HOPSENeurIPS 2026

NeurIPS 2026

HOPSE

Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations

Guillermo Bernárdez1*, Marco Montagna2*, Louis Van Langendonck3*, Martin Carrasco4, Amirreza Akbari5, Louisa Cornelis1, Mathilde Papillon1, Pere Barlet-Ros3, Nina Miolane1, Lev Telyatnikov6

1 UC Santa Barbara · 2 Sapienza University of Rome · 3 Universitat Politècnica de Catalunya · 4 University of Fribourg · 5 Aalto University · 6 Capital One · * Equal contribution

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Overview

Higher-order expressivity, without higher-order message passing

Higher-order message passing (HOMP) models multi-way interactions but scales poorly. HOPSE computes topologically informed positional and structural encodings once, on Hasse graphs, then trains message-passing-free.

Abstract

While Graph Neural Networks (GNNs) have proven highly effective at modeling relational data, pairwise connections cannot fully capture multi-way relationships naturally present in complex real-world systems. In response to this, Topological Deep Learning (TDL) leverages more general combinatorial representations —such as simplicial or cellular complexes— to accommodate higher-order interactions. Existing TDL methods often extend GNNs through Higher-Order Message Passing (HOMP), but face critical scalability challenges due to the steep complexity overhead of propagating messages through combinatorial structures.

To overcome this limitation, we propose HOPSE (Higher-Order Positional and Structural Encoder), a framework free of learnable message passing layers that uses Hasse graph decompositions to derive efficient and expressive encodings over arbitrary higher-order domains. Notably, HOPSE training and inference scale linearly with the size of combinatorial representations while preserving the expressive power and permutation equivariance of the HOMP approaches. Experiments on molecular and topological benchmarks show that it matches or surpasses state-of-the-art performance while consistently achieving speedups over HOMP-based models, opening a new path for scalable TDL. The code is available at github.com/geometric-intelligence/topobench.git.

Research questions

  1. Q1Does HOPSE match or exceed the predictive performance of state-of-the-art HOMP networks on standard and fine-grained molecular benchmarks?See results →
  2. Q2Can a message-passing-free architecture effectively capture complex, pure topological properties—such as Betti numbers and orientability—on synthetic benchmarking tasks?See results →
  3. Q3To what extent does the reduced complexity of HOPSE, relative to HOMP-based methods, yield practical improvements in execution cost and scalability on standard TDL benchmarks?See results →
  4. Q4How do the choices of neighborhood configurations (NC\mathcal{N}_C) and specific positional/structural encodings influence the framework’s performance across different topological domains?See results →

Background

Complexes, neighborhoods and Hasse graphs

Combinatorial complexes hold cells of several ranks (nodes, edges, faces). Each neighborhood function turns a complex into an ordinary directed graph, its strictly augmented Hasse graph, on which any graph encoding can run.

Combinatorial complex

A triple T=(V,C,rk)\mathcal{T}=(\mathcal{V},\mathcal{C},\mathrm{rk}): nodes V\mathcal{V}, cells C⊆P(V)∖∅\mathcal{C}\subseteq\mathcal{P}(\mathcal{V})\setminus\emptyset, and an order-preserving rank function. A cell of rank kk is a kk-cell; it subsumes graphs, simplicial and cell complexes [1].

Neighborhoods

A neighborhood function N:C→P(C)\mathcal{N}:\mathcal{C}\to\mathcal{P}(\mathcal{C}) maps each cell to its neighbors: adjacencies At,s\mathcal{A}_{t,s} link cells of the same rank tt through cells of rank ss; incidences Is→t\mathcal{I}_{s\to t} link rank ss to rank tt [1].

Strictly augmented Hasse graph

GN=(CN,EN)G_{\mathcal{N}}=(\mathcal{C}_{\mathcal{N}},\mathcal{E}_{\mathcal{N}}) with CN={σ∣N(σ)≠∅}\mathcal{C}_{\mathcal{N}}=\{\sigma\mid\mathcal{N}(\sigma)\neq\emptyset\} and EN={(τ,σ)∣τ∈N(σ)}\mathcal{E}_{\mathcal{N}}=\{(\tau,\sigma)\mid\tau\in\mathcal{N}(\sigma)\} [2]. Strictly: only cells with at least one neighbor are kept.

Target: nodes

Target: edges

Target: faces

nodeedgeface

I0→1\mathcal{I}_{0\to 1}nodes → edges

12 directed edges6 of 6 target edges have neighbors

In neighborhood sets Inc-1Inc-2used in the Method pipeline

Why higher-order message passing scales poorly

A rank-RR complex admits 2(R+1)R2(R+1)R neighborhood functions, and HOMP models must choose how to compose them into message passing routes [3, 4, 5]. Restricting to minimal routes, which update every rank exactly once, already leaves (R+1)!(R+1)! options; on top of this search space, propagating messages is computationally demanding. HOPSE avoids composing routes: it selects an unordered set of neighborhoods, encodes each Hasse graph once before training, and needs no message passing during training.

Complex rank R
12neighborhood functions, 2(R+1)R2(R+1)R
6minimal message passing routes, (R+1)!(R+1)!

R = 2 is the 2-dimensional case of the paper’s Figure 3: 12 neighborhoods.

Method

Encode Hasse graphs of selected neighborhoods once, then train only MLPs

Every structure-dependent computation happens before training. Steps A–C run once per dataset with no gradients; steps D and E are MLPs applied to each cell independently, and step F is the task readout.

Step

Select a step, or click a label under the figure.

Figure: the HOPSE pipeline (paper Figure 2).

Two realizations

Handcrafted or pretrained encoders

The choice of encoders G defines the two realizations, trading preprocessing cost against representational richness. Both aggregate with concatenation and use MLPs for every learnable map.

HOPSE-M

Handcrafted graph encodings: LapPE, ElectrostaticPE, RWSE, HKdiagSE (connectivity-only) or HKFE, KHopFE, PPRFE (feature-aware).

HOPSE-M-C connectivity-only

  • LapPE [6]Laplacian eigenvectorsThe first ii nontrivial eigenvectors of the graph Laplacian, providing spectral positional information.[ϕ1,…,ϕi]∈Rn×i[\phi_1,\dots,\phi_i]\in\mathbb{R}^{n\times i}
  • ElectrostaticPE [7]Electrostatic potentialComputed by solving the discrete Poisson equation on GG, capturing global relative node positioning.
  • RWSE [8]Random-walk structureDiagonal entries of the tt-step transition matrices PtP^t for t=1,…,Kt = 1,\dots,K, capturing local cluster structures.[(Pt)vv]t=1,…,K\big[(P^t)_{vv}\big]_{t=1,\dots,K}
  • HKdiagSE [9]Heat-kernel diagonalCaptures multiscale diffusion properties via the Laplacian eigenvalues λj\lambda_j.[∑j=1ne−λjt ϕj2]t=1,…,K\Big[\textstyle\sum_{j=1}^{n} e^{-\lambda_j t}\,\phi_j^2\Big]_{t=1,\dots,K}

HOPSE-M-F feature-aware

  • HKFE [10]Heat-kernel featuresApplies the heat-kernel diffusion operator to the node features across a range of diffusion scales, smoothing features over spectral neighborhoods.Xt=e−tLXX_t = e^{-tL}X
  • KHopFE [11]K-hop featuresPropagates node features through kk steps of the symmetric normalized adjacency, with K∈{1,2,3}K\in\{1,2,3\}.A^kX,    A^=D−1/2AD−1/2\hat A^{k}X,\;\; \hat A = D^{-1/2}AD^{-1/2}
  • PPRFE [12]Personalized PageRank featuresA multiscale approximate personalized PageRank that balances local feature preservation and global topology for various teleport probabilities α\alpha.(I−(1−α)A^) Z=αX(I-(1-\alpha)\hat A)\,Z = \alpha X

Cost: exact spectral encodings (LapPE, exact HKdiagSE) need an eigendecomposition, O(V3)\mathcal{O}(V^3); exact PPRFE needs a matrix inversion, also O(V3)\mathcal{O}(V^3). The Chebyshev approximation of HKFE and APPNP power iteration for PPRFE scale as O(K∣E∣)\mathcal{O}(K|E|).

HOPSE-GPSE

A pretrained GPSE encoder [13] supplies the encodings, avoiding eigendecompositions.

  • A single encoder, G={g1}\mathbf{G}=\{g_1\} with g1=GPSE(⋅)g_1=\mathrm{GPSE}(\cdot), a GNN pretrained to produce positional and structural representations with standard message passing.
  • Following GPSE’s pre-training protocol, Hasse-graph cells are fed random noise, Fr,k∼N(0,I)∈R20F_{r,k}\sim\mathcal{N}(0,I)\in\mathbb{R}^{20}; the cells’ original attributes are preserved and injected downstream (step D).
  • Encoders pretrained on ZINC and MolPCBA are evaluated; the corpus is treated as a categorical hyperparameter (ablation).
  • Suited to large complexes with many cells, offering a scalable alternative without significant loss in expressiveness. GPSE has its own limits in expressivity and generalization; it showcases pretrained models as encoders.

Learnable maps f, ε, Θ

LL-layer MLPs with independent parameters and ResNet-style layers, LayerNorm and LeakyReLU:

hr,k(ℓ)=σ(LayerNorm(hr,k(ℓ−1)Wr,k(ℓ)+br,k(ℓ))+hr,k(ℓ−1))h^{(\ell)}_{r,k} = \sigma\Big(\mathrm{LayerNorm}\big(h^{(\ell-1)}_{r,k}W^{(\ell)}_{r,k} + b^{(\ell)}_{r,k}\big) + h^{(\ell-1)}_{r,k}\Big)

Neighborhood sets

Which Hasse graphs to encode is a design choice. The paper evaluates adjacency-based and incidence-based sets of increasing size; the largest ones aggregate up and down relations, including second-order rank jumps.

Spectral encodings require symmetric operators, so Hasse graphs are made undirected before encoding: reciprocal incidences such as I0→1\mathcal{I}_{0\to1} and I1→0\mathcal{I}_{1\to0} collapse to the same graph.

Adj-1
{A0,1}\{\mathcal{A}_{0,1}\}
Adj-2
{A0,1,A0,2}\{\mathcal{A}_{0,1}, \mathcal{A}_{0,2}\}
Adj-3
{A0,1,A1,2,A0,2,A1,0,A2,1,A2,0}\{\mathcal{A}_{0,1}, \mathcal{A}_{1,2}, \mathcal{A}_{0,2}, \mathcal{A}_{1,0}, \mathcal{A}_{2,1}, \mathcal{A}_{2,0}\}
Inc-1
{I0→1,I0→2}\{\mathcal{I}_{0\to 1}, \mathcal{I}_{0\to 2}\}
Inc-2
{I0→1,I1→2,I0→2,I1→0,I2→1,I2→0}\{\mathcal{I}_{0\to 1}, \mathcal{I}_{1\to 2}, \mathcal{I}_{0\to 2}, \mathcal{I}_{1\to 0}, \mathcal{I}_{2\to 1}, \mathcal{I}_{2\to 0}\}

Theory

Linear in the number of cells, and more expressive than CCWL

Every structure-dependent computation happens once, before training, so a training step costs compute and memory in proportion to the number of cells, not to the relations between them. Under the conditions of Theorems 4.1 and 4.2, HOPSE also distinguishes non-isomorphic complexes at least as well as the CCWL test [2], sometimes strictly better, and is equivariant to cell permutations.

Complexity

Cost grows with cells, not with relations

CostMessage passingLGNNL_{\mathrm{GNN}} layers over the Hasse graphsHOPSELL-layer MLPs on precomputed encodings
Computeper training step
O(LGNN(∣E∣D+∣V∣D2))\mathcal{O}\big(L_{\mathrm{GNN}}(|E|D + |V|D^2)\big)

grows with the number of relations ∣E∣|E|

graph MPNN, Eq. 10

∑r=0RO(NrDr2(TK+1)L)\textstyle\sum_{r=0}^{R}\mathcal{O}\big(N_r D_r^2 (TK+1) L\big)

linear in the number of cells, independent of ∣E∣|E|

Eq. 7; on a graph O(L∣V∣D2K)\mathcal{O}(L|V|D^2K)

Memoryactivations kept for backpropagation
O(∑r=0R∣EGNr∣⋅Dr)\mathcal{O}\Big(\textstyle\sum_{r=0}^{R} |E_{G_{\mathcal{N}_r}}|\cdot D_r\Big)

one message per relation of the Hasse graphs

higher-order message passing, Appendix C

O(∑r=0RNr⋅Dr)\mathcal{O}\Big(\textstyle\sum_{r=0}^{R} N_r\cdot D_r\Big)

one activation per cell

Appendix C

NrN_r: number of rr-cells; ∣EGNr∣|E_{G_{\mathcal{N}_r}}|: relations (Hasse-graph edges) of the neighborhoods targeting rank rr; DrD_r: feature width; KK: encodings; TT: neighborhoods aggregated per rank.

On a graph HOPSE is cheaper than an MPNN whenever ∣E∣+∣V∣D>∣V∣DK⋅L/LGNN|E| + |V|D > |V|DK\cdot L/L_{\mathrm{GNN}} (Eq. 11): with a small number of encodings (e.g. K=4K=4 for HOPSE-M-C) it is significantly faster on dense graphs with ∣E∣≫∣V∣|E|\gg|V|. Its MLP depth is also independent of the graph diameter, whereas GNNs need depth that grows with the range of interactions.

In practice

Expressivity

Strictly more expressive than CCWL, and permutation equivariant

Theorem 4.1 Expressivity

HOPSE is more powerful than the CCWL test in distinguishing non-isomorphic combinatorial complexes if the functions εr(⋅)\varepsilon_r(\cdot), ψk(⋅)\psi_k(\cdot), fr,k(⋅)f_{r,k}(\cdot), and Θr(⋅)\Theta_r(\cdot) are injective for all k∈{1,…,K}k\in\{1,\dots,K\} and for all r∈{0,…,R}r\in\{0,\dots,R\} and gk(⋅)g_k(\cdot) is equally expressive as the WL test on the corresponding Hasse graph for at least one k∈{1,…,K}k\in\{1,\dots,K\}.

If some gk is as discriminative as WL on its Hasse graph, the injective maps carry every difference it detects through to the cell representations, so HOPSE is at least as powerful as CCWL. Strictness follows from a pair of complexes that CCWL cannot separate but HOPSE can (below).

Theorem 4.2 Equivariance

HOPSE is cell permutation equivariant if the functions εr(⋅)\varepsilon_r(\cdot), gk(⋅)g_k(\cdot), ψk(⋅)\psi_k(\cdot), fr,k(⋅)f_{r,k}(\cdot), and Θr(⋅)\Theta_r(\cdot) are cell permutation equivariant for all k∈{1,…,K}k\in\{1,\dots,K\} and all r∈{0,…,R}r\in\{0,\dots,R\}.

Compositions of equivariant functions are equivariant. In practice this holds because the MLPs are applied row-wise, with shared weights, across the rank-r cells.

A pair CCWL cannot tell apart

An icosahedron T1\mathcal{T}_1 and five disjoint tetrahedra T2\mathcal{T}_2. Under face adjacency A2,1\mathcal{A}_{2,1} both Hasse graphs are 3-regular on 20 nodes, so the WL test [14], and hence CCWL, assigns every node the same colour. A HOPSE encoding such as connected-component size separates them at once. Example adapted from Papillon et al. [2].

Colour nodes by

T1\mathcal{T}_1 · icosahedron

J1\mathcal{J}_1 · Hasse graph for A2,1\mathcal{A}_{2,1}

T2\mathcal{T}_2 · five tetrahedra

J2\mathcal{J}_2 · Hasse graph for A2,1\mathcal{A}_{2,1}

degree 3 everywhere: identical colours at every WL iterationNot distinguished

Beyond component counting: connected triangulated tori of equal area, such as T(4, 9) vs T(6, 6) and T(4, 12) vs T(6, 8), have 1-WL-equivalent 1-skeletons, yet LapPE and ElectrostaticPE computed on their higher-order Hasse graphs separate them across all eight standard neighborhoods, driven by differences in their Laplacian spectra.

Results

Competitive accuracy at a fraction of the training time

HOPSE is benchmarked under TopoBench’s standardized protocol [3] against GNNs (GCN [15], GAT [16], GIN [17]) and state-of-the-art HOMP networks (SCCNN [18], SANN [19], CWN [20], CCCN [21], TopoTune [2]) on eight molecular datasets [22, 23], lifted to simplicial and cell complexes, and on four MANTRA benchmarks [24] of purely topological tasks: homeomorphism type (NAME), orientability (ORIENT) and Betti numbers (β₁, β₂).

Accuracy

Best mean score on 7 of 12 benchmarks

Comparing the best HOPSE variant with the best baseline on each benchmark, HOPSE has the higher mean on MUTAG, NCI1, NCI109, BBB, Cl.Hep., β₁ and β₂. In the paper’s own markup, a HOPSE model is the top performer on MUTAG (HOPSE-M-C (cell)), NCI1 (HOPSE-GPSE (cell)), NCI109 (HOPSE-GPSE (cell)), BBB (HOPSE-GPSE (cell)), Cl.Hep. (HOPSE-GPSE (cell)), β₁ (HOPSE-M-C (simplicial)) and β₂ (HOPSE-M-C (simplicial)).

On the pure topological tasks HOPSE-M-C, a message-passing-free model, sets the best β₁ and β₂, is statistically tied with SCCNN on ORIENT and is a close second on NAME. GNN baselines were evaluated with and without the same encodings HOPSE uses, and Table 1 reports their best configuration (see ablation).

View
Show
Graph · molecular
MUTAG↑
78.1992.35
88.09±2.89
85.11±5.55
PROTEINS↑
74.5379.59
76.27±1.25
77.06±2.04
NCI1↑
75.9580.38
78.64±1.31
77.45±1.07
NCI109↑
75.6478.23
76.94±1.04
76.57±0.68
BBB↑
83.6587.33
86.11±0.86
85.07±1.06
CYP3A4↑
78.5680.46
79.03±0.29
79.71±0.57
Cl.Hep.↓
32.4134.76
33.07±0.43
33.93±0.60
Caco2↓
0.320.35
0.34±0.01
0.33±0.01
Simplicial · MANTRA
NAME↑
93.4196.13
93.96±0.29
95.43±0.44
ORIENT↑
80.4987.39
83.94±2.78
84.93±0.60
β1↑
89.3890.59
90.33±0.14
89.77±0.27
β2↑
78.0485.40
83.70±0.99
80.37±1.62

Best HOPSE variant vs best baseline (GNN or HOMP), test mean ± std over 5 seeds for the best validation configuration, on a per-benchmark scale. ↑ higher is better, ↓ lower is better. Hover a row for the models behind each value.

Training time vs HOMP

2.5× faster on average, up to 20×

HOPSE pays a one-time preprocessing cost (see preprocessing), then trains MLPs only. Each mark below is a HOMP model’s training time divided by that of the fastest HOPSE variant in the same domain; the geometric mean over all 40 comparisons is 2.55× end-to-end (2.25× per epoch), and the largest gap is SCCNN on ORIENT at 20.3×.

Measured against every model in the domain, a HOPSE variant is the fastest or not significantly slower in 16 of 20 dataset–domain columns of end-to-end time (fastest outright in 13), and in 20 of 20 for time per epoch. The end-to-end exceptions are PROTEINS (simplicial, fastest: TopoTune); Cl.Hep. (simplicial, fastest: TopoTune); BBB (cell, fastest: TopoTune); Cl.Hep. (cell, fastest: TopoTune).

View
Time
HOPSE
Graph · molecular
MUTAG
2.1×
PROTEINS
1.6×
NCI1
2.0×
NCI109
2.0×
BBB
1.9×
CYP3A4
4.2×
Cl.Hep.
1.1×
Caco2
2.2×
Simplicial · MANTRA
NAME
6.9×
ORIENT
11.4×
β1
4.8×
β2
4.8×
Fastest HOPSE Best TopoTuneSCCNNCWNCCCN

Time ratio to the fastest HOPSE variant in the same domain (simplicial or cell), log scale; the circle is the faster of TopoTune’s simplicial and cell runs; the right column is the geometric mean of a row’s ratios. Ratios use the paper’s reported means (Table 3 end-to-end, excluding preprocessing; Table 4 per epoch). β₁ and β₂ come from one training run, so their timings repeat.

Efficiency

A one-time preprocessing cost, and a parameter trade-off

HOPSE moves every structure-dependent computation into preprocessing and replaces message passing with MLPs over richer inputs. Both choices have a price, reported here in full.

Preprocessing

Encoding cost grows with the neighborhood set

Encodings are extracted once per dataset, before training. Moving from Adj-1 to Adj-3 or Inc-2 generally raises the overhead; in the paper’s markup Adj-1 is the fastest configuration in 20 of 20 dataset–domain columns. On the large MANTRA complexes, preprocessing takes between 203 s and 1690 s depending on variant and neighborhood set: a one-time overhead that the paper finds is heavily offset during training, increasingly so on larger datasets.

View
Variant
Domain
MUTAG
Adj-1 2.23
Adj-2 2.45
Adj-3 3.37
Inc-1 2.29
Inc-2 3.48
PROTEINS
Adj-1 20.45
Adj-2 26.07
Adj-3 52.61
Inc-1 120.36
Inc-2 355.84
NCI1
Adj-1 41.86
Adj-2 47.18
Adj-3 70.39
Inc-1 78.79
Inc-2 142.45
NCI109
Adj-1 41.71
Adj-2 47.22
Adj-3 70.40
Inc-1 80.29
Inc-2 144.25
BBB
Adj-1 20.24
Adj-2 22.65
Adj-3 33.46
Inc-1 24.93
Inc-2 42.48
CYP3A4
Adj-1 120.73
Adj-2 136.20
Adj-3 204.20
Inc-1 133.20
Inc-2 222.65
Cl.Hep.
Adj-1 12.95
Adj-2 14.68
Adj-3 21.51
Inc-1 15.96
Inc-2 26.65
Caco2
Adj-1 10.07
Adj-2 11.19
Adj-3 16.58
Inc-1 11.57
Inc-2 19.09
NAME
Adj-1 203.74
Adj-2 373.93
Adj-3 1141.67
Inc-1 395.12
Inc-2 1114.51
ORIENT
Adj-1 203.24
Adj-2 378.69
Adj-3 1143.81
Inc-1 394.77
Inc-2 1110.81
β1
Adj-1 203.81
Adj-2 372.46
Adj-3 1138.54
Inc-1 393.15
Inc-2 1112.50
β2
Adj-1 203.81
Adj-2 372.46
Adj-3 1138.54
Inc-1 393.15
Inc-2 1112.50

Total preprocessing time in seconds per neighborhood set (linear scale within each panel). The accent bar is the fastest set for the selected variant; a ring marks the configuration the paper bolds as fastest across all variants in the domain. MANTRA panels appear for the simplicial domain only.

Parameters

More parameters, less computation

Replacing neighborhood aggregation with rich encodings means HOPSE relies on more expressive, parameter-dense MLPs. In the simplicial domain HOPSE-M-C uses a median of 904k parameters against 774k for SCCNN. In the cell domain the gap is wider: HOPSE medians range from 565k to 910k against 349k for CWN, although all three variants use fewer parameters than CWN on NCI1 (HOPSE-M-C 916k, HOPSE-M-F 571k, HOPSE-GPSE 740k; CWN 1.15M), and HOPSE-M-C also does on Cl.Hep. Every model is evaluated at its own performance-optimized capacity, and GNN baselines tend to select smaller configurations.

View
Graph
GCN
74.0k
GAT
76.2k
GIN
77.7k
Simplicial
TopoTune
268.9k
SCCNN
773.8k
SANN
609.2k
HOPSE-M-F
1.15M
HOPSE-M-C
903.8k
HOPSE-GPSE
1.03M
Cell
TopoTune
148.8k
CWN
348.9k
CCCN
428.0k
HOPSE-M-F
565.3k
HOPSE-M-C
910.0k
HOPSE-GPSE
734.3k
HOPSE (one dot per dataset)baselinesmedian

Parameters of the best configuration found by grid search, log scale; β₁ and β₂ share one model. Values as printed in the paper’s Table 6 (GCN on β₁, β₂ is listed as 67.0).

Ablations

What matters, and where

Five studies isolate the design choices: which Hasse graphs to encode, which encodings, which GPSE pre-training corpus, and whether the same encodings would suffice to lift GNN or HOMP baselines.

Every plot shows the paper’s significance markup:
best (bold in the paper) not significantly different from best other ± std

Neighborhood sets · Table 7

Topological tasks need the full adjacency set

On NAME, ORIENT and the Betti numbers the richest adjacency set matters most: with Adj-3, which aggregates up and down relations across dimensions including second-order rank jumps, HOPSE-M-C holds the best mean on all four MANTRA benchmarks (93.96 on NAME, 83.94 on ORIENT), while its incidence-only sets fall far behind (Inc-2: 76.04 and 58.34). For molecular tasks, simpler sets such as Adj-1 and Adj-2 often perform competitively, acting as a regularizer while staying cheap; HOPSE-M-C reaches 88.09 on MUTAG (cell) with Adj-1 alone.

View
Model
Domain
MUTAG↑
Adj-1 84.68
Adj-2 86.38
Adj-3 85.11
Inc-1 77.02
Inc-2 77.02
PROTEINS↑
Adj-1 74.84
Adj-2 72.97
Adj-3 73.41
Inc-1 74.62
Inc-2 74.27
NCI1↑
Adj-1 72.67
Adj-2 72.90
Adj-3 72.74
Inc-1 67.30
Inc-2 68.05
NCI109↑
Adj-1 71.66
Adj-2 72.02
Adj-3 71.46
Inc-1 68.58
Inc-2 67.57
BBB↑
Adj-1 81.53
Adj-2 82.96
Adj-3 82.81
Inc-1 82.66
Inc-2 82.66
CYP3A4↑
Adj-1 74.27
Adj-2 74.21
Adj-3 75.00
Inc-1 74.20
Inc-2 74.73
Cl.Hep.↓
Adj-1 36.22
Adj-2 37.41
Adj-3 34.47
Inc-1 37.73
Inc-2 36.26
Caco2↓
Adj-1 0.41
Adj-2 0.42
Adj-3 0.47
Inc-1 0.37
Inc-2 0.37
NAME↑
Adj-1 91.13
Adj-2 90.98
Adj-3 93.96
Inc-1 76.08
Inc-2 76.04
ORIENT↑
Adj-1 79.96
Adj-2 80.43
Adj-3 83.94
Inc-1 56.26
Inc-2 58.34
β1↑
Adj-1 89.47
Adj-2 89.59
Adj-3 90.33
Inc-1 88.13
Inc-2 88.12
β2↑
Adj-1 78.60
Adj-2 82.62
Adj-3 83.70
Inc-1 56.37
Inc-2 56.37

Test mean ± std per neighborhood set, one panel per benchmark on its own scale; significance is computed within each model–domain block. TopoTune is included as a HOMP reference.

Single encodings · Table 8

LapPE unlocks the topological tasks; no encoding wins everywhere

Evaluated one at a time, encodings show dataset-dependent behaviour. On the pure topological tasks spectral information is decisive: with Adj-3, LapPE has the best mean on all four MANTRA benchmarks, reaching 92.39 on NAME, about 16 points above the feature-based encodings. On molecular data RWSE and ElectrostaticPE are strong; ElectrostaticPE is best on MUTAG and CYP3A4 with Adj-1, and PPRFE peaks on BBB with Adj-3. Single encodings occasionally beat combined ones, so encoding sets should be calibrated to the task.

View
Neighborhoods
MUTAG↑
LapPE 82.55
ElectrostaticPE 85.96
RWSE 86.38
HKdiagSE 81.28
HKFE 76.17
KHopFE 74.47
PPRFE 70.21
PROTEINS↑
LapPE 73.69
ElectrostaticPE 75.13
RWSE 73.69
HKdiagSE 76.34
HKFE 74.84
KHopFE 75.05
PPRFE 75.34
NCI1↑
LapPE 68.60
ElectrostaticPE 74.36
RWSE 76.30
HKdiagSE 73.81
HKFE 72.72
KHopFE 71.93
PPRFE 72.06
NCI109↑
LapPE 67.53
ElectrostaticPE 74.87
RWSE 75.51
HKdiagSE 71.15
HKFE 71.27
KHopFE 71.42
PPRFE 71.35
BBB↑
LapPE 80.74
ElectrostaticPE 83.30
RWSE 84.48
HKdiagSE 84.24
HKFE 84.98
KHopFE 84.88
PPRFE 85.12
CYP3A4↑
LapPE 74.56
ElectrostaticPE 77.74
RWSE 78.27
HKdiagSE 76.43
HKFE 74.62
KHopFE 73.83
PPRFE 75.83
Cl.Hep.↓
LapPE 36.47
ElectrostaticPE 34.52
RWSE 34.85
HKdiagSE 36.91
HKFE 36.68
KHopFE 37.60
PPRFE 35.65
Caco2↓
LapPE 0.46
ElectrostaticPE 0.36
RWSE 0.36
HKdiagSE 0.39
HKFE 0.39
KHopFE 0.38
PPRFE 0.39
NAME↑
LapPE 92.39
ElectrostaticPE 76.14
RWSE 76.04
HKdiagSE 76.08
HKFE 76.08
KHopFE 76.11
PPRFE 76.11
ORIENT↑
LapPE 74.87
ElectrostaticPE 56.29
RWSE 62.71
HKdiagSE 56.29
HKFE 56.29
KHopFE 56.29
PPRFE 56.29
β1↑
LapPE 90.06
ElectrostaticPE 88.63
RWSE 88.13
HKdiagSE 88.13
HKFE 88.13
KHopFE 88.13
PPRFE 88.13
β2↑
LapPE 82.74
ElectrostaticPE 68.10
RWSE 56.38
HKdiagSE 56.38
HKFE 56.38
KHopFE 56.38
PPRFE 56.38

Each panel compares the seven encodings used alone with the selected neighborhood set; significance is computed within each neighborhood block.

GPSE pre-training corpus · Table 9

No universal pre-training corpus

GPSE encoders [13] pre-trained on MolPCBA and on ZINC both provide useful encodings, but neither dominates: MolPCBA has the higher mean on 7 benchmarks, ZINC on 3, with 2 exact ties, and only on Cl.Hep. is the difference statistically significant. This task dependence is why the corpus is treated as a categorical hyperparameter.

View
MUTAG↑
MolPCBA 83.83
ZINC 81.28
PROTEINS↑
MolPCBA 75.13
ZINC 75.63
NCI1↑
MolPCBA 77.45
ZINC 78.64
NCI109↑
MolPCBA 76.05
ZINC 75.41
BBB↑
MolPCBA 84.68
ZINC 84.53
CYP3A4↑
MolPCBA 78.35
ZINC 79.03
Cl.Hep.↓
MolPCBA 33.07
ZINC 37.04
Caco2↓
MolPCBA 0.36
ZINC 0.37
NAME↑
MolPCBA 80.23
ZINC 79.98
ORIENT↑
MolPCBA 61.88
ZINC 59.41
β1↑
MolPCBA 88.13
ZINC 88.13
β2↑
MolPCBA 56.38
ZINC 56.38

PSEs in GNN baselines · Table 10

Encodings help GNNs, case by case

For a fair comparison, GCN [15], GAT [16] and GIN [17] are run without encodings (×), with feature encodings (F) and with connectivity encodings (C), and Table 1 reports their best configuration. Encodings boost the baselines on several datasets such as MUTAG and NCI1, but there is no single dominant type: C gives the largest MUTAG gains for GAT and GIN, while F works best for GCN on the same dataset.

View
Model
MUTAG↑
× none 68.94
F feature 85.53
C connectivity 85.11
PROTEINS↑
× none 75.13
F feature 74.05
C connectivity 75.41
NCI1↑
× none 70.86
F feature 72.51
C connectivity 74.69
NCI109↑
× none 71.21
F feature 73.44
C connectivity 73.34
BBB↑
× none 85.47
F feature 83.69
C connectivity 79.61
CYP3A4↑
× none 77.42
F feature 78.68
C connectivity 77.42
Cl.Hep.↓
× none 37.42
F feature 34.76
C connectivity 36.88
Caco2↓
× none 1.21
F feature 1.03
C connectivity 0.81
NAME↑
× none 22.80
F feature 21.59
C connectivity 21.76
ORIENT↑
× none 47.94
F feature 47.94
C connectivity 47.94
β1↑
× none 36.67
F feature 29.01
C connectivity 19.71
β2↑
× none 47.94
F feature 48.09
C connectivity 47.94

GIN is reported on the molecular benchmarks only in this table.

PSEs in HOMP baselines · Table 11

Adding the encodings to HOMP helps in 13 of 40 cases

Each HOMP baseline was re-run with its best-validation hyperparameters and rank-0 features augmented with CombinedPE. Gains are dataset-dependent rather than systematic: the mean improves for all five models on Caco2 and four of five on MUTAG, but rarely elsewhere, and no augmented baseline overtakes the best method of any column in Table 1, indicating that HOPSE’s advantage stems from its architecture, not merely from the encodings.

View
Model
MUTAG ↑
PROTEINS ↑
NCI1 ↑
NCI109 ↑
BBB ↑
CYP3A4 ↑
Cl.Hep. ↓
Caco2 ↓
CCCN
+3.83
−0.86
−2.14
−1.17
+1.28
−0.94
+0.75
−0.18
CWN
+2.98
−3.51
−2.39
−5.07
−0.69
−0.62
−0.62
−0.24
SCCNN
+8.51
−3.65
+1.48
−1.01
−0.39
−3.97
+0.92
−0.33
TopoTune (cell)
+5.11
−3.01
−0.90
−2.17
−2.22
−2.06
+0.32
−0.14
TopoTune (simplicial)
−2.98
−5.01
−1.94
−2.42
−1.82
+0.06
+0.57
−0.11

Change in test mean from Original to +All PSEs, in each benchmark’s own units; ▲ marks an improvement in the benchmark’s direction (↓ means lower is better), and stronger tint means a larger relative change. Hover for both values with their standard deviations.

Setup

Experimental protocol and reproducibility

All models run under TopoBench’s standardized protocol [3], with common encoders and readouts. HOPSE is integrated into the open-source TopoBench, including the hyperparameter search configurations and ablation scripts.

Datasets

TUDataset [22]
MUTAG, PROTEINS, NCI1, NCI109Inductive graph classification, kept for comparability with prior work.
Therapeutics Data Commons [23]
BBB, CYP3A4, Cl.Hep., Caco2Clinically relevant molecular property prediction (BBB_Martins, CYP3A4_Veith, Clearance_Hepatocyte_AZ, Caco2_Wang); Cl.Hep. and Caco2 are reported as errors (↓).
MANTRA [24]
NAME, ORIENT, β₁, β₂Purely topological tasks on 43,000+ manifold triangulations: homeomorphism type, orientability and Betti numbers.

Graph datasets are lifted to higher-order domains with standard cycle-based (cell) and clique-complex (simplicial) liftings.

Protocol

  • Five stratified splits per dataset with a 50/25/25 train/validation/test ratio.
  • The configuration with the best average validation score over five seeds is selected; test mean ± std is reported.
  • Metrics follow each task’s optimization direction: ↑ accuracy/F1, ↓ error.
  • Significance: best mean in bold; blue marks no statistically significant difference at the 95% level (two-sided Z-test for times).
  • Hardware: Linux, 256 logical cores, 1.5 TB RAM, 4 × NVIDIA H100 (94 GB).

Hyperparameter grid · all models, Adam

Encoder hidden dimension{128, 256}
Encoder dropout{0.25, 0.5}
Number of backbone layers{1, 2, 4}
Learning rate{0.01, 0.001}
Batch size{128, 256}
HOPSE-GPSE pre-training corpus{ZINC, MolPCBA}
SANN hop count{1, 2, 3}

Where each step lives in TopoBench

CWN under TopoBench vs its original protocol · Table 12

AspectOriginal CWNTopoBench (this paper)
Data split10-fold CV (90% train)50/25/25 train/val/test
Hyperparameter searchPer-dataset, bespokeShared 5-axis grid, all models
Learning rate scheduleModel-specificFixed across models
Encoders / readoutsModel-specificShared across models

Reported CWN [20] numbers match TopoBench’s reproducibility reports within one standard deviation. Reverting only NCI1’s split to CWN’s 10-fold cross-validation raised accuracy from 74.51 to 79.08 ± 2.50, recovering most of the gap to the originally reported 83.6; the constraints apply uniformly to HOPSE and every baseline.

Limitations

  1. Exact manual encodings such as LapPE can be expensive on massive datasets; sparse approximations or scalable neural encoders like GPSE are then preferable.
  2. Replacing topological message passing with independent MLPs yields a larger parameter footprint than baseline GNNs.
  3. The expressivity guarantees assume strict injectivity; practical instantiations with standard pooling or bounded MLPs approximate it.
  4. Spectral PSEs need symmetric operators, so Hasse graphs are made undirected and reciprocal incidences lose their direction, an effect not quantified here.
  5. The strict separation over CCWL is established analytically, as no controlled expressivity benchmark for combinatorial complexes exists yet.

Citation

Cite HOPSE

If HOPSE or its encodings are useful in your work, please cite the paper. Code, configurations and ablation scripts are part of TopoBench.

BibTeX
@inproceedings{bernardez2026hopse,
  title     = {{HOPSE}: Scalable Higher-Order Positional and Structural
               Encoder for Combinatorial Representations},
  author    = {Bern{\'a}rdez, Guillermo and Montagna, Marco and
               Van Langendonck, Louis and Carrasco, Martin and
               Akbari, Amirreza and Cornelis, Louisa and Papillon, Mathilde and
               Barlet-Ros, Pere and Miolane, Nina and Telyatnikov, Lev},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2026}
}
Paper on arXivNeurIPS 2026 on OpenReviewTopoBench on GitHub

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