BiGG-E: Scalable Weighted Graph Generation

• BiGG-E is a scalable autoregressive generative model for weighted graphs that jointly learns graph topology and continuous edge weights for efficient sparse generation.
• It extends the original BiGG model by incorporating a joint merge of topology and weight states, enabling simultaneous conditioning of edge existence and weight prediction.
• The model achieves near-linear generation time, O((n+m) log n), by leveraging efficient Fenwick trees and binary-tree algorithms to handle sparsity.

Searching arXiv for papers on BiGG-E and related BiGG graph generative models. BiGG-E is a scalable autoregressive generative model for weighted graphs that learns a joint distribution over graph topology and continuous edge weights while exploiting sparsity to retain the near-linear generation-time behavior of the original BiGG model. In the formulation given in "Scalable Generative Modeling of Weighted Graphs" (Williams et al., 30 Jul 2025), BiGG-E generates a weighted graph with $n$ nodes and $m$ edges in $O((n+m)\log n)$ time, extending BiGG’s sparse-graph machinery from unweighted graphs to weighted topologies. Relative to BiGG, which was developed for unweighted sparse graphs and achieved the same asymptotic scaling by avoiding explicit $n\times n$ adjacency construction (Dai et al., 2020), BiGG-E introduces explicit state for weights and a joint topology–weight conditioning mechanism rather than treating weights as a post hoc annotation.

1. Problem setting and conceptual scope

BiGG-E is defined for weighted graphs $G=(V,E,W)$, where $W(v_i,v_j)=w_{ij}$ if $(v_i,v_j)\in E$, and $0$ otherwise (Williams et al., 30 Jul 2025). The model is motivated by two difficulties that do not arise in the unweighted setting in the same form. First, weighted-graph generation requires modeling both edge existence and the continuous value of each edge weight, yielding a mixed discrete-continuous likelihood. Second, topology and weights are often mutually dependent: edge weights can influence which edges appear, and the topology can in turn affect the weight distribution.

As in BiGG, permutation invariance is handled through a canonical ordering rather than exact marginalization over all node permutations, which the paper describes as intractable. Under this approximation,

$p(G) \simeq p(|V|=n)p(\mathbf{W}^{\pi(G)}).$

This places BiGG-E within the class of ordered autoregressive graph models, but its ordering dependence is inherited from the same practical compromise used in earlier sparse graph generation work (Dai et al., 2020).

The paper characterizes BiGG-E as a nontrivial extension of BiGG rather than a superficial weighted variant. That distinction is central: the design objective is not merely to append a weight regressor to a sparse topology generator, but to learn a joint distribution in which topology and weights remain intertwined throughout generation (Williams et al., 30 Jul 2025).

2. Probabilistic formulation

The key factorization in BiGG-E is at the level of a weighted edge $(e,w)$: $m$0 Here $m$1 is a Bernoulli probability that an edge exists, and $m$2 is a continuous density used only when $m$3 (Williams et al., 30 Jul 2025).

For positive weights, the paper uses a softplus-transformed normal: $m$4 The resulting conditional density is written, up to a constant factor, as

$m$5

The parameters $m$6 and $m$7 are outputs of neural networks. Because this choice enforces positivity, a plausible implication is that the formulation as presented is tailored to strictly positive edge-weight regimes.

The paper provides two equivalent likelihood views. An entrywise lower-triangular factorization is

$m$8

with

$m$9

More directly aligned with BiGG’s sparse representation, the weighted graph is factorized as a weighted edge set: $O((n+m)\log n)$0 Training maximizes the corresponding log-likelihood

$O((n+m)\log n)$1

This formulation preserves the sparse edge-set viewpoint of BiGG while augmenting each autoregressive step with a continuous conditional density (Williams et al., 30 Jul 2025).

3. Architecture and joint topology–weight state

BiGG-E retains BiGG’s two-level autoregressive design for sparse topology. In BiGG, each node’s incident edges are generated using a binary decision tree $O((n+m)\log n)$2, and previous rows are summarized efficiently through a Fenwick-style structure rather than a long sequential recurrence (Dai et al., 2020). BiGG-E keeps that architecture and adds a second summary structure for weights (Williams et al., 30 Jul 2025).

Within a row/tree $O((n+m)\log n)$3, BiGG uses a top-down state $O((n+m)\log n)$4 that encodes the path so far and a bottom-up summary of the already-generated left subtree. The right-child state is formed by a Tree-LSTM merge: $O((n+m)\log n)$5 This yields the tree factorization

$O((n+m)\log n)$6

BiGG-E’s distinctive addition is a second Fenwick tree for weights. Each generated weight $O((n+m)\log n)$7 is embedded by an LSTM and recursively aggregated via Tree-LSTM nodes, producing a weight summary $O((n+m)\log n)$8 in $O((n+m)\log n)$9 time. That hidden state parameterizes the next weight distribution: $n\times n$0

The central extension is the joint merge of topology and weight states: $n\times n$1 BiGG-E uses this merged state both to condition the next edge-existence decision in $n\times n$2 and to condition the next weight distribution at a leaf: $n\times n$3 and

$n\times n$4

This architecture directly addresses a common misunderstanding: BiGG-E is not described as “BiGG plus a weight head.” The paper explicitly distinguishes it from BiGG-MLP, a naive extension that injects weight information using an MLP but does not preserve separate topology and weight states, and from BiGG+GCN, a two-stage topology-first approach (Williams et al., 30 Jul 2025).

4. Autoregressive generation procedure and sparsity exploitation

Generation proceeds autoregressively over nodes $n\times n$5. For each row/tree $n\times n$6, the model recursively samples left/right edge-existence decisions. Whenever a singleton interval indicates an actual edge, it samples a weight from the softplus-normal conditional distribution, then immediately embeds that new weight and updates the Fenwick weight tree (Williams et al., 30 Jul 2025).

The paper’s pseudocode makes the update path explicit: after sampling a new weight $n\times n$7, the model calls embed_weight, passes the scalar through an LSTM, adds the embedding to the Fenwick weight tree, updates the tree with Tree-LSTM merges, and recomputes the current summary state $n\times n$8. As a result, the model never needs to revisit all previously generated edges or weights.

This generation strategy inherits the main sparse-structure insight of BiGG. In the original unweighted setting, BiGG avoided explicit adjacency-matrix generation by representing neighbor selection through binary splits and by paying computational cost for actual edges rather than for absent entries (Dai et al., 2020). BiGG-E preserves that principle but extends the state space so that every topology decision may depend on weight history and every weight prediction may depend on topology history (Williams et al., 30 Jul 2025).

The architectural consequence is that the model remains edge-centric rather than matrix-centric. This is why the weighted extension does not collapse back to $n\times n$9 behavior even though it models continuous edge attributes jointly with sparse topology.

5. Complexity and scalability

The main asymptotic claim is that BiGG-E generates a sparse weighted graph with $G=(V,E,W)$0 nodes and $G=(V,E,W)$1 edges in

$G=(V,E,W)$2

time (Williams et al., 30 Jul 2025). The argument parallels the one for BiGG (Dai et al., 2020). The Fenwick topology tree contributes $G=(V,E,W)$3 overall for row and structure handling, and the construction of the edge trees costs $G=(V,E,W)$4 because each actual edge induces only logarithmic work in the binary-tree and Fenwick updates.

The paper further states that weight-tree maintenance adds only $G=(V,E,W)$5 training time and $G=(V,E,W)$6 sampling time, and treats this as asymptotically negligible when $G=(V,E,W)$7. Under the same sparsity assumptions used by BiGG, the weighted extension therefore preserves the original model’s near-linear scaling in graph size (Williams et al., 30 Jul 2025).

This computational profile is important because the principal alternatives in the paper occupy less favorable regimes. Adj-LSTM is described as fully expressive but computationally expensive, SparseDiff is slower and less scalable in the reported experiments, and dense lower-triangular modeling remains tied to $G=(V,E,W)$8-style dependence on the number of node pairs. BiGG-E, by contrast, is reported to remain practical on large graphs while maintaining strong fidelity. The paper summarizes the empirical scaling as training time $G=(V,E,W)$9 and sampling time $W(v_i,v_j)=w_{ij}$0.

6. Empirical behavior, baselines, and relation to BiGG

The evaluation is designed around three questions: whether the models capture diverse weighted-graph distributions, whether joint modeling of topology and weights helps, and whether the methods scale to large graphs (Williams et al., 30 Jul 2025). The baselines are Adj-LSTM, BiGG-MLP, BiGG+GCN, and an Erdős–Rényi baseline.

The benchmark datasets are Erdős–Rényi graphs, tree graphs, 3D Point Cloud graphs, lobster graphs, and joint trees. For scalability, the paper also trains on weighted trees of sizes from $W(v_i,v_j)=w_{ij}$1 up to $W(v_i,v_j)=w_{ij}$2 nodes and compares against SparseDiff. Evaluation uses MMD over degree distributions, clustering coefficients, orbit statistics, and the normalized unweighted Laplacian spectrum for topology; mean, standard deviation, and MMD over the marginal weight distribution for weights; and MMD over weighted Laplacian spectrum and weighted degree distribution for joint structure. For tree and lobster data, it additionally reports an error rate defined as the proportion of generated graphs that do not have the correct structural form.

Empirically, BiGG-E is reported as the strongest overall method. On topological metrics it is consistently among the best and often the best across datasets, including trees, lobsters, and joint trees. The paper highlights that BiGG-MLP degrades substantially topologically, which it interprets as evidence that simply injecting weights into BiGG through a single mixed representation is not sufficient. On 3D point clouds, BiGG+GCN can match topology fairly well but produces implausible weights with inflated variance, whereas BiGG-E is presented as avoiding that instability because weights and topology condition on each other through the joint state.

The clearest separation appears on the joint trees dataset, where topology and weights are explicitly coupled. There BiGG-E is reported to dominate all competitors across degree, weighted degree, spectrum, weighted spectrum, orbit, and joint MMD metrics, and to achieve zero structural error (Williams et al., 30 Jul 2025). This is the paper’s most direct evidence against the misconception that a topology-first, weight-second pipeline is generally sufficient.

In historical relation, BiGG-E should be understood as extending the sparse autoregressive program introduced by BiGG rather than replacing it. BiGG established that sparse graph generation need not build a full adjacency matrix and can be reduced to $W(v_i,v_j)=w_{ij}$3 time for unweighted graphs (Dai et al., 2020). BiGG-E preserves that sparse binary-tree/Fenwick-tree backbone while adding a second explicitly modeled state for weights and a joint merge between the two modalities (Williams et al., 30 Jul 2025). The resulting model occupies a specific technical niche: scalable joint generation of sparse weighted graphs, with topology and continuous edge weights learned as a single autoregressive object rather than as decoupled stages.