Skip to contents

Durational Event Model (DEM) Estimation

Source: R/dem.R
dem.Rd

Estimates a Durational Event Model (DEM) for relational event sequences where interactions have a duration.

Usage

dem(
  events,
  training_start = 0,
  exogenous_end = NULL,
  formula_0_1 = NULL,
  formula_1_0 = NULL,
  n_nodes = NULL,
  directed = FALSE,
  estimate_0_1 = NULL,
  estimate_1_0 = NULL,
  coef_0_1 = NULL,
  coef_1_0 = NULL,
  semiparametric = FALSE,
  simultaneous_interactions = TRUE,
  control = control.redeem()
)

Arguments

events

A matrix of events with columns time, from, to, and type (1 for start, 0 for end, 3 for exogenous changes).

training_start

Numeric; the time point at which to start the estimation. Defaults to 0.

exogenous_end

Numeric; the exogenous time point at which the observational period ends. Defaults to NULL, which implies that time when the final event was observed is taken as the end of the observational period.

formula_0_1

A one-sided formula specifying the sufficient statistics for the formation process (\(0 \rightarrow 1\)). The right-hand side must be composed of terms from redeem_terms. For example: ~ inertia() + degree. An intercept (~ 1) is the minimal specification. Defaults to NULL.

formula_1_0

A one-sided formula specifying the sufficient statistics for the dissolution process (\(1 \rightarrow 0\)). The right-hand side must be composed of terms from redeem_terms. An intercept (~ 1) is the minimal specification. Defaults to NULL.

n_nodes

Integer; the total number of actors in the network. If NULL (default), it is automatically identified based on the actors in the events set.

directed

Logical; whether the interaction events are directed. Defaults to FALSE.

estimate_0_1

Logical; whether to estimate the formation process. Defaults to NULL, in which case it is estimated if formula_0_1 is provided.

estimate_1_0

Logical; whether to estimate the dissolution process. Defaults to NULL, in which case it is estimated if formula_1_0 is provided.

coef_0_1

Numeric vector; initial coefficients for the formation model. If provided, this must be a concatenated vector of:

  1. Core coefficients: values for sufficient statistics in the formula.

  2. Degree coefficients (if degree is in the formula): a vector of length n_nodes (undirected) or 2 * n_nodes (directed, sender effects first then receiver effects).

  3. Baseline coefficients (if temporal changepoints are present): a vector of length equal to the number of baseline intervals (equal to number of changepoints if an intercept/degree is present, or changepoints + 1 if neither is present).

Defaults to NULL, in which case default starting values are automatically computed.

coef_1_0

Numeric vector; initial coefficients for the dissolution model. If provided, this must be a concatenated vector of:

  1. Core coefficients: values for sufficient statistics in the formula.

  2. Degree coefficients (if degree is in the formula): a vector of length n_nodes (undirected) or 2 * n_nodes (directed, sender effects first then receiver effects).

  3. Baseline coefficients (if temporal changepoints are present): a vector of length equal to the number of baseline intervals (equal to number of changepoints if an intercept/degree is present, or changepoints + 1 if neither is present).

Defaults to NULL, in which case default starting values are automatically computed.

semiparametric

Logical; whether to use a semiparametric baseline. Defaults to FALSE. See the 'Semiparametric Baseline' section for details.

simultaneous_interactions

Logical; whether to allow simultaneous interactions (i.e. multiple active events for the same actor or dyad at the same time). Defaults to TRUE.

control

A list of control parameters from control.redeem. Defaults to control.redeem().

Value

An object of class dem_object containing model estimates, log-likelihoods, and preprocessed data. See dem_object for details on the components of the returned object and S3 methods.

Details

The Durational Event Model (DEM) is a general framework for analyzing durational events, extending standard Relational Event Models (REM) by decoupling the modeling of event incidence from event duration. It characterizes the dynamics via two separate continuous-time counting processes:

Formation Process (\(0 \rightarrow 1\))

Counts the number of times that actor pair \((i,j)\) starts an interaction up to time \(t\). The incidence intensity is denoted by \(\lambda_{i,j}^{0\rightarrow 1}(t | \mathscr{H}_t)\).

Dissolution Process (\(1 \rightarrow 0\))

Counts the number of times that actor pair \((i,j)\) stops interacting up to time \(t\). The dissolution intensity is denoted by \(\lambda_{i,j}^{1\rightarrow 0}(t | \mathscr{H}_t)\).

Under the assumption that the processes are non-homogeneous Poisson processes, the intensities are modeled as: $$\lambda_{i,j}^{0\rightarrow 1}(t | \mathscr{H}_t, \theta^{0\rightarrow 1}) = \exp(s_{i,j}^{0\rightarrow 1}(\mathscr{H}_t)^\top \alpha^{0\rightarrow 1} + \beta_i^{0\rightarrow 1} + \beta_j^{0\rightarrow 1} + f(t, \gamma^{0\rightarrow 1}))$$ $$\lambda_{i,j}^{1\rightarrow 0}(t | \mathscr{H}_t, \theta^{1\rightarrow 0}) = \exp(s_{i,j}^{1\rightarrow 0}(\mathscr{H}_t)^\top \alpha^{1\rightarrow 0} + \beta_i^{1\rightarrow 0} + \beta_j^{1\rightarrow 0} + f(t, \gamma^{1\rightarrow 0}))$$ where:

  • \(s_{i,j}(\mathscr{H}_t)\) is a vector of dynamic network statistics capturing the history of past interactions \(\mathscr{H}_t\).

  • \(\alpha\) is a parameter vector determining the covariate effects.

  • \(\beta_i\) and \(\beta_j\) are actor-specific sociality/popularity parameters (degree correction) capturing actor heterogeneity.

  • \(f(t, \gamma)\) is a piecewise-constant step function modeling temporal baseline fluctuations across a set of changepoints.

To satisfy the Feller criterion and ensure that the continuous-time counting process remains non-explosive, count-based network statistics (such as inertia or common partners) are typically log-transformed on the \(\log(x + 1)\) scale.

Scalable Estimation Algorithm

The likelihood of the model is separable with respect to \(\theta^{0\rightarrow 1}\) and \(\theta^{1\rightarrow 0}\), allowing independent estimation of the incidence and duration components. Traditional maximum likelihood estimation via standard Newton-Raphson requires computing and inverting an \(O(N^2)\) Hessian matrix, which is computationally prohibitive for larger networks. To bypass this, the redeem package implements a highly scalable block-coordinate ascent algorithm that separates parameter updates:

  1. Step 1: Update covariate parameters \(\alpha\) using a standard Newton-Raphson update.

  2. Step 2: Update high-dimensional actor popularity baselines \(\beta\) using Minorization-Maximization (MM) steps, avoiding explicit matrix inversion.

  3. Step 3: Update baseline step function parameters \(\gamma\) via a closed-form step.

More information is provided in Fritz et at. (2026).

Semiparametric Baseline

When semiparametric = TRUE, the baseline rates for both the formation (\(0 \rightarrow 1\)) and dissolution (\(1 \rightarrow 0\)) processes are left completely unspecified. Both processes are estimated as separate Cox proportional hazards models using the survival package. In this path:

  • For the formation process, each start event (1) is treated as a failure, and all inactive dyads at that time constitute the risk set.

  • For the dissolution process, each end event (0) is treated as a failure, and all currently active interactions constitute the risk set.

  • The exact waiting times (durations of the active and inactive states) are conditioned out, and estimation is based solely on the ordering of events and the relative dyadic intensities at each transition time.

  • This approach is highly robust to arbitrary temporal fluctuations in baseline rates since no piecewise-constant temporal baselines or changepoints need to be specified.

  • Limitations: This path does not support the specialized scalable estimation of sender/receiver popularity effects (degree) or piecewise-constant temporal baselines.

References

Fritz, C., Rastelli, R., Fop, M., & Caimo, A. (2026). Scalable Durational Event Models: Application to Physical and Digital Interactions. arXiv:2504.00049.

Examples

# Simulate some durational data
n <- 20
events <- matrix(c(
  1.2, 1, 5, 1,
  2.5, 1, 5, 0,
  3.1, 2, 8, 1,
  4.4, 2, 8, 0
), ncol = 4, byrow = TRUE)
colnames(events) <- c("time", "from", "to", "type")

# Estimate a simple DEM
fit <- dem(
  events = events,
  n_nodes = n,
  formula_0_1 = ~1,
  formula_1_0 = ~1,
  control = control.redeem(estimate = "Blockwise")
)
summary(fit)
#> Call:
#> dem(events = events, formula_0_1 = ~1, formula_1_0 = ~1, n_nodes = n, 
#>     control = control.redeem(estimate = "Blockwise"))
#> 
#> Results for Incidence Intensity (0 -> 1): 
#> Fixed Effects:
#>           Estimate Std. Error t value  Pr(>|t|)    
#> Intercept  -6.0324     0.7071 -8.5311 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Log-likelihood: -14.065 
#> 
#> Results for Duration Intensity (1 -> 0): 
#> Fixed Effects:
#>           Estimate Std. Error t value Pr(>|t|)
#> Intercept -0.26236    0.70711  -0.371   0.7106
#> 
#> Log-likelihood: -2.525 
#> 
#> Combined Model Fit:
#>   AIC: 37.17892 
#>   BIC: 35.95151 
#> 
#> Total estimation time: 0.009922266 secs