}{D+2p-1}$$

and

$$\frac{p_{\text{new}}-p}{p}=\frac{-p}{D+p}.$$

We would like to identify for what values of $p$ and $D$ we have

$$\left|p_{\text{new}}-p\right|<\left|p_{\text{old}}-p\right|$$

or, equivalently,

$$\left|\frac{p_{\text{new}}-p}{p}\right|<\left|\frac{p_{\text{old}}-p}{p}\right|.$$

From the formulas above, this condition is equivalent to

$$pD+2p^2-p<|-2p+1|(D+p).$$

If $p>1/2$, this becomes

$$pD+2p^2-p<\left(2p-1\right)\left(D+p\right)=2pD+2p^2-D-p$$

i.e.

$$0<(p-1)D$$

which has no solutions as $p<1$ and $D>1$. If instead $p\le 1/2$, the condition is

$$pD+2p^2-p<\left(1-2p\right)\left(D+p\right)=D+p-2pD-2p^2,$$

i.e.

$$4p^2-p(2-3D)-D<0.$$

The quadratic equation

$$4p^2-p\left(2-3D\right)-D=0$$

has roots

$$\frac{2-3D\pm \sqrt{4+4D+9D^2}}{8}$$

and $4p^2-p\left(2-3D\right)-D<0$ precisely when $p$ lies strictly between these roots, as the parabola opens upwards. It is clear that the lower root

$$\frac{2-3D-\sqrt{4+4D+9D^2}}{8}$$

is negative, whereas $p$ is constrained to be positive. The formula for the upper root

$$\frac{2-3D+\sqrt{4+4D+9D^2}}{8}$$

defines a positive, monotonically decreasing function of $D\in (1,\mathrm{\infty })$, with limiting values $\frac{\sqrt{17}-1}{8}$ as $D\to 1^{+}$ and $\frac{1}{3}$ as $D\to +\mathrm{\infty }.$

In other words, the error (relative or absolute) in the equilibrium edge probability is smaller with the new approximation precisely when $p<\frac{2-3D+\sqrt{4+4D+9D^2}}{8}$, which in particular includes the range $p\le \frac{1}{3}$ for any value of $D$.

3.2. Empirical Behavior of the Approximations for Dyad-Dependent Models

In this section, we utilize simulations to examine how the old and new EDAs behave on models with some commonly used dyad-dependent terms. Using the syntax from the ergm package, we will specify an ERGM with the degree term, that counts the number of nodes with a particular degree, and the gwesp term, that computes a geometrically weighted measure of the number of edgewise shared partners. The gwesp term has been shown to possess certain desirable properties over a simple homogeneous measure of triangles (Goodreau et al., 2008; Hunter, 2007), and can be interpreted as a triad bias: two nodes are more or less likely to have an edge between them based on the number of other nodes to which they both have an edge (i.e., the number of shared partners).

3.2.1. Simulation Setup

To exhibit the behavior of the old and new approximations near dyad independence, we performed a series of simulations of edges + degree(1) models on a 1000 node undirected network. For the target statistics, we selected ranges that reflected those seen in applied research.

We used mean degrees of 0.7, 1.0, 1.3, and 2.0. The range 0.7–1.3 is taken from applied research on HIV, where STERGMs are used to summarize and simulate sexual transmission networks (Goodreau et al., 2017; Weiss et al., 2020), and the value 2.0 matches that used for the simulation in Carnegie et al. (2015). We used degree(1) targets of 200 to 600 in steps of 100. This range includes the mean degree(1) value for an edges-only model of each mean degree used. A dyad-independent model is therefore within the range of models we simulate for each mean degree. We included durations of 15, 50, and 100, to exhibit how errors change with duration when all other variables are held fixed. These durations include the range used for the simulation in Carnegie et al. (2015), and are on the order of durations used in applied work (Goldstein et al., 2017; Ezenwa et al., 2016; Goodreau et al., 2017; Weiss et al., 2020).

We also performed simulations for various edges + degree(1) + degree(2) + gwesp(0.5, fixed = TRUE) models on a 1000 node undirected network, analogous to those of Carnegie et al. (2015), but using gwesp throughout (rather than a mix of triangle and gwesp). We used a mean degree of 2.0, a degree(1) target of 200, a degree(2) target of 350, and gwesp(0.5, fixed = TRUE) targets ranging from 3 to 300 (considering that an isolated triangle counts as three gwesp). We again included durations of 15, 50, and 100.

For all models used, we examined the distribution of simulated model statistics for possible evidence of model degeneracy, and found none.

3.2.2. Simulation Results

From the theoretical results of Section 3.1, we expect the following general trends:

• 1errors with the new approximation should be small for sparse models near dyad independence, and
• 2we lose control of the errors for either approximation as we move away from dyad independence.

Anticipating the results of Section 3.3, we also expect that in general

• 3all else being equal, errors should tend to decrease as duration increases, for either approximation.

The results of the edges + degree(1) simulations described in Section 3.2.1 are shown in Figure 1. Expectation (1) is supported by the fact that at the dyad-independent value of degree(1) (dotted vertical line), the error with the new approximation (line with crosses) is very close to zero (dashed horizontal line), regardless of duration. Expectation (2) is supported by the generally larger errors near at least one margin of each plot, as compared to the errors at the dyad-independent value of degree(1). (Since various sources of error may partially cancel out rather than reinforcing each other, the errors in the old approximation (line with circles) may noticeably decrease for awhile as we move away from dyad independence in a particular direction.) Expectation (3) is supported by the observation that errors for both the new and old approximations tend to decrease as duration increases, for given target values of mean degree and degree(1).

Fig. 1.

Relative errors in the mean values of the edges and degree(1) statistics for the ~edges + degree(1) model on a 1000 node undirected network with mean degree targets of 0.7–2.0 and a range of degree(1) targets and durations.

The results of the edges + degree(1) + degree(2) + gwesp(0.5, fixed = TRUE) simulations are shown in Figure 2.⁴ There is no exhibition of the general trends (1) and (2) for this family of models, as there is no dyad-independent reference model in the range of models used. The general trend (3) is exhibited in these simulations by the tendency for errors to decrease as duration increases for a fixed target value of gwesp. Overall, the errors show mixed results, with the new approximation generally outperforming the old for edges, degree(2), and gwesp, but underperforming the old for degree(1). This illustrates the point that for an arbitrary dyad-dependent model, there is no guarantee which approximation will be better, regardless of duration; the new approximation tends to do better than the old approximation for sparse models near dyad independence, but if we stray sufficiently far from dyad-independent models, we cannot predict which approximation will be better, and the results may be mixed even within a single model.

Fig. 2.

Relative errors in the mean values of the edges, degree(1), degree(2), and gwesp(0.5, fixed = TRUE) statistics for the ~edges + degree(1) + degree(2) + gwesp(0.5, fixed = TRUE) model on a 1000 node undirected network with mean degree 2.0, degree(1) target 200, degree(2) target 350, and a range of gwesp(0.5, fixed = TRUE) targets and durations.

3.2.3. Interpretation

Intuitively, we may think of the EDA errors specific to dyad-dependent models as arising from the differences between the networks used to specify the ERGM and the STERGM: ERGMs use the cross-sectional network to compute the model statistics, as seen in (1); STERGMs use the union network to compute the formation model statistics and the intersection network to compute the dissolution model statistics, as seen in (2). For any dyad-independent network statistics $g$ and any transition $x\to y$ between valid network states, the cross-sectional change statistics $g\left(y\right)-g\left(x\right)$ are simply the sum of the union change statistics $g\left(x\cup y\right)-g\left(x\right)$ and the intersection change statistics $g(x\cap y)-g(x)$. This is not true in the dyad-dependent case, where the cross-sectional change statistics for a general multi-toggle (i.e., $\left|x\mathrm{\Delta }y\right|>1$) transition are not easily expressed in terms of the union and intersection change statistics. However, when the transition is single-toggle (i.e., $\left|x\mathrm{\Delta }y\right|=1$), for any network statistics $h$ we have $h(y)-h(x)=h(x\cup y)-h(x)+h(x\cap y)-h(x)$, even if $h$ is dyad-dependent.

This suggests that reducing the amount of change per time step may reduce the EDA errors, which is empirically supported by the simulations in Figures 1 and 2. We prove a general result along these lines in the next section.

3.3. Theoretical Results for Dyad-Dependent Models

Our goal in this section is to prove that when the ERGM is allowed to have arbitrary dyad-dependent terms and some dyad-dependent constraints, both the old and new approximations are asymptotically exact as the size of the STERGM time step goes to zero. In order to do this, we introduce in Section 3.3.3 a discrete-time process, denoted $R$, that is related to the continuous-time limit of the EDA STERGMs, and that has the desired cross-sectional and durational behavior not just asymptotically but at any sufficiently small time step. In Section 3.3.4, we obtain the desired asymptotic results for the EDA by comparing the discrete-time EDA STERGMs to $R$ as the time step size goes to zero.⁵

3.3.1. General Setup and Notation

Suppose we are given an ERGM with arbitrary terms and constraints. To prove cross-sectional exactness results, we will need the following assumption on the constraints:

• igiven any valid network states $x$ and $y$, there is a sequence $z^{\left(1\right)},\dots ,z^{\left(m\right)}$ of valid network states such that $z^{\left(1\right)}=x$, $z^{\left(m\right)}=y$, and $|z^{\left(i\right)}\mathrm{\Delta }{z}^{\left(i+1\right)}|=1$ for all $i\in \left\{1,\dots ,m-1\right\}$.

To prove durational exactness results, we will need the following additional assumption on the constraints:

• iigiven any valid network state, any edge in that network that is not unconditionally fixed by the constraints (i.e., any free edge) can be toggled off without violating the constraints.

We will consider edge duration only for free dyads.

Suppose we are also given a vector of positive durational targets ${\overrightarrow{D}}_0$ of length $L$, and a mapping $\varphi$ from dyads to $\{1,\dots ,L\}$. We define $\overrightarrow{D}=\lambda {\overrightarrow{D}}_0$, where $\lambda$ is a positive scalar whose value we will regard as a variable parameter. The interpretation is that ${\overrightarrow{D}}_0$ is given in some specific units (say days, weeks, years,...) and ${\lambda }^{-1}$ signifies the fraction of that time unit that is represented by a single STERGM time step, making $\overrightarrow{D}$ the vector of durational targets in units of the STERGM time step. We require $\lambda$ to be large enough that each component of $\overrightarrow{D}$ is strictly greater than 1. We will frequently use the asymptotic notation $𝒪$; the limit being taken is $\lambda \to +\mathrm{\infty }$, and the implied bounds may be interpreted as holding componentwise when stated for entire vectors or matrices.

The notation $\parallel \cdot \parallel$ denotes the Euclidean norm for vectors.

3.3.2. Leading Order Behavior of Discrete-Time EDAs at Small Time Step Sizes

Let $T$ denote the discrete-time EDA STERGM transition probability matrix with duration $\overrightarrow{D}$ (and dyad mapping $\varphi$). We will use the old convention (4) in writing out the details below, but the conclusions apply just as well to the new convention (6), whose transition probability matrix differs from that of the old by $𝒪\left(1/{\lambda }^2\right)$.

Recalling (3), if $x$ and $y$ are any valid network states (including the possibility that $x=y$), we have that

$$T_{xy}=\frac{1}{C_x}\cdot \frac{\pi (x\cup y)}{\pi (x)}\cdot \prod _k \frac{1}{{\left(D_k-1\right)}^{\left|(x\mathrm{\Delta }y{)}_k\right|}}$$

where

$$C_x=\sum _z \frac{\pi (x\cup z)}{\pi (x)}\cdot \prod _k \frac{1}{{\left(D_k-1\right)}^{\left|(x\mathrm{\Delta }z{)}_k\right|}}.$$

Note that for any $k$, we have $\frac{1}{D_k-1}=𝒪(1/\lambda )$, so that

$$\frac{\pi (x\cup z)}{\pi (x)}\cdot \prod _k \frac{1}{{\left(D_k-1\right)}^{\left|(x\mathrm{\Delta }z{)}_k\right|}}=𝒪(1/{\lambda }^{\left|x\mathrm{\Delta }z\right|}).$$

Thus

$$C_x=1+𝒪(1/\lambda ),$$

the 1 coming from the case $z=x$, and the $𝒪(1/\lambda )$ encompassing all cases $z\ne x$. Consequently,

$$T_{xy}=𝒪(1/{\lambda }^{\left|x\mathrm{\Delta }y\right|}).$$

Thus, we may write

$$T=I+\frac{A}{\lambda }+𝒪\left(1/{\lambda }^2\right)$$

where the matrix $A$ is given by

$$A_{xy}=\{\begin{array}{ll}0\hfill & \text{if}\left|x\text{Δ}y\right|\ge 2\hfill \\ \frac{\pi (x\cup y)}{\pi (x)}\cdot \frac{1}{D_{0,k}}\hfill & \text{if}\left|x\text{Δ}y\right|=\left|{(x\text{Δ}y)}_k\right|=1.\hfill \\ -\sum _{z\ne x}{A}_{xz}\hfill & \text{if}x=y\hfill \end{array}$$

Note that the transition rate matrix of the continuous-time limit of $T$ is proportional to $A$.

3.3.3. Exactness of “Infinitesimal Time Step EDA STERGMs”

For sufficiently large values of the parameter $\lambda$, we now define another discrete-time process by declaring that its transition probability matrix, denoted $R$, is given by $R=I+\frac{A}{\lambda }$, where $I$ is the identity. For $\lambda$ sufficiently large, this $R$ is a well-defined transition probability matrix on the state space of valid networks; we will assume that $\lambda$ is large enough that all diagonal entries in $R$ are strictly positive (so that $R$ is aperiodic). From our formulas in Section 3.3.2, we see that if we expand $T$ in powers of the small parameter ${\lambda }^{-1}$, then $R$ corresponds exactly to keeping the 0th and 1st order terms in this expansion, dropping all higher order terms. Thus, if we express the STERGM time step size $dt$ as $dt\propto {\lambda }^{-1}$, then we obtain $R$ from $T$ by regarding $dt$ as a formal infinitesimal: $dt$ itself is not zero, but $(dt{)}^2=0$. This explains why we (quite loosely) call $R$ an “infinitesimal time step EDA STERGM” and also shows how $R$ is related to the continuous-time limit of $T$. We emphasize that $R$ is a discrete-time process, and for a given (sufficiently large) value of $\lambda$, we think of $R$ and $T$ as having the same time step size; references to infinitesimals are purely motivational.

The transition probabilities for $R$ prohibit direct transitions between networks that differ in edge state on two or more dyads; in this sense, multiple, simultaneous changes are forbidden. Note that if $y$ equals $x$ plus one edge of type $k$, then $R_{xy}=\frac{\pi (y)}{\pi (x)}\cdot \frac{1}{D_k}$ and $R_{yx}=\frac{1}{D_k}$. It follows that $R$ satisfies detailed balance with respect to the ERGM distribution $\pi$. Since $R$ is finite and (under the assumptions stated in Section 3.3.1 for cross-sectional exactness) irreducible, $\pi$ is the unique stationary distribution of $R$. Since $R$ is also aperiodic, $\pi$ is the equilibrium distribution of $R$.

Observe that under the assumptions stated in Section 3.3.1 for durational exactness, given any valid network state $x$ and any free edge of type $k$ in $x$, the probability under $R$ to transition from $x$ to the state $y$ that equals $x$ minus the edge in question is exactly $\frac{1}{D_k}$. Under $R$, there are no other (positive probability) transitions out of $x$ that involve dissolving the edge in question, so this immediately implies that the edge duration distribution on any free dyad of type $k$ is geometric with mean $D_k$ under $R$.

We have thus shown that $R$ has precisely the cross-sectional and durational behavior we desire of the EDA: it reproduces the ERGM’s cross-sectional network distribution and the specified mean edge durations with geometric distributions, even though the ERGM may have dyad dependence. We will use $R$ to approximate the discrete-time EDA STERGM transition probability matrix with duration $\overrightarrow{D}$ so as to show that this latter transition probability matrix has the desired cross-sectional and durational behavior asymptotically as $\lambda \to +\mathrm{\infty }$.

From Section 3.3.2, we have that the continuous-time limit of the EDA STERGMs has transition rate matrix proportional to $A$. It follows that the continuous-time limit also has cross-sectional equilibrium distribution $\pi$, and has exponentially distributed tie durations with the desired means (up to an overall scaling of time).

3.3.4. Asymptotic Exactness of Discrete-Time EDAs

We will use the notation and results from Sections 3.3.2 and 3.3.3. Recalling that

$$R=I+\frac{A}{\lambda }$$

and

$$T=I+\frac{A}{\lambda }+𝒪\left(1/{\lambda }^2\right)$$

we define a $\lambda$-dependent matrix $\delta$ by

$$T=R+\delta$$

so that $\delta =𝒪\left(1/{\lambda }^2\right)$.

Since $\pi$ is the unique stationary distribution of $R$, the left nullspace of $A$ is precisely $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left\{\pi \right\}$. (If there were a (real) vector $\sigma \notin \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left\{\pi \right\}$ with $\sigma A=0$, then as ${\pi }_x>0$ for all valid $x$ we will have all entries of $\pi +\alpha \sigma$ positive for sufficiently small positive $\alpha$. Renormalizing $\pi +\alpha \sigma$ to a probability vector produces a stationary distribution of $R$ that is distinct from $\pi$, contradicting uniqueness.) The function $v\mapsto \parallel vA\parallel$, defined on the compact set $\{v\in (\mathrm{span}\left\{\pi \right\}{)}^{\perp }:\parallel v\parallel =1\}$, is positive and continuous, and thus admits a positive lower bound $c>0$.

Since the transition probability matrix $T$ is finite and irreducible, it possesses a unique $\lambda$-dependent stationary distribution which we choose to write as $\pi +ϵ$ where $ϵ$ is a $\lambda$-dependent perturbation to the ERGM distribution $\pi$. We have

$$\pi +ϵ=(\pi +ϵ)T=(\pi +ϵ)(R+\delta )=\pi +ϵ+ϵ\frac{A}{\lambda }+(\pi +ϵ)\delta .$$

Rearranging,

$$ϵA=-\lambda (\pi +ϵ)\delta =𝒪(1/\lambda )$$

since $\delta$ is $𝒪\left(1/{\lambda }^2\right)$ and $\pi +ϵ$ is $𝒪(1)$ (being a probability vector). We now write

$$ϵ=\beta \pi +{\pi }^{\perp }$$

where $\beta$ is a $\lambda$-dependent scalar and ${\pi }^{\perp }$ is a $\lambda$-dependent element of $(\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{\pi \}{)}^{\perp }$. By our observations above about the left nullspace of $A$, we have

$$c∥{\pi }^{\perp }∥\le ∥{\pi }^{\perp }A∥=\parallel ϵA\parallel =𝒪(1/\lambda )$$

thus showing that $∥{\pi }^{\perp }∥=𝒪(1/\lambda )$ since $c>0$. Letting $\overrightarrow{1}$ denote the vector of 1 s, we have

$$1=\overrightarrow{1}\cdot (\pi +ϵ)=\overrightarrow{1}\cdot \pi +\overrightarrow{1}\cdot ϵ=1+\overrightarrow{1}\cdot \left(\beta \pi +{\pi }^{\perp }\right)=1+\beta +\overrightarrow{1}\cdot {\pi }^{\perp }$$

thus showing

$$\beta =-\overrightarrow{1}\cdot {\pi }^{\perp }=𝒪(1/\lambda ).$$

Since $ϵ=\beta \pi +{\pi }^{\perp }$ and both $\beta$ and ${\pi }^{\perp }$ are $𝒪(1/\lambda )$ (while $\pi$ is of course $𝒪(1)$), we have $ϵ=𝒪(1/\lambda )$. Since $\pi +ϵ$ is the stationary distribution of $T$, this shows that the stationary distribution of $T$ converges to $\pi$ as $\lambda \to +\mathrm{\infty }$. For sufficiently large $\lambda$, all diagonal entries in $T$ will be positive, so that $\pi +ϵ$ is also the equilibrium distribution of $T$, proving asymptotic cross-sectional exactness for discrete-time EDAs with arbitrary dyad-dependent ERGM model terms and some dyad-dependent constraints.

We still need to show that tie duration distributions are asymptotically correct, under the required assumptions stated in Section 3.3.1. This is true in the relative sense: for any $0<ϵ<1$ (having nothing to do with the $ϵ$ above), there is a ${\lambda }_0(ϵ)$ such that $\lambda >{\lambda }_0(ϵ)$ implies any free dyad of type $k$ has cumulative distribution function for tie durations under $T$ sandwiched between those of a geometric distribution with mean $(1+ϵ)D_k$ and a geometric distribution with mean $(1-ϵ)D_k$. The reason is that free edges of type $k$ have dissolution probability exactly $\frac{1}{D_k}$ under $R$ and $T=R+𝒪\left(1/{\lambda }^2\right)$; taking $\lambda$ sufficiently large (depending on the given $ϵ$), we can guarantee that the dissolution probability under $T$ of any free edge of type $k$ in any valid network state is between $\frac{1}{(1+ϵ)D_k}$ and $\frac{1}{(1-ϵ)D_k}$, which gives the desired distributional inequalities.

3.4. Extension to Hypergraphs

The theoretical results of this paper extend directly to hypergraph models. To sketch this extension, we first introduce some terminology. A hypergraph consists of a set of vertices $V$ and a set of hyperedges $E$. The hypergraph may be directed (in which case $E$ is a set of ordered pairs of subsets of $V$) or undirected (in which case $E$ is a set of subsets of $V$). We will use the term hyperdyad to refer to a potential hyperedge, i.e. a subset of the vertex set in the undirected case and an ordered pair of subsets of the vertex set in the directed case. The notions of constraints, network statistics, ERGMs, STERGMs, and EDA STERGMs from Section 2.2 all generalize immediately to hypergraphs. We also adopt the obvious notions of hyperdyad (in)dependence for hypergraph model terms and constraints, corresponding to those of dyad (in)dependence in the non-hyper case. Graphs in the “ordinary” (non-hyper) sense of Section 2.2 will be referred to as ordinary graphs in this subsection.

The dyad-independent material of Section 3.1 simply regards the free dyads of a dyad-independent ordinary graph model as mutually independent two-state systems, and this point of view also applies to free hyperdyads in a hyperdyad-independent hypergraph model. The asymptotic exactness results of Section 3.3 also generalize to hypergraphs, because we can set up a correspondence between a given hypergraph model and an ordinary graph model while preserving the technical conditions on the constraints required in Section 3.3. Since arbitrary model terms are allowed in Section 3.3, any injective mapping from hyperdyads in a hypergraph to dyads in an ordinary graph suffices to obtain the result for hypergraphs. A natural (but by no means necessary) identification is the following. Given a hypergraph on a vertex set $V$, we consider an ordinary graph whose vertex set is the power set of $V$. For any $X\subseteq V$, let $v_X$ denote the corresponding node in the ordinary graph. If the hypergraph is directed, so is the ordinary graph, and for any $X,Y\subseteq V$, the hyperdyad ($X,Y$) corresponds to the ordinary dyad $(v_X,\left.v_Y\right)$. If the hypergraph is undirected, so is the ordinary graph, and for any $X\subseteq V$, the hyperdyad $X$ corresponds to the ordinary dyad $\left\{v_X\right\}$ (i.e., the unordered pair of $v_X$ with itself). Dyads in the ordinary graph that do not correspond to hyperdyads in the hypergraph are fixed as non-edges via a dyad-independent constraint on the ordinary graph model. Any constraints on the hypergraph model are then translated into additional constraints on the ordinary graph model in the obvious way, and the hypergraph model terms are likewise translated into functions on the space of ordinary networks of the same directedness whose node set is the power set of $V$.⁶ It is straightforward to verify that the constraints on the ordinary graph model satisfy the conditions of Section 3.3 if and only if the constraints on the hypergraph model satisfy the obvious hyper-analogues of those conditions. Thus, the asymptotic exactness results for ordinary graph models imply the corresponding asymptotic exactness results for hypergraph models.⁷

3.5. Proposed Extension of the EDA to Age-Dependent Dissolution Hazards

A major drawback of the EDA presented thus far is that it cannot accommodate an edge dissolution hazard that depends on the age of the active edge. It is straightforward to specify a generalization of the EDA that allows for this greater flexibility.

First, we need to generalize our definition of STERGM (initially given in Section 2.2) to allow for age-dependent dissolution hazards. We augment the network state with the ages of extant ties, allowing $g^{+}$ and $g^{-}$ to be functions not only of the edges in their network arguments but also of those edges’ ages. We continue to assume that the constraints are cross-sectional (i.e., depending only on presence or absence of edges, not edge ages).

Given a valid augmented network state $x$, we let $x^{\prime }$ denote the augmented network state whose edges are the same as $x$ but with all ages increased by 1. We also let $F(x)$ denote all valid augmented network states compatible with $x$ being the previous timestep’s network. This means a valid augmented network state $y$ belongs to $F(x)$ if and only if every edge in both $y$ and $x$ has age 1 greater in $y$ than in $x$, and every edge in $y$ and not in $x$ has age 1 in $y$. When taking unions and intersections of augmented network states, age is included as part of the edge state.

We can then say that if the network at time $t$ is $x$, then the probability that the network at time $t+1$ is $y\in F(x)$ is given by

$$T_{xy}=\frac{1}{C_x}\cdot \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\theta }^{+}\cdot g^{+}\left(x^{\prime }\cup y\right)\right)}{\mathrm{e}\mathrm{x}\mathrm{p}\left({\theta }^{+}\cdot g^{+}\left(x^{\prime }\right)\right)}\cdot \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\theta }^{-}\cdot g^{-}\left(x^{\prime }\cap y\right)\right)}{\mathrm{e}\mathrm{x}\mathrm{p}\left({\theta }^{-}\cdot g^{-}\left(x^{\prime }\right)\right)}$$ (7)

where

$$C_x=\sum _{z\in F(x)} \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\theta }^{+}\cdot g^{+}\left(x^{\prime }\cup z\right)\right)}{\mathrm{e}\mathrm{x}\mathrm{p}\left({\theta }^{+}\cdot g^{+}\left(x^{\prime }\right)\right)}\cdot \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\theta }^{-}\cdot g^{-}\left(x^{\prime }\cap z\right)\right)}{\mathrm{e}\mathrm{x}\mathrm{p}\left({\theta }^{-}\cdot g^{-}\left(x^{\prime }\right)\right)}$$

and ${\theta }^{+}$, ${\theta }^{-}$ are the formation and dissolution coefficients, respectively.

Now, we turn to EDAs for age-dependent dissolution hazards. Starting with the dyad-independent case, for a fixed but arbitrary free dyad (which will be suppressed in the notation), let $f$ be any map from the positive integers to (0, 1) satisfying $\prod _{i=1}^{\infty }\left[1-f(i)\right]=0$. We interpret $f(i)$ as the probability that an active edge of age $i$ on this dyad will dissolve on the next time step; the condition $\prod _{i=1}^{\infty }[1-f(i)]=0$ says that a tie dissolves in finite time almost surely. Define

$$D=\sum _{n=1}^{\mathrm{\infty }} n\cdot f(n)\cdot \prod _{i=1}^{n-1} [1-f(i)]$$

(with the empty product interpreted as 1), which is the mean duration of completed ties for dissolution hazard $f$, and assume that $D<\mathrm{\infty }$. Let $p\in (0,1)$ denote the ERGM edge probability for the dyad in question, and $\eta =\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(p)$ the corresponding linear predictor.

We require the consistency condition $\frac{p}{(1-p)D}<1$. Define the formation linear predictor ${\eta }^{+}$ by

$${\eta }^{+}=\eta -\mathrm{l}\mathrm{o}\mathrm{g}(D-\mathrm{e}\mathrm{x}\mathrm{p}(\eta ))$$ (8)

and the dissolution linear predictor for an active edge of age $i$ in the transitioned-from network, denoted ${\eta }_i^{-}$, by

$${\eta }_i^{-}=\mathrm{logit}\left(1-f\left(i\right)\right).$$ (9)

We label states on this dyad by non-negative integers, with 0 corresponding to no edge and $i>0$ corresponding to an active edge of age $i$. Under the conditions $D<\mathrm{\infty }$ and $\frac{p}{(1-p)D}<1$, it is straightforward to confirm that the distribution $\pi$ on dyad states defined by

$${\pi }_0=1-p$$

and, for any $i>0$,

$${\pi }_i=\frac{p}{D}\cdot \prod _{j=1}^{i-1}\left[1-f(j)\right]$$

is the unique stationary distribution of the age-dependent EDA STERGM on this dyad defined by (8) and (9), and that it is also the equilibrium distribution. (It is useful to recall that $D$ can be equivalently expressed as $\sum _{n=1}^{\infty }\prod _{i=1}^{n-1}\left[1-f(i)\right]$, and ${\eta }^{+}$ can be equivalently expressed as $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left(\frac{p}{(1-p)D}\right)$.) This equilibrium distribution has precisely the cross-sectional behavior defined by the ERGM and the tie age (and thus tie duration) behavior defined by $f$. This is the “exact” EDA for age-dependent dissolution hazards; the formula for ${\eta }^{+}$ is the same as that in (5) for constant hazard models, provided we interpret $D$ as the mean duration of completed ties for the given hazard.

Both the old and new approximations to the formation linear predictors continue to be well-defined in this context, using the mean completed tie duration $D$ for the given hazard $f$ (which may vary from one free dyad to the next) in the formulas (4) and (6) for ${\eta }^{+}$. (The age-dependent dissolution linear predictors are given by ${\eta }_i^{-}=\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}(1-f(i)$), as above.) Translating these approximations into model terms and coefficients (analogous to the presentation of the EDA STERGM for constant hazards in Section 2.2) gives us candidate EDAs to use in the dyad-dependent formation, age-dependent dissolution context. Determining whether or not these approximations have the desired behavior asymptotically, analogous to the results in Section 3.3, could be an avenue for future research.

4. Discussion

The work of Carnegie et al. (2015) established a reliable method for estimating a nontrivial class of STERGMs from a single cross-sectional network observation (or equivalent target statistics) and edge duration information (typically the ages of extant edges in the sample), by combining cross-sectional ERGM estimation with adjustments for edge duration. In doing so it opened up a rich temporal network modeling framework that is compatible with practical study designs for data collection, removing some key obstacles to the use of STERGMs in applied settings. We have improved and extended the approximation of Carnegie et al. in multiple ways, including its theoretical foundations and its scope of application.

First, we proposed a new approximation for sparse models in Section 3.1, and proved that it has smaller errors than the original approximation when applied to sparse, dyad-independent models. The correction $-\mathrm{l}\mathrm{o}\mathrm{g}(D-1)$ for the formation linear predictor in the original approximation increases without bound as $D$ goes to 1, which is generally not appropriate for sparse models. By contrast, the correction $-\mathrm{l}\mathrm{o}\mathrm{g}(D)$ for the formation linear predictor in the new approximation remains bounded as $D$ goes to 1. While the new approximation is not categorically better than the old approximation for sparse, dyad-dependent models, we recommend its use in general for sparse models, to limit this undesirable behavior at short durations. In practice, sparse models are the rule in nearly all of the applied examples we are aware of (e.g., Goldstein et al. (2017); Ezenwa et al. (2016); Jenness et al. (2021)), so it is appropriate to have an approximation that is tailored to them.

Second, we established the validity of the EDA in the dyad-dependent case. More specifically, we showed that when the ERGM is allowed to have any dyad-dependent terms and some dyad-dependent constraints, the approximation errors go to zero as the STERGM time step size goes to zero. For network research, dyad dependence is the most interesting use case, because the feedback introduced by these effects is responsible for the counter-intuitive patterns and emergent properties that make network dynamics unique (e.g., Goldstein et al. (2017); Jenness et al. (2016); Morris et al. (2009)).

Combining these results, we have the following general recommendations for reducing the approximation error: use the sparse approximation for sparse models, and decrease the time step size as much as necessary to obtain satisfactory results. In practice, the errors for a given time step size are checked by simulating the EDA STERGM and comparing the simulated network statistics to their desired means. As noted in the introduction, the errors due to the approximation can be on the order of the uncertainty in the data. Since getting better data tends to be expensive, while reducing the time step size is relatively cheap, it is useful to know that this method of error reduction is available in the general, dyad-dependent case.

Third, we showed that the above results extend to hypergraphs. This opens up an entirely new application area for the EDA. Temporal hypergraphs are an effective way to represent the dynamics of clustered networks, and are increasingly used in research applications, both in epidemiology (where the EDA for non-hyper graphs has already seen broad use) and elsewhere (John Higham and de Kergorlay, 2022; Antelmi et al., 2020; Schwob et al., 2021). Extending the EDA to hypergraphs allows researchers working with these models to benefit from reduced burdens for data collection and improved estimation efficiency.

Fourth, we proposed an extension of the EDA to age-dependent dissolution hazards, proving the validity of the exact form of the EDA for dyad-independent models in this context. Age-dependent dissolution hazards allow each free dyad to have an essentially arbitrary tie duration distribution, rather than being constrained to follow a geometric distribution as in the constant-hazard case. This allows for more realistic durational modeling in a variety of applications. For example, in Goldstein et al. (2017), a mean edge duration of 12 is used to model a typical 12-hour shift for a healthcare worker, with the “at-risk” period (i.e., edge presence) lasting for the entire shift. Using a constant-hazard model to represent the risk of a shift ending produces a geometric distribution of shift lengths (i.e., tie durations) with more 1-hour shifts than 12-hour shifts. That is clearly not a desirable outcome. A general age-dependent hazard function allows the durational distribution to instead be centered around 12 hours, for example. When applying the EDA with age-dependent dissolution hazards, there is a need to estimate the hazard function(s) from available data on tie age and/or tie duration; the extensive survival analysis literature provides methods for doing this.

In proving asymptotic exactness of the EDA for dyad-dependent models in Section 3.3, we showed that under broad conditions, the cross-sectional equilibrium distribution of the EDA STERGM converges to an ERGM distribution in the continuous-time limit. This helps to place EDA STERGMs into the wider literature of temporal network models. There are many different types of temporal network models, both discrete and continuous time, and their equilibrium properties are of considerable interest when they are used for simulation and projection over time. In a recent review paper (Butts, 2023) Butts notes that there are several continuous-time models with known general cross-sectional ERGM equilibria, including LERGMs (Snijders and Koskinen, 2012) and certain SAOMs (Snijders, 2001). What distinguishes the behavior of the EDA from that of these LERGMs and SAOMs is that the EDA reproduces (asymptotically) not only the cross-sectional ERGM equilibrium distribution but also independently specified edge duration targets, which can be useful in a range of applied research settings.

Obtaining satisfactory results when applying the EDA to real data requires first finding an ERGM that provides a reasonable model for the distribution of cross-sectional network states. There is by now an extensive literature on using ERGMs to model cross-sectional data arising from various network processes; we refer the reader to Ghafouri and Khasteh (2020), Goodreau et al. (2009), and Krivitsky and Morris (2017). The processes of interest must be reflected in the ERGM in order to be captured by the EDA STERGM. The EDA also carries an equilibrium assumption that will not be tenable in all cases, but this assumption is not an issue for many applications, as evidenced by the existing uses of the EDA cited in the introduction.

Ultimately, EDA STERGMs are less general than arbitrary TERGMs (or even arbitrary STERGMs), but they achieve an increasingly useful balance of theoretical model flexibility and practical model estimability from limited data. We hope the present work encourages their broader use and further exploration.