---
kind: concept
slug: hypergraph-neural-network
title: Hypergraph neural network
summary: >-
  A hypergraph neural network passes messages over hyperedges, sets of nodes of any size, instead of only node pairs. Covered: one variant, GroupNet, which builds agent groups at several sizes for trajectory prediction. The trade-off: one paper, author-reported numbers, and a released code that covers one dataset only.
tags: [hypergraph, relational-reasoning, trajectory-prediction, multi-agent, message-passing]
papers: ["2204.08770"]
related: [graph-neural-network, motion-prediction, variational-autoencoder, relational-graph-convolution, graph-structure-learning]
status: draft
updated_at: 2026-11-05
---

## Overview

A hypergraph generalises a graph. An ordinary edge joins two nodes; a
hyperedge joins any number of nodes at once. A hypergraph neural network runs
message passing over those hyperedges, so one interaction can involve a whole
group rather than being split into pairs [arxiv:2204.08770 §1] [arxiv:2204.08770 §4.1].

The core idea shared by the variants is a two-way exchange. Nodes send their
states into each hyperedge, the hyperedge computes one interaction embedding
from its members, and the embedding is sent back to the nodes [arxiv:2204.08770 §4.2].
Only one variant is on wikillm so far, so the invariant below comes from that
single paper and is not yet checked across several.

## Variants

### GroupNet: multiscale hyperedges over agents

GroupNet infers hyperedges over the agents in a traffic or sports scene. The
hyperedges are built from pairwise affinities, which are cosine similarities
between agent embeddings [arxiv:2204.08770 §4.1] [arxiv:2204.08770 eq. 1]. For
each agent it creates one hyperedge per scale. A hyperedge at scale s holds that
agent plus the most affine agents, up to the scale's size. The exact choice
maximises the total affinity inside the group. It is solved by enumeration when
the number of agents is small, and by a greedy step otherwise
[arxiv:2204.08770 §4.1] [arxiv:2204.08770 eq. 2].

Each hyperedge then gives a strength r and a soft category c, produced from a
Gumbel-softmax. The interaction embedding is r times the category-weighted sum
of one MLP per category [arxiv:2204.08770 §4.2] [arxiv:2204.08770 eq. 3]. The
categories are learned without labels. The paper calls this "unsupervised
relational reasoning" [arxiv:2204.08770 §3].

GroupNet is used as the social encoder inside a CVAE trajectory predictor.
It can also be attached to existing predictors [arxiv:2204.08770 §5] [arxiv:2204.08770 §6.3].
Its card: [GroupNet: Multiscale Hypergraph Neural Networks for Trajectory Prediction with Relational Reasoning](/papers/2204.08770).

## When to use which

With one variant on the page, the choice is between GroupNet and not using a
hypergraph at all. The paper's own comparisons are the only evidence.

| Situation | Use | Why |
|---|---|---|
| Interactions are naturally groups, and you want the group structure learned from data | GroupNet-style hyperedges | The paper reports hyperedges as the best of its compared encoders on NBA at 4.0 s, 1.13/1.69 m against 1.17/1.76 m for a pairwise graph [arxiv:2204.08770 tab. 5] |
| Pairwise interactions are enough, or the scene has few agents | A pairwise graph or attention module | Pairwise and attention encoders are within about 0.04 m of GroupNet on NBA at 4.0 s in the same table [arxiv:2204.08770 tab. 5] |
| You need results on a dataset other than NBA, with official code | Check the card first | Released code covers NBA only [arxiv:2204.08770 §6.1]; SDD and ETH-UCY code is not in the release |
| You need latency or memory numbers | Measure yourself | Not stated in the cited sections |

Scale count matters. On NBA, adding scales improves error and then levels off.
One scale gives 1.27/1.95 m at 4.0 s. Scales 1, 2, 5 and 11 give 1.13/1.69 m
[arxiv:2204.08770 tab. 6] [arxiv:2204.08770 §6.4].

## Implementation sketch

Core of the hypergraph step, for one scene with N agents. It follows
[arxiv:2204.08770 §4.1] and [arxiv:2204.08770 §4.2]. Variant-specific parts
(the MLPs and the Gumbel temperature) are not given numerically in the paper.

```pseudocode
# Build one hyperedge per agent per scale  [arxiv:2204.08770 §4.1]
# Input:  queries (N, d); scales = [K_1, ..., K_S]
# Output: one 0/1 incidence matrix per scale

def build_hyperedges(queries, scales):
    # Cosine affinity between agents  [arxiv:2204.08770 eq. 1]
    unit = queries / norm(queries, axis=-1)                # (N, d)
    affinity = unit @ transpose(unit)                      # (N, N)
    incidences = []
    for k in scales:
        # Greedy form of eq. 2: each agent plus its k-1 most affine agents
        incidences.append(top_k_per_row(affinity, k))      # (N, N)
    return incidences
```

```pseudocode
# One message round over the hyperedges  [arxiv:2204.08770 §4.2] [eq. 3]
# Input:  agents (N, d); incidence (E, N); categories = L
# Output: updated agents (N, d)

def message_round(agents, incidence, category_fns, tau):
    # Pool members of each hyperedge  [arxiv:2204.08770 §4.2]
    member_sum = incidence @ agents                        # (E, d)
    # Strength r in (0, 1) and Gumbel-softmax category c  [§4.2]
    collective = pool_members(agents, member_sum, incidence)
    strength = sigmoid(mlp_r(collective))                  # (E,)
    category = gumbel_softmax(mlp_c(collective), tau)      # (E, L)
    # Interaction embedding r * sum_l c_l F_l(members)  [eq. 3]
    edge = zeros(E, d)
    for l in range(L):
        edge += category[:, l, None] * category_fns[l](member_sum)
    edge = strength[:, None] * edge                        # (E, d)
    # Hyperedge-to-node update  [arxiv:2204.08770 §4.2]
    edge_sum = transpose(incidence) @ edge                 # (N, d)
    return update_fn(concat(agents, edge_sum))             # (N, d)
```

Implementation notes for the sketch. The paper says message passing repeats
"several times" per scale without a count [arxiv:2204.08770 §4.2]. Its
Gumbel temperature and number of categories are not stated in the cited
sections. The categories are sampled again at every forward pass, so
predictions are random at test time unless the implementation fixes the noise.
The released code shows this behaviour; see the card for the code citation.

## Hyperparameters

Values stated in the paper for the NBA setup. Anything else is not stated in
the cited sections.

| Name | Value | Where |
|---|---|---|
| hyperedge sizes, NBA | 2, 5, 11 agents | [arxiv:2204.08770 §6.1] |
| hyperedge scales, ETH-UCY | half of the agents, and all agents | [arxiv:2204.08770 §6.1] |
| hyperedge scale, SDD | all agents within 100 px | [arxiv:2204.08770 §6.1] |
| optimiser, learning rate | Adam, 1e-4, decayed every 10 epochs | [arxiv:2204.08770 §6.1] |
| α, β, γ, λ | 1.0 each | [arxiv:2204.08770 §6.1] |
| latent dimension | 32 | [arxiv:2204.08770 §6.1] |
| MLP depth | 3 layers, ReLU | [arxiv:2204.08770 §6.1] |
| NBA input and output | 2.0 s observed, 4.0 s predicted | [arxiv:2204.08770 §6.3] |
| Gumbel temperature, categories, epochs, batch size | not stated | [arxiv:2204.08770 §4.2] [arxiv:2204.08770 §6.1] |

## Contradictions

- **Scale-0 links.** §4.1 says each node links to the nodes it is most affine
  to, which gives pairwise edges [arxiv:2204.08770 §4.1]. §6.1 says scale 0
  links each node to all other agents [arxiv:2204.08770 §6.1]. The two readings
  give different graphs, and the paper does not say which one its results used.
- **Scale identity.** §6.1 gives NBA scales 2, 5 and 11 [arxiv:2204.08770 §6.1].
  Table 6's best row is 1, 2, 5, 11, and the paper does not say which module
  corresponds to scale 2 [arxiv:2204.08770 tab. 6].
- **Relative gains.** The text reports GroupNet+CVAE lowers minADE20 by 6.5%
  against PECNet on SDD [arxiv:2204.08770 §6.3]. On the same table its minFDE20
  is higher than PECNet's [arxiv:2204.08770 tab. 3]. Read both numbers together.
- **Loss weighting.** §5 writes the three loss terms without normalisation
  [arxiv:2204.08770 §5]. The released code divides them by different counts,
  so the effective weights differ from the formula. See the card's Paper vs
  code section for the details and the code citations.

## Papers

- [GroupNet: Multiscale Hypergraph Neural Networks for Trajectory Prediction with Relational Reasoning](/papers/2204.08770): builds agent hyperedges at several sizes, learns each one's strength and category without labels, and uses them as the social encoder in a CVAE for trajectory prediction.