Automated Residential Bubble Diagram Generation Based on Dual-Branch Graph Neural Network and Variational Encoding

Abstract

Bubble diagrams containing key features and information are used for generative design of floor plans. The lack of reliable methods for automatically generating bubble diagrams significantly affects the smoothness of layout generation systems. To improve the time-consuming and unstable acquisition process, a novel method based on graph neural networks (GNNs) is proposed to generate various residential bubble diagrams. First, a dual-branch graph neural network (DBGNN) is introduced to learn the feature patterns of heterogeneous links, including connectivity and adjacency relations. Then, decentralized node sampling (DNS) and centralized node sampling (CNS) are proposed to enhance the local feature learning of DBGNN. Subsequently, a variational graph autoencoder (VGAE) is used to learn the implicit distribution of topological patterns, enabling the model to generate diverse outputs. Experimental results show that the proposed model performs excellently in two link prediction tasks, achieving 92.39% ACC-Door and 78.84% ACC-Wall, while also generating 50 distinct bubble diagrams, validating the effectiveness of the proposed method and demonstrating its outstanding application value.

1. Introduction

The architectural floor plan is a crucial component of architectural design, reflecting in detail the functional requirements of the building and the relationships between its spaces [1]. Traditional floor plan design typically relies on experienced architects, which can be time-consuming and may lead to variations in efficiency [2]. With advancements in artificial intelligence, neural network-based floor plan generation methods have shown potential in improving design efficiency, enhancing and assisting traditional labor-intensive approaches [3]. These floor plan generation models are based on sparse design inputs, such as bubble diagrams and area constraints, to automatically create the desired layout [4,5]. For these generation models, the bubble diagram plays a key role, as it effectively represents the functional spaces of the floor plan while also describing the organizational relationships between different functional areas [6]. Floor plan generation models progressively reason from the information conveyed by the bubble diagram to achieve a topologically consistent layout. Therefore, the accuracy and structure of the bubble diagram directly influence the feasibility and effectiveness of the generated layout.

However, ensuring the diversity and reliability of bubble diagrams in current floor plan generation models remains challenging. These bubble diagrams are either extracted from floor plan datasets [7], or manually designed by users [8]. For example, Nauata et al. [9] obtained bubble diagrams from floor plan datasets and used a generative adversarial network (GAN) to map these bubble diagrams to layouts. Shabani et al. [10] extracted bubble diagrams from the RPLAN dataset and used a diffusion model to generate floor plans. Hu et al. [11] pre-builds a bubble diagram database and retrieves bubble diagrams based on user-provided boundaries, which are then input into GNN and convolutional neural networks (CNNs) to generate layouts. IPLAN [12] allows users to interactively design bubble diagrams and progressively generate layouts using a variational autoencoder (VAE). Nevertheless, these bubble diagram acquisition strategies may limit the application of floor plan generation models in real-world design scenarios. On the one hand, extracting bubble diagrams from datasets does not create new bubble diagrams, and the styles of the obtained bubble diagrams are limited, which may hinder the generation of innovative solutions and lead to challenges in layout diversity [13]. On the other hand, bubble diagrams designed by non-professional users may contain errors, and the model may fail to satisfy valid topological constraints during the generation process, leading to issues such as spatial conflicts or illogical functionality in the generated floor plans. Additionally, the extra steps of manually creating or retrieving bubble diagrams may reduce the usability of software [14]. Therefore, there is a need to propose a method capable of automatically generating reliable bubble diagrams.

However, automatically creating reliable bubble diagrams is challenging, particularly due to the complexity of the topological structures. A bubble diagram is an abstract schematic of the architectural floor plan, describing the functional attributes and spatial organization of the building [15]. In a bubble diagram, there are links between various functional nodes that represent spatial relationships, including connectivity and adjacency [16]. The ways in which these functional nodes are connected are numerous, and even between nodes of the same type, there are various forms of connectivity or adjacency. Additionally, the degree of each functional node must be controlled within a reasonable range to avoid incorrect topological structures [17]. Furthermore, due to the dynamic nature of the link forms and quantities, the bubble diagram topology created with the same nodes can vary greatly. These characteristics present fundamental requirements for bubble diagram generation models.

The successful application of GNNs on various forms of topological structures has inspired solutions to the bubble diagram generation problem [18]. GNNs excel at processing various structured graph data, including social networks [19], molecular structures [20], and architectural structures [14]. For example, Fan et al. used GNNs to model social graphs and perform social recommendations [21]. Hao et al. used GNNs to learn semi-supervised representations of a large number of molecules, achieving accurate molecular property prediction [22]. Zhao et al. used graph data to represent shear wall structures and applied GNNs for shear wall layout design [23]. However, previous GNNs cannot be directly applied to bubble diagrams for three main reasons. First, some GNNs are typically designed for large graphs with hundreds or thousands of nodes, making their architecture unsuitable for learning on smaller graphs with only a few nodes [24,25]. Second, the multi-hop neighbor sampling strategies developed for large graphs are not suitable for bubble diagrams with limited nodes [26]. Applying these strategies to bubble diagrams may result in random sampling that quickly traverses all global nodes and generates excessive redundant subgraphs, which could hinder the model’s ability to learn structural knowledge. Third, achieving diversity in bubble diagram generation is difficult. A simple solution to produce different prediction results is to train multiple models, each independently generating predictions, but this approach incurs additional computational and financial costs.

In order to address these challenges, this paper proposes a novel method for bubble diagram generation to create accurate and diverse bubble diagrams. Firstly, we present a dual-branch graph neural network (DBGNN) that effectively captures the characteristic patterns of heterogeneous links. Second, we propose decentralized node sampling (DNS) and centralized node sampling (CNS) to enhance the learning of subtle local features. Finally, we introduce a variational graph auto-encoder (VGAE) to learn the latent space of bubble diagrams, enabling the model to sample from a Gaussian distribution to generate a large variety of bubble diagrams. Overall, by combining GNN, subgraph sampling strategy, and variational encoding, this approach provides a solution for automated bubble diagram generation, advancing the development of floorplan generation technologies.

The structure of this article is organized as follows: Section 2 introduces the bubble diagram link prediction problem. Section 3 describes the details of the proposed method. Section 4 presents comprehensive experiments. Section 5 summarizes this paper.

2. Preliminary

In a bubble diagram, each bubble represents a functional room, while the links between them indicate spatial relationships [27]. These links are categorized into two types: door links and wall links. Door links define room connectivity, indicating whether a door exists between two rooms, such as kitchen and living room. Wall links represent spatial adjacency, signifying whether two rooms share a common wall. For example, a bedroom and a bathroom may be adjacent but separated by a wall. Figure 1 illustrates the mutual conversion between a floorplan and a bubble diagram.

From a graph theory perspective, a residential bubble diagram can be modeled as an undirected graph, since spatial relationships in floor plans are bidirectional. This graph is defined as $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$, where $\mathcal{V}={\left\{v^n\right\}}_{n=0}^N$ represents the set of room nodes, corresponding to different functional spaces. The edge set $\mathcal{E}=\{{\mathcal{E}}^d,{\mathcal{E}}^w\}$ consists of two types of links: door links ${\mathcal{E}}^d=\{e_{v^i,v^j}^d=(v^i,v^j\left)\right|v^i,v^j\in \mathcal{V}\}$, representing spatial accessibility; and wall links ${\mathcal{E}}^w=\{e_{v^i,v^j}^w=(v^i,v^j\left)\right|v^i,v^j\in \mathcal{V}\}$, indicating adjacency where a wall separates two rooms. For each node, $v^n$, its neighboring nodes form a set, $\mathcal{N}\left(v^n\right)$. The bubble diagram generation problem is formulated as a link prediction problem. For a given input graph, $\mathcal{G}=\{\mathcal{V},{\mathcal{E}}_a,{\mathcal{E}}_b\}$, $\mathcal{V}$ is the user-defined room nodes, ${\mathcal{E}}_a$ is the existing links between rooms, and ${\mathcal{E}}_b$ is the links to be predicted. We employ a neural network to learn an implicit function, $\varphi$, to represent the likelihood, $y_{(v^i,v^j)}$, of connectivity between two nodes, $v^i$ and $v^j$. Given a pair of nodes, an implicit function, $\varphi$, maps the nodes to embedding vectors and determines whether there are door or wall links between the nodes. And $y_{(v^i,v^j)}$ can be expressed as follows:

$$y_{{(v}^i,v^j)}=\varphi \left(v^i,v^j\right)=\left\{\begin{array}{c}1,e_{v^i,v^j}^{type}\in {\mathcal{E}}_b^{type}\\ 0,e_{v^i,v^j}^{type}\notin {\mathcal{E}}_b^{type}\end{array}\right.$$

(1)

where 1 indicates that there is a link between the nodes, while 0 indicates that there is no link between the nodes, $e_{v^i,v^j}^{type}$ represents the edge between nodes $v^i$ and $v^j$, which has two types: $e_{v^i,v^j}^d$ and $e_{v^i,v^j}^w$.

3. Methodology

3.1. Overall Framework

The proposed model consists of two parts: link prediction and diversity generation, as illustrated in Figure 2. First, DBGNN is used to fully learn the topological knowledge from the bubble diagram dataset, enabling the establishment of reasonable connectivity or adjacency relationships between various types of nodes, thereby achieving accurate link prediction, i.e., the generation of a single bubble diagram. Then, VGAE is concatenated with DBGNN for training to implicitly encode the features of the bubble diagram, enabling the model to generate multiple distinct bubble diagrams from a given input, thus achieving diversity generation.

3.2. Dual-Branch Graph Neural Network for Bubble Diagram Link Prediction

The DBGNN model has two crucial task branches, and each branch consists of graph normalization (GN), a task feature encoder, a link prediction decoder, a Gaussian error linear unit (GELU) activation function [28], and a skip connection [29], as shown in Figure 2. A weight-shared fully connected layer is used to extract common features from the input bubble diagram. The architecture of task branch is developed based on the typical prediction model [18,30]. The model architecture and the modules play an equal role in prediction performance [31], so we referred to these architectures to design the link prediction task branch.

3.2.1. Graph Normalization

The update of the neural network hidden layer parameters can lead to internal covariance shift. After feature layer mapping, the distribution of the graph’s embedding data changes. Therefore, it is necessary to perform uniform distribution processing on the feature data of the middle layer. To ensure that the model learns correct structure information, the normalization module is effective [32]. We employ graph-wise normalization to maintain the global topology information of the entire bubble diagram [32]. Each node in the bubble diagram is crucial because there are explicit topological constraints between them. GN considers the features of all nodes for normalization, and the expression is described as follows:

$${\widehat{h}}_{v_i}=\lambda {\displaystyle \frac{h_{v_i}-\mu }{\sigma }}+\beta$$

(2)

$$\mu ={\displaystyle \frac{1}{\left|\mathcal{G}\right|}}{\sum }_{v_i\in \mathcal{G}}{h}_{v_i}$$

(3)

$$\sigma =\sqrt{{\displaystyle \frac{1}{\left|\mathcal{G}\right|}}\sum _{v_i\in \mathcal{G}}{\left(h_{v_i}-\mu \right)}^2}$$

(4)

where $h_{v_i}$ is $i$-th node feature, $\mu$ is the mean vector, $\sigma$ is standard deviation vector, and $\lambda$ and $\beta$ are scale and shift.

3.2.2. Decentralized Node Subgraph Sampling and Centralized Node Subgraph Sampling

We propose DNS and CNS to obtain multiple local subgraphs with common features. By learning the characteristics of these local subgraphs, the model’s prediction performance can be enhanced. Both task feature encoders involve two steps: sampling and aggregation, utilizing the same SAGEConv aggregator but differing in their sampling strategies. SAGEConv is the feature encoder of GraphSAGE, which is an inductive learning model designed for large graphs [26]. The core idea behind GraphSAGE is to generate embeddings for nodes by sampling and aggregating information from their local neighborhoods. While GraphSAGE is optimized for large graphs using neighbor sampling to address memory limitations and global feature learning issues, bubble diagrams are smaller graphs. Multi-hop neighbor sampling in this context may yield subgraphs that are too similar to the input, increasing the risk of overfitting. Additionally, the diversity and inherent patterns of bubble diagrams, such as varying graph orders or random subgraph structures, can complicate feature learning.

As shown in Figure 3, the nodes in the bubble diagram are divided into two categories: central node (living room node) and non-central nodes (bedrooms, toilets, kitchens, etc.). We observe that the bubble diagram has significant centrality, and the living room node usually connects to all other room nodes and has the highest degree. Additionally, non-central nodes are more likely to have wall links between them. The details of DNS and CNS are illustrated in Figure 4. DNS removes the central node and its edges, and it samples multiple subgraphs from the graph composed of non-central nodes. To enhance the embedding learning for non-central nodes, the subgraphs without a central node are used to train the wall link prediction branch. CNS samples multiple subgraphs based on the central node, and each subgraph must contain the central node. The subgraph containing the central node is used for training the door link prediction branch to fully perceive the intrinsic patterns of local subgraphs.

All subgraphs are input into SAGEConv to learn node embeddings. GCN and GAT are not good at handling flexible bubble diagrams and tend to learn fixed node embeddings. Therefore, SAGEConv is used as graph feature encoder to extract the topology information of bubble diagrams. SAGEConv can be described as follows:

$$h_{N\left(v\right)}^k={\mathrm{A}\mathrm{G}\mathrm{G}\mathrm{R}\mathrm{E}\mathrm{G}\mathrm{A}\mathrm{T}\mathrm{E}}_{\mathrm{k}}\left(\{h_u^{k-1},\forall u\in \mathcal{N}\left(v\right)\}\right)$$

(5)

$$h_v^k=W^k\cdot \mathrm{C}\mathrm{O}\mathrm{N}\mathrm{C}\mathrm{A}\mathrm{T}\left(h_v^{k-1},h_{N\left(v\right)}^k\right)$$

(6)

where $k$ denotes depth, $v$ denotes the destination node, $u$ is the neighbor of node $v$, $W$ is the weight matrix, $h_{N\left(v\right)}$ is the neighborhood vector, and $h_v$ is the node’s embedding.

3.2.3. Link Prediction Decoder

The link decoder predicts the probability of link existence based on the node representations learned by the feature encoder. We leverage a dual-branch network to separately model the adjacency and connectivity of the bubble diagram in spatial structure and employ two independent decoders to predict wall links and door links, respectively. After feature extraction by DBGNN, the node embeddings inherently capture implicit topological information, eliminating the need for additional parameter learning in the link prediction decoder. The embeddings of two nodes are directly computed, and the link prediction decoder is formulated as follows:

$$\widehat{y}=s\left(h_u^k\cdot h_v^k\right)$$

(7)

where $s$ is the sigmoid function. The sigmoid function maps the latent embeddings to a probability range between 0 and 1, When the predicted value $\widehat{y}$ exceeds 0.5, it indicates the existence of a link between the nodes.

3.3. Variational Encoding for Diverse Bubble Diagram Generation

Users expect to receive a variety of distinct solutions when designing bubble diagrams [33], allowing for diversity in the design process. However, current link prediction models can only generate a single result for a given input at a time. Re-running the model for additional predictions can lead to increased computational costs and extended usage time. To address this issue, we integrated VGAE with DBGNN. During the training process, VGAE is employed to convert the bubble diagram into a low-dimensional embedding that represents a probability distribution in latent space. This low-dimensional embedding is expressed as $\mathcal{N}(\mu ,\sigma )$, consisting of a mean vector ($\mu$) and a variance vector ($\sigma$). Directly sampling latent variables ($z$) from this probability distribution can lead to the vanishing gradient problem, hindering the ability to update the model parameters during the forward propagation phase. To overcome this issue, the reparameterization technique is used to sample a noise term, $ϵ$, from a standard normal distribution. The latent variable, $z$, is then constructed as $z=\mu +\sigma ⨀ϵ$, where $⨀$ denotes element-wise multiplication. This latent variable ($z$) is concatenated with the feature matrix, $X$. Finally, the concatenated vectors are fed into DBGNN, which functions as a decoder, reconstructing the implicit variables and predicting the structural information

Specifically, VGAE consists of three GCN encoders, which aim to minimize the distribution difference between the generation vector distribution and the normal distribution [34]. For an input bubble diagram, $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$, with $N$ nodes, $A$ is the adjacency matrix, $D$ is the degree matrix, and $X\in N\times D$ is the feature matrix. The encoding process of VGAE can be described as follows:

$$q\left(\mathrm{Z}∣\mathrm{X},\mathrm{A}\right)=\prod _{i=1}^N q\left({\mathrm{z}}_i∣\mathrm{X},\mathrm{A}\right)$$

(8)

where $q({\mathrm{z}}_i\mid \mathrm{X},\mathrm{A})=\mathcal{N}({\mathrm{z}}_i\mid {\mu }_i,\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\sigma }_i^2\left)\right)$, $z_i$ is the latent vectors, ${\mu }_i={GCN}_{\mu }(\mathrm{X},\mathrm{A})$ is the mean vectors, and log $\sigma ={GCN}_{\sigma }(\mathrm{X},\mathrm{A})$ is the variance vectors. The GCN is expressed as follows:

$$\mathrm{G}\mathrm{C}\mathrm{N}(\mathrm{X},\mathrm{A})=\tilde{A}\mathrm{R}\mathrm{e}\mathrm{L}\mathrm{U}\left({\tilde{A}\mathrm{X}\mathrm{W}}_0\right){\mathrm{W}}_1$$

(9)

where $\tilde{A}=D^{-1/2}A{D}^{-1/2}$, and $\mathrm{W}$ is the weight matrix.

3.4. Multi-Task Loss Functions

The proposed model focuses on predicting two types of links between nodes, with the goal of generating multiple bubble diagrams that have a reasonable overall structure and local links. To train the model, a multi-task loss function that comprehensively considers global information, local information, and regularization is designed. The loss function is defined as follows:

$$\mathcal{L}={\mathcal{L}}_W+{\mathcal{L}}_D+{\mathcal{L}}_{DS}+{\mathcal{L}}_{CS}+{\mathcal{L}}_V$$

(10)

where ${\mathcal{L}}_W$ is the wall link prediction loss; ${\mathcal{L}}_D$ is the door link prediction loss; ${\mathcal{L}}_{DS}$ is the link prediction loss of the subgraph without central node; ${\mathcal{L}}_{CS}$ is the link prediction loss of the subgraph with central node; and ${\mathcal{L}}_V$ is the distance loss between the latent variable, $z$, distribution and the normal distribution.

The bubble diagram should have a reasonable global topological structure. Specifically, for all input nodes, both wall links and door links should be established accurately. The link prediction problem is a binary classification problem, so we use binary cross-entropy loss function. The wall link prediction loss, ${\mathcal{L}}_W$, and the door link prediction loss ${\mathcal{L}}_D$ can be defined as follows:

$${\mathcal{L}}_W=-\sum _i{y}_i^{wall}log{\widehat{y}}_i^{wall}+(1-y_i^{wall})log(1-{\widehat{y}}_i^{wall})$$

(11)

$${\mathcal{L}}_D=-\sum _i{y}_i^{door}log{\widehat{y}}_i^{door}+(1-y_i^{door})log(1-{\widehat{y}}_i^{door})$$

(12)

where $y_i$ is the true label, and ${\widehat{y}}_i$ is the prediction value.

The link between two nodes should be consistent with the actual situation; for example, the living room node and bedroom node cannot be connected solely by wall links. We sample from the bubble diagram to obtain subgraphs with a central node and subgraphs without a central node for model training. We use the model to predict the links between these subgraphs, calculate the cumulative error loss, and use it to update the model parameters. Suppose there are $k$ subgraphs, and ${\mathcal{L}}_{CS}$ and ${\mathcal{L}}_{DS}$ can be calculated as follows:

$${\mathcal{L}}_{CS}=\sum _k{\mathcal{L}}_{CS\_wall}^k+{\mathcal{L}}_{CS\_door}^k$$

(13)

$${\mathcal{L}}_{DS}=\sum _k{\mathcal{L}}_{DS\_wall}^k+{\mathcal{L}}_{DS\_door}^k$$

(14)

where ${\mathcal{L}}_{CS\_wall}^k$ is the wall link prediction loss of the subgraph with central node, ${\mathcal{L}}_{CS\_door}^k$ is the door link prediction loss of the subgraph with central node, ${\mathcal{L}}_{DS\_wall}^k$ is the wall link prediction loss of the subgraph without central node, and ${\mathcal{L}}_{DS\_door}^k$ is the door link prediction loss of the subgraph without central node.

The regularization loss ${\mathcal{L}}_V$, is used to regularize the latent space. The distribution of latent variable $z$, encoded by VGAE is enforced to normal distribution. ${\mathcal{L}}_V$ is expressed as the Kulback–Leibler (KL) divergence between these two distributions, which can be calculated as follows:

$${\mathcal{L}}_V=KL\left[q\right(\mathrm{Z}|\mathrm{X},\mathrm{A})\left|\right|p\left(\mathrm{Z}\right)]$$

(15)

$$KL=\sum _{i}p\left(i\right)\mathrm{l}\mathrm{o}\mathrm{g}\left({\displaystyle \frac{p\left(i\right)}{q\left(i\right)}}\right)$$

(16)

where $q\left(\mathrm{Z}\right|\mathrm{X},\mathrm{A})$ is the latent variable distribution, and $p\left(\mathrm{Z}\right)$ is the normal distribution.

3.5. Evaluation Metrics

We conduct both quantitative and qualitative assessments of link prediction and diversity generation. Link prediction is essentially a binary classification task that determines whether there is a door link or a wall link between two nodes. Therefore, we use accuracy (ACC) and average precision (AP) to quantitatively evaluate the differences between the link prediction results and the ground truth. Additionally, considering the need for user editability in practical applications, we employ graph edit distance (GED) to qualitatively assess the editing complexity of the predicted results. In terms of diversity generation, we conduct experiments, including the presentation of generated results and diversity quantitative evaluation based on GED.

The ACC metric is applied to measure the proportion of correct prediction links among the total number of labeled links. ACC assesses the general predictive performance of the model but does not reflect the model’s predictive performance for a particular category. It can be calculated by the following formula:

$$ACC={\displaystyle \frac{TP+TN}{TP+FN+TN+FP}}$$

(17)

where true positive (TP) is the number of samples correctly identified as positive class, true negative (TN) is the number of samples correctly identified as negative class, false positive (FP) is the number of samples incorrectly identified as positive class, and false negative (FN) is the number of samples incorrectly identified as negative class.

The AP metric is used to estimate the classification performance of the model when the proportion of positive and negative samples is not balanced. AP measures the quality of the ranked list of predicted positive samples by computing the area under the precision–recall curve. Precision is the ratio of true positive samples to the total number of predicted positive samples. The precision can be expressed as follows:

$$Precision={\displaystyle \frac{TP}{TP+FP}}$$

(18)

$$AP=\sum _{k=1}^{n}P\left(k\right)\mathsf{\Delta }{R}_k$$

(19)

where $P\left(k\right)$ is the precision at the $k$-th position in the ranked list, and $\mathsf{\Delta }{R}_k$ is the change in recall between the $k-1$-th and $k$-th predictions for positive samples.

The GED metric is employed to measure the minimum number of edit steps required to complete the transformation from the bubble diagram and another bubble diagram [35]. The nodes are deterministic; only edge editing operations such as insertion, deletion, and substitution are required. GED is defined as follows:

$$GED({\mathcal{G}}_1,{\mathcal{G}}_2)=\underset{(e_1,...,e_k)\in \mathcal{P}({\mathcal{G}}_1,{\mathcal{G}}_2)}{\mathrm{m}\mathrm{i}\mathrm{n}}\sum _{k=1}^{k}c\left(e_i\right)$$

(20)

where $e_i$ denotes graph edit operation, and $\mathcal{P}({\mathcal{G}}_1,{\mathcal{G}}_2)$ represents the set of edit paths from ${\mathcal{G}}_1$ to ${\mathcal{G}}_2$.

4. Experiments

In this section, comprehensive experiments were conducted to verify the proposed method. The proposed method is trained and tested on a computer with the following configuration: Intel i7-12700F 2.10 GHz CPU, 16.0 GB RAM, and NVIDIA GeForce RTX 2060 SUPER GPU. PyTorch 2.0.0 [36] and Deep Graph Library [37] were employed to implement the proposed method. Adam optimizer was used to train the model. To eliminate potential bias, the intermediate layer dimension was set to 50 for all models in this study.

4.1. Bubble Diagram Dataset Establishment

Currently, there is no readily available dataset for residential bubble diagrams; thus, we developed a residential bubble diagram dataset based on the RPLAN dataset. Following the survey conducted by Pizarro et al. [38], the RPLAN dataset is chosen [39] for its quality, quantity, labeling, and accessibility. The RPLAN dataset contains over 80,000 residential building floorplans, sufficiently meeting our data requirements.

As shown in Figure 5, each floorplan image features four channels: the boundary mask channel, the inside mask channel, the wall mask channel, and the instance mask channel. We extracted structural information from the floorplan using various image-processing techniques, as shown in Figure 6. First, we identify room instances based on the pixel encoding of instance mask channels to construct room graph nodes. Next, the corners of each room are detected using the Harris corner detector [40]. Each room is then constructed by connecting the detected corner points into line segments, ensuring that each room instance is paired with the corresponding surrounding wall segments. The link relationships between room nodes are established by calculating the distances between parallel wall segments of room pairs. If the distance between two parallel wall segments is below a specified threshold, it indicates a wall link. Additionally, if door pixels are present between two rooms, a door link is established. Importantly, door links and wall links cannot coexist; when both conditions are satisfied, the door link takes precedence over the wall link. We traverse all room pairs to identify all potential wall and door links. Finally, the resulting bubble diagram is created and stored in a CSV file.

Typically, a residential floorplan must include essential components, such as a bedroom, living room, kitchen, and bathroom. However, some floorplans in the RPLAN dataset lack these critical elements. To prevent the model from learning incorrect structural information and to enhance the reliability of the dataset, we filtered out non-compliant data, retaining only reasonable configurations [17]. The resulting bubble diagram dataset consists of 72,678 graphs, featuring nine node types and two link types, with a total of 494,240 nodes and 1,058,478 edges. The detailed data statistics are shown in Figure 7.

4.2. Model Training and Performance Comparison

An evaluation of link prediction performance was conducted to assess the effectiveness of DBGNN on the bubble diagram link prediction task. Several baseline models, including GCN, GAT, and GraphSAGE, are selected for comparison. GCN is constructed based on the node’s spatial relations, propagating node information along edges [41]. GAT introduces the attention mechanism to learn the relative weights between two connected nodes because it has greater expressive capability. GraphSAGE is an inductive learning method based on node features and structural information, which can efficiently generate node embeddings. We used 1500 graphs for training and 500 graphs for testing these baseline models and DBGNN.

Table 1. Training time of DBGNN and baseline models.

Model	Training Time
GCN	1240.9033 s
GAT	5714.1277 s
GraphSAGE	1119.8463 s
DBGNN	1452.7707 s

presents the training time of four models. Figure 8 shows the loss and accuracy curves of these models during the training process. According to the data in Table 1, GraphSAGE has the shortest training time (1119.85 s), followed by GCN (1240.90 s). Our model requires 1452.77 s for training, which is slightly higher than GCN and GraphSAGE but significantly lower than GAT (5714.13 s). The extended training time of GAT is mainly due to the computational overhead introduced by its attention mechanism. Overall, our method strikes a balance between training cost and predictive accuracy.

As illustrated in Figure 8, GCN showed relatively steady performance, with a slow decrease in loss and noticeable fluctuations during training, resulting in unstable prediction performance, with a maximum accuracy of only 64%. This suggests that GCN struggles to handle the complex topological patterns in bubble diagrams. GAT showed some improvement over GCN, with reduced training fluctuations and a slight improvement in the rate of loss decrease, achieving a maximum accuracy of 75%. However, this indicates that GAT struggles to learn the implicit knowledge contained in the bubble diagram. GraphSAGE rapidly converged in the early stages of training, reaching a low loss and steadily improving prediction accuracy, demonstrating an advantage in both prediction tasks, suggesting that GraphSAGE is capable of learning different topological knowledge correctly. Among them, DBGNN performed the best, with rapid and smooth loss convergence and a quick improvement in prediction accuracy, reaching a maximum of 93%, demonstrating DBGNN’s exceptional ability in handling both link prediction tasks.

The test results are shown in

Table 2. Test results of ACC and AP of DBGNN and baseline models.

Method	ACC (↑)		AP (↑)
	Door	Wall	Door	Wall
GCN	61.96%	65.93%	45.63%	69.32%
GAT	69.52%	70.77%	49.52%	74.08%
GraphSAGE	86.63%	76.24%	87.33%	86.43%
DBGNN	92.39%	78.84%	91.35%	88.36%

. GCN achieves low prediction accuracy in wall and door link prediction tasks, with 65.93% ACC-Wall and 45.63% AP-Door. It can be argued that GCN fails to effectively learn the local and global patterns of bubble diagrams. GCN can encode structural information, but its low-pass filtering characteristics cause feature signals to converge, limiting its ability to learn bubble diagram patterns, resulting in poor link prediction accuracy. Similar to GCN, GAT performs poorly regarding door link prediction, achieving only 49.52% AP-Door. GAT improves the contribution of neighboring nodes’ features by incorporating self-attention; however, it suffers from over-smoothing, which affects prediction accuracy. GraphSAGE performs better than GCN and GAT, with 86.63% ACC-Door and 76.24% ACC-Wall. GraphSAGE generates node embeddings by sampling and aggregating neighboring nodes’ features, making it advantageous for handling large graph features, yet it is less suited for small bubble diagrams. DBGNN achieves the best prediction performance, with 92.39% ACC-Door and 78.84% ACC-Wall, surpassing all baseline models in both wall and door link prediction tasks, demonstrating remarkable accuracy in bubble diagram link prediction. The proposed model employs a dual-branch approach to decouple bubble diagrams and leverages subgraph sampling strategies to enhance local feature learning. As a result, DBGNN achieves the highest accuracy in both sub-tasks, confirming its effectiveness for the unique requirements of bubble diagram link prediction.

Moreover, we visualized the link prediction results of the four models in Figure 9. For easy comparison, real floorplans and bubble diagrams are displayed on the left. Due to insufficient learning of the structural information within bubble diagram data, GCN and GAT generated a significant number of incorrect links. GraphSAGE produced bubble diagrams of relatively higher quality, but there were still issues; for instance, the 3-3 balcony node only had a wall link and no door link, implying no accessible path to the balcony from other spaces. DBGNN provided satisfactory predictions with no incorrect door or wall links. Our dual-branch network architecture effectively decouples the bubble diagram, allowing for targeted handling of door and wall links. The central node (living room) is highly correlated with non-central nodes, as most rooms should have a door link to the living room for accessibility, which is enhanced by CNS. Additionally, local links between non-central nodes should follow basic spatial logic; for example, a bathroom should not only connect to a kitchen via a door. We used DNS to improve the learning of spatial adjacency. Furthermore, while maintaining logical local topology, the global number of links must be controlled within a certain range. We addressed this with a multi-task loss function that balances global and local link structures. As shown in the visual results, DBGNN accurately created local links and optimal global structures without extraneous links, regardless of the number of input nodes. These results demonstrate the model’s notable predictive accuracy and robustness, affirming the superiority of our approach in the bubble diagram generation task.

4.3. Ablation Experiment

A comprehensive ablation experiment was conducted to evaluate the impact of various components of DBGNN. Fully armed DBGNN is used as the baseline, which consists of GN, GELU activation, skip connection, and subgraph sampling. The variants represent the architecture component that has been replaced or removed in the fully armed DBGNN. The variants were trained and tested using the same data as before. The experimental results are shown in Figure 10.

Fully armed DBGNN demonstrated the highest accuracy, while the accuracy of ACC-Door and AP-Door reached over 91%. Removing the activation function caused a significant drop in prediction performance, especially in wall classification accuracy (ACC-Wall), which decreased by about 6 percentage points, indicating that the activation function has a significant impact on performance. Additionally, GELU was replaced with the commonly used ReLU [42] and the recently introduced SELU [43] to find the optimal activation functions. Both transformations resulted in a decrease in accuracy, confirming the effectiveness of GELU. Removing normalization led to a noticeable performance drop, particularly in AP-Door and ACC-Door. Furthermore, we compared several graph normalization techniques, including node normalization and batch normalization [32]. Batch normalization performed slightly better than node normalization, but both had lower accuracy than GN, with ACC-Door dropping by 5.55% and 2.36%, respectively, indicating that GN is more suitable for bubble diagram generation tasks. The inclusion of skip connections significantly improved all four metrics. This is because skip connections effectively address the gradient vanishing problem in graph embedding learning, ensuring the efficient transfer of topological information. Subgraph sampling drove the model to achieve optimal performance, increasing ACC-Door by 2.04% and ACC-Wall by 1.33%, as this strategy enhances feature learning for both non-central and central subgraphs, thereby improving the prediction accuracy of local links and ultimately enhancing overall prediction performance. Ablation experiments showed that normalization, activation functions, and skip connections are key factors in improving model performance, and removing any of these would lead to a decrease in accuracy. Subgraph sampling is crucial for perceiving local features in bubble diagrams; it further promotes the learning of complex patterns, thus improving prediction performance.

4.4. Diverse Bubble Diagrams Generation

This section will validate the diversity of the results generated by DBGNN-VGAE and assess their effectiveness. Figure 11 shows the training curve of DBGNN-VGAE, with good convergence characteristics in all losses. Based on the trained model, we generated bubble diagrams using seven common room combinations, varying the number of bedrooms, bathrooms, and balconies. The generated bubble diagrams with varying input nodes are illustrated in Figure 12. The results indicate that the proposed method successfully creates bubble diagrams based on these varying inputs, demonstrating its superior adaptability. Upon examining each bubble diagram, we observe that the results contain correct door and wall links, with no invalid links. These reasonable outputs confirm that the model has learned effective topological knowledge, allowing it to provide multiple valid bubble diagrams.

Furthermore, we sampled latent variables from a normal distribution to generate dozens of bubble diagrams, as shown in Figure 13. It is evident that, based on different latent variables, the model generates bubble diagrams with significant differences. This demonstrates that VGAE successfully encoded the implicit distribution of bubble diagrams. When combined with DBGNN, the model gained the ability for multi-prediction, effectively addressing the issue of diversity generation.

4.5. Diversity Evaluation

This section aims to quantitatively evaluate the diversity of the generated results from both single and multiple experiments. Firstly, we evaluate the differences among the results generated in a single experiment based on one-shot generation. A total of 100 bubble diagrams are generated based on the input nodes, and the GED is calculated for each pair of these results. The GED values are normalized to represent similarity coefficients, where a GED value closer to 1 indicates a greater structural difference, and a value closer to 0 indicates a higher structural similarity between two bubble diagrams. The similarity distribution is illustrated in Figure 14a, where most differences exceed 0.2, indicating significant distinction among the generated results. We visualized a similarity matrix to intuitively display the variation between the generated outputs, as shown in Figure 14b. The results demonstrate that VGAE successfully projects bubble diagrams from high-dimensional to low-dimensional space, achieving an abstract representation of structural information. Furthermore, the dissimilar outputs confirm the model’s capacity for diverse generation, effectively addressing the challenge of producing diverse bubble diagrams.

We conduct one hundred generation experiments to verify how the diversity of generated results varies under a high-intensity continuous generation. We test one hundred groups of input room nodes and generate one hundred bubble diagrams for each group by randomly sampling latent variables. In each experiment, we calculate the GED between each generated bubble diagram and the reference diagram, and then we sum these values to measure the total diversity for a single experiment. Higher total GED values indicate greater diversity, while lower values suggest less differentiation. The reference GED value is set to 100, signifying that, on average, each generated diagram has at least one link differing from the ground truth. The diversity trend curve is shown in Figure 15a, revealing a fluctuation with a peak reaching up to 1600 total GED, and most experiments display diversity above the mean line. Additionally, we recorded the bubble diagram with the lowest GED for each experiment, as illustrated in Figure 15b. Remarkably, 55% of the generated diagrams exhibit a GED of 0, affirming their alignment with the ground truth. As the sampling of latent variables increases, the proposed model can produce outputs fully consistent with actual layouts. The experiments demonstrate that the model maintains robust diversity even under high-intensity continuous use, verifying its reliability in practical applications. Furthermore, the excellent diversity results confirm the effectiveness of integrating VGAE.

4.6. Usability Evaluation

In this section, we assess the usability of the generated results through qualitative evaluation. Specifically, we compare the generated bubble diagrams with real reference diagrams to analyze the modifications needed to refine the generated outputs into practical and usable designs. This assessment approach allows us to observe the ease with which users can adjust the generated diagrams to meet practical needs, highlighting the efficiency and adaptability of the model’s output in real-world scenarios. The GED metric is employed to assess the difference between the generated bubble diagrams and the actual bubble diagrams. A smaller GED indicates that fewer edits are required to align the generated diagram with real data. The GED range is divided into four intervals, each representing different evaluation levels, as shown in

Table 3. Evaluation levels of bubble diagram generation results.

GED Range	Level	Description
0	Outstanding	No additional modifications needed; fully compatible with real data and entirely usable.
(0,5)	Superior	Modifications can be completed in a short time, demonstrating high efficiency and usability.
[5,10)	Acceptable	Edits can be made within an acceptable timeframe, maintaining usability.
[10,+∞)	Terrible	Excessive editing required cannot be completed within a tolerable timeframe, resulting in low practical value.

. We first trained four models using the bubble diagram dataset and saved their parameters. Subsequently, the models were tested with 500 samples for clear comparison. The GED distributions of the four models are presented in Figure 16.

The proposed method demonstrates high competitiveness in practical applications. DBGNN achieved a remarkable 93.8% of generated bubble diagrams requiring fewer than five graph edits, while other models produced fewer than 52%. Additionally, DBGNN generated 299 bubble diagrams with a GED of 0, significantly surpassing other models. This indicates that DBGNN’s results have exceptional utility, allowing users to quickly obtain satisfactory bubble diagrams. These experimental results emphasize the advanced performance of our model in creating bubble diagrams with precise local links and coherent global structure, highlighting its usability and efficiency.

4.7. Time Efficiency Study

The time efficiency comparison is shown in

Table 4. Comparison of time efficiency for different methods.

Input Nodes	Inference Time	Retrieve from Database	Manual Design (Expert)	Manual Design (Non-Expert)
4	0.0155 s	0.0058 s	8.5 s	16.1 s
5	0.0158 s	0.0063 s	15.6 s	43.7 s
6	0.0160 s	0.0060 s	25.4 s	61.3 s
7	0.0163 s	0.0060 s	34.7 s	85.9 s
8	0.0172 s	0.0070 s	44.1 s	115.7 s

. As observed from the data, our method maintains a stable inference time across different input nodes, ranging from 0.0155 s to 0.0172 s. In contrast, the retrieve-from-database approach achieves a shorter computation time (0.0058 s to 0.0070 s), but its applicability may be limited by the database size. Manual design consumes significantly more time than automated methods and exhibits an increasing trend as the number of input nodes grows. Experts, who are familiar with the topological patterns of bubble diagrams, understand which nodes should be connected by door links or wall links. As a result, their primary time consumption comes from clicking on nodes to establish links, which takes at least two seconds per adjacent node pair. On the other hand, non-expert users are less aware of the necessary links between nodes, requiring additional thinking time, leading to longer completion times (16.1 s to 115.7 s). If they need to create bubble diagrams for new structures, even more time is required for design considerations. Overall, our method outperforms manual design in computational efficiency and does not rely on a database, making it more versatile and practical for various applications.

We use Miller’s response time model [44] to quantify user experience, with response time classifications shown in

Table 5. Definitions of response time.

Time Delay	Description
0.1 s	The user perceives no delay.
1 s	The maximum acceptable response time for the user to feel the system responds immediately.
10 s	The limit of the user’s attention span to complete the current task. If no effective feedback is received beyond this threshold, the user will switch to other tasks while waiting for the computer to finish the current operation.

. Combining the data from Table 4 with response time thresholds, our method completes all test cases within 0.02 s, which is well below the 0.1 s perceptual delay threshold. As a result, users do not perceive any delay, ensuring a smooth experience. The retrieval-from-database approach also maintains response times below 0.1 s, but its effectiveness depends on the comprehensiveness of the pre-stored data. In contrast, manual design significantly exceeds the 10 s acceptable threshold, making users likely to experience frustration due to prolonged waiting times. Overall, compared to manual design, our method greatly enhances interactive experience and is better suited for real-time generation and interactive design systems.

5. Discussion

5.1. Application

The proposed method can quickly produce dozens of structurally diverse bubble diagrams within seconds, whereas manually creating the same number typically takes users several minutes. Integrating our method into intelligent design systems facilitates the generation of constrained layouts, streamlining the workflow and lowering the learning curve for users. HouseDiffusion [19] is used as the floorplan generation model. As shown in Figure 17, HouseDiffusion correctly infers a topologically consistent floor plan based on the generated bubble diagram, demonstrating the effectiveness of the bubble diagram.

In addition, the proposed method can also be applied to build a floorplan retrieval system. Architectural companies often maintain dedicated layout databases [45], where architects select floorplans for standard floor designs. However, quickly finding a satisfactory layout within a vast floorplan database poses a challenge. Our method can be integrated into the floor plan retrieval system to further enhance architects’ efficiency. Designers can simply input a few nodes to generate bubble diagrams using our method. With straightforward graph-matching techniques, they can swiftly retrieve desired layouts from the enterprise floorplan database, simplifying the design workflow, as illustrated in Figure 18.

5.2. Limitations

During model testing, we identified several failure cases. To facilitate analysis, we visualized these results using a fixed node layout, as shown in Figure 19. Some bubble diagrams exhibited abnormal link patterns, such as nodes lacking door links, having excessive door links, or displaying an excessive number of global links. These results suggest that the model’s learning of both global link count features and individual node degree features remains insufficient. Developing an attention mechanism that integrates both global and local link count features may help address this issue.

Additionally, the proposed method has certain limitations. Due to dataset constraints, it can only generate topologies aligned with Asian residential floor plans and cannot create bubble diagrams for other regions or different building types, such as apartments or public buildings. Moreover, since the model is designed for bubble diagrams without three-dimensional spatial information, it cannot handle space topologies with vertical coupling relationships, such as multi-story villas.

6. Conclusions and Future Works

In this study, a method for automatically constructing connectivity and adjacency relationships in bubble diagrams is proposed. First, in response to the heterogeneous nature of bubble diagrams, a DBGNN with two branches is designed. Then, the DNS and CNS strategies are introduced to enhance the learning of local links. Finally, VGAE is incorporated to address the diversity issue of the generated results. Based on the research results, the conclusions are as follows: The DBGNN can learn more accurate connectivity and adjacency features of bubble diagrams. Compared to the baseline models, this model significantly improves prediction accuracy, achieving 92.39% ACC-Door and 78.84% ACC-Wall. To enhance the feature learning of both central and non-central subgraphs, the DNS and CNS strategies are proposed to improve the model. After incorporating subgraph sampling, prediction accuracy increased by 2.04% ACC-Door and 1.33% ACC-Wall, achieving optimal link prediction performance. Additionally, after introducing VGAE, DBGNN can generate 50 distinct bubble diagrams for various input node combinations simultaneously, demonstrating its exceptional diversity and stability. The floor plan generation application based on HouseDiffusion confirmed the effectiveness of the generated bubble diagrams.

Future work will further develop a hybrid attention mechanism that considers the number of node links to enhance both local and global link prediction capabilities. Additionally, to maximize the proposed method’s potential, future research will extend its application to other building types, such as public buildings, to support the development of more universal intelligent design systems. Moreover, we plan to integrate the method with layout generation models and architectural design software like AutoCAD 2024, enabling designers to manually define nodes and edit links, thereby improving design efficiency.