Hyperedge models

amorem supports relational hyper event models (RHEMs) — the extension of REMs from dyadic (sender, receiver, time) events to set-valued (I, J, time) hyperedges, where I is a set of senders and J is a set of receivers. Following Boschi, Lerner & Wit (2025), this lets a single event involve multiple actors per axis — e.g. a paper authored by a co-author set citing a set of references, or a multi-actor meeting (J = ∅).

Driver script for the experiments below: paper/wiki/experiments/hyper_demo.R.

Data model

hl <- hyperedge_log(
  I    = list(c("alice", "bob"), c("alice", "carol"), c("bob", "carol")),
  J    = list(c("paperA"),        c("paperA", "paperB"), c("paperB")),
  time = c(1.0, 2.5, 4.0))

is_hyperedge_log(hl)         # TRUE
hyperedge_sizes(hl)
#>              I              J time size_I size_J
#> 1   alice, bob         paperA  1.0      2      1
#> 2 alice, carol paperA, paperB  2.5      2      2
#> 3   bob, carol         paperB  4.0      2      1

Function	Purpose
`hyperedge_log(I, J, time)`	Validating constructor; sorts by time
`is_hyperedge_log(x)`	Structural predicate
`as_hyperedge_log(event_log)`	Promote a dyadic log (each row becomes a singleton on each axis)
`as_dyadic_log(h)`	Inverse — succeeds only when `\|I\| = \|J\| = 1` everywhere
`hyperedge_sizes(h)`	Adds `size_I` and `size_J` columns

For undirected hyperevents (e.g. multi-actor meetings), pass J = list(character(0), ...) and use the actor-only simulator below.

Subset repetition

The canonical hyperedge-native covariates from the paper (eq. 3-4):

hyperedge_activity(hl, I = c("alice", "bob"), J = "paperA", t = 5)
#> [1] 1                # one past event contains {alice, bob} and {paperA}

hyperedge_subrep(hl, I = c("alice", "bob", "carol"), J = "paperA",
                 t = 5, rho = 2, l = 1)
#> [1] 0.6666667        # 2 of the 3 size-2 subsets of I have co-fired with paperA

For dyadic events with |I| = |J| = 1, hyperedge_subrep(rho = 1, l = 1) reduces to the standard dyad event count.

Feature engine

hyperedge_features() is the hyperedge analogue of endogenous_features():

feat <- hyperedge_features(hl,
  stats = c("activity", "subrep_1_1", "subrep_2_1"))

Stat name	Meaning
`"activity"`	`hyperedge_activity` on each row’s `(I, J)`
`"subrep_<rho>_<l>"`	directed subset repetition
`"subrep_<rho>"`	undirected subset repetition (`l = 0`)
any name in the 68-stat catalogue	delegated to `endogenous_features()` after `as_dyadic_log()` — requires every row to be dyadic

Two-mode directed simulator

For citation-style data (Boschi et al. 2025 Section 5):

hl <- simulate_directed_hyperedge_events(
  n_events  = 50,
  senders   = paste0("author", 1:5),
  receivers = paste0("paper",  1:5),
  max_size_I = 2, max_size_J = 2,
  baseline_rate = 0.3,
  endogenous_stats   = c("subrep_1_1", "size_I"),
  endogenous_effects = c(subrep_1_1 = 0.8, size_I = -0.4))

Supported endogenous stats: "size_I", "size_J", "activity", "subrep_<rho>_<l>". Output is a hyperedge_log() with non-empty I and J on every row.

What the two endogenous effects do

A 150-event sweep on 4 senders × 4 receivers, varying the endogenous terms one at a time:

Specification	mean `\|I\|`	mean `\|J\|`
No penalty	1.97	1.60
`size_I = -0.6`	1.66	1.59
`subrep_1_1 = 0.8`	1.19	1.05
Both	1.07	1.14

Cardinality distributions

size_I < 0 broadens the size distribution toward smaller sender sets (|I| = 1 jumps from 33 % to 46 %).
subrep_1_1 > 0 collapses the distribution onto pairs the log has already seen — and most repeating pairs are singletons, so |I| = 1 dominates (85 % under the subrep-only spec).
Both combine: 94 % singleton sender sets.

This is exactly the levers the Boschi et al. (2025) framework exposes for shaping a citation-style stream.

Per-event cost is exponential in |V^I|, |V^J| because the simulator enumerates every candidate (I, J) per event; practical for small universes (≤ ~10 each side, max_size ≤ 3).

Undirected meeting simulator

For undirected multi-actor meetings (Section 4 of the paper):

hl <- simulate_hyperedge_events(
  n_events           = 150,
  actors             = LETTERS[1:5],
  max_size           = 3,
  baseline_rate      = 0.3,
  endogenous_stats   = c("subrep_2", "size"),
  endogenous_effects = c(subrep_2 = 0.6, size = -0.3))

The simulator enumerates every subset of actors of size min_size..max_size, scores each via the rolling event history, draws an event proportional to its exp-score, and waits an exponential time with rate equal to the total score. Output: a hyperedge_log() with empty receiver sets.

Preferential attachment from `subrep`

When subrep > 0, a positive feedback loop locks in: a subset that fires once becomes more likely to fire again. Tracking the running hyperedge_activity of two focal pairs across a 150-event meeting stream:

Running activity of a focal pair

One 2-actor pair ({C, E}) accumulates almost all the co-attendance mass; the other ({A, B}) gets one event in the entire stream. Under subrep_2 = 0.6, 146 of the 150 meetings contain the same pair. This is the kind of dynamic the subrep_<rho> family is designed to detect and the kind of runaway you should expect (or design against) when the effect is strong.

Smooth effects on hyperedge data

Once features are computed, you can fit linear / TV / NL / TVNL specifications via compare_models_smooth() — see Estimation. The case-control likelihood used by compare_models_smooth() matches Boschi et al. (2025) equation 8.

Hyperedge models