Generating Directed Graphs with Dual Attention and Asymmetric Encoding
Authors: Alba Carballo-Castro
Manuel Madeira
Yiming QIN
Dorina Thanou
Pascal Frossard
Venue: ICLR 2026 Poster (2026)
Keywords: graph generation
directed graphs
flow matching
discrete diffusion
Primary area: generative models

Abstract: Directed graphs naturally model systems with asymmetric, ordered relationships, essential to applications in biology, transportation, social networks, or visual understanding. Generating such graphs enables simulation, data augmentation and novel instance discovery; however, this task remains underexplored. We identify two key reasons: first, modeling edge directionality introduces a substantially larger dependency space, making the underlying distribution harder to learn; second, the absence of standardized benchmarks hinders rigorous evaluation. Addressing the former limitation requires more expressive models that are sensitive to directional topologies. Thus, we propose Directo, the first generative model for directed graphs built upon the discrete flow matching framework. Our approach combines: (i) a dual-attention mechanism distinctly capturing incoming and outgoing dependencies, (ii) a robust, discrete generative framework, and (iii) principled positional encodings tailored to asymmetric pairwise relations. To address the second limitation and support evaluation, we introduce a novel and extensive benchmark suite covering synthetic and real-world datasets. Experiments show that our method outperforms existing directed graph generation approaches across diverse settings and competes with specialized models for particular classes, such as directed acyclic graphs. These results highlight the effectiveness and generality of our approach, establishing a solid foundation for future research in directed graph generation.

Published as a conference paper at ICLR 2026
GENERATING DIRECTED GRAPHS WITH DUAL ATTENTION AND ASYMMETRIC ENCODING
Alba Carballo-Castrg, Manuel Madeira, Yiming Qin, Dorina Thanou & Pascal Frossard LTS4, EPFL., Lausanne, Switzerland
ABSTRACT
Directed graphs naturally model systems with asymmetric, ordered relationships, essential to applications in biology. transportation, social networks, or visual under- standing. Generating such graphs enables simulation, data augmentation and novel instance discovery; however, this task remains underexplored. We identify two key reasons: first, modeling edge directionality introduces a substantially larger dependency space, making the underlying distribution harder to learn; second, the absence of standardized benchmarks hinders rigorous evaluation. Addressing the former limitation requires more expressive models that are sensitive to directional topologies. Thus, we propose DIRECTO, the first generative model for directed graphs built upon the discrete flow matching framework. Our approach combines: (i) dual-attention mechanism distinctly capturing incoming and outgoing depen- dencies, (ii) robust, discrete generative framework, and (iii) principled positional encodings tailored to asymmetric pairwise relations. To address the second limita- tion and support evaluation, we introduce novel and extensive benchmark suite covering synthetic and real-world datasets. Experiments show that our method out- performs existing directed graph generation approaches across diverse settings and competes with specialized models for particular classes, such as directed acyclic graphs. These results highlight the effectiveness and generality of our approach, establishing a solid foundation for future research in directed graph generation.
INTRODUCTION
Directed graphs (digraphs) model systems with asymmetric relationships, capturing essential struc- tures such as flows, dependencies, and hicrarchies that arise in many real-world applications. This makes digraphs particularly well-suited for problems in diverse domains including biology (Li et al., 2006; Takane et al., 2023; Wei et al., 2024), transportation (Concas et al., 2022), social dynam- ies (Schweimer et al., 2022), and, more recently, image and video understanding (Chang et al., 2023; Rodin et al., 2023), where structured and directional representations are critical for interpretation and reasoning. Consequently, generating digraphs central to tasks such as simulation, data augmentation and novel instance discovery in domains with directional structure.
Graph generative models have shown strong potential in diverse applications. including drug dis- covery (Mercado et al., 2020; Vignac et al., 2023b), finance (Li et al., 2023), social network model- ing (T'sai et al., 2023), and medicine (Nikolentzos et al., 2023). However, most generative models focus on undirected graphs (Liao et al., 2019; Martinkus et al., 2022; Vignac et al., 2023a; Siraudin et al., 2024; Xu et al., 2024; Eijkelboom et al., 2024; Qin et al., 2025a), with only limited efforts on directed settings where recent works propose auto-regressive models for DAGs (Zhang et al., 2019; Li et al., 2025) and, more recently, for general digraphs (Law et al., 2025). We identify two main factors limiting progress in this direction. First, at the modeling level, the directed setting is inherently more challenging than its undirected counterpart, as edge directionality greatly enlarges the learnale space (see Figure la: for graphs with 4 nodes, there are 218 possible digraphs, compared to only 11 undirected ones). We will see that simply extending undirected architectures without explicit directional components fails on this larger space of digraphs. This is further amplified in settings where domain-specifet structural constraints must be respected, such as directed acyclic graphs (DAGs). Second, beyond these technical challenges, we identify the lack of standardized benchmarks for directed graph generation as a barrier to rigorous evaluation and fair comparison. Current DAG methods (Li al., 2025) typically rely only on proxy metrics derived from downstream tasks, with no established benchmarks to evaluate digraph generative quality sy systematically.
"Correspondence ba carbal locastro@epf1...

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Published as a conference paper at ICLR 2026
Unlineeted Directed
Logg graph structans]
10-
6-
Noised sample
R/ Caar
Original sample
Entrapy
Tretuing
Sampling
Directed Pocitional Enooding
Dual Attention Tranformer
Prediction
nodes
-*(-1 G.)
(a) Comparison on # structures.
(b) Overview of DIRECTO.
Figure 1: (a) The learnable space increases drastically with the number of nodes for digraphs compared to undirected graphs (Harary & Palmer, 1973), highlighting challenges in extending graph generative models to directed structures. (b) Overview our generative model for directed graphs (with node colors representing different classes). During training, the dual attention transformer (denoising network), parameterized by enhanced with asymmetric positional encoding, learning to reverse predictions using cross-entropy loss. During inference, compute the rate matrix Re (G,, G(+ which governs the evolution of the generative process over finite intervals At, based on the model prediction p² Gt).
To address the first challenge, introduce DIRECTO', the first iterative refinement-based generative model for directed graphs of small to medium scale (from few up to 200 nodes on average).
Our approach brings together three core components of expressive graph generative modeling: (i) an expressive neural architecture, (ii) a robust generative framework, and (iii) informative input employ dual attention mechanism that performs cross cross-attents attention
features. For the architecture, between edge features and their reversed counterparts, capturing both source-to-target and target- to-source information flow. All this is integrated within robust discrete-state flow matching framework that enables efficient generation with state-of- tf-----rttt-ttttt performance and more efficiently than diffusion- basedd alternatives, and which is also readily extendable conditional generation tasks through classifier-fres free guidance (Ho & Salimans, 2022). Finally, the network's 's input features include positional encodings designed for asymmetric adjacency matrices, which we show outperform directionality-agnooii alternatives. Figure 1b gives an overview of method and its components.
To tackle the second challenge, we propose new benchmarking framework for rigorous evaluation of directed graph generation, spanning both synthetic and real-world datasets. The synthetic bench- marks cover different distributions, including acyelic and community-bon based graphs, while real- wollw benchmarks include DAG dataset for Neural Architecture Search (NAS) (Phothilimthana et al., 2023) and a scene understanding dataset (Krishna et al., 2017). The evaluation metrics are systematic and tailored to directed graphs, including distributional distances for fidelity, and dataset-specificc metrics reflecting the graph types used in different tasks. We evaluate DIRECTO on the proposed benchmarks, where it consistently outperforms prior approaches across all settings, underscoring its effectiveness on structurally diverse datasets. Extensive ablations further reveal that the dual attention mechanism is critical for modeling directional dependencies, while positional encodings play a role in enhancing overall generation quality.
Our main contributions are summarized as follows:
We propose DIRECTO, the first flow-based digraph generative model, combining a direction- aware dual attention block with positional encodings that capture asymmetric structure.
(ii) We address the lack of benchmarks by releasing an evaluation suite covering synthetic graphs with relevant structural properties and real-world datasets with meaningful applications.
(iii) We conduct an extensive empirical analysis demonstrating that DIRECTO achieves state-of- the-art performance structurally diverse synthetic and real-world graphs, while showing the need for a dedicated directed architecture.
Code available at: https:://///. httpa:://ggttbbtoooee rbal oocastro RRRC
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Overall, by addressing the two main bottlenecks in directed graph generation, we propose an effective, new generative framework and provide standardized benchmarks, aiming to lay the groundwork for future research and to support real-world applications.
BACKGROUND: DISCRETE FLOW MATCHING FOR GRAPH GENERATION
Notation We denote by G= (z(1::eN), e(Is))):)) directed graphs with N nodes. The of nodes N is denoted by (a(") In=1, and the set of directed edges by {e(1.) his$55N. Both nodes and edges are categorical variables, where z.(n) E (1,.... } and e(+d) E (1,..., ......F) F). Inspired by standard practice iterative refinement for undirected graphs (Vignac al., 2023a; Xu et al., 2024; Siraudin et al.. 2024; Qin et al., 2025a), we assume that every ordered pair of distinct nodes corresponds directed edge. Each edge is treated separately and always belongs to one of several possible categorical classes, including a class representing an absent edge. This formulation allows edges (i.j) and (j.t) to take different classes: both may be absent (no connection between nodes and j), only one may exist, or both may exist, even with different classes. Finally, we use subscript to denote variables time e.g., ax()) or Eq (i.5)
(n)
Problem statement Graph generative models aim to Iearn the probability distribution of a set of observed graphs, enabling the generation of new samples that preserve structural and statistical properties. Discrete Flow Matching (DFM))based models (Gat al., 2024; Campbell et al., 2024) have recently achieved state-of- ofte-offthhe-rt performance in undirected graph generation (Qin et al., 2025a). DFM belongs to a broader class of discrete state-space methods that recover the original graph distribution pl by progressively denoising samples from a pre-specified noise distribution Puoise (Vignac al., 2023a; Xu et al., 2024: Siraudin al., 2024). It distinguishes itself from other methods in this class by decoupling sampling from training, allowing for post-training sampling optimization. Additionally, by operating directly in the discrete spaces, these methods naturally align with the discreteness of graphs and are well-suited to capture complex dependencies through iterative refinement. We hypothesize that this added expressivity especially valuable for modeling the added complexity of digraphs. We now introduce the two main processes of DFM: noising and denoising.
Noising process The noising process runs from / = to t = 0. progressively corrupting the original input graphs through linear interpolation between the data distribution and the limit distribution Po = Pnoise (as detailed in Figure lb). This interpolation is applied independently to each variable following discrete iterative refinement frameworks (Austin et al., 2021b; Campbell et al., 2022; Gat et al., 2024; Campbell et al., 2024), which corresponds to each node and edge in the case of
graphs Qin al. (2025a). For example. for each node (n) its noisy distribution at step is defined as:
(n) Piji Pti (a)
| (x(²) = t 1(x( (m)) + (1 t) Phoise (
where -(:-) denotes the Kronecker delta and Phoise is the limit distribution over node categories. similar construction is used for the edges (Qin et al., 2025a).
Denoising process to = 1. It is formulated as a Continuous-Tine Markov Chain (CTMC), which starts sampling noisy graph Go ~ Po and evolves according to:
The denoising process aims to reverse the noising trajectory, running from t =
Pe (G G.) = 5(G,, G(+A)) + (G,, G++at) dt,
(2)
where R, is the CTMC rate matrix (Camphell et al., 2024). practice, this update rule is approx- imated in two ways: first, by discretizing time with finite step size At, in Euler method step; and second, by approximating the ground-truth rate matrix with an estimate R. computed from the predictions of denoising neural network p Ge) parametrized by f. The denoising neural
network, typically a denoising graph transformer, is trained to predict the clean node Pils
(.(n) and edge
0.(6.5) distributions given a noisy graph, aggregated as: Pitt
()G()) = (11
(00 (x(")) (G.)) ISnSN' (Pile (1 (Gi) (5SHSSN)
(3)
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Intuitively, the denoising graph transformer predicts larger probability of the clean graph plae( G:) given the noisy graph Gt, which in turn is used to compute higher transition rate in the rate matrix R. (see Equation 43 for further clarification). This means that the network predicts the most likely transitions required denoise the graph and recover the clean target, directly influencing the estimated rate of change.
This network is trained using cross-entropy loss independently each node and each edge:
(3.5) Gi) [
L = Fv,
Σ n
9.(m) log (Pije
[1 (n)
Gr Gi)) -
log (p Pile
(.(5.)
(4)
where the expectation is taken time f, sampled from predefined distribution over (0, 1] (e.g.. uniform); G1 ~ Pr (G1) is a clean graph from the dataset; and G2 ~ (G is its noised version time & (Qin al., 2025a). The hyperparameter node and edge reconstruction losses. Further details are provided in Appendix I.
R* controls the relative weighting between
GENERATING DIGRAPHS WITH ASYMMETRIC ENCODING AND DUAL ATTENTION
Building the discrete diffusion setting previously described, first how-based digraph generative model. We begin by outlining the overall generative framework, followed by detailed description of the directionality of DIRECTO: the use of directed graph positional encodings and the dual attention mechanism to enhance the edge directionality awareness of the denoising neural network in our framework.
introduce DIRECTO, the
components that form the
3.1 DIRECTED GRAPH GENERATION VIA DISCRETE FLOW MATCHING
Building on the formulation of Section 2, we extend DFM to the directed graph setting. To address the added complexity of asymmetric adjacency matrices in digraphs (Figure la), we exploit key strength of DFM: the decoupling of training and sampling. This enables post-training optimizations such as time-adaptive schedules and custom CTMC rate matrices (Qin et al., 2025a), which we can extend to directed graphs (details in Appendix C). While these strategies improve performance (see Appendix H.9), they remain insufficient to capture the structural properties of digraphs (see Section 5), We therefore introduce architectural components explicitly tailored for directionality-awar graph generation, providing the complete training and sampling algorithms in Appendix D.
3.2
ASYMMETRIC POSITIONAL ENCODING
Our generative framework employs denoising GNN, which inherently struggles to capture global higher-order patterns due to the locality of message passing (Xu et al., 2019; Morris et al., 2019). To maximize the capacity of our method, we augment our GNN with positional encodings (PEs) that inject structural information beyond local neighborhoods (Beaini et al., 2021; Bouritsas et al., 2020). Common choices for undirected graphs include Laplacian eigenvectors or shortest-patte distances, effective in enriching node and edge representations (Vignac et al., 2023a; Qin et al., 2025a). However, these encodings overlook edge directionality, failing distinguish asymmetric structural roles. To overcome this, we adopt direction-an -aree aware positional encodings (Geisler et al., 2023; Huang et al., 2025) that account for both outgoing and incoming connectivity. We append them to node and edge features, allowing our model to better capture the structural dependencies unique to digraphs.
Specifically, we experiment with three families of encodings: (i) the Magnetic Laplacian (MagLap) (Geisler et al., 2023), which introduces complex-valued phase shifts into the standard Laplacian to retain edge orientation; (ii) its Multi-g MagLap extension (Huang et al., 2025), which stacks Q Multiple Magnetic Laplacians with different complex potentials provide richer representa- tions; and (iii) Directed Relative Random Walk Probabilities (RRWP) (Geisler et al., 2023), which combine forward and reverse transition probabilities to capture outgoing and incoming asymmetric flows. These encodings are designed to provide direction-sensitive structural signals that better reflect the properties of directed graphs, and are described in detail in Appendix B.
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Figure 2: Network architecture of DIRECTO. We stack L dual attention layers that account for both source-to-taie tortggett and the stacked input node, edge, and global features. X', E' and y' are the output of the model, i.e., predicted clean node and edge distribution, and graph feature. FiLM (Perez et al., 2018) and PNA pooling layers (Corso al., 2020) are incorporated to enable flexible modulation between node, edge, and graph-level features. Full technical details are provided in Appendix A.
information via cross-attention mechanisms. X, E and y denote
We hypothesize that incorporating directionality- erative performance. To test this, we design an ablation study (see Section 5.2, with further details in Appendix H) evaluating which of the described encodings provide the most informative struc- tural signals, while also considering their computational cost, to identify the best tradcoff between performance and efficiency.
positional encodings improves digraph gen-
3.3 GRAPH TRANSFORMER WITH DUAL ATTENTION
Graph Transformers (Dwivedi & Bresson, 2021; Rampášek et al., 2022) effectively model node interactions by leveraging the adjacency matrix encode structural information. However, in directed graphs, it is crucial for the model to distinguish the different semantics carried by incoming and outgoing edges, which makes the standard graph transformer insufficiently expressive. Generating digraphs thus requires models that (i) capture bidirectional information flow to learn how source and target nodes influence each other distinctly, and (ii) integrate structural signals across nodes, edges, and the global graph to preserve relational dependencies and ensure coherent outputs.
To address this, our model jointly processes stacked node features X, edge features E, and global features through graph transformer composed of T. layers of a dual attention block (see Figure 2). This dual attention mechanism introduces two key components: (i) an attention- based aggregation scheme that captures directional information, and (ii) modulation layers to incorporate information at different levels of granularity across nodes, edges, and the global graph structure.
Bidirectional information flow via dual attention aggregation To model directional dependen- cies more effectively, we use a cross-attentiin mechanism between cource-to-targe (outgoing) edge features EsT and their reversed farget-to-soures (incoming) counterparts ETs. By explicitly attending to both directions, our architecture, which builds on ideas from Wang et al. (2024), processes infor- mation bidirectionally and better captures directed relationships, improving accuracy and resulting in better generations. Concretely, we compute two directional attention maps between ooeesfccciiic node projections:
Qs/i] KTb] Vd,
Yis QT(1] Ksbi ddg
Ys
(5)
where Qs. QT. Ks. and Ky are the role-specific projections for nodes that act as the source or the target of an edge, respectively, and dy is the query feature dimension. These attention weights modulated through a FiLM layer (see Equation (8)) using edge features to enable effective reasoning, producing the updated attention maps Ys- and Y
To further consolidate directional information, we introduce an attention aggregation mechanism. Instead of treating the attentions from the two directions independently, concatenate the modulated
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attention maps and apply single softmax operation obtain unified attention weights:
Auger soflmax ST (ss) ) E R==22,,
V ager
concal (V' V1 2 (6)
where h is the hidden dimension, Vs. VT are the value vectors for source and target nodes, and the concatenation operation is performed along the node dimensions. These unified weights are then used to compute convex (i.e., softmax -ofthax-weighted) combination of the value vectors:
Aagar Vager E n
This aggregation allows the model to assign asymmetric importance to the outgoing and the incoming directions. The node features are then updated using a gated residual connection, which adaptively combines the original and updated features by learning gate that controls how much of the new information should be integrated the update.
Multi-scale information modulation for graph denoising A graph denoising model should predict clean node and edge types from noisy inputs, which requires integrating both local interactions and global structure from node, edge, and graph-level features. Building on standard graph generation architectures (Vignac et al., 2023a; Siraudin et al., 2024; Qin al., 2025a), we incorporate Feature- wise Linear Modulation (FiLM) (Perez et al., 201 8) and Principal Neighbourhood Aggregation (PNA) (Corso et al., 2020) layers to enhance multi-scale feature fusion. FiLM adaptively modulates edge features using attention signals: given the edge feature matrix E, the attention matrix Eatm and the learnable weights WHLM W
FLLM
computes
FELM Emm) EW FLLM + (EW? FILM) O Eatn
E
(8)
where @ denotes element-win wise multiplication. These layers are also used to integrate global graph- level signals y into the construction of edge features.
To complement this, PNA layers aggregates multi-scale neighborhood information via pooling operations: given node (or edge) features X Rn*h and a learnable weight matrix WPNA:
PNA(X) concat (max( X) min(X),,,..... mean ( X), std(X)) WPNA E R
(9)
with concatenation performed along the feature dimension. We PNA layers update global graph signals based on local node- and edge-level information at each attention pass. Together, these components enable expressive and scalable integration of local and global features, allowing the model to capture higher-order structure which critical for accurate graph denoising.
BENCHMARKING DIRECTED GRAPH GENERATION
Despite growing interest in digraphs (Sun et al., 2024), benchmarking in this area remains under- developed Bechher-Speicherr al., 2025). Existing approaches typically rely downstream task performance, failing to directly evaluate generative quality. With the objective of establishing standard datasets and metrics, we propose comprehensive benchmark that covers synthetic and real-world digraphs, along with tailored metrics that directly assess key generative qualities, including sample validity, diversity, and distributional alignment.
4.1 DATASETS
To enable systematic evaluation, we introduce a suite of synthetic directed datasets alongside two real- world use cases: (i) neural architecture search (NAS), where DAGs model computational pipelines, and (ii) scene graphs, where directed edges encode semantic relations in visual scenes. Further details and statistics for all datasets are available in Appendix
Synthetic datasets We sample from the directed and DAG Erdös Rényi (binomial) model; from Price's 's model, directed analogue of the Barabási-Albert model that produces DAGs; and from directed version the Stochastic Block Model (SBM) dataset proposed in Martinkus et al. (2022). Each dataset comprises 200 graphs, split into 128 training, 32 validation, and 40 test graphs.

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TPU Tiles A relevant application of DAGs is Neural Architecture Search, which aims generate novel instances to achieve an automated design of efficient Neural Networks (Elsken et al., 2019). To support this task, the TPU Tiles dataset (Phothilimthana et al., 2023) contains computational graphs extracted from Tensor Processing Unit workloads. We split into 5,040 training, 630 validation, and 631 test graphs following the preprocessing from Li al. (2025).
Visual Genome Scene graphs represent the semantics of visual scenes, supporting tasks such as image retrieval, captioning, or visual question answering. Generating such graphs is useful for data augmentation, scenario simulation, or robustness, complementing recent scene-to-image generation work (Johnson et al., 2018; Yang et al., 2022). We use the Visual Genome dataset (Krishna al., 2017), a widely adopted benchmark of graphs that are not acyclic but instead represent general directed structures, with nodes corresponding to objects (entities), relationships (directed interactions), or attributes (object properties). We extract 203 training, 51 validation, and 63 test graphs.
4.2 METRICS
To evaluate digraph generations, we build upon the evaluation metrics from the undirected graph generation literature (Martinkus et al., 2022; Bergmeister al., 2024; Thompson al., 2022) and further extend the ones already present in the directed setting (Law et al., 2025).
Validity, uniqueness, and novelty For synthetic datasets, validity assessed via statistical tests of adherence to the generative distribution and, for DAG datasets, by ensuring acyclicity. On the TPU Tiles dataset, we measure acyclicity, while for Visual Genome we check typed structural constraints (e.g., edges go from objects to attributes and relationships, and from relationships to objects). To measure generative diversity while avoiding memorization, we compute uniqueness (fraction of non-isomorphic generated graphs) and novelty (fraction of generated graphs not in the training set). The V.U.N. ratio reports the proportion of samples that are simultaneously valid, unique, and novel.
Structural distributional alignment To evaluate proximity to the original distribution, we compute the Maximum Mean Discrepancy (MMD) (Martinkus et al., 2022) between generated and test graphs across several statistics: outgoing and incoming degree distributions, directed clustering coefficients, spectral, and wavelet-basee features. Spectral features are derived from the directed Laplacian (Chung, 2005), simplifying computations compared to the Magnetic Laplacian. Following standard practice, we first normalize results as the ratio from the generated samples MMD to that of the training data and then average the ratios for summarization. Values per MMD descriptor are detailed in the extended results in Appendix H..
Joint node-edge distributional alignment To better capture performance on attributed graphs, extend evaluation beyond structural statisties joint node-edge distributions. Specifically, measure MMD over label histograms to assess coverage of categorical proportions. We also compute triplet-based precision and recall by matching semantic tuples (c.g.. (object, relation, object)) between generated and test graphs. Finally, we evaluate embedding based distances (MMD with RBF kernel and FID) over GeN-derived representations that encode both labels and structure (Thompson et al., 2022). These metrics provide a comprehensive view of alignment with the reference distribution.
Among all metrics, we consider V.U.N. and the average ratio as the most informative for structural distributional alignment, and the RBF MMD for the joint node-label distributional alignment in real-world datasets. Additional details on the evaluation metrics are provided in Appendix
EXPERIMENTS
In this Section, we first evaluate the flexibility of our method to generate digraphs across synthetic and real-world datasets. Then, we analyze the role of the proposed architectural improvements on generative performance and analyze scalability and conditional generation.
Baselines In the general directed setting, we compare DIRECTO against performing Maximum Likelihood Estimation (MLE) baseline of the node count, node types, and edge types, and sampling from the resulting distribution generate new graph instances. For DAG settings, we also include
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Table Directed graph generation performance for different configurations of DIRECTO. Results are the mean + standard deviation across five sampling runs. We considered Q = 10 for MagLap in the synthetic datasets and Q = 5 in the real-world ones (due to the computational cost of this positional encoding). OOT indicates that the model could not be run within reasonable timeframe.
ER-DAG
SRM
TPUTiles
Visual Genome
Model
Ratio
VUN.
Ratio
Ratio
V.U.N.
Ratio J
V.N..
Training set
1.0
0.0
1.0
0.0
1.0
0.0
0.002
1.0
0.0
0.021
MLE
15.1 8:02
0.0 200
11
0.0
149.8 a:
24.7 on
L.099 4:0
17.0
0.0
0.618 wes
D-VAE
134
0.0 =00
OOT
OOT
00T
LayerDAG
42
=:
DiGres
19
34.0=41
39 209
4:51
$7.5 1:1
70.9 2.4
0.07 4.0
17.0
0.3 2.
0.232 = 0008
DeFati
75.0
4.3 a:
$7.0 2:66
63.7 4.74
72.0 AM
0.099 1.11
10.8.00 1:07
50.8 1:44
0.085 : 0003
DiaEmn-DD RRWP
1:
79.0 17
1,7 101
81.5 1:13
6..0
76.8
D.088 1
15.3
72.7
0.839
DIRETIDDD Ma
1.5
95.5
64.3
61.9 2.6
0.042
DIRECTO
1.7
99.5 1:
75.4
0.044
DIRECTO Magl.ap
1.3 8:02
92.0 537
2.0 0:3
96.5 1:23
44.0 4:23
80.5 46
0.042 4:00
6.2 0:03
67.0 4:43
0.051 = 0012
the only two publicly available baselines: D-VAE (Zhang et al., 2019) and LAYERDAG (Li et al., 2025). Notably, the architectural improvements in Sections 3.2 and 3.3 are agnostic to the underlying iterative refinement framework. To demonstrate versatility and enable wider experimentation, extend DIRECTO to a discrete diffusion (Vignac et al., 2023a) backbone (DIRECTO-DD), described in Appendix I.3. We also include DEFOG (Qin et al.. 2025a) and DIGRESS (Vignac et al., 2023a), discrete flow matching and diffusion methods in the undirected setting, respectively, adapting both to the directed case by removing the edge symmetrization operations. For each experiment, we highlight the best result and second- best method, with further experimental details in Appendix G, including Table 8 for a fairness comparison of resources and runtime of each of the models.
5.1 GENERATIVE PERFORMANCE EVALUATION
Synthetic datasets Table 1 reports the results on two synthetic datasets: ER-DAG and SBM. In both settings, DIRECTO consistently achieves the best trade-off between sample quality and validity. For ER-DAG, it achieves the highest validity score using directed RRWP as positional encoding, with a V.U.N. ratio of 94%. Notably, while LAYERDAG enforces acyclicity by design, fails to capture the ER structure of the target graph distribution, as evidenced by its low V.U.N. score (21.5%) see Table 17 for details). In the SBM case, DIRECTO also successfully captures the distribution and consistently outperforms the baselines, achieving up to 99.5% in V.U.N. ratio and keeping low average graph statistics ratios. We provide the complete results for the distributional alignment metrics and the results for the other two synthetic datasets in Appendices H.1 and H.4.
Real-world datasets Table shows the performance for the two real-world datasets. In TPU tiles, all DIRECTO variants significantly outperform DAG generation baselines in terms of structure-aware metrics. The ratio scores for DIRECTO are consistently and substantially lower than those observed in MLE, and especially LayerDAG, which indicates a more realistic distribution. Again, DIRECTO variants achieve higher V.U.N. than all baselines, with the exception of LAVERDAG, which benefits from enforcing acyclicity, the only validity constraint evaluated this case. For Visual Genome, results reinforce the versatility of DIRECTO. DIRECTO RRWP achieves the highest V.U.N., indicating a superior ability to generate diverse and novel graphs while maintaining structural fidelity. Finally, we also see that DIRECTO achieves the lowest RBF MMD score for both datasets, demonstrating that the model is successful in capturing the joint node-label dependencies of the distribution in the original dataset. Complete results available in Tables 10 and 11 in Appendix H.2.
Conditional generation As groundwork for directed graph generation, we prioritize strong un- conditional generation, which we consider key prerequisite for effective digraph modeling. At the same time, steering generation with graph-level properties is important for many downstream tasks. To illustrate the flexibility of our framework, we show that DIRECTO can be extended to conditional generation without major architectural changes via classifier-free guidance (Ho & Sali- mans, 2022; Nisonoff et al., 2025). On the TPU Tiles dataset, we condition on the execution time of the computational graph, aiming at low execution times. DIRECTO performs competitively with specialized autoregressive models like GraphRNN and LayerDAG (Appendix H.8), even surpassing the OneShotDAG variant of Li et al. (2025).
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5.2 ABLATION STUDY
In the following, we present the results for ablations on the role of dual attention and the different positional encodings. We complement this with a scalability study and results on conditional generation. Complete results and further ablations on other model components can be found in Appendix H.
Impact of dual attention We now analyze the influence of the dual attention mechanism on the performance of DIRECTO. In Figure 3, we compare the V.U.N metric and the average graph statistics ratios ("Ratio") score across both synthetic datasets (ER- -DR and SBM), highlighting the impact of dual attention on generation quality and graph realism. We observe that the dual attention mechanism consistently improves generative performance, regardless of the positional encoding it is combined with. Notably, even in the absence of any positional encoding ("No PE"), dual attention still achieves non-zero V.U.N., highlighting its capacity independently capture directionality-reiee information. See Appendix H.3 for the full ablation tables, both for DIRECTO and DIRECTO-DI
Impact of positional encodings We also evaluate the sensitivity of DIRECTO different positional encodings, including a direction-agnostic baseline with the Laplacian ("Lap") on synthetic datasets (see Appendix for details on how is computed using the symmetrized adjacency matrix). In Figure 3, we observe consistent trends across datasets: integrating positional encodings improves V.U.N. and Ratio, mirroring the undirected setting; moreover, direction-aware encodings outperform agnostic ones, supporting our modeling hypothesis. Among the encodings tested, Directed RRWP achieves the best overall performance on ER-DAG, and MultiMagLapPE with Q = 10 on SBM. Nevertheless, remark that RRWP demonstrates clear superiority in both scalability and runtime efficiency (see Appendix G.3). Overall, while positional encodings influence performance, the dual attention mechanism remains the most critical component for capturing directional dependencies. Full tables for both DIRECTO and DIRECTO-DD can be found in Appendix H.4.
15
Ratio
10
40
2.5
15
20
-15
Maglag
Magl-ap
No PE
Lap
Mod Megtop (Q Iu)
Mastaa
3)
No PE
Lay Thi W
Madt ap Mbedo (Q in)
Staelap
Figure 3: Ablation results for dual attention (left) and positional encodings (right). Each plot shows
results on the ER-DAG (left bars/lines) and SBM (right bars/lines) datasets. Better performance corresponds to V.U.N. bars and Ratio lines appearing closer the top of each subplot.
Scalability study We investigate scalability across three axes: dataset size, parameter efficiency, and graph size (Appendix H.7). Increasing the amount of training data improves performance but comes with longer training time (Table 18). For parame- ter efficiency, our dual attention mechanism clearly demonstrates better performance than simply enlarg- ing the adapted undirected model, as can be seen in Table 2. Finally, while larger graphs pose challenges, particularly for enforcing strict acyclicity, DIRECTO achieves strong generation quality (Ratio) across all graph sizes commonly used in the literature for the applications explored in this paper (Table 21).
Table 2: Directed graph generation perfor-
mance for the dual attention mechanism versus doubling the depth of the standard network.
Dataset
Model
Ratio
V.U.N. 1
RRWP Double
4,9 0.2
72.0 + 9,8
RRWP Dual
1.7
94.0 -
ER-DAG
MagLap Double
80.0 0.7
MagLap Dual
1.3
91.0
RRWP Double
26.3 2.5
(1.0 0.0
RRWP Dual
1.8 0.5
99.5 100
SBM
MagLap Double
14.2 1.5
8.0
Magl.ap Dual
1.+ 0.3
77.0

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6 RELATED WORK
Graph generative methods Initial methods include aulo-regressive models (You et al., 2018; Liao et al., 2019), which grow the graph progressively by inserting nodes and edges, but require node ordering for feasibility. On the other hand, one-shot graph generative models such as VAEs (Kipf & Welling, 2016; Simonovsky & Komodakis, 2018: Jin et al., 2018; Ma & Zhang. 2021; Vignac & Frossard, 2022), GANs (De Cao & Kipf. 2018: Krawezuk al., 2021; 1; Martinkus et al., 2022), and normalizing flows (Kaushalya et al., 2019; Wehenkel & Louppe, 2021; Lippe & Gavves, 2021; Luo et al., 2021b) can directly generate full graphs in single forward pass, thereby eliminating the need for predefined node ordering. Graph diffusion models also belong to this class, with the initial approaches consisting of adaptations of continuous state-space discrete-time diffusion frameworks (Sohl-Dicke -Schh-Dickstein et al., 2015; Ho et al., 2020; Song & Ermon, 2019) to the graph setting. These include works such as EDP-GNN (Niu et al., 2020), DGSM (Luo et al., 2021a) 1)), and GeoDiff (Xu et al., 2022). Later, the discrete state-space diffusion framework (Austin al., 2021a) was adopted, with DiGress (Vignac et al., 2023a) and MCD (Haefeli et al., 2022) operating in discrete time. On the other hand, methods that operate in continuous time (Song et al., 2021: Campbell et al., 2022) were introduced, including GDSS (Jo et al., 2022) and GruM (Jo et al., 2024) in continuous state-spaces and DisCo (Xu et al.. 2024) and Cometh (Siraudin et al., 2024) for discrete state-spaces. Recent approaches combine the sequential modeling of ruto-segeessive methods with the global denoising of diffusion (Zhao et al., 2024). Finally, DeFoG (Qin al., 2025a) recently achieves state-of-the-ar performance by leveraging discrete flow matching (Camphell et al., 2024; Gat et al., 2024).
Directed graph generation The vast majority of works in this area focus solely on DAG generation, with research including autoencoders (Zhang et al., 2019), autoregressive methods (Li et al., 2022) and, more recently, combining diffusion in different steps of an auto-regressive process (Li et al., 2025). However, they require the input DAGs to be topologically ordered, which can result in higher computational cost. Other approaches (Asthana al., 2024; An et al., 2024) use ideas from discrete diffusion for Neural Architecture Search (NAS). but they are tailored to the specific application and, therefore, to DAGs. Finally, Law et al. (2025) are the first to propose general digraph generation method from an auto-regressive perspective. Unfortunately. to the best of our knowledge, its implementation is not publicly available, limiting its accessibility for study and evaluation.
7 CONCLUSION AND FUTURE DIRECTIONS
We propose DIRECTO, a discrete flow matching-bai method for directed graph generation. It combines directionality positional encodings with dual-attention mechanism that handles both source-to-target and target-to-source dependencies, overcoming the limitations of naive extensions from undirected models. We also introduce a dedicated benchmarking framework to address the lack of standardized evaluation. Empirical results show that DIRECTO outperforms existing baselines on synthetic and real-world datasets, successfully generating directed structures and preserving key properties, such as acyclicity, without explicit constraints, highlighting its generality and robustness.
Limitations and future directions Despite these promising results, limitations remain. First, scalability can be further improved in handling larger datasets and graph sizes (a direct result of the combinatorial complexity of digraphs). Incorporating scalable strategies, such as sparsity-aware attention (Qin et al., 2025b), hierarchical (Bergmeister et al., 2024; Jang et al., 2024), or latent diffusion (Yang et al., 2024), could help extend DIRECTO at scale. Second, although our current setup includes classifier- conditional generation, exploring alternative conditional strategies could further improve performance. Third, while DIRECTO can learn structural properties such as acyclicity implicitly, enforcing such constraints (e.g., through PRODIGY (Sharma et al., 2024) or ConStruct (Madeira et al., 2024)) could further improve controllability and domain-specific validity. Finally, although the proposed graph transformer is used here for graph-to-graph generation, it can be readily adapted to discriminative tasks such as link prediction or node classification. Given its performance in the generative setting, it would be interesting study how the architecture transfers to these alternative setups. Exploring these directions would enhance the applicability and impact of digraph generative models and our proposed architecture, particularly on new real-world scenarios. Appendix K includes a detailed discussion of limitations and future work.
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ACKNOWLEDGEMENTS
This work was supported by the Swiss National Science Foundation (SNSF) under grant 10001445.
REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our results, we have taken the following steps. The implementation of our method is provided in a GitHub repository, including all code and scripts necessary to reproduce the experiments and details on how to run it. In addition, we detail the proposed algorithm in Appendix D. while Appendix G provides full descriptions of the baselines, training setup, and hyperparameters employed. We also report the computational resources used and the runtime required for all experiments. Together, these materials are intended to enable researchers fully replicate and build upon our work.
ETHICS STATEMENT
The objective of this work is to advance methodologies for the generation of directed graphs by introducing architectural mechanisms that explicitly model directionality and asymmetric relation- ships. Directed graphs are central to wide range of applications, including causal inference, traffic modeling, and biological network analysis. Accordingly, improvements directed graph generation have the potential to support progress in scientific research, decision-making systems, infrastructure design, or causal discovery.
The proposed framework strengthens the ability of generative models to handle directed graphs, and, at the moment, we do not identify any immediate societal risks associated with its deployment in its current form. Moreover, although the method is capable of generating structured and semantically meaningful graphs, the scale and complexity these outputs remain limited, which may difficult direct applicability in high-impact domains such clinical research policy modeling this stage.
USE OF LARGE LANGUAGE MODELS
Large Language Models (LLMs)
used writing assistant tool in the preparation of this manuscript. Specifically, they were employed to aid in writing, checking spelling, and editing for clarity. Nonetheless, all text produced with the assistance of LLMS carefully reviewed, verified, and revised by the authors to ensure accuracy and appropriateness. The use of LLMs was limited to these support functions, and the authors take full responsibility for the final content of the paper.
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Ling Yang. Zhilin Huang, Zhilong Zhang, Zhongyi Liu, Shenda Hong, Wentao Zhang, Wenming Yang. Bin Cui, and Luxia Zhang. Graphusion: Latent Diffusion for Graph Generation. IEEE Transactions Knowledge and Data Engineering, 2024. 10, 48
Jiaxuan You, Rex Ying, Xiang Ren, William L.. Hamilton, and Jure Leskovec. GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models. In International Conference on Machine Learning (ICML), Proceedings of Machine Learning Research, 2018. 10
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Appendix
Table of Contents
A Dual Attention
18
B Positional Encodings
C Sampling Optimization in DIRECTO
21
D DIRECTO Training and Sampling Algorithms
22
E Dataset descriptions
23
E.1 Synthetic datasets
23
TPU Tiles
................................
24
E.3
Visual Genome
24
F
Further details evaluation metries
25
F.1
Validity metrics
25
F2
Uniqueness and novelty
26
F3
Maximum Mean
................................
metries
27
Joint node-edge distributional metrics
27
F5
Downstream tasks metrics
G Experimental details
28
G.I Details baselines
28
G.2 Training setup
29
G.3 Resources and runtime
30
Additional results
31
H.1
Extended results for synthetic datasets
................................
31
H.2
Extended results for real-world datasets
................................
31
H.3 The role of dual attention
32
H14
The role of positional encodings
33
H.5
Ablation of other model components
34
H.6 ER vs DAG performance
35
H.7
Scalability experiments
H.8 Conditional generation
37
H.9 Impact of sampling optimization using discrete flow matching
38
I Iterative Refinement Methods for Graph Generation
39
L11
Graph iterative refinement methods
39
I.2 Discrete flow matching for graph generation
40
1.3 Discrete diffusion graph generation
41
Visualizations
42
K Limitations and future work
48
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A DUAL ATTENTION
In this section, we provide a detailed description the dual attention mechanism used in our model, which effectively integrates both target-to-source and mourc-tto-targer information enhance the graph generation process.
We start with the stacked tensors of node X Bx edge E E BB and global E RBdd features, where B is the batch size, N the number of nodes of the biggest graph in the batch, and dz. de. and dy are the node, edge, and global feature dimensions respectively. To handle different node sizes batch, we use binary node mask M E {0, 10 BxNx1 that accounts for the presence of nodes in each graph.
First, we begin by projecting the queries, keys, and values for the source and the target directions. These are then projected and reshaped into n, heads of dimension dg dz/n/, thus obtaining for each node:
Qs reshape (M X)
Ks = (eshape(MM W X)
Vs = reshapee M O W W X), (10)
Q reshape W X)),
KT reshape(
Vi reshape ( O W X), (11)
where W W W WT WK. WT are the learned linear projections. The final shape of each of the elements after the reshaping is N nı, x dg.
Then we compute the dual attention, both from source to target and from target to source:
YsT(i.]] Qs/i] KrLs
Qrli] Ksbi
Yis(i,
(12)
Vdg
ddg
with j the index nodes and each Y E BBBNXXXX xd,
To incorporate edge features into the attention mechanism, we apply FiLM-style modulation (Perez et al., 2018) using additive and multiplicative projections Wmnl. Wadi:
E
mal = reshape W mul (E)),
E Edd reshape W
add [E),
(13)
ET mal reshape( WT mull E T);
E reshape W add E )),
(14)
where we reshape the edge features to E E Bx.Nxx.XX.. xd, This is done so that they can be added or multiplied to the attention scores YST and Yis to modify them:
YST YST (ES mul + 1) + ES add-
(15)
YTS + YrS -E + 1) + E
(16)
Edge Feature Update
To get the final update of the edge features, the attention attention-weighhee edge representations are flattened and modulated by global features y using FiLM with additive and multiplicative projections W add and W.E
mul-
E'
W (y) + (W
W ( (y) + 1) @ flatten (YsT),
(17)
where we flatten over the last dimension. We use YsT as it captures the directional flow from source to target, avoiding ambiguity in the directional influence, and still effectively encoding the influence of source nodes on their targets. Then, the result is followed by an output projection and residual connection
E W
E E+E
(18)
resulting in the updated E' BxxNNxx
Node Feature Update To perform the node feature update, first we concatenate the dual attention
scores and apply softmax:
A aggr = softmax (concat (osoc YTs)) E BBBxXx2XXN xRxxNaNNxmxxd
(19)
V aggr concat VT, V+) BB82Nxm.xd, xnxxd,
(20)
with concat() being the concatenation over the third dimension for Auger and over the second dimension for V Viggr-
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After this, the unified weights are aggregated through weighted summation and the result flattened over the second dimension to produce:
aggr att aggr Vager E IR BxNx.. xd=
(21)
We then FiLM --moullateee the attended features with the global vector y with additive and multiplicative projections W and W X mal to produce:
add
mnd
W + (W mull (y) + 1) X aggrr
(22)
To allow the model to adaptively balance between preserving the original node features and incorpo- rating the updated ones, we introduce learnable gating mechanism g.X [0, 1]:
gx = (W O
concat [X, Vaggr)),
(23)
gule
followed by gated residual connection:
X' = (1+ @ X + (1 9x) X X
(24)
where the weights assigned to X and Mod sum up to 2, ensuring that the overall magnitude remains consistent with the ungated case.
Finally, the result is followed by an output projection:
X' = W (X'),
(25)
resulting in the updated X' E BxNxd
Global Features Update and edge E projections:
The updated global vector is computed by aggregating from the node X
y - y + W.( + X + (E),
(26)
where X+y and E+y are PNA layers (Corso et al., 2020) that given node features X E IR BxXxd, or edge features E E IR Bx xd. and a learnable weight matrix WPNA, computes
PNA(X) concat (max X) mi ( mean X), std(()) WPNa E Redd=
(27)
PNA (E) concat (max E), mi min(())))))))) mean [),, std(E)) WINA E RR
(28)
where the different operations are performed over the features and then concatenation is per- formed along the feature dimension. Both results then pass through a linear layer resulting in X X-,((X) E E Rd
Once again, we use an output projection
y' WW (3/), which results in the updated global features y' E Bxd,
(29)
This dual attention block is applied times within the graph transformer.
B POSITIONAL. ENCODINGS
To effectively capture the structural properties of directed graphs, we explore range of positional encodings (PEs) that can be incorporated into the transformer architecture. These encodings provide nodes with sense of position and orientation within the digraph, enabling the model to better reason about directionality. Below, we detail the specific PEs evaluated in our work.
Laplacian (Lap) Traditionally in the undirected case, the cigenvectors of the combinatorial Lapla- cian are widely used to encode graph structure, relying on the spectral decomposition = FAr-1 where L is the combinatorial Laplacian, A is the diagonal matrix of eigenvalues, and I is the matrix of eigenvectors. The unnormalized and normalized Laplacians are defined as
Lu= = D - A., L = D 1/2 A.DA 1/2
(30)
where A, is the adjacency matrix (symmetrized when working with digraphs) and D is the degree matrix. However, when dealing with directed graphs. this symmetrization discards directionality information, as it enforces A, = but is necessary to guarantee that Ly and L are symmetric and positive semi-definite.
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Directed Laplacian (DirLap) The directed Laplacian Chung (2005) is defined as
L - P + p
(31)
where I is the identity matrix, P is the transition matrix of the directed graph, and + is diagonal matrix containing the Perron vector of P. Depending on the chosen walk type, P may correspond to standard random walk, lazy random walk, or random walk with teleportation (PageRank). This construction results in real eigenvalues and eigenvectors and, as a consequence, it cannot jointly encode magnitude and phase information, limiting its ability capture directional patterns.
Magnetic Laplacian (MagLap) To retain directional information while preserving desirable spectral properties, Geisler et al. (2023) propose the Magnetic Laplacian, which introduces complex- valued phase encoding into the adjacency matrix. Specifically, the (unnormalized and normalized) Magnetic Laplacians are given by:
(9) Ui = D (A, o cxp(iel))))
L dir = I - 1/2 A, 1/2) -D-
c exp(ie(m)).
(32)
where ④ denotes the element-wise (Hadamard) product, i is the imaginary unit, and the phase matrix ((a) is defined as:
(4) ( Au,u).
(33)
The potential parameter q 2 0 controls the strength of the phase shift. For g = 0, the Magnetic Lapla- cian reduces to the classical combinatorial Laplacian. The resulting complex-valued eigenvectors can effectively encode directionality patterns in the graph. We leverage the information by concatenating the real and complex parts of the eigenvectors to the node features, and the eigenvalues to the global graph features.
Multi-g Magnetic Laplacian Recently, Huang al. (2025) extends the Magnetic Laplacian by introducing a multi-q formulation, which considers as positional encoding stacking together the eigenvalues and eigenvectors of Q distinct Magnetic Laplacians with potentials 01.....9(2 This approach enables the recovery of the bidirectional walk profile, a generalization of walk counting in undirected graphs that captures a broader range of directional relationships.
By incorporating information from multiple potentials, the multi-q formulation enhances the repre- sentational power for directed structures. In particular, they show that the method is robust to the chosen potentials chosen. In our model, we leverage this information by concatenating the first k eigenvectors (real and imaginary parts) of each of the Q magnetic Laplacians to the global features, and the corresponding first & eigenvalues to the node features, providing direction-awarc aware structural signals to both levels of representation. As a drawback, this positional encoding results in higher computational costs, as it is necessary to compute several eigenvalue decompositions and, in addition, results in a higher dimensional positional encoding to be processed by the denoising network.
Directed Relative Random Walk Probabilities (RRWP) For undirected graphs, RRWP encodes the likelihood of arriving from one node to another through k-step random walks. This is usually expressed via transition matrix T with T = AD- that encodes the transition probabilities.
For directed graphs, is possible to consider T = -AD-1 outs where we take into account the degree matrix of the outgoing edges D (Geisler et al., 2023). The K-step transition probabilities are then T, captured by powers of T. producing the sequence T, T .r. -
Additionally, in the directed setting, Geisler et al. (2023) suggest it is also informative to model reverse random walks, starting from incoming edges. For this, matrix R = A D where Dmn is the in-degree matrix. This would represent the likelihood of arriving given edge starting from another through k-step random walks. Analogously, RK-1).
define the reverse transition
compute I, R, R²
To have final positional encoding, following Geisler al. (2023), ensure that graphs not nilpotent and that the probabilities encoded in the matrices sum up to 1 by adding self-loops to sink nodes during pre- mreprocesssngg Finally, the full positional encoding concatenates both forward and reverse walk features for pre-defined K, yielding:
RRPW(G) [I. T
T
I. I. R2
RK-1
(34)
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We concatenate the PE the edge features, and additionally concatenate the diagonal elements of each matrix to the node features to further infuse them with the information.
Additionally, to mitigate that for large the probabilities tend to converge sink nodes, optionally incorporate Personalized PageRank (PPR) features. The PPR matrix is defined in closed form as:
PPR P -P))T)
(35)
where p, denotes the restart probability.
C SAMPLING OPTIMIZATION IN DIRECTO
The decoupling of training and sampling in discrete flow matching enables principled, training-free modifications to the sampling procedure, offering a flexible design space to enhance generation quality across digraph distributions (Qin et al., 2025a). In this section, detail the three parameters that govern sampling behavior: (i) time distortion functions, (ii) target guidance intensity, and (iii) stochasticity strength. Each plays distinct role in shaping the denoising trajectory and contributes to different aspects of generation quality and structural fidelity.
Time Distortion Functions A central part for sampling optimization lies in the application of time distortion functions. These functions transform the time variable & [0, 1] through bijective, monotonic mapping [(t), yielding a distorted time variable /' = f(t). While used in both training and sampling. their objectives differ. In training, time distortions skew the sampling distribution over time, concentrating the model's learning capacity on specific denoising intervals. In sampling, they induce variable step sizes along the denoising trajectory, with finer resolution where structural sensitivity is highest (e.g., late-stage edits near 1).
Formally, the probability density function (PDF) of the transformed time variable t' is:
de f (t')
d
(t)(t) di(()
where (t) 1 for t [0, 1].
(36)
We consider the same five representative time distortion functions from Qin al. (2025a) that yield diverse distributions for t':
Identity: f(t) = t Uniform time density, baseline behavior.
Polydec: f(t) 2t t
Increasing density over time, emphasizing late-stage denoising.
Cos: f(f 1 1-cos( mt)
Concentrates density at both boundaries.
2
Revcos: [(1) = 21
l-cos(xt) Peaks at intermediate times.
2
Polyine: (t) = 1 Decreasing density over time, emphasizes early steps.
In the sampling phase, the induced variable step sizes, particularly from functions like polydec, allow for finer resolution the clean data, where structural coherence is most fragile. Empirically, these functions are selected based on dataset-specific properties to improve fidelity and structural constraints without need for retraining.
Target Guidance Another axis for improving sampling efficiency is the modification of the condi- tional rate matrix RG to better guide the generation trajectory toward the clean target graph ø1. This achieved by incorporating an additive guidance term:
8(21+211 21)
Re(z,, Z++ (122,24443
21) = R2 (Ee, Z+ 21) + w. 2 Peji (2, 21))
(37)
Here, E R4 controls the strength of the guidance, Z,20 is discrete variable with Z possible positive values, and 6(..) is the Kronecker delta. Intuitively, this formulation prioritizes transitions that directly align with the clean graph.
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Stochasticity Control The final campling-time parameter is the stochasticity coefficient 1). This parameter scales an auxiliary rate matrix R that satisfies the detailed balance condition (Campbell et al., 2024) and adds it to the optimal rate matrix Rç:
R = R + 1 R.
(38)
The role of 1/ is to regulate trajectory stochasticity: higher values promote broader exploration during denoising, potentially correcting suboptimal states. Setting !/ = 0 recovers the deterministic path prescribed by R2 minimizing the expected number of transitions.
Sampling Optimization Together, these sampling parameters constitute flexible, principled toolkit for tailoring the sampling procedure diverse graph domains and structural requirements. In our case, we optimize these parameters at sampling time to improve the performance of DIRECTO. In particular, individually optimize each of them for the main results, and perform the final sampling using the best performing parameters terms of V.U.N (a detailed description is available in Appendix G.2). and ratio performance. A study of the effect of these parameters sampling performance can be found in H.9.
D DIRECTO TRAINING AND SAMPLING ALGORITHMS
This appendix provides detailed pseudocode for the training and sampling procedures used in the DIRECTO framework. Alg. outlines the training loop, where digraph is progressively noised and denoising model is optimized to reconstruct the original graph.
We consider the original data distribution pı as well as the time distribution which our case is set to be the uniform distribution. In addition, f refers to the denoising network, in our case the graph transformer with dual attention.
Algorithm DIRECTO Training
Input: Graph dataset (G'....,,CH ~ Pı
2: while fo not converged do
3: Sample ~ Pt
4: Sample t ~ T
5: Iteratively sample G. ~ Pin(
P Noising process Positional encoding
6: + PosEne(GG))
7:
(1G.) + Jo(G,,
Denoising prediction with dual attention
8: loss < (f.
9: end while
Alg. describes the generative sampling process in which new digraphs are synthesized by iteratively denoising from an initill random digraph structure. Again, we consider po Proise the prede- fined noise distribution, f the denoising graph transformer with dual attention, Pi the denoising predictions, and Pf as the update rule, see Equation (42).
Algorithm 2 DIRECTO Sampling
1: Input: graphs to sample S
2: for to S do
3: Sample N from train set
# nodes
4: Sample Go ~ Po (Gg)
5: for f to 1 At with step At do
6:
h (--PosEncGG))
b Positional encoding Denoising prediction with dual attention
7:
(.G.) + ho(Gi,
8:
Getar - PG
Update rule
9:
end for
Store G1 as G
11:
end for
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E DATASET DESCRIPTIONS
In this section, we provide fürther detailed information about the datasets described in Section 4 and used for the experiments in Section 5.
E.I SYNTHETIC DATASETS
We outline the mathematical formulation of each graph generation strategy used in experiments. All graphs are generated as adjacency matrices A E (0, 1 (where A.j = denotes a directed edge from node i node j), and preprocessed adapt the structure required by the model.
For all datasets, we generate total of 200 graphs, split as: 128 train, 32 validation, 40 test. study on the effect of the dataset size on the quality of the generations can be found in Appendix H.7. Furthermore, the dataset statistics can be found in Table 3.
Table 3: Synthetic dataset statistics.
Dataset
Min. nodes
Max. nodes
Avg. nodes
Min. edges
Max. edges
Avg. edges
WTrain
#Val
#Test
Erdos-Renyi (ER)
20
80
46
SBM
44
175
106
340
3008
1426
128
32
40
ER (DAG)
20
80
49
103
1892
762
128
32
40
Price (DAG)
64
64
64
132
246
197
Erdós-Renyi (Erdös & Rényi, 1959) We generate graphs with variable number of nodes between n = 20 and n 80. For number of nodes n, edges are sampled independently with a fixed probability p 0.6. The adjacency matrix A is generated by:
Aij Bernouli(p), Vi }
To generate a second dataset of DAGs, we enforce a lower-triangular structure (i.e., edges only from higher-indexed lower-indexed nodes), which results in graphs generated with a probability p 0.3.
Price's model (de Solla Price, 1965) This is the directed version Barabási-Albert or preferen- tial attachment model, which simulates the growth of citation networks. We build DAGs sequentially for a fixed number of nodes N = 64. Each new node forms m = Iog)(N) edges to existing nodes j with probability proportional to their degree:
Pii + j) x deg(ī).
Since the number of nodes is fixed to N = 64, mean out-degree results in m = 6. In practice, we implement this "bag" of nodes that replicates existing connections. For each node i, sample m destination nodes from the bag. Then, for each selected node j. we set A;j 1 and add and the selected nodes to the bag. This ensures a directed acyclic graph (DAG) due the order of node addition.
Stochastic Block Model (Holland et al., 1983) We create K communities with sizes (ni, ..,,..,, ]- We set the number of communities between = and K = 5, and the number of nodes per community between 20 and 40. The probability of an edge between any two nodes depends on their community assignments
Aij (ermoulliPPsss))
where E (1,...,K) is the block assignment of node and P E [0, Kxk is matrix of intra- and inter-community connection probabilities. particular, we use
Pek = Pinira = 0.3. Pai Pinter = 0.05 for k
generate block-structured, directed graph.
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E.2 TPU TILES
In this dataset introduced by Phothilimthana et al. (2023), each DAG represents a computation within a machine learning workload, such as a training epoch or an inference step. Each of the 6301 datapoints includes computational graph, specific compilation configuration, and the execution time of the graph when compiled with that configuration. The dataset features different model architectures, such as ResNets, EfficientNets, Masked R-CNNs, and Transformers, with graphs of up to ~ 400 nodes.
Table 4: TPU Tiles dataset statistics.
Dataset
Min. nodes
Max. nodes
Avg. nodes
Min. edges
Max. edges
Avg. edges
# Graphs
Train
2
394
41
711
43
5040
Validation
2
113
41
1
123
43
630
Test
154
41
1
249
44
631
adopt the processing and splits from et (2025). Details on dataset statistics for the three different splits can be found in Table 4.
E.3
VISUAL GENOME
The Visual Genome dataset (Krishna et al., 2017) aims bridging computer vision and natural language understanding by providing richly annotated images. It comprises over 100,000 images annotated with object labels, attributes, relationships, which allows to capture the presence of objects but also their attributes and interactions. It has become standard benchmark for tasks such as scene graph generation, visual question answering, and grounded language understanding.
To build the directed graph dataset, we set 3 types of nodes: objects, attributes, and relationships, with the actual node class being the text label. Then, we link them via directed edges according to the visual rules provided in the original data and taking into account that:
1. Objects have outgoing edges to attributes and relationships and incoming edges from relationships.
2. Relationships have outgoing edges to objects and incoming edges from objects.
3. Attributes have outgoing edges and incoming edges from objects.
Once we built the digraphs, we select relevant subset from this raw dataset by keeping graphs with 20 to 40 nodes, and with minimum of 25 edges per graph. Then, for each graph, we only consider objects, attributes, and relationships that are in the top-20 in terms of prevalence in the full dataset, ending with 60 node classes. We randomly split the graphs into 203 train, 52 validation and 63 test graphs, with detailed statistics of each split available in Table 5. The total number of cyclic graph in the dataset is 119 (~~37.5% of the digraphs).
Table 5: Visual Genome dataset statisties.
Dataset
Min. nodes
Max. nodes
Avg. nodes
Min. edges
Max. edges
Avg. edges
# Graphs
Global
21
40
34
25
47
28
317
Train
21
40
33
25
47
28
203
Validation
40
25
40
Test
40
36
The selected top-20 objects, relationships, and attributes are:
l'window', 'tree', "man', 'shirt', 'wall', 'person', "building', 'ground', 'sign', 'light', 'sky', 'head', (Leaf', "eg', "hand', 'pole', 'grass', Khair', 'car', Yeloudr]
Objects:
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Relationships: I'on', 'has', 'in', 'ot', 'wearing', 'with', 'behind', holding', 'n top of', 'on a', 'near', 'next to', haas a', 'on', 'under,, 'by', 'cf a', 'wears', 'above', 'sitting on
Attributes: l'white', 'black', 'blue', 'green', 'red', 'brown', yellow', 'small', 'large', 'wooden', 'gray', 'silver', 'metal', ''rangee, 'grey', 'tall' Mong' 'dark', 'pink', rclear
F FURTHER DETAILS ON EVALUATION METRICS
In this section, we detail the statistical procedures used to evaluate the graphs generated by our model. We detail how we compute the validity metries for the different datasets, as well as how we make the validity and uniqueness computation more efficient.
F.1 VALIDITY METRICS
Erdös-Renyi Given a graph G = EV with n = |V] nodes and m [ edges, we want to determine whether the graph likely originates from Brdōs-Rénç model G(n, p) with edge probability p using a Wald test (Held & Sabanés Bové, 2013). For that, we follow the following steps distinguishing the cases in which the graphs are DAGs not:
1. Empirical edge probability: we compute the number of possible edges
n(n-1) 2 if the graph acyelic
Mmaxx
ninn 1)
if the graph is not acyclic
and then estimate the empirical probability of edge presence ş
2. Wald test statistic: the Wald statistic tests the null hypothesis Ho P p. where p is the expected edge probability:
W (p-p)2
p) +
and where we add a small regularization = 10 to prevent division by zero.
3. p-value computation: assuming the null hypothesis, the Wald statistic W asymptotically follows a Chi-squared distribution with 1 degree of freedom:
p-value 1 Fx>= (W; df 1)
where F22 is the cumulative distribution function (CDF) of the Chi-squared distribution.
In the rare limit case where graphs contain only one node, we consider that the graph does follow the distribution.
Stochastic Block Model To assess whether graph = (V, conforms to a Stochastic Block Model (SBM), we recover the block structure and probability parameters. The test then computes Wald statisties for intra- and inter-block edge probabilities.
1. Block assignment via model inference: Given the adjacency matrix A {0, 10xx of graph G, we use an algorithm (Peixoto, 2014) to infer the stochastic block model (block assignment) from given network:
Infer 2 V fI....,B) B) using min L(G,
where B is the number of non-empty blocks found and L is the description length. The model can be further refined using Markov- -hhin Monte Carlo (MCMC) for timesteps. 2. Estimating intra- and inter-block probabilities: let N, be number of nodes in block and E,j the number of edges between block and block recovered by the algorithm in the previous step. Then the estimated probabilities for directed SBM graphs are:
E
Pintra = N.(N. - +
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Ey
i it)
N +
where 10-6 6 is small regularization constant.
3. Wald statistie: we compare the empirical estimates to expected values Puuta and Pinter using Wald statistic:
(Puatra Pintra)
W
P (1 - Pintra, i) +
(pinter,k) Pinter) )
Wi Pinter, Pinter,ij)
+
4. p-value computation: assuming the null hypothesis (i.e., estimated probabilities match expected ones), each Wald statistic follows Chi-squared distribution with I degree of freedom:
Pij 1
F2a (Wayi of 1),
and therefore the overall p value can be computed the mean of all Pij
B
B
p-value
Pai
B
Price's model To assess whether a graph follows Price's model, we use nonparametric two- sample Kolmogorov-Smirovv (KS) test:
1. Computing degree distribution: let D) = (d1, .2...., dn) } be the degree sequence of the nodes G. To reduce noise and ensure more stable distribution, ignore nodes with degree <. 1 and end with the distribution D' {d; E D d: > 1}.
2. Synthetic graph generation: we construct synthetic graph GBA using the Barabási-Alber model with the same number of nodes: GBA BA(n, m). where n V the number of nodes and m 6 is the number of edges each new node adds during attachment, fixed to the same value as in our data generation. The degree sequence of the synthetic graph is DBA.
3. Saanoggorv--mmiroo Two-Sample test: perform two-sample Kolmogorov-Smirnov (KS) test to compare the empirical distribution functions of D' and DBA
p-value- 1 KS2 sampled DBA)
which evaluates the null hypothesis: Ho D DBA-
TPU Tiles For the TPU Tiles dataset, we report the percentage of valid Directed Acyclic Graphs (DAGs) as our primary validity metric, reflecting our focus on accurately reconstructing acyelic computational graphs.
Visual Genome As seen in Appendix E.3, by construction, object nodes can have outgoing edges to relationship attribute nodes but only receive incoming edges from relationship nodes. Relationship nodes only have incoming edges from object nodes and send outgoing edges to objects. Finally, attributes only receive edges from object nodes and do not have any outgoing edges. Therefore, to measure digraph validity we evaluate that the generated graphs verify these constraints.
F.2 UNIQUENESS AND NOVELTY
To evaluate the quality and novelty of generated graphs, we employ the usual metrics based on graph isomorphism. In particular we measure the fraction of isomorphic graphs from the sampled set the train set (uniqueness) and the fraction of graphs from the sampled set that are not isomorphic to any other in the same sampled set (novelty).
However, since exact graph isomorphism testing can be computationally expensive, particularly for large or dense graphs such the one in the TPU Tiles dataset, we incorporate timeout mechanism. Specifically, if an isomorphism check does not complete within 5 seconds, is treated as a timeout and the graph is conservatively assumed to be non-isomorphic.
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In Table 10, that in three of the model configurations the computations timed out. We report the average percentage of graphs that could not be tested, which was 2.5% for DIGRESS, 0.8% for DEFoG, and 2% for DIRECTO RRWP.
F.3 MAXIMUM MEAN DISCREPANCY METRICS
For the different metrics, we adapt previous work from Martinkus et al. (2022); Law al. (2025). We propose to measure both out-degree and in-degree distributions, as well as clustering, spectre and walvelet. To measure the clustering coefficient, clustering coefficients using network rxx function c clusterine g (), which supports digraphs. For the spectral features (spectre and wavelet), derive the spectral features from the directed Laplacian, as described in Section 4.
compute the distribution of directed local
In addition, the orbil metric that was also computed in Martinkus al. (2022) has not been adapted to the directed setting as it relies on the Orbit Counting Algorithm (ORCA) which counts graphlets in networks but is only implemented in the undirected setting. Therefore, adapting this metric the directed setting remains an open challenge.
F.4 JOINT NODE-EDGE DISTRIBUTIONAL METRICS
Downstream evaluations in graph generative models usually assume conditional graph generation and alignment with auxiliary inputs. Moreover, standard evaluations in domains such as Scene Graph Generation (Li et al., 2024; Lorenz et al., 2024) assume access to input images to compare generated graphs against ground truth images using retrieval metries such Precision and Recall @ K. These require alignment with specific input conditions, which is not possible in formulation as there are no ground truth images for the generated graphs. Instead, we propose metrics to directly evaluate generative performance by assessing joint node-edge distributional coverage, essential aspect for labeled graphs such as NAS and scene graphs.
To evaluate coverage of node class distributions, we compute normalized histograms over node labels for each graph and calculate the MMD score between generated and reference sets, which captures how well categorical proportions are preserved across generated graphs.
Node type distribution MMD:
Triplet-based precision and recall: Inspired by retrieval-style evaluations in Scene Graph Generation (Li al., 2024; Lorenz et al., 2024), we extract semantic tuples from graphs. For scene graphs, we use (subject, attribute) pairs and (object, relation, object) triplets. For the TPU Tiles dataset, we extract element element-element element pairs. We compare the tuples from generated graphs with those in the test set and compute:
correctly generated tuples] Igenerated tuples)
ccorrectly generated tuples)
Precision
Recall
Itest tuples)
Embedding- distances (FID and RBF): Following Thompson et al. (2022), we compute RBF-kernel MMD and Fréchet Inception Distance (FID) over embeddings extracted from a random GNN trunk. These embeddings encode both node labels and structural information, thus capturing joint node-edge distributions.
F.5 DOWNSTREAM TASKS METRICS
In addition to the distributional alignment evaluations, we also provide evaluations downstream tasks for the TPU Tiles dataset. Here, we train a conditional generative model and follow the evaluation protocol of LayerDAG (Li al., 2025). Specifically, employ their ML-based surrogate cost models to approximate the computational performance of generated architectures. This ensures direct comparability with prior work.
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G EXPERIMENTAL DETAILS
G.1 DETAILS ON BASELINES
Maximum Likelihood Estimation (MLE) We define a simple baseline by estimating empirical probabilities from the training dataset via maximum likelihood estimation. The following distributions are computed:
1. Number of Nodes Distribution: Let n denote the number of nodes in a graph. The empirical distribution over n is given by:
Number of training graphs with n nodes
P(n) =
Total number of training graphs
2. Node Class Distribution: Let c be node class. The empirical node class distribution is: Number of nodes of class c P(u = c) Total number of nodes in all training graphs
3. Edge Type Distribution Conditioned on Node Pairs: Let (e,, 22) be pair of nodes such that v; has class c, and vy has class c₁. Let eij E ,I,...,KK K} denote the edge type between them (with K total edge types). The empirical conditional distribution is:
Number of edges of type & between (Ca: Cb)
P(ey = 02 cas Uj = cb)
K
Number of edges of type ke between (Ca. (b)
These distributions are stored and later used to construct graphs by first sampling the number of nodes, then sampling node types independently, and finally sampling edge types between each node pair according to the conditional edge distribution.
D-VAE (Zhang et al., 2019) This autoregressive method encodes and decodes directed acyelic graphs (DAGs) using an asynchronous message passing scheme. Unlike standard graph neural networks (GNNs), which apply simultaneous message passing across all nodes and updates them all at once, D-VAE updates nodes sequentially, allowing it to capture the computational flow within the graph rather than just its structure. During generation, the DAG is built one node time in topological order, with edges always directed toward newly added nodes ensuring the resulting graph remains acyclic. To enable this process, input graphs must be topologically sorted before being processed by the model.
To reproduce this baseline, we adapted the code available in their GitHub and arranged both the TPU Tiles and ER-DAG datasets to match the input format expected by the original model. We used the same hyperparameter configurations as suggested in the original paper to ensure consistency in training, which happened for 500 epochs. After generation, converted the output DAGs back into the required format for compatibility with our evaluation metrics.
For ER-DAG, tested two different learning rates 10 and 10-5, and ended using the model with 10 10-5, which resulted in better V.U.N. results with a training time of ~3mi per epoch. For TPU Tiles, due to the large size of the dataset and some of the graphs in it, was impossible to perform training epoch in a reasonable time, as each of them was predicted to take 12h.
LayerDAG (Li et al., 2025) This autoregressive diffusion model generates DAGs by converting them into sequences of bipartite graphs, effectively enabling a layer-wise tokenization suitable for autoregressive generation. At each step, a layer-wise diffusion process captures dependencies between nodes that not directly comparable (i.c., not connected by path). As with other autoregressive DAG models, the input graphs must be topologically sorted to ensure correct processing.
To reproduce this baseline, we adopted the implementation from GitHub adapted the ER-DAG data to the correct format ordering the graphs topologically, as TPU Tiles was already in the correct format. We kept the hyperparameters from the default configurations, including the number of epochs for each of the elements of the model. After generation, we converted the output DAGs back into the required format for compatibility with our evaluation metrics.
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DiGress (Vignac et al., 2023a) To benchmark against DIGRESS, consider its directed version by simply removing the symmetrization operations. This corresponds to DIRECTO-DE without dual attention and using the Laplacian positional encoding.
DeFoG (Qin et al., 2025a) To benchmark against DEFOG, we consider its directed version by again removing the symmetrization operations. In this case this corresponds DIRECTO without dual attention and using the RRWP Positional encoding (without PPR).
G.2 TRAINING SETUP
This section details the training hyperparameters used across experiments presented in the main text. Unless otherwise specified, the values reported below were used uniformly across all positional encoding variants and ablation settings. In particular, we report the choices for the training, for the positional encodings employed, and for the sampling.
Training The training hyperparameters used in our experiments are summarized Table These settings were consistently applied across all training runs for both DIRECTO and DIRECTO-DD (with discrete diffusion). We trained each model for up to 10,000 epochs, stopping the runs based on validation performance account for variations in convergence behavior across datasets.
Table Training and model hyperparameters used in all experiments.
Hyperparameter
Value
Training Settings
Learning rate Weight decay Optimizer
0,0002 x 10 10-12
AdamW
Model Scttings
Initial distribution
Marginal Identity 500 Cosine
Train distortion
Number of diffusion steps
Noise schedule
Number of layers
Hidden MLP dimensions
X: 256, E: 128, y: 128 dg: 256, dy: Rhead:
Transformer hidden dimensions
Feedforward dims
dimyfx: 256, dim z 128, dimyr, 128
Training loss weights (Auai))
[5. 0]
Positional encodings For RRWP, we set the walk length parameter K 20. For MagLap with Q = 5, we used equidistant potentials qs- = (0, 0.1, 0.2, 0.3, 0.4). For Q = 10, we selected again 10 equidistant potentials Qı0 = (0.01, ..... 0.1). ). all cases, we then kept the k- 10 first eigenvalues and eigenvectors, padding with null values whenever the graphs had less than 10 nodes.
Conditional generation To train with classifier-fret free guidance conditioning, we first conducted hyperparameter search to determine the guidance strength with values from 0 to 4, in particular Woond E (0, 0.25, 0.5, 1, 1.5, 2, 4). We ultimately set the parameter to Weond = 0.25.
Sampling To perform sampling optimization, conducted a targeted search over the three key hyperparameters: the time distortion function, the target guidance factor (w), and the stochasticity coefficient (1)). Each hyperparameter was varied independently, while the others were held their default values (identity distortion and w = y 0). For computational efficiency, searches were performed using 100 sampling steps for each configuration with five independent sampling runs to ensure robustness. An exception run was performed per configuration due to the large size of the dataset and some individual graphs.
made for the TPU Tiles dataset, where only single sampling
Based on the outcome of this optimization, selected the bert-performing configurations for each combination of dataset and positional encoding, balancing the trade-ofi off between V.U.N and ratio. The optimal configurations of the results reported in the main paper are summarized in Table 7. For the final evaluation, we performed five sampling runs per configuration using 1000 sampling steps. In all cases, the number of generated graphs matched the size of the test set (40 for synthetic datasets, 63
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for Visual Genome), except for TPU Tiles, where again limited the number of sampled digraphs to 40 due to computational constraints.
Table 7: Optimal sampling hyperparameters for each dataset and positional encoding variant.
ER-DAG
SBM
TPU Tiles
Visual Genome
Encoding
Distortion
w
Distortion
Distortion
Distortion
w
Baseline
Polydec
0.3
10
Revcos
0
0
Polydec
0
Cos
0.1
0
RRWP
Polydec
0.1
Polyine
0.1
Polyine
10
Polydec
0.3
10
MagLap
Polydec
0
100
Polydec
0.01
50
Polydec
100
Polydoc
0.1
50
Finally, for the ablations on dual attention and positional encodings, we did not perform sampling optimization 1) and avoid the consequent computational cost, but searched for the optimal time distortion. For the scalability experiments, we also did not perform sampling optimization but chose the relevant optimal configuration (from Table 7). For DIRECTO-DD we performed 500 sampling steps in all the different model configurations.
G.3 RESOURCES AND RUNTIME
All experiments were conducted on a single NVIDIA A A100SXMM-80GGB GPU. Table summarizes the runtime of the training and sampling stages across the different datasets. These timings reflect the training time until best performance and the average sampling time per sample, for all the configurations reported in the main results for DIRECTO.
Table 8: Training and sampling runtimes, and VRAM usage, across datasets and models.
Model
Train Graphs
Train Time
# Sampled
Sample Time (min/'sample)
VRAM (GB)
ER-DAG
MLE
128
0,001
40
0.02
0.15
D-VAE
128
31.8
40
1.2
9.07
LayerDAG
128
0.2
40
0.1
1,21
DiGress
DaFoG
128
13.6
40
5.71
Directo-DD DD (RRWF)
128
11,5
40
0.2
5.1
Directo DD (MagLap @-1)
128
40
0.4
5.21
Directo (MagLap (=5)
Directo-DD (MagLap Q=10)
128
26.5
40
5.30
Directo (RRWP)
128
13.5
40
0.3
5.72
Directo (Maglap (=:1)
128
40
0.5
5.72
MMglaap
Directo MMaglap Q-
128
40
1.5
5.71
SBM
MLE
128
0.002
40
0.01
0.15
DiGress
40
DeFaG
128
23.7
40
0.3
28.46
DirectoDD (RRWP)
128
21.3
40
0.4
26.83
DirectoDDD (Magl.gg ap Q-1)
128
Directo-DD (Maglap Q-5)
DirectoDD (Maglap Q=10)
128
20
40
3.1
26
Directo (RRWP)
128
13.7
40
0.5
28.34
(Maglap Q=1)
Directo (Maglap Q==5)
128
40
28.49
Directo (Magl ap Q-10)
128
40
6.0
28.46
TPU Tiles
D-VAE
5010
0OM
OOM
00M
0OM
LayerDAG
5040
40
0.5
1.75
DiGress
5010
26.3
40
1.9
30.56
DaFoG
Directo (RRWP)
5040
24,3
40
2.2
30.62
Directo DD (MagLap @=1)
5010
25
40
2.4
30.56
Directo (MagLap Q-5)
5040
40
2.9
30.60
Directo
Directo (Maglap Q=1)
5010
40
2.8
30.58
Directo (Maglap Q-5)
5040
30
40
3.2
30,61
Visual Genome
DiCrews
203
11
63
1.0
11.03
DeFaCi
203
13.6
63
0.9
11.22
Directo-DD DD (RRWP)
203
63
1.0
11.25
Directo-DD (Magl.m ap @=1)
11.03
Directo-DD (Maglap @ Q-5)
203
63
1.5
11.06
Directo (RRWP)
203
63
1.1
11.23
Directo ap
Directo (Magl ap (-5)
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H ADDITIONAL RESULTS
In this section, we present additional results that offer deeper insights into our model's performance and design choices. Specifically, we analyze the impact of the dual attention mechanism and the choice of positional encoding, compare performance on ER and DAG accuracy metrics, examine how dataset size influences generation quality, provide detailed results for conditional generation. and evaluate the effect of the sampling optimization step in discrete flow matching.
H.1 EXTENDED RESULTS FOR SYNTHETIC DATASETS
Table 9 reports the performance of DIRECTO across different configurations on synthetic datasets. We observe that the choice of architecture and positional encodings leads noticeable differences in the distributional alignment metrics (MMD), highlighting the strong performance of DIRECTO. At the same time, uniqueness and novelty remain consistently 100% across all models, confirming that the V.U.N. ratio is driven uniquely by the validity metric.
Table 9: Directed graph generation performance on synthetic graphs for different configurations of DIRECTO. Results are presented as mean + standard deviation across five sampling runs. We considered Q 10 for the Maglap variants.
Model
Out Out-degree
In degree
Clustering:
Spectre 1
Wavelet
Ratio
Valid 1
Unique
Novel
V.U.
Follo-Reryi Dimnsted Acyelic (FR--DD))
Training se1
0,0113
00/003
00355
0,0038
0,0024
1,0
99.2
100
0.0
0.0
0.063 c30s
0,0099
0.3315 -0007)
0.0635
0.1162
=5.1=x4
0.0
100 -:0000
-:0
D-VAE
0.615% 6319
1,509
C000
0.6160
63815
2:3000
106.6 4:54
0.0
100 -:0000
-00
LAVERDAG
0U0751 63397
68722
0.3842 L0060
0.0159
6.020
0,0218
20100
21.5
99.9-190
-90
4:
DIGRESS
0.0138 1330
1.000
0,1074 Lo009
0.0073
34.0
100 -:0
-90
DeFoG
0.0407 4300
0.0441
0.0460
-0000
0.0061
5/00
0.0008
2000
1.6
75
100
-0
0.0065
DiaEcro MagLap
0.0145 10304)
1.00
-0003
0.0063
8.063
1.2
55.0 92
00
RRWP
0.0107 10300
0.1219
000094
0.0064
8.000
1.11
-0
DIRECTO MapLap
40117.03044
0,071
500022
0,005
80006
0.0026
92.0
100
100
Stuclasiic Block Model (SBM)
Training
0,0031
00001
00274
0,0027
0
97.7
100
0.0
0.0
MLE
0.002 6
0U2020 cam
0.3973
0.808T
6.003
0,0508 10000
11.6 9:2
0,0
100
100
0.0
DICREESS DEFoG
0.0057 4
cams
0,0722
40009
0,0045
8.006
0.0142
1.000
3.9 4.30
41.5
100 -000
100 -10
41.5 4.1
0.0026 1.39
QU0022
1.00
0,0013
L0005
0.0045
0.0110 2.005
4,3
100 200
100
37.0 1.66
DIaEcTO- DD RKWP
0.006 2:39
0.0016
0,0892
2
0.0008
0.0027
2 2:000
1.7 2.-
51.5
100 200
100 00
81.5
Drassmoo-D DD Magl ap
0.007 0.007 2.39
0.0088 man
0.0450
200e
0.0098
0.0021
2.00
1.5 2.
95.5 2:17
100
100 10
05.5 2.17
Diaiseto RRWP
00011
01008
mmmm
0.0994
2000
0.0095
0.0027
: 2:00
1.8
99.5 1:0
100
100 100
99.5 =
DIRECTO MagLap
0.0039 1.302
0.0088
1.00
0,0654
1o00
0,0008
8.00
0.0039
1.00
2.3
96.5 1.5
100
100
05.5
H.2 EXTENDED RESULTS FOR REAL-WORLD DATASETS
Table 10 reports the performance of DIRECTO across different configurations real-world datasets. While some configurations show evidence of partial memorization, particularly in node and edge distributions, this does not negatively affect the overall generative quality: V.U.N. remains high, and DIRECTO consistently achieves the best or second-bes -est performance across the majority of metrics. These results highlight that, despite occasional replication of training structures, the model maintains strong distributional alignment and generates diverse, valid, and novel graphs
Table 11 reports the node-label distributional alignment metrics for the two real-world datasets considered. The results highlight that DIRECTO consistently manages to capture node-label distri- butions accurately. On TPU Tiles, DIRECTO achieve the lowest node-type MMD, indicating strong alignment with the training set, while also maintaining high precision and competitive recall. On Visual Genome, although DIRECTO exhibits slightly lower node-type MMD compared to DeFoG, attains the highest precision, recall, and RBF MMD, reflecting its ability to capture the underlying distribution. Overall, these results indicate that DIRECTO effectively balances reproducing node label distributions while capturing richer graph structure, with the different positional encoding strategies providing complementary strengths in distributional alignment.
It is relevant to note that the low precision achieved in the training sets is due to the fact that the ground truth set is much larger than the generated set, and therefore there are many pairs and triplets present in the ground truth dataset that have not been generated.
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Table 10: Directed graph generation performance on real-world graphs for different configurations of DIRECTO. Results are presented as mean + standard deviation across five sampling runs. We considered Q = 5 for the MagLap variants. OOT indicates that the model could not be run within a reasonable timeframe.
(+) indicates that isomorphism tests occasionally timed out, due to the large size of some graphs in this dataset; such cases were excluded from the uniqueness computation.
Model
Out-dearce]
In degreee.
Clustering
Warclt
Ratio
Unique
Novel
VUN.
TPU Tiles
Training
BUB003
0,0000
0,0077
000006
0,0022
1,0
100
100
0.0
0.0
MLE
0,0354 opes
0.0578
00014
(.014]
1000
0,0659 a 6303
0,0407 0008
149.5 Lo3
24.3 1oz
99.9 20
100 -0
24.7 c
VAB
OUT
OOT
OOT
OCT
OUT
OUT
OCT
O0T
OUT
OCT
LAYeRDAG
0.1933 1 obes
0.2225
-0005
0.1512
1082
0.0501
1 60336
0.076 Lo02
415. 1.11
100 1o0
99.5 1:3
98.5 -30
98.5 1.23
DIGREES
0.0054 euee
0.0725
E10020
1000
0.0033
à 030
0.0015
00000
57.5 m1
56.1 4:23
87.1 -:5
99.4 cm
70.9 4:14
DeFoO
0,0099 0000
0.0794
10821 IU8042
0,0040
0.0017
-0000
63.7 26
56.3 2:22
87.7
23.3-22
72.0 1:4
DIRECTOR 10- DD- RKW
0,0099 2 0300
(11167
018015
30011
0,0085
6:307
0.0015
2008
61.0 29
56.5 1:15
90.5
99.6 =m
76.8 1:15
DIRECTO-BE DD MagLap
0.0115 a epus
0.0703
00833
IL0103
10621
0.0076
à 6:30
0.0043
00014
64.3 -55
$6.5 4:51
90.5 4:33
100 -00
77.0 6.1
DIRECTO RRWP
0,0133
0.0859
00012
IU0136
10675
0,0056
0.300
0.0355
0000
75.4
97.0 1:
81.8%%
96.5 -12
77.0 1.2
DIRECTO MagLap
0,0039
10007
0.1775
00821
110211
1017
0,0126
0.302
0,0002
0000
44.0
90.5 1:33
90.5 1.45
97.5-.33
30.5 4.4
Visual Genoene
Training set
0.0088
0,00.0
0,0000
0.0022
00006
1.0
100
100
0.0
0.0
MILB
Le00s
0.474 -0003
015542 1000
0,0355
1.004
200077
17.0
0.143
100
-000
010006
0.3
DEFOG
0,0467
0.0002
1:300
c000
10.
100
DIRECTO DD REWY
0,0424 oowc
0.4113
0,0000 2/00
0,0132
0.0074
15.3..0
1:35
100
100
12.7
DIsecTo- DD Maglap
0,0245
01059
0.0000
0,0183
0,0085
100
619
DIRECTO RKWP
0,0494 o00
0.0255
0.0000
0,0226
0/0228
12
56.0
9844
99.3
83.8
DIRECTO Magi
0.01 80
0.002
0.0000
0,0142
0:00
0.0099
62
676
99.7
99.7-m
Table 11: Directed graph generation performance node-label distributional alignment metrics for the two real-world datasets. Results are presented as mean + standard deviation across five different sampling runs.
Model
Node type (MMD)
Precision f
Recall f
FID.
RBF (MMI)
TPU Tiles
Training set
0.0001
0.64
0.99
0.0854
0.0021
MLE
0,4573 0.00m
0.10 n:0
0,22 0001
105776.8 = 126763
1.0392
+00020
D-VAE
OOT
OOT
O0T
OOT
OOT
LAYERDAG
0,0130 0000
0.31 #
0,24 0.10
2864,1
17156
1.0208
000255
DIGRESS
0,0037 00083
0.95 # 0.03
0.46 008
887.7 # 3133
0.0969
066331
DEFOG
0,0031
0.90
0.45
159.5 1959
0,0579
DIRECTO-DD RRWP
0.0028 0,006
0.92 a 0.0
0.48 0:92
336.6 +: 1639
0.0587 06325
DIRECTO-DD MagLap
0,0051
0.42
000
482.2 1243
0.0792
0/266
DIRECTO RRWP
0.0018 0.000
0.97 0:0
0.44 0:01
293.0
=: 1191
0.0438
00179
DIRECTO MagLap
0,0086
0.96
0.42 0.02
96.0
0.0421
Visual Genome
Training set
0.0072
0.34
0.81
0.0214
MLE DIGRESS DEPoG
0,0807
0.25
0.13
48941.0 1998.5
0.6179
0.0206 00004
0.46 # 0.04
0.35
0.0
42,6 1
0.2321
#00276
0.0181
0.51
0.39
19.9
0.0845
00033
DIRECTO- DD RRWP
0,0231 0.002
0.48 1 1.0
0.48 0.0
4.8 10
0.0391
DIRECTO- MagLap
0,0288
0u0099
8.4
0.0420
000066
DIRECTO RRWP
0.0272
0.00%6
0.53 1 0.0
0.45
0.00
4.7 -M
0.0383
0.C0MM
DIRECTO Maglap
0.0299 000077
000
9.3
0.0507
00118
H.3 THE ROLE OF DUAL ATTENTION
To assess the impact of the dual-attention mechanism in our model, conduct an ablation study in which we remove the cross-attention between edge features and their transposes. This mechanism is designed to capture both source-to-target and target-to-source interactions, which are critical for modeling directional dependencies in directed graphs. By disabling dual attention, we isolate its contribution to generation quality, particularly in terms of validity, expressiveness, and generalization. We compare the full model with its ablated variant across model combinations to quantify the importance of this component.
DIRECTO Table 16 presents detailed ablation results highlighting the impact of the dual attention mechanism in our method. The results are presented for two datasets: ER-DAG and SBM. In particular, we observe that, across both datasets, the inclusion of the dual attention mechanism consistently improves the validity of the generated graphs and reduces structural discrepancies as measured by clustering, spectral, and wavelet distances. In the ER-DAG setting, models with dual
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attention (e.g.. RRWP--Dull and MeggLapDuall significantly outperform their non-dual counterparts in validity, achieving up to 94% validity with RRWP-Dual and 91 % with MagLap-Dudl compared to just 67.5% and 74%, respectively. These models also demonstrate better structural fidelity, particularly in clustering and spectral metrics. A similar trend is observed in the SBM dataset, where dual attention boosts validity (e.g.. RRWP-Dual: 87%, Magl.ap-Dual: 77%). These results demonstrate how incorporating bidirectional information flow into the attention mechanism contributes to improved model performance and provides insight into the effectiveness of this architectural component.
Table 12: Directed Graph Generation performance across different transformer architectures using discrete flow matching, MagLap considers Q = 1.
Dutuset
Model
Out- degree
In-degree 1
Clustering
Spectre
Warelet 1
Radio
Valid
Unique 1
Novel
V.U.11
010078 1.00
0.1293 10000
0.0735 uoad
0.0874
12.2
2.00
4:43
No
0,0109 50012
0,0110
0,0807
1.0147
0.0073 Loae7
0,0215
0007
47.0
100 1.00
6:63
47.0-122
FR-DAG
RRWP
010109
0.0108
0,0540
0.0061
0.003
1.3
67.5
100
67.5
RRWP. Dual
0,0107
0.1299
0,0054
0,0042
100
94.0
Mag
0.0113 80624
0.0130
0.119
63327
0.0061
0,0029
00.00
1.7 4.
14.0 6.4
100 4.
100 6.
74.0
MagLap- Dual
000100
30017
0,0166
0,0509
0.0062
0,0027
02085
1.3 4.02
91.0
100
100 6.03
91.0 4:35
No PE
010035
20015
0.0031
+ 5:805
0.1782
60017
0.0044
0,0655
00000
20.6 4:20
0.0 m/
100 =0
100
6.00
8.00
No PE- Dead
0.0038
5.011
0.0057
6 8:866
0.0638
6.809
0.0057
0.0127
a 0000s
3.5 4.04
48.5 6.64
100 430
100 6.6
48.5 6.64
SBM
RRWP
0.0026+
0/0022
0:0813
0.0048
0.0110
100 =30
37.0
RRWP- Dual
0.0031
5:014
0,0028
+ 5.003
0.0594
0.0033
0.0027
1.8 mz
99.5 1.
100 430
6.
99.5
Maglag
0.0032
0.0068
0.0195
100
Maglap- Dual
0.0079
0,0653
0,0033
0.0038
100
77.0
DIRECTO-DD Table 13 shows a second ablation assessing the contribution of the dual attention mechanism within the discrete diffusion framework. In this case, we see a similar pattern to that observed under discrete flow matching: the incorporation of dual attention leads to consistent gains in graph validity across both ER-DAG and SBM datasets. Models augmented with dual attention not only produce more valid graphs but also exhibit improved alignment with structural statistics such as clustering, spectre, and wavelet.
Table 13: Directed Graph Generation performance across different transformer architectures using discrete diffusion.. MagLap considers Q = 1.
Datavet
Model
Out-degree
Indlgree
Clustering
Spectre 1
Wavelet |
Ratio
Valid
Unique 1
Novd 1
V.U.N.
0,0095
0.1221 10112
0.0814
0,1094 Looe
14.4 13.
0.0.i.a
100
100 1:00
NL
0,0149
0.0138
0,1256 10085
0,0105
26.
53.0-22
1.0
53.0
RRWP
0.0147
0.0199
0,0985
0.0067
0.0088
21.
42.0
100
100
BRWP Dual
0.0199
0,0405
0.0055
0,0043
100
100
Maglap
B.0144 c.00
0.0355
0,0792
0.0071
5.012
0.0056
1646
59.0
100 a:00
100 4.00
59.0 1:3
Maglip- Dual
0,0142
0.0135
04054
0,0438
10001
0.0057
0.0066
10913
1.4
09.0
100 100
100 1.00
69.0 1.4
No PE
0.008
0.0040
0.1912
211177
0.0120
0.0975
20.5 239
0.0.2.
100
100 2:00
1.0
No Dual
000057 mmam
0.0099
20101
0.0721
1:00 0.0055
0,040
8.8 1.13
35.0
100
100 2:00
35.0 251
SBM
RKWP
01008
1 E:300
0.0038
0001
0,10K5
0.0088
2:000
0,0059
85.
4.0.38
100
100
4.0
KKWWP- Dual
010016
2 m000
0.0055
2000
0,0692
0.0088
1:00
0,0027
1.7
81.5
100
100:200
81.5 2.22
Magl-ap Maglaap
0.0155
0/08:3
13.3
10.0:14:
100
100 1:08
1:22
20000
0,0033
2000
66.5
200
100 1:00
H.4 THE ROLE OF POSITIONAL ENCODINGS
To evaluate the importance of positional encodings in model, we perform an ablation study comparing different options. Directed graphs lack a canonical node ordering, making positional information crucial for capturing structural context. We test several variants, including positional encoding, encodings that do not take into accounts directed information, and then different directed positional encodings. This analysis isolates how each encoding contributes to the model's ability to generate valid and semantically meaningful graphs. We report performance across synthetic datasets for both discrete diffusion and flow matching.
DIRECTO Table 14 presents an ablation study analyzing the impact of various positional encoding (PE) strategies on directed graph generation performance under the discrete flow matching framework. This evaluation highlights the critical role of positional information in enabling models to capture the asymmetric and hierarchical structure inherent to directed graphs.
Overall, the inclusion of any form of positional encoding significantly improves generation validity compared to the baseline with no PE, as expected. Among the methods tested, directed encodings like RRWP and magnetic Laplacian- variants demonstrate particularly strong performance. These
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encodings consistently enhance graph validity and reduce MMD ratio in both datasets. Notably, the MagLap variant with multiple potentials (Q = 10) achieves the best results in terms of validity and structural fidelity, although resulting in a higher computational cost.
Table 14: Directed Graph Generation performance across different positional encodings using discrete flow matching.
Duiaset
Model
Out-dearee
In-degree
Clustering
Wardet
Ratio
Valid
Unique
Novel
V.U.N.
No PE
0L0309 -000
0
6.362
0,0807
-0547
0,0073
0.0215
10017
3.0
4.2
47.0 12.1
100
100 -00
44.0 12.1
Lap DuLap
190119 10000
0,0115
6.300
0.1272
-0424
0.0047
L0000
0.0648
1.005
1.8 1:50
54.0
1:5
100
100 -00
54.00
1.49
BR-DAG
REWP
EL0045 L.0000
0.0107 --0000
0,0656
0.0104
1.342
0,2508
0.1209
-000
0
0.0054
Lowes
0,0036
0.0043
1.0002
1.7 4:
16: 1:2
59.0 14
94.0 10
100 190
100 mm
100 -000
100 m100
04.0
55.0
1.43
6:1
Maglap
D.0110 --00
0.
0,2509
-0807
0.0062
0.0027
1.3 8:2
91.0
100 -m0
100 -1
91.0
4:25
Maglaap (Q = 53
01012
2-0015
0.0107
0
2000
0U005
2.0000
010032
2:00
8.5
91.5 2:35
100 2:0
100 2:00
51.3
8:35
Magi Spgl-pppp (2) 100
08117
2-00m
0.01
0.711
0,0055
0/0026
2000
27.0
100
2000
02.00 8.17
010127
100
Lap
230
SBM
REWF
[U003]
0.006]
0.0057
0.0028
0,0594
0.0027
63.0
99.5
100
100
99.5
B10039
0,0036
0.0038
77.0
100
77.0
Maglay (2) 50
0.0002
0.0025
0.0620
0.0385
0.0034
2.1
95.5 6:58
100
100
05.5
Maglap (2) 1.0
0.0089
0,0038
0,0654
0.0338
0.0039
2.0
96.5
100
100
96.3
Interestingly, although classical Laplacian encodings (Lap and Dirl.ap) also play a role in increasing the V.U.N. and overall ratio, they obtain worse results than the variants that explicitly encode position and directionality. This emphasizes that, for directed generation tasks, the choice of positional encoding is not merely design detail, but relevant architectural decision that affects model success.
DIRECTO-DD Table 15 presents an ablation study assessing the influence of various positional encodings within the discrete diffusion framework across four synthetic datasets: SBM, ER-DAG, ER, and Price's ' model.
Table 15: Directed Graph Generation performance across different positional encodings using discrete diffusion.
Dataset
Model
degree
degree
Clastering
Spectre
Wavelet
Ratlo
Volid
Novel
V.U.N.
No FE
0,0149 2083
0.0138
Lo00s
0.125 came
0.0599
Locis
0,0106
10014
2.6 2:3
53.0 22
100 1:00
100 1:0
53.0 22
Lap
0.0138 3uee
00144
200007
01942
anam
0.0155
0:052
1.5
62.0 o2.0=
100 sm
100
57.0 ='
ER-DAG
RRWP
Maglap
0.0142 3001
0,0142
20044
00129
00135
0.8455
0.8488
cam
0.0055
0.0557
0.048
0.046
1101
21012
1.
14
69.0 4m
70.0 k7
100 a:
100 =:
100
100+
50.0 a44
79.0 -3
Maglapp eQ 53
0.035:
3002
00140
0000
01895
2.0m)
0.061
10007
00007
2001
1.4
78.5 2:5
100 1:0
100
18.5 235
Mig 100)
1004 0.045
00134
200m
0,582
2008
0.0053
00004
1.5
25.0
100 2:00
1.0
R3D 232
2000
1.0000
00003
1:00
100 130
Lap
00000
2:37
2:0
SBM
0.0035
0.0655
0.003
1:7
2:00
100
Miglap (12 5)
0,0039
0,005
92.0
100
Magl (Q)
0.0037
0.006
0.045
0.0055
0/001
1,5
95.5
100
100
05.5
No PE
0,0039 30622
0.0088
-000)
0.0485
C
0,0399
1E005
0,0030 SH01A
2.5 2.4
100
100 1:00
100 1.00
100 L03
Lap
0,002 3011
00001
200011
0.479
0.0057
D0017
31015
1.8 39
99.5.11
100 am
100
99.5 =11
FR
RRWP
0,0021
+ 20011
00001
200014
0,88S
0.0033
0.018
34012
1.8
9 mm
100 8:
100 +00
99.5 10
Maglasp
0,0021
20011
00001
200014
0.0515
0.0020
1:00
0.0000
1.8 1.5
90.0 =12
100 4:00
100 +:0
99.0 11
Maglap QQ) = 5)
0,0022
20014
00001
000114
0,0555
0,0029
10:0
0,007
2 50013
1.8 2:.3
100 8.0
100 1:00
100
100 08
0.0149: 2111
0,001
20002
0.0534 (1
0.044
0:007
31000
2.0 2:1
92.5
4:10
100- a00
=11
Lap
30655
1000
0.0124
2.
99.0 1.1
1:00
Phoce
30554
1::0
0.125
99.0 1:2
1:00
0.0232
100
Maglap 0,0191
0,0044
100
We observe that that ER and Price graphs, due to their simplicity and highly regular generative processes, pose a relatively easy modeling challenge. As a result, performance consistently high across all variants, including the baseline with no positional encoding. In contrast, the SBM and ER-DAG datasets present more intricate structural patterns, which amplify the benefits of informed positional encodings. In these cases, clear performance gaps emerge between the baseline, non- directed encodings (e.g., Lap), and airected-aware methods such as RRWP and MagLapPE.
These differences highlight the importance of encoding directionality faithfully capture the dis- tribution of more structurally diverse directed graphs. For this reason, in the main paper we focus our synthetic evaluations SBM and ER-DAG, where the modeling challenge is sufficiently rich to reveal the comparative strengths of different modeling strategies.
H.5 ABLATION OF OTHER MODEL COMPONENTS
To evaluate the importance of other parts of our proposed architecture, we have performed an ablation on the role of FiLM layers to enrich the attention with edge information, the use of the softmax in
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the aggregation of the source and target attention maps, and the gated residual connection for the node update. The results in Table support the addition these model components, since it results in improved model performance.
Table 16: Ablation results for different model components. All trainings use dual attention and RRWP the positional encoding.
Dataset
Model
Ratio +
Valid
Unique f
Novelty
V.U.N.
Original
2.
=00
100
1:
No FILM
6.9
4 1:30
100
100
4
ER-DAG
No softmax
1.7 0.22
74.5 1919
100
100 1.00
74.5 1.9
No FILM + No softmax
6.3 0.22
0.0 a 0.0
100 0:
100 =
0.0 Do
No gated residual connection
2.5 0.2
66.5 6.
0.0
= 0.0
66.5 + 6.0
Original No FILM
1.8 =
99.5 37
100 00
100 0.0
99.5 = 37
15.8
0.0
100
100
0.0
SBM
No softmax
2.4
89.0
100
100
89.0
No FILM No softmax
16.7 2:28
0.0 0:00
100 =0
100 2.00
0.0
No gated residuul connection
2.7 2.
91.5 1:7
100
100 1:
91.5 1.7
H.6 ER VS DAG PERFORMANCE
To evaluate the structural quality of generated synthetic directed acyclic graphs, we defined composite validity metric that captures two key properties: (i) the percentage of generated graphs that are Directed Acyclic Graphs (DAGs), and (ii) the percentage that follow the target structural distribution (e.g. Frdōs-Renyi). The validity score is computed as the proportion of generated graphs that satisfy both criteria. For completeness, we also report each component individually to disentangle the contributions of acyclicity and distributional alignment.
Table 17: ER vs DAG accuracy
Model
% DAG
% ER
Valid
Training set
100
99.2
99.2
MLE
0.0
97.0 211
0.0
D-VAR
100 =00
0.. 20
0.0
LAYERDAG
100 0.0
21.5 2:27
21.5 277
DIGRESS
35.0 422
98.5 20
34.0 4:1
DRFOG
21.5 2.7
100 +.00
21.5 2.7
DIRECTO-DD RRWP
79.0 1.3
100 1.00
79.0 1.37
DIRECTO-DD Magl.ap (Q= 10)
86.0 9.
99 1.2
85.0 = 102
DIRECTO RRWP
94.0
100
94.0 10
DIRECTO MagLap (Q 10)
99.5 +
92.5 #:
92.0
As shown in Table 17, baselines such as MLE and LAYERDAG exhibit stark imbalance: while LAYERDAG guarantees acyelicity, both fail to model the target ER distribution, resulting in low overall validity. On the other hand, MLE closely aligns with the distribution but fails to generate acyelic graphs.
In contrast, diffusion- and flow matching-based methods paired with dual attention and relevant directed positional encodings achieve strong performance on both fronts, demonstrating their ability to generate structurally valid DAGs that also align with the underlying data distribution. This highlights the limitations of purely autoregressive or heuristic DAG-specific approaches when used in isolation.
H.7 SCALABILITY EXPERIMENTS
In this section, investigate the scalability of our method with respect to three key axes: dataset size, parameter efficiency, and graph size. These experiments are designed to disentangle the contributions of data availability, architectural choices, and input complexity to the overall generative performance. providing a clearer understanding of the trade-offs between accuracy, efficiency, and computational cost in our framework.
Effect of dataset size To evaluate the effect on the number of graphs available at training time, we conduct ablation study using the synthetic ER-DAG dataset in two configurations: the standard
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size and variant with 10x more training, test, and validation samples. This setup allows us to assess whether the method encodings benefit from increased data. We compare the different positional encodings for discrete diffusion using dual attention, with results in Table 18.
Table 18: Effect of the dataset size (using DIRECTO-DD),
Dutuset Model
Out-degrce
In-dogree
Clustectng
Spectre
Warelet
Ratio :
Valid
Unique
Novel 1
0.0144
0J0052
67.0
100
Standrr
RSWF
0,0142
0.0129
0.0408
0.004S
1.4
79,0
100
0,0142
0.0135
100057
09.0-64
100
Maglapp (Q
0.0155
0,0140
0
D 110061
0.0037
1.400
-8.5
100
100
78.5
Train sec
0,0002
0,0003
0,0021
0,0004
0,0000
1.0
99,9
100
0.0
0,0
Lap
0000555 --mm
0.0032
0.0261 SAETS
0.0024 -0000
0.0066 -000
+0.5
85.5=24
100 =0
+00
65.5 =14
10
REWP
0.0050 vonex
0.0051
0,0277
BARSS
0.0024
0.0006
--0000
10.1 4:22
94.0 04
100 mm
+00
94.0 +44
Magl-ap
0,0050 cunes
0.0013
5.800
0,0505
EL0025
0.0007 0000
14.5 4:24
69.5-As
100 -00
100 50
59.5 +.3
Maglap (Q) 59
010051 2.00
0,0028
8:000
0,0237
= RORES
010002
2 9008
0.0008
2.000
9.5
93.522
100 80
100 5.00
V3.3 2:3
Overall, we see that increasing the dataset size significantly improves generation performance across all positional encoding variants, confirming that additional training data helps models better capture structural patterns. However, this performance gain comes at the cost of substantially longer training times and greater computational demand (~12h for the standard dataset vs ~32h for the bigger version). These trade-offs motivate our focus on architectural innovations such as employing discrete flow matching as more efficient alternatives scaling purely through data and compute.
Effect of model size (parameter efficiency) We next evaluate parameter efficiency by contrasting our dual-attention mechanism with alternative approach that scales model capacity by doubling the number of parameters. Dual attention increases the parameter space by introducing two adjacency- based attention heads, each specialized for handling edge directionality. To ablate for model size, we compare against a baseline in which the architecture is widened to match the parameter count, but without dual-attention.
Table 19: Directed Graph Generation performance for the dual attention architecture versus doubling the depth of the network without using the dual attention mechanism.
Dutuset
Model
Out-dearee 1
In-degree
Clastering
Spectre
Wavdet
Ratio 1
Valid
Unique
Novel 1
0,1741
2.2
2:13
ER-DAG
RRWP-
0.0107
0.0104
1.7
100
Double
00131
1.
1,52
0,0055
4.3
100
Maglap- Duad
0.010
00007
0.0116
4001
0,0509
0,0062
0.0027
6:000
1.3
91.0
100 mm
100 03
91.0 1.
REWP Double
0.005 00025
0.0081
0.00
0,1758
1.0215
0,0078
0,0816
26.3 2:5
0.0 1:00
100 -00
100 8:00
0.0 00
SRM
RRWP- Dual
0.031
2000
0.0025
0.0594
0,0035
0.0027
8..00
1.8 6.6
99.5
100 mm
100 100+000 60
99.5 L.
ap- Deable
0.086 2000
0.0081
0.0776
-00
Maglap- Duai
0.0099 110
0.0036
0000
0,0653
0.0038
0.0000
1.9
27.0
100 m3
As shown in Table 19, this baseline fails to recover the correct edge distribution, particularly in more challenging regimes such as SBM graphs. In contrast, the dual-attention model achieves substantially higher fidelity, demonstrating that our architecture leverages parameters more effectively by embedding structural inductive bias rather than merely increasing model scale.
Effect of node size We finally investigate how the method scales with increasing graph size by conducting experiments on ER-DAG datasets ranging from 80-150 up to 200-250 nodes (see Table 20 for dataset statistics). Due to the poor scalability of MuitMagLap encodings, we restrict this ablation to RRWP and MagLap positional encodings, with the largest graphs (200-250 nodes) evaluated only under RRWP. Training and sampling times, together with generation quality, are reported in Table 21. Importantly, the computational overhead introduced by dual attention remains bounded by constant factor of two and does not affect the overall asymptotic complexity, which matches that of standard graph transformers.
Table 20: Dataset statistics for the synthetic experiments with larger graph size (nodes and edges).
Min. nodes
Max. nodes
Avg. nodes
Min. edges
Max, edges
Avg. edges
80
150
117
1866
6699
4170
150
200
173
6643
11851
9046
200
250
225
11982
18645
15148
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In terms of positional encodings, MagLap exhibits clear scalability limitations, both in runtime and performance, whereas RRWP features scale considerably better, consistent with prior findings (e.g., Appendix G.4 of DeFoG (Qin et al., 2025a)). Regarding generative performance, DIRECTO demonstrates strong distributional fidelity across node sizes, as reflected the Ratio and distributional validity scores. However, ensuring strict acyelicity becomes increasingly challenging at scale: single misplaced edge can violate the constraint entirely, and the risk grows quadratically with the number of nodes. This highlights the inherent difficulty of enforcing global constraints in large directed graphs motivating future work on more robust inductive biases for acyclicity preservation.
Table 21: Scaling performance of DIRECTO on ER-DAG datasets of increasing node size.
Mldd
Oui-dearce
Io-degrec 5
Clustring 1
Spectre 1
Warckt
Ratio
Volid
Unique
Novel
V.UN..
Trais (k)
Min /stmple
90-
0,0040
010034 0.00
0.3582
0.005 5042
0.0007 s.0m0
4.2
65.0
100
2:940
150-300 (NUWW)
00024
2002
[U0K2] cane
0.1570
cam
0.003
0.0000 1.0000
1.2 4.33
60.0 -:3
100 cmo
100 u.
60.0 30
28
D.55S
19--000 (Me
01031
2:000
0.6001
010006
010001 1:000
km
100:200 2.1
21.52:0
1.365
H.8 CONDITIONAL GENERATION
Table 22 shows the results of the experiments with classifier-free guidance conditional generation. We followed the setup of Li et al. (2025), training on the TPU Tiles dataset with the provided conditional information and evaluating with their available suite and surrogate models.
Table 22: Averaged conditional generation results across 3 runs Baseline results were obtained from Li et al. (2025), where Spearman correlation was not reported.
Model
Pearson
Spearman
MAE
Real graphs
0.75 = 0.1
0.83 0.01
0.96 = 0.00
D-VAE
0,50 =: uo1
N/A
0.000
GRAPHRNN
0.62 00z
N/A
3.00
GRAPHPNAS
0.24 a: uno
N/A
2,1 1:00
ONESHOTDAG
0.56 =: 0uz
N/A
1.4 0.00
(T=1)
0.37 :
N/A
LAYERDAG
0.65
N/A
1.2 200
DIRECTO
0.59 0.63
1,3
Unlike methods explicitly designed for conditional DAG generation, our framework is general and primarily optimized for unconditional digraph generation. In contrast, autoregressive models such as GraphRNN and LayerDAG (especially the latter, where conditional generation is the default training setup) naturally more suited for this task and thus represent strong baselines. Moreover, our evaluation follows their official suite, which is tailored to their own method and may not fully align with our formulation.
Despite this, DIRECTO achieves competitive performance. In Pearson correlation, DIRECTO ranks just behind LayerDAG and GraphRNN, while performing on par with them mean absolute error (MAE). Notably, it also outperforms LAYERDAG' one-shot variant ONESHOTDAG, We further report Spearman correlation, which is implemented in the evaluation suite but was not included in prior work. These results show that even though conditional generation is not our main focus, DIRECTO achieves a strong balance across all metrics, demonstrating that our framework can readily extend to conditional tasks without specialized architectural modifications.
Finally, our results were obtained using classifier-free guidance, a straightforward extension that may not be optimal for conditional training in our setting. We expect that more dedicated strategies for conditioning could further improve performance. Overall, these findings highlight the versatility of DIRECTO: while unconditional generation remains our primary focus, its ability to adapt effectively to conditional generation underscores the promise of diffusion-basse formulation. for directed graph generation.
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H.9 IMPACT OF SAMPLING OPTIMIZATION USING DISCRETE FLOW MATCHING
To better understand the influence of the three sampling hyperparameters discussed in Appendix C: time distortion, stochasticity coefficient (17), and target guidance factor (w) on generative performance, analyze their effect the V.U.N. and the structural ratio. Specifically, curves for the four primary datasets reported in the main results table: SBM, RRWP, TPU Tiles, and Visual Genome. For each dataset, evaluate the impact of these parameters under positional encoding strategies: RRWP and MagLap. We report the results for 100 sampling steps and different sampling runs (except for TPU Tiles where only run was performed due computational constraints).
present performance
The results for the synthetic datasets in Figure illustrate that the sampling hyperparameters' influence the generative performance. We observe that different values can lead to variations in both V.U.N. and the structural ratio. However, higher V.U.N. does necessarily correspond to lower ratio. While certain configurations tend to perform well across both RRWP and MagLap positional encodings, the improvements are generally modest, suggesting that optimal settings may vary slightly depending on the dataset and positional encoding strategy used.
12-
Plodic
Tenoe
631 622 655
11
12 03 04
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(a) ER-DAG dataset with RRWP positional encoding
12
Stzapla Ditterkon
Duget Guddaaee
Stockastitty
(b) ER-DAG dataset with Maglaap (Q 10) positional encoding
YUN
42
12.400
Scom
301 MMS
SBM dataset with RRWP positional encoding
100
125
23
13
134
2.00
225
Sumppe Dimartiee
Stockneticky
(d) SBM dataset with MagLap (Q 10) positional encoding
Figure 4: Sampling optimization curves for the synthetic datasets with 100 sampling steps and sampling runs. We represent V.U.N. (blue) and MMD ratio (purple) and optimize for best trade-off for each of the three parameters individually.
A similar pattern is observed in the results for the real-world datasets (TPU Tiles and Visual Genome), as shown in Figure 5. The sampling hyperparameters continue to affect generation performance, and there is clear one-to-one relationship between V.U.N. and ratio. As with the synthetic
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datasets, some hyperparameter combinations show slightly consistent behavior across positional encodings, but overall, the best hyperparameters combination remain dependent on the dataset and positional encoding used.
MX
23
03
kmddy
Podde
On
03
TPU Tiles dataset with RRWP positional encoding
Piñdao
Oo
Pilyao
Sample Datorton
Darger Costers
Biockerlitty
(b) TPU Tiles dataset with MagLap (Q = 10) positional encoding
22
12
Pobcko
On
Bono
Polae
E41 Lic 01
05 05. 64
Searph
Teget
Stechanldtty
(c) Visual Genome dalaset with RRWP positional encoding
Secgle Dasirtiin
Grddaane
Stadarritt
(d) Visual Genome dataset with MagLap (Q 10) positional encoding
Figure 5: Sampling optimization curves for the synthetic datasets with 100 sampling steps and sampling runs. We represent V.U.N. (blue) and MMD ratio (purple) and optimize for best trade-off for each of the three parameters individually.
1 ITERATIVE REFINEMENT METHODS FOR GRAPH GENERATION
I.1 GRAPH ITERATIVE REFINEMENT METHODS
Within the realm of undirected graph generation, discrete space iterative refinement ap- proaches (Vignac et al., 2023a; Xu et al., 2024; Siraudin et al., 2024; Qin et al., 2025a) have emerged as powerful framework for capturing the intricate dependencies that govern target graph distributions. These methods achieve state-of-the-an performance by naturally aligning with the structure of graphs: they operate directly in discrete state spaces, matching the inherent discreteness of adjacency matrices, and respect node permutation equivariances due their one-shot prediction formulation. These characteristics make iterative refinement approaches particularly well-suited for the task of directed graph generation.
These methods comprise two main processes: a noising process and a denoising process. For consistency, define 0 as corresponding fully noised graphs and / clean graphs. The
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noising process runs from f = 1 to 0, progressively corrupting the original graphs, while the denoising process learns how to reverse this trajectory, running from t = 0 to ==
Denoising is performed using neural network parametrized by 8.
(+G.) -(Pij) (x) G.) ISnSN"
(Pile (e
(id) Gi) 15H4JNN) (5
(39)
which predicts categorical distributions over nodes and edges given noised graph G.. In practice, Pīje(-GGi is parameterized using a graph transformer such as the one from Dwivedi Bresson (2021), which allows for expressive modeling of complex graph structures. The network is trained using a cross-entropy loss applied independently to each node and each edge:
(c(a0) (i)))
L= C.
(,(m) Elog log Pile
((i))) - log (b.d) Pill
(40)
where the expectation is taken over time I sampled from a predefined distribution over [0, 1] (e.g.. uniform); G1 ~ pi (G1) is a clean graph from the dataset; and G. ~ (r.GG( is its noised version time t. The hyperparameter
E E* controls the relative weighting between node and edge reconstruction losses. Once the denoising network is trained, different graph iterative refinement methods vary how they leverage the predicted pf (-(G,) to progressively recover the clean graphs.
I.2 DISCRETE FLOW MATCHING FOR GRAPH GENERATION
Among graph iterative refinement methods, Discrete Flow Matching (DFM) (Campbell et al., 2024; Gat et al., 2024) has recently emerged as particularly powerful framework. By decoupling the training and sampling processes, DFM enables a broader and more flexible design space, which has been shown improve generative performance. This framework has proven particularly effective for undirected graph generation (Qin et al., 2025a).
The noising process in DFM is defined as a linear interpolation between the data distribution and a pre-specified noise distribution. This interpolation is applied independently to each variable, corresponding to each node and each edge in the of graphs:
Pijp (x)
= t t((x(") (") (1()) + (1 t) ) P (a/")) (n)),
(41)
where -): is the Kronecker delta and pi a reference distribution over node categories. A similar construction is used for edges. Therefore, the complete noising process corresponds to independently noising each node and edge through Pij1 and p respectively.
To reverse this process, DFM models denoising as Continuous-Time Markov Chain (CTMC). Starting from an initial distribution Po, the generative process evolves according to:
(
G,) 5(G,, GI+L1) + (G,,, Ge+ dt,
(42)
where R2 is the CTMC rate matrix. In practice, this update is approximated over finite interval At, in an Euler method step, with the rate matrix estimated from the network predictions Pf (- G.) via:
(,,,,, Gi-a))= = Σ 5(G) Gi
E PiB
(2()))))))))) Pil
(at (n) (n) (R(")
Ttt x (m)))
(43)
R
+ Σ B(G)
(6a))
l lefte)]] (44)
Pite
where the ('dd) denote the full graph G except variable d (node or edge). The Knonecker deltas ensure that each variable-specific rate matrix, R(m) R( is applied independently each node and edge, respectively. Finally, each node-specific rate matrix is defined as:
(n) (n) Re( FHA (mm))= ReLU (x
(=()) OrPpy (x)
("))]]]]]
(45)
x2- Pej1 (x( |x(())
for each node ,(n), where x²0 ( ( (()) 0} for edge rate matrices.
An analogous definition applics
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1.3 DISCRETE DIFFUSION FOR GRAPH GENERATION
Among discrete diffusion-based models for undirected graphs, DIGRESS (Vignac et al., 2023a), grounded in the structured discrete diffusion framework (Austin al., 2021a), has been particularly influential. This approach mostly differs from the DFM-based formulation described in Appendix 1.2, as it operates in discrete time, with the denoising process modeled Discrete-Tine Markov Chain (DTMC), in contrast to the continuous-
formulation used in DFM.
The forward, noising, process modeled as Markov noise process q, which generates sequence of progressively noised graphs Gt. for t I....,.. At each timestep, node and edge tensors are perturbed using categorical transition matrices [Q = g( = j -t-1 = i) and [Q,]((3) = q(e; = f-11 = i), respectively. This process induces structural changes such as edge addition or deletion, and edits to node and edge categories. The transition probabilities can be summarized as:
g(G, G+-1) (X,- E. 12) and g(G, G))
G)
x III 1=1 1=1
of E 0
(46)
In practice, these noise matrices are implemented based on the marginal noise model, defined as:
Q = I (1 x') and Q = (1 (1-a') g
where at transitions from to O with according to the popular cosine scheduling (Nichol & Dhariwal, 2021). The vectors 1x [1]x and 1g E [1] are filled with ones, and mx A and mg E E++ are vectors filled with the marginal node and edge distributions, respectively".
This noising process can be rewritten using the analogous notation the one used for DFM in Equation (47) as:
P (1
5(a47) (n) | ( a(t')((a(")) 22 ( + (1 a(t'))()))))))))))) (z)
(z(()),
(47)
with /' 1 & and a(l") = cos ( with small s. Importantly, this function only evaluated in the discrete values of time considered for the DTMC associated to the diffusion model. For the remaining of this section, we consider the transformed variable t' the reference time variable.
In the reverse, or denoising, process, a clean graph is progressively built leveraging the denoising neural network predictions pf (-IG),, analogously to the DFM setting. In particular, the model begins from noise sample Go and iteratively predicts the clean graph G1 by modeling node and edge distributions conditioned on the full graph structure. The reverse transition is defined as:
(Crt Ge) = II v ("F4Av Gr) II
=I
(i.) Pe+ (ertse (e
Ge). (48)
ISi<JEN
with each of the node and edge denoising terms are computer through the following marginalization:
p (E
Gr)
Σ
(2
(m) (x
(x)) Ge P1 0.(m)
(z) (a) (G)) (49)
a ( X)
and similarly for edges. To compute the missing posterior term in Equation (49), Gr), we equate the posterior term of the forward process:
(2 (1
Petaeii
(n) (2)) Pox
if if (n) (2)))) 0,
Pe4sUT1 (a (art G+)
(50)
otherwise,
where x((") and ((") denote the vectorized versions of (n) and (("), respectivelly Crucially, and contrarily to DFM, the sampling strategy with discrete diffusion is fixed at training time, which
restricts the design space of this generative framework.
We denote the probability simplex of the state-space of cardinality Z by A
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J VISUALIZATIONS
In this section, we present visualizations of the digraphs generated with DIRECTO for the different synthetic and real-world datasets and combinations of positional encodings reported the main paper. To allow for a fair comparison between original and generated digraphs, we additionally report digraphs from the train splits of the original datasets.
Original digraphs for the SBM dataset are shown in Figure 6. Figure 7 and Figure 8 show examples of digraphs from the Stochastic Block Model (SBM), where clear community structures are visible, consistent with the underlying block partitioning. These visualizations help illustrate how the positional encodings preserve modularity and inter-cluster connectivity. Similarly, Figure shows original digraphs from the ER-DAG dataset, with Figure 10 and Figure 11 displaying generated digraphs from the Rényi (ER-DAG) model. In this case, digraphs are plotted with nodes in topological order, to highlight the acyclicity.
For the real-world datasets, in particular for the TPU Tiles dataset, which consists directed acyelic graphs (DAGs) representing hardware execution plans, From the visualizations in Figure 13 and Figure 14 we see that the model is able to capture and generalize across structural patterns that emerge in computational workloads. In similar way as before, digraphs are represented with nodes in topological order to highlight acyclicity.
see the original DAGs in Figure 12
Finally, for the Visual Genome dataset, represent the three different node types: objects (blue), relationships (red) and attributes (green), alongside their respective node label. Figure 15 shows the original graphs, while Figure 16 and Figure 17 report the generations. They show how the model effectively captures the semantic and relational structure across these node categories, learning meaningful patterns that align with human understanding (i.e. person wears a yellow shirt, white cloud in the sky).
Figure 6: Visualizations of original synthetic digraphs from the train splits of the SBM dataset.
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Figure 7: Visualizations of four generated digraphs for the SBM dataset with RRWP positional encoding.
Figure 8: Visualizations of four generated digraphs for the SBM dataset with MagLap (Q = 10) positional encoding.
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Figure 9: Visualizations of original synthetic digraphs from the train splits of the ER-DAG datasets. Nodes are in topological ordering to highlight the acyclic structure.
Figure 10: Visualizations of four generated digraphs for the ER-DAG dataset with RRWP positional encoding. Nodes are in topological ordering highlight the acyclic structure.
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Figure 11: Visualizations four generated digraphs for the ER-DAG dataset with MagLap (Q 10) positional encoding. Nodes are in topological ordering to highlight the acyclic structure.
Figure 12: Visualizations of original real-world digraphs from the train splits of the TPU Tiles dataset. Nodes are topological ordering to highlight the acyclic structure.
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88
36
88
29
Figure 13: Visualizations of four generated dgraphs for the TPU Tiles dataset with RRWP positional encoding. Nodes are in topological ordering highlight the acyelic structure.
28-29
29
Figure 14: Visualizations of four generated digraphs for the TPU Tiles dataset with MagLap (Q = 5) positional encoding. Nodes topological ordering to highlight the acyclic structure.
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Figure 15: Visualizations of original real-world digraphs from the train splits of the Visual Genome dataset. Nodes represent objects (blue), relationships (red), and attributes (green).
Figure 16: Visualizations of four generated digraphs for the Visual Genome dataset with RRWP positional encoding. Nodes represent objects (blue), relationships (red), and attributes (green).
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Figure 17: Visualizations generated digraphs for the Visual Genome dataset with Maglap (Q = 5)
positional encoding. Nodes represent objects (blue), relationships (red), and attributes (green).
K LIMITATIONS AND FUTURE WORK
Although DIRECTO shows strong performance in the generation of directed graphs, it suffers from inheren Imitiations, that open several avenues for future work. These are mostly addressed at further enhancing its scalability, controllability, and applicability.
Directed graphs present a combinatorial explosion in the space of possible edges due to asymmetry, posing challenges in both training and sampling efficiency. Future work could explore sparsity-aware attention mechanisms (Qin et al., 2025b) to reduce quadratic memory and computational complexity. Hierarchical or latent diffusion ap- proaches (Bergmeister et al., 2024; Jang et al., 2024; Yang et al., 2024) may allow the model to efficiently generate large-scale graphs by operating at multiple levels of granularity or lower- dive--dimesionall latent space. Investigating adaptive sampling schedules and dataset subsampling strategies could also help manage computational costs without sacrificing model quality.
Scalability to Large Graphs and Datasets:
Conditional Graph Generation: Many real-world applications require generating graphs con- ditioned on specific properties or context, such as node features, initial subgraphs, or global graph characteristics. DIRECTO already supports classifier-free conditional generation, which allows flexible conditioning without requiring additional networks. However, this approach may not be optimal for all tasks, and exploring alternative conditional strategies (such as learned conditional embeddings or auxiliary networks) could further improve performance. Such extensions would enable targeted graph design, e.g., generating traffic networks with predefined entry/exit nodes, molecular graphs with given scaffolds, or DAGS adhering to specific causal structures
Explicit Structural Constraints: While DIRECTO can implicitly learn structural constraints such acyclicity, certain domains require strict adherence properties like DAG structure, planarity, or node degree limits. Integrating ideas from methods such as ConStrucy (Madeira et al., 2024) or PRODIGY (Sharma al., 2024) could enforce these constraints explicitly, enhancing control over
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generated graphs. Future research could also explore soft vs. hard constraint integration, allowing trade-off between flexibility and domain -mmain- validity.
Integration with Downstream Tasks: Beyond purely generative evaluation, integrating DIRECTO with downstream tasks such as causal inference, traffic simulation, or molecular optimization-coull highlight practical benefits and guide model improvements. Extending the metrics the benchmark to further reflect task-specific objectives could be relevant to quantify real-world impact.
Interpretabilityy and Controllability: Understanding which components (e.g., dual attention vs. directional positional encodings) drive specific structural properties in the generated graphs is an open research question. Future work could investigate interpretable latent representations and mechanisms to control generation in a predictable manner.
49


Reviews:
[{"review_id": "lWX0PnbmxU", "reviewer": "Reviewer_A2gN", "summary": "The authors present DIRECTO, a new method for generating directed graphs, an area that has been underexplored in graph generative modeling. The approach builds upon the discrete flow matching framework and introduces two key architectural components: (1) a dual attention mechanism that captures both source-to-target and target-to-source representations, and (2) direction-aware positional encodings. The authors indicate that the dual attention component is more critical to performance than the positional encodings. Additionally, they contribute a new benchmark suite containing synthetic and real-world datasets, along with metrics tailored for evaluating directed graph generation quality. The paper demonstrates strong empirical results across diverse settings. While the work provides a solid foundation for directed graph generation, conditional generation remains relatively underexplored and is identified as an important direction for future work.", "questions": "- Positional Encoding Design Choice: The authors concatenate positional encodings to node and edge features rather than using more integrated approaches common in modern transformers (e.g., addition, rotational embeddings like RoPE, or ALiBi). What is the rationale for choosing concatenation? Has the impact on parameter efficiency been considered, given that concatenation increases the input dimensionality? Were alternative integration methods explored?\n\n- Figure 1b Clarification: What do the node colors represent in Figure 1b? This visual element should be explained or removed if purely aesthetic.\n\n- Independent Interpolation Assumption: Why is independent interpolation used in the noising process (Eq. 1)? Is this a simplifying assumption for computational tractability, or is there theoretical justification for treating nodes and edges independently during the noising process?\n\n- Architectural Choices: What is the motivation for the gated residual connection in the node feature update (Eq. 23-24) when standard residual connections are used elsewhere? Has ablation been performed on this component?\n\n- Section 3.2 Reference: The last paragraph mentions ablation studies but doesn't specify where results are presented. Could you add a forward reference?\n\n- TPU Tiles Dataset: For the TPU Tiles dataset, was the preprocessing identical to Li et al. (2025), or were there modifications? This should be clarified.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- Novel and Important Problem: The paper addresses a significantly underexplored area in graph generation. The focus on directed graphs is well-motivated, with clear applications in biology, transportation networks, and scene understanding.\n- Comprehensive Technical Approach: The dual attention mechanism is presented as a solution to capture bidirectional dependencies inherent in directed graphs. The integration with discrete flow matching provides a principled generative framework with theoretical grounding.\n- Extensive Empirical Evaluation: The paper presents thorough experiments across multiple datasets, baselines, and ablation studies. The results consistently demonstrate the effectiveness of the proposed approach.\n- Benchmark Contribution: The introduction of standardized benchmarks for directed graph generation is a valuable contribution that will facilitate future research in this area. This work can provide a solid evaluation suite to later directions in the field.\n- Clear Presentation: The paper is well-written with clear motivation, technical exposition, and comprehensive experimental analysis. All visualizations seem relevant and convey information clearly.\n- Thorough Ablations: The systematic ablation studies effectively isolate the contributions of different components.", "weaknesses": "- Unclear Claims on Expressiveness: While the authors claim their method is \"expressive\" and \"robust,\" these claims lack formal theoretical justification or empirical evidence. The notion of expressiveness in the context of directed graph generation needs clearer definition and supporting arguments.\n\n- Scalability Concerns vs. Efficiency Claims: The paper mentions \"efficient generation\" but simultaneously acknowledges scalability limitations. This contradiction needs clarification. Table 20 shows significant performance degradation for larger graphs (200-250 nodes), which undermines efficiency claims.\n\n- Conditional Generation Underexplored: While acknowledged as future work, the conditional generation experiments (Section H.7) are limited. Given the practical importance of conditional generation for real-world applications, this might deserve more attention.\n\nLimited Architectural Justification: Several design choices appear ad-hoc:\n- Why is independent interpolation used in the noising process (Eq. 1). It is this unclear if this is a simplifying assumption or theoretically motivated.\n- The gated residual connection for node features (Eq. 23-24) lacks justification compared to standard residual connections used elsewhere.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tQvPeK7DyJ", "reviewer": "Reviewer_pVm7", "summary": "This work proposes a model for generating directed graphs. Its main contributions are in (i) the use of direction-aware positional encodings, (ii) a dual attention mechanism that again takes edge directions into account and (iii) the introduction of new benchmarks for evaluating directed graph generative models. The performance is measured across different synthetic and true graphs and seemingly outperforms other undirected GGM or specialized in generating DAG models. An ablation study shows that the dual attention mechanism is critical in the increased performance.", "questions": "Q1: are global features utilized/ part of the training loss? what is the categorical modeling for edges for the datasets the model is evaluated against?\nQ2: What is the benefit of MagLap encoding vs the directed laplacian of chung?\nQ3: Ratio comes from averaging the MMD for different descriptors. Is it though a good practice to average MMD for different descriptors when it is unbounded and lacks scale comparison?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "* The model proposed in this work succeeds in generating graphs with specific constraints (acyclic, or edges only across certain types) even if those constrains are not explicitly stated (e.g., through some regularization parameter) during the training process. \n* The architecture is quite generic in that it can be used to generate directed graphs of virtually any type as edges are encoding through categorical variables, while it can also accomodate node and graph features.", "weaknesses": "*  The model itself, at least as presented, is not self-contained. Section 2 which provides a background on diffusion models and past work seems disconnected to Section 3 that describes the attention mechanism employed. For example, it is not specified how edges are modeled through categorical variables (e.g, is it representing one of the 4 possible classes -- edges in both directions, in one of the two (x2), or absent?), how the rate matrix is parameterized though the proposed architecture, while fig. 2 employs global features that are not part of the loss function in eq. 4.\n* The complexity of the proposed model seems to make it a bit susceptible to choice of hyperparameters (see fluctuation in performance in fig 4).\n* The novelty of the work is on the dual attention mechanism, but it does not extend to other parts of it (e.g., new positional encoding, diffusion process is of DFM, evaluation metrics is MMD).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DA8kGojHwP", "reviewer": "Reviewer_RPrZ", "summary": "The paper proposes Directo, a method for generating directed graphs based on discrete flow matching. Great care is given to the denoising model arch. To enhance the capability of the method to handle edge directionality, 1) a tailored attention mechanism called \"dual attention\" is proposed; and 2) asymmetric position encodings, such as those based on directional Laplacians, are used. Other GNN components from prior work, FiLM and PNA, are also incorporated into the arch. Besides the method, the paper also creates a benchmark for directed graph generation, with several synthetic and real-world datasets. Directo is shown to have strong performance on this benchmark relative to other methods, based on evaluation metrics adapted to the setting. Finally, further experiments are discussed that explore the impact of ablating the major model arch components, as well as the scalability of the method.", "questions": "- Much of the main paper describes a complex GNN arch tailored for directed graphs, which could be applied outside of the graph generation setting, e.g., to link prediction or node classification. Has this been attempted? It seems worthwhile to evaluate the arch on the more standardized benchmarks for those tasks. If the results are weak on other tasks but strong for generation, that is also an interesting finding.\n- Relatedly, there is little discussion of the flow matching part in the main paper. Were there any interesting findings related to the flow matching as opposed to the arch that have not been discussed in prior work?\n- Regarding the time distortion function, it is stated that \"these functions are selected based on dataset-specific properties to improve fidelity and structural constraints without need for retraining.\" Could you please clarify the selection procedure, given that there is not a unique objective unlike in typical hyperparameter selection via cross-validation, e.g., for node classification?\n- For graph generation papers in general, a core theme is the trade-off between generating diverse graphs (U.N. from V.U.N.) and graphs that match the training distribution (\"ratio\" in this paper). Is there some way to tune the proposed method to favor one or the other? How does the ease of such tuning compare to other methods?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The area of directed graph generation seems important but relatively unexplored given the many works for undirected graph generation.\n- The writing is clear, grammatical, and well-organized.\n- The paper introduces not only a new method, but also a benchmark for the area of directed graph generation. A code repository is included for reproducibility.\n- Ablations are included to justify major new components of the model arch.", "weaknesses": "- The proposed arch is complex, and while there are ablation studies for some important components, this is not true of all components, e.g., use of FiLM.\n- As the authors note, the proposed method can fail to maintain validity when scaling up (e.g., failing to maintain strict acyclicity beyond 200 nodes).\n- Ideally the broad setting of the paper in terms of the scale of generated graphs would be clarified earlier in the paper. Graph generation papers and algorithms roughly cluster on two categories, those for 10s-100s of nodes (this paper), and others for 1000s-10000s and up.\n- (nit) Some notation could be easier to read, e.g., the use of Kronecker delta in Eq 1 could be replaced with piecewise notation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wmv35YInjC", "reviewer": "Reviewer_gox2", "summary": "DIRECTO is a discrete-state, iterative-refinement generator specifically designed for directed graphs. It introduces two key enhancements: (1) direction-aware positional encodings, and (2) a dual-attention transformer that explicitly combines source-to-target and target-to-source channels. The training process employs discrete flow matching, and the authors also present a discrete-diffusion variant. The generated graphs are evaluated on various benchmarks, including synthetic directed/DAG distributions, TPU compute DAGs, and Visual Genome scene graphs. The evaluation metrics focus on validity, uniqueness, and novelty, as well as a normalized MMD Ratio and label-aware distances on real-world data.", "questions": "How much of the performance improvement is attributed to (a) splitting the roles (S/T) versus (b) employing the aggregation trick (concat two maps and then apply one softmax)? Please provide an ablation study that distinguishes between (i) role-splitting alone, (ii) role-splitting with FiLM edge modulation, and (iii) role-splitting with a unified softmax.\n\nWere the sampling steps, time, and VRAM used in each model (including DIRECTO-DD) consistent? Please include a fairness table.\n\nThe authors consider every ordered pair as a categorical “edge (including absent).” How is the graph size N determined at the generation time, and do the metrics account for N?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Treating directionality as a first-class citizen (both in PEs and attention) is long overdue. Dual attention is a clean, architecture-level bias that addresses asymmetry.\n\nThe comparison between “dual vs double depth” demonstrates a genuine architectural advantage, not merely an increase in capacity.\n\nEvaluating typed constraints on Visual Genome and acyclicity on DAGs is the appropriate approach. The V.U.N. effectively penalizes memorization.\n\nThe same directional concepts also apply to discrete diffusion (DIRECTO-DD), indicating that they are not exclusive to DFM-specific techniques.", "weaknesses": "1. Directed attention doubles attention maps, while MagLap/Multi-q PEs are expensive. Additionally, CTMC sampling requires numerous steps to achieve quality. The paper claims decent scaling, but we need to compare it to strong autoregressive digraph models on larger graphs, considering wall-clock time and VRAM usage versus.\n\n2. The benchmarks are relatively small and close to the training support. We need out-of-distribution (OOD) tests, such as altered degree exponents, flipped community asymmetry, or different label marginals. This will help us understand how brittle dual attention is when arrow statistics shift.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)