BD-GNN: Integrating Spatial and Administrative Boundaries in Property Valuation Using Graph Neural Networks

Abstract

GNN approaches to property valuation typically rely on spatial proximity, assuming that nearby properties exhibit similar price patterns. In practice, this assumption often fails as neighborhood and administrative boundaries create sharp price discontinuities, a form of spatial heterophily. This study proposes a Boundary-Aware Dual-Path Graph Neural Network (BD-GNN), a heterophily-oriented GNN specifically designed for continuous regression tasks. The model uses a dual and adaptive message passing design, separating inter- and intra-boundary pathways and combining them through a learnable gating parameter $\alpha$. This allows it to capture boundary effects while preserving spatial continuity. Experiments conducted on three structurally contrasting housing datasets, namely Bangkok, King County (USA), and Singapore, demonstrate consistent performance improvements over strong baselines. The proposed BD-GNN reduces MAPE by 7.9%, 4.4%, and 4.5% and increases $R^2$ by 3.2%, 0.7%, and 5.0% for the respective datasets. Beyond predictive performance, $\alpha$ provides a clear picture of how spatial and administrative factors interact across urban scales. GNN Explainer provides local interpretability by showing which neighbors and features shape each prediction. BD-GNN bridges predictive accuracy and structural insight, offering a practical, interpretable framework for applications such as property valuation, taxation, mortgage risk assessment, and urban planning.

1. Introduction

Property valuation represents a fundamental component of economic systems worldwide, directly influencing monetary policy decisions, urban development strategies, and household wealth dynamics. Housing markets play a major role in shaping economic activity. Rising property values can boost household wealth and encourage consumer spending, while sharp fluctuations in prices may worsen affordability problems and widen social inequality. For governments, financial institutions, and individual stakeholders, accurate house price prediction serves as a critical tool for informing investment decisions, taxation frameworks, and urban planning initiatives.

Accurate property valuation is a central challenge in urban analytics, real-estate economics, and spatial decision support. Estimating land and housing prices requires understanding how geographic proximity, functional amenities, and boundary-induced segmentation jointly influence market dynamics. Traditional hedonic and spatial econometric models such as spatial autoregressive (SAR) or geographically weighted regression (GWR) have long provided interpretable spatial insights through distance-based dependence structures [1,2]. However, these methods rely on predefined spatial weight matrices, limiting their ability to capture complex spatial heterogeneity or sharp discontinuities caused by segmentation effects. As cities become more fragmented by zoning regulations, school district boundaries, and developer-defined communities, these rigid assumptions can no longer capture the complex spatial logic that shapes modern property markets.

In parallel, machine learning (ML) approaches including support-vector machines, tree-based ensembles, and deep neural networks have achieved higher predictive accuracy through data-driven learning [3,4,5]. However, these methods typically treat properties as independent observations and therefore overlook spatial interactions altogether.

Graph Neural Networks (GNNs) have recently emerged as powerful alternatives that learn spatial relationships directly from graph structure rather than from fixed kernels [6,7,8]. Their message-passing mechanisms allow flexible aggregation of neighborhood information and have advanced spatial modeling by enabling complex, data-driven relationships between nodes to be learned directly from observed interactions. Recent graph-based valuation models [9,10,11,12] and scalable extensions incorporating transformer layers [13] show that GNNs can match or even surpass traditional spatial models in predictive performance.

Despite their promise, classical GNNs assume homophily—that spatially or topologically connected nodes share similar attributes and outcomes. In housing markets, spatial heterogeneity and heterophily are common rather than exceptional. Boundary-induced segmentation, such as zoning regimes or neighborhood communities, can produce sharp price discontinuities between adjacent parcels. When the homophily assumption fails, message passing across such boundaries propagates misleading signals, thereby degrading predictive performance and obscuring the underlying spatial structure.

To overcome this limitation, a growing body of research on heterophily-aware GNNs has sought to relax the homophily constraint through enhanced message passing, multi-hop aggregation, adaptive filtering, and attention-based mechanisms [14,15,16,17,18,19,20,21,22]. Although these approaches improve learning in weak-homophily contexts, they tend to treat all heterophilous relationships uniformly, overlooking the distinction between structured boundary effects and random variation. Moreover, most existing heterophily-aware GNNs address node-classification problems, whereas property valuation is a continuous regression task that demands finer sensitivity to both spatial smoothness and boundary contrast.

The critical gap in current research lies in the treatment of boundary-induced segmentation. Existing graph-based valuation models [9,10,11,12] typically build graphs using only geographic distance or simple spatial adjacency. As a result, segmentation created by administrative or functional boundaries remains outside the learning process. This is problematic because decades of urban economics research have shown that such boundaries consistently produce clear and persistent price differences [23,24,25,26,27,28].

The key limitation is twofold: First, traditional spatial models and recent GNN approaches fail to explicitly differentiate between intra-boundary similarity (properties within the same administrative unit subject to similar regulations) and inter-boundary discontinuity (neighboring properties belonging to different administrative units with distinct market regimes). Second, while heterophily-aware GNNs address node heterogeneity, they treat all heterophilous relationships uniformly without distinguishing structured boundary effects from random variation. They cannot differentiate between systematic price discontinuities caused by administrative boundaries and random price variations within the same administrative context.

This leads to a fundamental question: How can spatial learning models incorporate boundary-induced segmentation directly into the learning process, while maintaining the flexibility of data-driven representation learning?

To address this gap, this study proposes the Boundary-Aware Dual-Path Graph Neural Network (BD-GNN). The key innovation is the use of two separate learning paths: one for capturing spatial proximity relationships among geographically nearby properties, and another for modeling administrative boundary effects. BD-GNN learns when to emphasize spatial similarity within administrative units and when to capture price discontinuities across boundaries, enabling more accurate and interpretable property valuation in heterogeneous urban markets.

To comprehensively evaluate the proposed framework, we conduct experiments on three datasets representing diverse developmental contexts and spatial configurations across Bangkok (Thailand), King County (USA), and Singapore. These settings differ markedly in urban morphology, data richness, and segmentation structures, providing a rigorous basis for assessing both predictive generalizability and interpretability robustness. The results show that BD-GNN not only improves predictive accuracy but also reveals interpretable spatial–administrative trade-offs consistent with established urban-economic theory.

The main contributions of this paper are threefold:

1.

Conceptual integration: We formalize boundary-induced segmentation as structured sources of geospatial heterophily and embed them directly into a learnable dual-path graph representation, explicitly distinguishing intra-boundary and inter-boundary relationships.

2.

Methodological advancement: We integrate an adaptive gating mechanism that seamlessly fuses spatial and boundary pathways within a unified dual-path framework, enabling the model to dynamically balance information flow between proximity-based and segmentation-based relationships during message passing.

3.

Empirical validation and interpretability: We evaluate the proposed BD-GNN across three cities—Bangkok, King County, and Singapore—demonstrating consistent performance gains and interpretable boundary effects aligned with urban-economic theory.

Together, these contributions advance boundary-aware spatial learning and offer a new direction for interpretable, policy-relevant property valuation in complex urban environments.

The remainder of this paper is organized as follows. Section 2 reviews related work in traditional approaches, ML, GNNs, heterophily-aware GNNs, and real estate-specific GNNs, including explainability. Section 3 describes the experimental setups, datasets, and the proposed architecture, including evaluation metrics. Section 4 presents results and analysis across different urban contexts. Section 5 discusses the findings and their practical implications. Section 6 concludes with future research directions.

2. Related Work

Research on property valuation has progressed through several methodological phases, from traditional econometric models to modern machine learning and graph-based deep learning approaches. Each stage provides a different perspective on how spatial, structural, and institutional factors influence property prices. The hedonic pricing model [29] established the theoretical foundation for linking property values to structural and locational attributes. Spatial econometric models such as the Spatial Autoregressive (SAR) Model and the Spatial Error Model (SEM) extended this framework by explicitly modeling spatial dependence among nearby properties [1,30,31]. Geographically Weighted Regression (GWR) [2] and its multiscale extension, MGWR [32], further enhanced this approach by allowing model parameters to vary locally across space. Although these models remain interpretable and widely used, they rely on predefined spatial weight matrices that impose rigid neighborhood structures. Such assumptions limit their ability to capture complex spatial heterogeneity or sharp price contrasts at neighborhood and administrative boundaries, motivating a shift toward more flexible, data-driven frameworks that can learn spatial relationships directly from data.

Machine learning (ML) approaches address this need for flexibility by prioritizing predictive performance over structural constraints. Ensemble methods such as Random Forests, Gradient Boosting, and XGBoost have demonstrated strong predictive power in property valuation and mass appraisal tasks [33,34], while deep learning techniques have enabled the integration of multiple data modalities, including imagery and tabular information [35,36]. However, most ML models treat each property as an independent observation, overlooking spatial dependence among nearby locations. Although these approaches capture complex nonlinear relationships among features, they fail to model neighborhood interactions that strongly influence price formation in real markets. This limitation has motivated growing interest in frameworks capable of learning spatial relationships directly from observed data rather than relying on fixed distance-based kernels.

Graph Neural Networks (GNNs) provide a flexible way to model spatial dependencies by learning directly from graph structures instead of fixed spatial kernels. Core architectures such as GCN [6], GraphSAGE [7], and GAT [8] employ message passing mechanisms that aggregate neighborhood information adaptively to learn spatial relationships. However, these models rely on the homophily assumption, which suggests that connected nodes tend to share similar features or outcomes. In housing markets, this assumption is often violated, as zoning policies, administrative boundaries, or developer-defined communities create sharp discontinuities between adjacent properties. When message passing occurs across such dissimilar regions, it propagates noisy signals that reduce predictive performance and obscure meaningful spatial patterns.

To address this issue, heterophily-aware GNNs have been proposed to improve representation learning under weak-homophily conditions. MixHop [14] and GPR-GNN [15] extend information aggregation through multi-hop or weighted propagation, while H2GCN [16] separates ego and neighbor embeddings to mitigate noisy neighbor effects. Geom-GCN [17] redefines neighborhoods based on geometric or structural similarity, and FAGCN [18] applies adaptive filtering to balance low- and high-frequency signals. PNA [19] enhances model expressiveness through multiple aggregation functions with degree scaling, whereas ACM [20] and LinkX [21] decouple structural and feature representations, improving generalization to non-homophilous and large-scale graphs. SHGNN [22] explicitly targets spatial heterophily in urban graphs by introducing a Spatial Diversity Score and heterophily-sensitive aggregation modules. Despite these advances, most models treat heterophilous relationships in a uniform way and do not distinguish between structured spatial heterogeneity, such as boundary-induced segmentation, and random local variation. A recent study indicates that the reported progress of heterophily-aware GNNs may be overstated, as many evaluations rely on biased benchmarks and inconsistent experimental setups [37]. This highlights the need for models that can explicitly represent structured spatial heterogeneity. Furthermore, they are primarily developed for node classification tasks rather than continuous regression, which requires finer sensitivity to spatial smoothness and boundary contrast.

Several studies have applied GNNs directly to property valuation, demonstrating their ability to integrate spatial and nonlinear relationships. Spatial Regression GCNs (SRGCNN and A–SRGCNN) [9,10], geo-spatial network embeddings (GSNEs) [11], lifelong valuation over heterogeneous information networks (LUCE) [12], and scalable architectures for large urban markets [13] show that GNNs can outperform traditional spatial econometric models. Nevertheless, these approaches often rely on homophily-based message passing and construct graphs solely from Euclidean proximity, without incorporating administrative or functional boundaries into graph design. Extensive evidence from urban economics shows that institutional boundaries, such as school districts, zoning areas, and developer-defined communities, lead to persistent and measurable price discontinuities [23,24,25,26,27,28]. When boundary effects are used only as additional features instead of being modeled as part of the graph structure, the model becomes less interpretable and its ability to represent local neighborhood patterns is reduced. Explicitly embedding boundary structures into spatial learning can better align model behavior with real-world market segmentation.

Interpretability has long been a core consideration in property valuation, given that valuation models directly inform taxation, investment, and urban planning decisions. Traditional econometric models provide transparent, coefficient-based explanations linking property characteristics to price outcomes [29]. Machine learning models, while offering higher predictive accuracy, rely on post hoc tools such as feature importance [3] and SHAP values [38,39] to interpret predictions. However, these methods largely focus on individual feature effects and do not explain how spatial dependencies shape property prices. Although GNNs are effective at capturing relational structures, they often function as black boxes because their learned spatial embeddings are difficult to interpret. Explainable AI (XAI) techniques such as GNNExplainer [40], PGExplainer [41], and SubgraphX [42] attempt to reveal the most influential nodes, edges, or subgraphs. Recent work by Karamanou et al. [43] applied explainable GNNs to housing markets, illustrating the potential of graph-based interpretation in real estate analysis. Yet, these explanation techniques remain largely data-driven and detached from economic reasoning. Incorporating domain-relevant structures, such as market or administrative boundaries, directly into GNN learning could enhance both predictive performance and interpretability by aligning spatial representations with economic and institutional realities.

In summary, studies across econometric, machine learning, and graph-based approaches reveal several unresolved challenges in modeling spatial dynamics of property markets. The main research gaps that emerge from this body of work are outlined as follows:

• Econometric models emphasize interpretability through explicit coefficient estimates but impose rigid spatial assumptions.
• Machine learning models improve predictive accuracy but ignore spatial dependence.
• GNN-based models learn spatial relationships but often overlook structured heterophily caused by administrative or functional boundaries.
• Heterophily-aware GNNs represent progress toward handling spatial complexity but still fail to differentiate systematic boundary effects from random noise.
• Existing property valuation GNNs similarly omit boundary segmentation from their learning architecture.

This gap motivates the present study by embedding structured spatial heterogeneity directly into the graph representation. The proposed framework enhances both predictive performance and interpretability for property valuation in complex, heterophilic urban environments.

3. Materials and Methods

3.1. Study Area and Datasets

This study utilizes three urban housing datasets representing diverse developmental contexts and spatial configurations across Bangkok (Thailand), King County (USA), and Singapore, as shown in Figure 1. The Bangkok dataset is proprietary and subject to data-sharing restrictions, while the King County and Singapore datasets are derived from publicly available sources. Detailed information on data accessibility and reproducibility is provided in the Data Availability declaration statement.

3.1.1. Bangkok Dataset (Proprietary)

Bangkok, the capital city of Thailand, represents the spatial characteristics of horizontal residential developments in a rapidly urbanizing, middle-income economy. The dataset was provided courtesy of the Thai Government Housing Bank (G H BANK), which utilizes housing price information to support credit assessment for home loan applications. It contains 16,522 properties spanning the period 2018–2023 across 50 districts, aggregated at the village level covering 711 villages. Each village corresponds to a private residential development characterized by relatively uniform design and pricing structures, a pattern commonly observed in Southeast Asian urban expansion. Statistical summaries of key housing attributes are provided in

Table 1. Descriptive statistics of Bangkok house data.

Feature Name	Unit	Type	Min	Max	Mean	Median	Std Dev.
land area	Wah2 *	Float	10.30	76.30	20.34	18.80	5.10
usage area	m2	Float	50.50	276.00	119.27	114.00	28.52
age	Year	Int	4.00	48.00	20.30	23.00	9.39
num floors	–	Int	1.00	5.00	2.01	2.00	0.40
price (log1p)	–	Float	13.31	15.69	13.80	14.40	0.35

Note. * 1 Wah² = 4 Meters².

. The spatial distribution of house locations and an example of village boundary units are shown in Figure 1a and Figure 2, respectively.

3.1.2. King County Dataset

King County, Washington, USA, provides an international benchmark for comparative housing analysis in developed economies. The King County dataset (https://www.kaggle.com/datasets/astronautelvis/kc-house-data (accessed on 15 September 2025)) serves as a widely used benchmark for housing price prediction and spatial analysis studies in the real estate domain. It contains residential property sales records covering approximately 21,613 transactions primarily from 2014–2015, representing horizontal residential properties utilizing zip code boundaries as administrative divisions for spatial aggregation. This mature market serves as the standard reference for global real estate research. The spatial distribution of housing transactions is illustrated in Figure 1b, highlighting the spatial extent and administrative boundary structure used in this study. Descriptive statistics of key housing attributes are provided in

Table 2. Descriptive statistics of King County house data.

Feature Name	Type	Min	Max	Mean	Median	Std Dev.
bedrooms	Int	0.00	33.00	3.37	3.00	0.93
bathrooms	Int	0.00	8.00	2.11	2.25	0.77
sqft living	Float	290.00	13,540.00	2079.90	1910.00	918.44
sqft lot	Float	520.00	1,651,359.00	15,106.97	7618.00	41,420.51
floors	Float	1.00	3.5	1.49	1.50	0.54
waterfront	Int	0.00	1.00	0.01	0.00	0.09
view	Int	0.00	4.00	0.23	0.00	0.77
condition	Int	1.00	5.00	3.41	3.00	0.65
grade	Int	1.00	13.00	7.66	7.00	1.18
sqft above	Float	290.00	9410.00	1788.39	1560.00	828.09
sqft basement	Float	0.00	4820.00	291.51	0.00	442.58
year built	Int	1900.00	2015.00	1971.00	1975.00	29.37
year renovated	Int	0.00	2015.00	84.40	0.00	401.68
price (log1p)	Float	11.23	15.86	13.05	13.02	0.53

.

3.1.3. Singapore Dataset

Singapore represents vertical residential developments in a highly urbanized, developed city-state in Southeast Asia. Housing data were obtained from https://data.gov.sg (accessed on 15 September 2025), the Singapore government’s open data platform, which provides public access to housing transaction records via API. The dataset contains 177,171 resale Housing and Development Board (HDB) units spanning the period from 2017 to mid-2024, distributed across 8887 individual blocks that serve as administrative boundary units. This configuration enables the examination of vertical spatial relationships through floor-level connectivity within high-rise residential complexes. Descriptive statistics of key housing attributes are provided in

Table 3. Descriptive statistics of Singapore house data.

Feature Name	Unit	Type	Min	Max	Mean	Median	Std Dev.
number of rooms	–	Int	1.00	6.00	4.13	4.00	0.91
floor area	m2	Float	31.00	243.00	97.35	93.00	23.90
remaining lease	Year	Float	41.50	97.75	74.87	74.92	13.93
num floor	–	Int	2.00	50.00	8.81	8.00	5.95
price (log1p)	–	Float	7.64	9.65	8.52	8.50	0.26

, and the spatial distribution of housing blocks is illustrated in Figure 1c.

The three datasets were selected to represent different stages of urban development and contrasting urban forms, allowing us to test the proposed model in varied real-world settings. Bangkok represents a fast-growing city in a developing economy, where housing is organized into private residential villages with horizontal layouts and fragmented administrative structures. King County (USA) reflects a mature housing market in a developed economy, shaped by government-defined zip code regions that capture suburban sprawl and mixed housing types. Singapore illustrates a highly structured urban environment, with state-planned vertical housing blocks (HDB) that create clear and consistent administrative boundaries. Together, these three cases capture a broad range of spatial patterns (horizontal vs. vertical) and boundary systems (village-based, zip code, and state-defined), providing a solid foundation for evaluating the robustness of our approach across different urban and socioeconomic contexts.

3.1.4. Spatial Homophily Analysis

Homophily and heterophily are fundamental concepts in spatial network analysis. Homophily refers to the tendency of spatially adjacent properties to exhibit similar characteristics, such as neighboring houses with comparable market values or physical attributes. In contrast, heterophily arises when adjacent properties differ substantially, producing weak local similarity and often introducing noisy spatial signals. This distinction is particularly relevant for graph-based learning models, since most traditional GNN architectures implicitly assume homophilic relationships among connected nodes [16,21].

To quantify the degree of spatial homophily and heterophily in our datasets, we use a set of complementary metrics. Spatial relationships are evaluated through K-nearest neighbor (kNN) graphs at two representative neighborhood sizes ($K=5$ for local neighborhoods and $K=80$ for extended spatial relationships), which are selected to reflect two distinct spatial regimes rather than tuned hyperparameters, as well as boundary-based graphs that reflect each dataset’s urban morphology: private developer villages in Bangkok, government zip code divisions in King County, and individual HDB blocks in Singapore.

Our analysis focuses on three established spatial metrics. First, the assortativity coefficient [44] measures the tendency of nodes with similar property values to be connected, with values ranging from −1 (perfect heterophily) to +1 (perfect homophily). Second, Moran’s I statistic [45], a classic measure of spatial autocorrelation, quantifies the extent to which similar values cluster spatially: positive values indicate clustering, while negative values suggest dispersion. Third, threshold analysis at 5% and 10% similarity levels captures the proportion of directly connected property pairs whose price differences fall below a given relative threshold. Higher values indicate stronger local similarity in property prices, whereas lower values signal greater variability within local neighborhoods.

Although both assortativity and Moran’s I describe spatial dependence, they emphasize different aspects of the underlying structure. Moran’s I focuses on how closely each property value aligns with the average of its neighbors, making it more sensitive to overall cluster cohesion. Assortativity, on the other hand, measures pairwise correlation between connected nodes, capturing finer variation within clusters. In areas where properties are densely connected within administrative boundaries but exhibit only modest differences in value, as seen in Bangkok’s village developments, Moran’s I tends to remain high, whereas assortativity is comparatively lower. This contrast shows how the two metrics complement one another: Moran’s I highlights broader spatial coherence, whereas assortativity captures localized variability.

Beyond metric-based analysis, examining the graph structure itself provides further insight. We distinguish between intra-boundary edges (connections within the same administrative unit) and inter-boundary edges (connections across different administrative units). This structural composition directly shapes spatial dependence patterns, which are reflected in homophily metrics (

Table 4. Homophily metrics under kNN and boundary graphs, showing how spatial dependence weakens as K increases.

Dataset (N)	Graph	K	Assort.	Moran’s I	Thresh@5%/10%
Bangkok (16,522)	kNN	5	0.727	0.715	19.51/36.56
	kNN	80	0.438	0.432	13.16/25.60
	Boundary	–	0.573	0.720	16.04/31.05
King County (21,436)	kNN	5	0.646	0.631	15.95/29.56
	kNN	80	0.515	0.502	10.88/21.33
	Boundary	–	0.376	0.404	9.22/18.25
Singapore (177,171)	kNN	5	0.887	0.760	30.53/51.56
	kNN	80	0.819	0.747	23.40/42.73
	Boundary	–	0.887	0.793	25.81/46.56

Note. Assortativity (pairwise edge correlation) and Moran’s I (correlation between values and their spatial lag under row-standardized weights) capture complementary aspects of spatial dependence. Their magnitudes need not match exactly, but both consistently reflect the same trend as neighborhood scale K increases.

). In Bangkok, for example, the dominance of intra-boundary edges leads to a Moran’s I of 0.720 compared to an assortativity of 0.573, indicating strong boundary-level coherence despite moderate pairwise similarity. In King County, both Moran’s I (0.404) and assortativity (0.376) are lower, reflecting weaker boundary alignment as connectivity expands. Singapore shows both high assortativity (0.887) and Moran’s I (0.793), consistent with its highly structured HDB estate design.

As neighborhood size (K) increases, inter-boundary connections become more prevalent, especially in Bangkok and Singapore. This pattern reflects the gradual expansion of spatial reach beyond administrative boundaries and contributes to the observed decline in homophily metrics at larger K levels. For example, in Bangkok, assortativity drops from 0.727 at $K=5$ to 0.438 at $K=80$. A complementary view of this underlying edge structure, including the proportion of intra- and inter-boundary connections, is presented in

Table 5. Edge connection statistics under kNN graphs with different K values, showing how intra-boundary links decrease and inter-boundary links increase as K grows. Bold values highlight representative low-K and high-K settings (e.g., $K=5$ and $K=80$) used to illustrate local and global neighborhoods, respectively.

Dataset (N)	Graph	K	Edges	Intra-Boundary (%)	Inter-Boundary (%)
Bangkok (16,522)	kNN	5	82,610	77.6	22.4
		10	165,220	70.7	29.3
		20	330,440	60.1	39.9
		40	660,880	47.0	53.0
		80	1,321,760	34.3	65.7
King County (21,436)	kNN	5	107,180	96.3	3.7
		10	214,360	94.8	5.2
		20	428,720	92.5	7.5
		40	857,440	89.1	10.9
		80	1,714,880	84.0	16.0
Singapore (177,171)	kNN	5	885,855	99.8	0.2
		10	1,771,710	97.9	2.1
		20	3,543,420	86.1	13.9
		40	7,086,840	57.9	42.1
		80	14,173,680	31.5	68.5

.

Taken together, these findings suggest that no single graph construction or spatial unit can universally guarantee high homophily. This highlights the need for GNN models that can adapt flexibly to both spatial and administrative heterogeneity, achieving robust performance across diverse urban settings while remaining efficient enough for real-world use.

3.2. Data Preprocessing

The preprocessing workflow was designed to ensure spatial consistency and data quality prior to model training. The steps were standardized across all datasets while accounting for dataset-specific issues.

3.2.1. Spatial Validation

Spatial validation prioritized coordinate accuracy to preserve meaningful graph relationships. In the Bangkok dataset, approximately 3300 residential units (about 20% of total records) were identified as being located anomalously far from their designated village clusters. These records were repositioned programmatically within the correct village boundaries using controlled random sampling to ensure spatial consistency with the underlying administrative structure. Nodes with unverifiable coordinates were excluded from further analysis.

For King County, around 220 units (approximately 1%) were found outside county or zip code boundaries. Given the small proportion, these records were excluded rather than repositioned. In contrast, the Singapore dataset required no adjustment, as all coordinates were already spatially consistent and matched their respective administrative units with high precision.

3.2.2. Feature Engineering

Feature engineering was deliberately minimal. The primary transformation involved applying a logarithmic scale to house prices to reduce skewness and approximate a Gaussian distribution, which is beneficial for regression. This normalization step ensures that extreme values do not dominate the learning process.

3.2.3. Missing Values and Outliers

Data quality was ensured through complete case analysis: records with missing values were removed using listwise deletion. Outliers were identified and removed using the Interquartile Range (IQR) method, thereby improving robustness of the regression models to anomalous values.

3.2.4. Train–Validation–Test Splits

All datasets were partitioned into 70% training, 15% validation, and 15% testing sets. Splits were performed randomly without temporal stratification, as the objective was comparative algorithm evaluation rather than forecasting. This consistent protocol ensured a fair and reproducible benchmark across all baseline and proposed methods.

3.3. Graph Construction

This subsection explains how neighborhood sizes are selected and why specific values are used to represent different spatial regimes. We construct k-nearest neighbor (kNN) graphs to represent spatial proximity among properties, with the neighborhood size K controlling the scale of connectivity. Figure 3 and Figure 4 illustrate examples of horizontal (2D) and vertical (3D) graph structures at $K=5$. In horizontal settings (e.g., Bangkok and King County), edges reflect geographic distance on a 2D plane, while in vertical settings (e.g., Singapore), floor information is incorporated into the distance metric to capture vertical spatial relations.

To systematically examine the effects of neighborhood size, we experiment with $K=5, 10, 20, 40,$ and 80. As K increases, the resulting graphs exhibit a clear shift from highly localized structures to broader spatial connectivity. Specifically, smaller K values emphasize fine-scale local neighborhood relations, with a high proportion of intra-boundary edges. Increasing K progressively introduces more inter-boundary edges, reflecting spatial interactions that extend beyond immediate neighborhood clusters. This pattern is evident in Table 5, where the share of intra-boundary edges declines substantially as K grows, indicating a transition from localized to increasingly diffuse spatial structures.

We select $K=80$ as the upper bound for our experiments based on both empirical and practical considerations. First, preliminary analyses (Table 4) indicate that homophily metrics such as assortativity and Moran’s I stabilize once this neighborhood size is reached. Beyond this point, increasing K adds little new spatial information but greatly enlarges the graph. For instance, in Singapore, the number of edges grows from 7 million at $K=40$ to over 14 million at $K=80$ with only limited gains in homophily. Second, this choice aligns with prior work on spatial graph learning, which suggests that overly large neighborhood sizes can dilute meaningful local structure and introduce noise due to excessive inter-boundary connections [16,46,47]. Thus, $K=80$ provides a practical balance between capturing extended spatial context and maintaining computational scalability for large-scale urban property datasets.

Table 5 reports edge connection statistics across different neighborhood sizes, demonstrating the evolving balance between intra- and inter-boundary connectivity as K increases.

3.4. The Proposed Architecture

BD-GNN is designed to address structured spatial heterogeneity in property valuation by explicitly separating and adaptively integrating two spatial processes: geometric proximity and boundary-induced segmentation. Instead of relying on a single spatial graph, the model employs two complementary pathways, inter- and intra-boundary, and combines them through an adaptive gating mechanism that dynamically adjusts their influence at each node.

At a glance, the architecture in Figure 5 consists of five key components: (1) spatial aggregation (Figure 5a), which constructs both inter- and intra-boundary graphs; (2) inter-boundary pathway (Figure 5b), which captures geometric proximity effects across boundaries; (3) intra-boundary pathway (Figure 5c), which encodes segmentation within administrative or functional units; (4) adaptive gating mechanism (Figure 5d), which learns context-dependent weights between the two pathways; and (5) prediction layer (Figure 5e), which produces the final price estimate based on the fused representation. This structure allows the model to differentiate between spatial continuity and administrative segmentation, and to adapt its focus according to local heterophily patterns.

3.4.1. Motivation and Design Rationale

Property valuation is governed by two interacting spatial processes: geometric proximity, which captures similarities arising from locational closeness, and boundary-induced segmentation, which introduces price discontinuities across administrative or functional zones. Conventional GNNs, built on a single proximity-based graph, assume homophily and therefore propagate signals indiscriminately across boundaries. Heterophily-aware variants partially relax this assumption but still model segmentation only implicitly. To overcome these limitations, BD-GNN explicitly separates spatial interactions into intra-boundary and inter-boundary pathways, which are adaptively fused through a learnable gating mechanism (Figure 5). This architecture maintains spatial continuity while respecting segmentation boundaries, allowing the model to dynamically prioritize informative signals at each node. As a result, it achieves higher predictive accuracy and more transparent interpretability in heterogeneous spatial environments.

3.4.2. Dual-Pathway Feature Learning

The architecture integrates two parallel three-layer GCN pathways to capture distinct spatial relationship patterns. The choice of three layers balances model expressiveness with computational efficiency, allowing sufficient receptive field expansion while avoiding over-smoothing effects common in deeper GCN architectures. In practice, the number of layers can be adjusted (e.g., 1–3 layers) depending on the scale of the study area, spatial resolution, and graph connectivity characteristics

For the inter-boundary pathway, which captures geometric neighborhood effects:

$${\mathbf{H}}_s^{\left(1\right)}=\mathrm{ReLU}({\mathbf{W}}_s^{\left(1\right)}·{\mathrm{AGG}}_{\mathrm{inter}}(\mathbf{X},{\mathbf{E}}_{\mathrm{inter}})+{\mathbf{b}}_s^{\left(1\right)})$$

(1)

$${\mathbf{H}}_s^{\left(2\right)}=\mathrm{ReLU}({\mathbf{W}}_s^{\left(2\right)}·{\mathrm{AGG}}_{\mathrm{inter}}({\mathbf{H}}_s^{\left(1\right)},{\mathbf{E}}_{\mathrm{inter}})+{\mathbf{b}}_s^{\left(2\right)})$$

(2)

$${\mathbf{H}}_s^{\left(3\right)}=\mathrm{ReLU}({\mathbf{W}}_s^{\left(3\right)}·{\mathrm{AGG}}_{\mathrm{inter}}({\mathbf{H}}_s^{\left(2\right)},{\mathbf{E}}_{\mathrm{inter}})+{\mathbf{b}}_s^{\left(3\right)})$$

(3)

Similarly, for the intra-boundary pathway, which models boundary-based spatial relationships:

$${\mathbf{H}}_v^{\left(1\right)}=\mathrm{ReLU}({\mathbf{W}}_v^{\left(1\right)}·{\mathrm{AGG}}_{\mathrm{intra}}(\mathbf{X},{\mathbf{E}}_{\mathrm{intra}})+{\mathbf{b}}_v^{\left(1\right)})$$

(4)

$${\mathbf{H}}_v^{\left(2\right)}=\mathrm{ReLU}({\mathbf{W}}_v^{\left(2\right)}·{\mathrm{AGG}}_{\mathrm{intra}}({\mathbf{H}}_v^{\left(1\right)},{\mathbf{E}}_{\mathrm{intra}})+{\mathbf{b}}_v^{\left(2\right)})$$

(5)

$${\mathbf{H}}_v^{\left(3\right)}=\mathrm{ReLU}({\mathbf{W}}_v^{\left(3\right)}·{\mathrm{AGG}}_{\mathrm{intra}}({\mathbf{H}}_v^{\left(2\right)},{\mathbf{E}}_{\mathrm{intra}})+{\mathbf{b}}_v^{\left(3\right)})$$

(6)

where:

• $\mathbf{X}\in {\mathbb{R}}^{n\times d}$ represents the input node features for n properties with d attributes;
• ${\mathbf{W}}_s^{\left(l\right)}\in {\mathbb{R}}^{h\times h}$ and ${\mathbf{W}}_v^{\left(l\right)}\in {\mathbb{R}}^{h\times h}$ are learnable weight matrices for layer l (with ${\mathbf{W}}_s^{\left(1\right)},{\mathbf{W}}_v^{\left(1\right)}\in {\mathbb{R}}^{h\times d}$ for the first layer);
• ${\mathbf{b}}_s^{\left(l\right)},{\mathbf{b}}_v^{\left(l\right)}\in {\mathbb{R}}^h$ are bias vectors for layer l;
• ${\mathbf{H}}_s^{\left(l\right)},{\mathbf{H}}_v^{\left(l\right)}\in {\mathbb{R}}^{n\times h}$ represent hidden node embeddings at layer l with hidden dimension h;
• $\mathrm{AGG}(·)$ denotes the graph convolution aggregation function following the standard GCN formulation [6];
• ${\mathbf{E}}_{\mathrm{inter}}\subseteq {\mathbb{N}}^{2\times |{\mathbf{E}}_{\mathrm{inter}}|}$ and ${\mathbf{E}}_{\mathrm{intra}}\subseteq {\mathbb{N}}^{2\times |{\mathbf{E}}_{\mathrm{intra}}|}$ represent edge index tensors for inter-boundary and intra-boundary graphs, respectively.

The two edge sets correspond to Figure 5b,c, which depict how geometric proximity and boundary-induced segmentation are modeled as separate graph structures.

3.4.3. Adaptive Gating Mechanism

The core innovation of BD-GNN lies in the adaptive gating mechanism (Figure 5d), which addresses the fundamental question: when should geometric proximity be prioritized over administrative boundaries, and vice versa? This mechanism employs cosine similarity as a coherence measure between the two pathway representations and is fully differentiable and trained end-to-end via backpropagation.

$$S_{\cos }^{\left(i\right)}={\displaystyle \frac{{\mathbf{h}}_{s,i}^{\left(3\right)}·{\mathbf{h}}_{v,i}^{\left(3\right)}}{\parallel {\mathbf{h}}_{s,i}^{\left(3\right)}{\parallel }_2{\parallel {\mathbf{h}}_{v,i}^{\left(3\right)}\parallel }_2}}$$

(7)

where ${\mathbf{h}}_{s,i}^{\left(3\right)},{\mathbf{h}}_{v,i}^{\left(3\right)}\in {\mathbb{R}}^h$ represent the final layer representations for node i in the spatial and administrative pathways, respectively, and ${\parallel ·\parallel }_2$ denotes the L2 norm. High cosine similarity ($S_{\cos }^{\left(i\right)}\approx 1$) indicates consistent signals between pathways, while low similarity ($S_{\cos }^{\left(i\right)}\approx 0$) reflects divergent spatial patterns across boundaries.

The gating network processes a comprehensive feature vector that includes original features, both pathway representations, and their coherence measure:

$${\mathbf{G}}_{\mathrm{input}}^{\left(i\right)}=[{\mathbf{x}}_i,{\mathbf{h}}_{s,i}^{\left(3\right)},{\mathbf{h}}_{v,i}^{\left(3\right)},S_{\cos }^{\left(i\right)}]$$

(8)

The gate weight is computed through a two-layer MLP with ReLU activation:

$${\alpha }^{\left(i\right)}=\sigma ({\mathbf{W}}_{\mathrm{gate}}^{\left(2\right)}·\mathrm{ReLU}({\mathbf{W}}_{\mathrm{gate}}^{\left(1\right)}·{\mathbf{G}}_{\mathrm{input}}^{\left(i\right)}+{\mathbf{b}}_{\mathrm{gate}}^{\left(1\right)})+{\mathbf{b}}_{\mathrm{gate}}^{\left(2\right)})$$

(9)

where:

• ${\mathbf{x}}_i\in {\mathbb{R}}^d$ is the original feature vector for node i;
• $[·]$ denotes vector concatenation operation;
• ${\mathbf{G}}_{\mathrm{input}}^{\left(i\right)}\in {\mathbb{R}}^{d+2h+1}$ is the concatenated input to the gating network;
• ${\mathbf{W}}_{\mathrm{gate}}^{\left(1\right)}\in {\mathbb{R}}^{h\times (d+2h+1)}$ and ${\mathbf{W}}_{\mathrm{gate}}^{\left(2\right)}\in {\mathbb{R}}^{1\times h}$ are gating network weight matrices;
• ${\mathbf{b}}_{\mathrm{gate}}^{\left(1\right)}\in {\mathbb{R}}^h$ and $b_{\mathrm{gate}}^{\left(2\right)}\in \mathbb{R}$ are gating network bias terms;
• $\sigma$ denotes the sigmoid activation function ensuring ${\alpha }^{\left(i\right)}\in [0,1]$.

3.4.4. Final Representation and Prediction

The final node representation combines both pathways through the learned adaptive weights:

$${\mathbf{h}}_{\mathrm{final}}^{\left(i\right)}={\alpha }^{\left(i\right)}{\mathbf{h}}_{s,i}^{\left(3\right)}+(1-{\alpha }^{\left(i\right)}){\mathbf{h}}_{v,i}^{\left(3\right)}$$

(10)

This weighted combination enables smooth interpolation between spatially continuous and administratively segmented representations. When ${\alpha }^{\left(i\right)}\approx 1$, the model prioritizes inter-boundary patterns; when ${\alpha }^{\left(i\right)}\approx 0$, intra-boundary patterns dominate; intermediate values reflect a balanced contribution of both.

The final prediction is obtained through linear transformation:

$${\widehat{y}}^{\left(i\right)}={\mathbf{w}}_{\mathrm{out}}^T{\mathbf{h}}_{\mathrm{final}}^{\left(i\right)}+b_{\mathrm{out}}$$

(11)

where ${\mathbf{w}}_{\mathrm{out}}\in {\mathbb{R}}^h$ is the output weight vector and $b_{\mathrm{out}}\in \mathbb{R}$ is the output bias term.

3.4.5. Complete Model Formulation and Theoretical Properties

The complete BD-GNN can be expressed as a composition of functions:

$$f(\mathbf{X},{\mathbf{E}}_{\mathrm{inter}},{\mathbf{E}}_{\mathrm{intra}})=\mathrm{Linear}\left(\mathbf{G}({\mathbf{F}}_s(\mathbf{X},{\mathbf{E}}_{\mathrm{inter}}),{\mathbf{F}}_v(\mathbf{X},{\mathbf{E}}_{\mathrm{intra}}))\right)$$

(12)

where ${\mathbf{F}}_s$ and ${\mathbf{F}}_v$ represent the spatial and administrative pathway functions, and $\mathbf{G}$ denotes the adaptive gating function.

This architecture possesses several desirable theoretical properties: (1) Permutation invariance: the model output is invariant to node ordering; (2) Spatial locality: the three-layer design ensures that information propagates within reasonable spatial neighborhoods; and (3) Adaptive expressiveness: the gating mechanism allows the model to automatically adjust its spatial bias based on local data characteristics.

The per-iteration computational complexity of BD-GNN is $\mathcal{O}\left(h\left(\right|{\mathbf{E}}_{\mathrm{inter}}|+|{\mathbf{E}}_{\mathrm{intra}}\left|\right)+ndh+n{h}^2\right)$, where h is the hidden dimension. The first term corresponds to sparse message passing over both inter-boundary and intra-boundary graphs, while the latter two terms arise from the node-wise linear projections in the GCN layers and the gating MLP. By ignoring constant factors from the number of layers and pathways, the model remains scalable for large, sparse geospatial graphs while retaining sufficient capacity to capture complex spatial dependencies that are crucial for accurate property valuation.

3.5. Experimental Setup

3.5.1. Hardware and Software Environment

All experiments were conducted on the Google Colab Pro platform.

For the GNN-based models, GPU acceleration was enabled using an NVIDIA A100 Tensor Core GPU with 80 GB memory (High-RAM environment). The implementation was developed in Python 3.12.11, with PyTorch 2.8.0 serving as the deep learning framework and PyTorch Geometric 2.7.0 used for graph neural network operations. All GNN computations were executed on the GPU unless otherwise specified.

In contrast, the SAR model was estimated using the standard CPU backend on Google Colab with a high-RAM configuration (51 GB). The SAR implementation was also executed in Python 3.12.11, but relied exclusively on CPU-based computation without GPU acceleration.

3.5.2. Model Configuration and Hyperparameters

All GNN architectures were configured with standardized parameter settings as detailed in

Table 6. Hyperparameter configuration for all GNN models.

Parameter	Value	Description
Network Architecture
Hidden dimensions	64	Number of neurons per GNN layer
Number of GNN layers	2–3 *	Graph Neural Network layers
Activation function	ReLU, eLU **	Non-linear activation
Dropout rate	0.2	Regularization parameter
Output dimension	1	Price prediction output
Graph Construction
k-NN neighbors	5, 10, 20, 40, 80	Spatial connectivity parameter
Edge weighting	Binary (0, 1)	Spatial weight function
Training Configuration
Learning rate	0.001	Initial optimizer learning rate
Optimizer	Adam	Gradient descent optimizer
Max epochs	1000	Maximum training iterations
Early stopping patience	30	Epochs without improvement

Note. * GNN baselines use 2 layers, while BD-GNN uses 3 layers. ** eLU is applied only for the GAT baseline.

. We deliberately chose not to optimize hyperparameters for individual models to ensure fair comparative evaluation across different architectures. This approach prioritizes empirical comparison over model-specific performance maximization, allowing us to assess the relative effectiveness of each architecture under equivalent conditions. The uniform hyperparameter configuration enables direct attribution of performance differences to architectural design choices rather than optimization advantages. This standardized setup provides a controlled environment for evaluating how different GNN architectures handle heterophily regression in real estate valuation tasks without the confounding effects of architecture-specific tuning. Although individual models may achieve higher accuracy when carefully optimized, our comparative framework provides more reliable insights into their fundamental capabilities for boundary-aware property valuation.

3.5.3. Baseline Models

We compare BD-GNN against two categories of baseline methods: traditional spatial econometric models and graph neural network (GNN) architectures.

Spatial Econometric Model:  We include the Spatial Autoregressive (SAR) model [1,48] as a representative and well-established baseline from spatial econometrics. SAR is widely used in applied property valuation to model global spatial dependence through a fixed spatial weight matrix, making it a natural reference point for positioning BD-GNN with respect to classical valuation approaches.

The SAR model is estimated using the maximum likelihood estimator (ML_Lag) implemented in the spreg Python library, which is a standard approach in spatial econometric analysis [48]. In general, spatial econometric models are primarily designed for explanatory analysis. When applied in a predictive setting, careful handling of spatial lag construction is required to avoid information leakage. In this study, SAR is implemented under a strict train–test split with kNN-based spatial weight matrices matched to the GNN baselines.

SAR is also sensitive to multicollinearity, as the joint estimation of regression coefficients ($\beta$) and the spatial autoregressive parameter ($\rho$) requires a nonsingular design matrix. To ensure stable estimation, we apply automated feature filtering using a pairwise correlation threshold of 0.8, resulting in a reduced feature set. This follows standard econometric practice and allows a stable evaluation of spatial dependence under the SAR framework.

GWR and its multiscale extension (MGWR) are not included in the quantitative benchmark because their spatial influence is determined endogenously through bandwidth selection rather than fixed neighborhood sizes. Their effective spatial support therefore differs fundamentally from the fixed-K graph construction used in this study, which would prevent a consistent and interpretable comparison.

Classical GNNs: GCN serves as the foundational spectral convolution baseline. GAT introduces attention mechanisms to learn adaptive neighbor weights, while GraphSAGE employs inductive, sampling-based aggregation and has demonstrated strong performance in real estate applications [34].

Heterophily-Aware GNNs: MixHop addresses heterophily through higher-order neighborhood mixing. H2GCN separates ego and multi-hop information to mitigate oversmoothing under heterophily. PNA combines multiple aggregators and scalers to capture diverse neighborhood structures, and FAGCN balances low- and high-frequency signals through learnable filters.

To benchmark against recent spatial heterophily-aware GNNs, we additionally include SHGNN, which model spatial heterogeneity through sector–ring graph decomposition and multi-view feature fusion. The original SHGNN implementation is provided in PaddlePaddle (https://github.com/PaddlePaddle/PaddleSpatial/tree/main/research/SHGNN, accessed on 15 September 2025). To ensure compatibility with our PyTorch-based experimental pipeline, we reimplemented SHGNN in PyTorch following the official architecture and training procedure as closely as possible. While minor implementation-level differences across frameworks are unavoidable, this reimplementation represents a faithful reproduction that enables a transparent and meaningful comparison.

All GNN baselines use kNN graphs without boundary information, isolating the architectural contribution of BD-GNN. The SAR model similarly relies on conventional spatial weight matrices without boundary modifications. We conduct 30 independent runs for each GNN with different random seeds and report mean performance with standard deviations to account for training stochasticity [21,49]. In contrast, SAR is deterministic given fixed weights and therefore requires a single estimation per configuration.

Transformer-based GNN variants (e.g., graph transformers) are not included in this study. Most existing graph transformers rely on global or near-global attention mechanisms with quadratic complexity in the number of nodes, which becomes computationally impractical for large spatial property datasets. Moreover, incorporating global attention would shift the focus away from the core objective of this work, which is to isolate the effect of boundary-aware graph construction within a message-passing framework. Extending BD-GNN to transformer-based architectures is a promising direction for future research but is beyond the scope of this study.

3.6. Evaluation Metrics

Given the actual house price $y_i$ and the predicted price ${\widehat{y}}_i$ for each sample i, we evaluate the model’s performance using three error metrics: coefficient of determination ($R^2$), Mean Absolute Percentage Error (MAPE), and General Error Rate (GER).

3.6.1. Coefficient of Determination ($R^2$)

The $R^2$ score quantifies how well the model captures variance in the target variable, and is defined as:

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}$$

(13)

where $\overline{y}$ is the mean of the ground truth values. A higher $R^2$ (closer to 1) indicates better prediction performance.

3.6.2. MAPE (Mean Absolute Percentage Error)

MAPE is a scale-independent metric that computes the average of absolute percentage errors:

$$\mathrm{MAPE}={\displaystyle \frac{100}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|$$

(14)

Lower MAPE values indicate more accurate predictions. However, MAPE can be sensitive to very small ground truth values ($y_i$ close to zero).

3.6.3. GER (General Error Rate)

Following the definition in LUCE [12], we define the signed percentage error as:

$${\mathrm{GER}}_i={\displaystyle \frac{{\widehat{y}}_i-y_i}{y_i}}$$

(15)

We then use the absolute value to construct the cumulative distribution of errors:

$$|{\mathrm{GER}}_i|=\left|{\displaystyle \frac{{\widehat{y}}_i-y_i}{y_i}}\right|$$

(16)

Instead of averaging, we report the percentage of predictions where $|{\mathrm{GER}}_i|$ is below predefined thresholds (e.g., $5\%, 10\%, 15\%, 20\%$), thereby assessing how often the model produces relatively small errors. This complements MAPE by revealing the distributional shape of error magnitudes across all samples.

4. Results

Across all datasets, BD-GNN consistently achieved the best predictive performance, improving MAPE by up to 7.92% and increasing $R^2$ by up to 6.05% compared with the best baselines. The model demonstrates particularly strong gains in highly fragmented administrative settings, such as Singapore and Bangkok, while maintaining stable improvements in King County. These results confirm that explicitly modeling boundary effects not only enhances prediction accuracy but also improves robustness under high inter-boundary connectivity.

We note that SAR could not be estimated on the Singapore dataset due to prohibitive memory requirements associated with constructing large spatial weight matrices, even under a high-RAM environment. This reflects a known scalability limitation of classical spatial econometric models in large-scale urban settings.

Table 7. Comparison of MAPE (%) at $K=5$ and $K=80$ across three datasets. Bold values indicate the best overall performance, and underlined values indicate the second best-performing method. The arrow (↓) indicates that lower MAPE values correspond to better predictive accuracy.

Model	Bangkok		King County		Singapore
	$\mathit{K}=\mathbf{5}$↓	$\mathit{K}=\mathbf{80}$↓	$\mathit{K}=\mathbf{5}$↓	$\mathit{K}=\mathbf{80}$↓	$\mathit{K}=\mathbf{5}$↓	$\mathit{K}=\mathbf{80}$↓
SAR (1988) [1]	15.3386	15.8238	18.0926	16.0462	-	-
GCN (2016) [6]	19.4718 ± 0.1001	22.3293 ± 0.2617	25.8342 ± 0.2665	25.7337 ± 0.3006	17.1562 ± 0.0721	17.1382 ± 0.0911
GAT (2017) [8]	19.6766 ± 0.4481	19.7184 ± 1.6340	25.7910 ± 0.1381	23.8540 ± 2.2480	17.9461 ± 0.1371	16.7682 ± 0.1569
GraphSAGE (2017) [7]	16.1375 ± 0.2665	14.3203 ± 0.2838	21.1287 ± 0.3074	16.2221 ± 0.3945	16.2632 ± 0.4125	14.8680 ± 0.7081
MixHop (2019) [14]	15.3097 ± 0.1512	13.3017 ± 0.1204	18.8092 ± 0.3389	13.2200 ± 0.3355	16.0210 ± 0.3629	14.2790 ± 0.3392
H2GCN (2020) [16]	16.8942 ± 0.4404	15.0402 ± 0.4219	22.5333 ± 0.5386	18.2042 ± 0.6190	17.3873 ± 0.2693	15.6753 ± 0.7471
PNA (2020) [19]	15.8619 ± 0.2301	15.2075 ± 0.4971	20.6827 ± 0.3104	15.6680 ± 0.8206	14.5124 ± 0.2107	12.8045 ± 0.8114
FAGCN (2021) [18]	16.6632 ± 0.3050	14.7351 ± 0.3298	21.7958 ± 0.2880	16.7397 ± 0.3543	17.0319 ± 0.2224	15.3650 ± 0.4185
SHGNN (2023) [22]	15.3039 ± 0.0665	12.8123 ± 0.0937	20.8523 ± 0.1384	13.2661 ± 0.1291	16.6109 ± 0.0922	12.1387 ± 0.2164
BD-GNN (ours)	14.9518 ± 0.0987	11.7971 ± 0.1293	17.8873 ± 0.2524	12.6354 ± 0.1347	14.3226 ± 0.0596	11.5873 ± 0.3977

Note: SAR results for the Singapore dataset are not reported because model estimation could not be completed due to memory constraints. Despite using a high-RAM environment (51 GB) on Google Colab, the spatial weight matrix required for SAR exceeded available memory at both K = 5 and K = 80.

summarizes the MAPE results across all datasets and connectivity levels. BD-GNN achieves the lowest MAPE in every configuration: Bangkok (14.95% at $K=5$, 11.80% at $K=80$), King County (17.89% at $K=5$, 12.64% at $K=80$), and Singapore (14.32% at $K=5$, 11.59% at $K=80$). Relative to the best baselines in case of $K=80$, the reductions in MAPE range from 4.42% to 7.92%, with the largest improvement observed in Bangkok (7.92%), while King County and Singapore exhibit comparable gains of 4.42% and 4.54% respectively. Pairwise t-tests at the 95% confidence level (p-value < 0.05) confirm that these improvements are statistically significant across all datasets and connectivity settings. These results indicate that BD-GNN effectively captures structured discontinuities in property values that conventional and heterophily-aware GNNs often fail to model.

Table 8. Comparison of $R^2$ at $K=5$ and $K=80$ across three datasets. Bold values indicate the best overall performance, and underlined values indicate the second best-performing method. The arrow (↑) indicates that higher $R^2$ values correspond to better explanatory power.

Model	Bangkok		King County		Singapore
	$\mathit{K}=\mathbf{5}$↑	$\mathit{K}=\mathbf{80}$↑	$\mathit{K}=\mathbf{5}$↑	$\mathit{K}=\mathbf{80}$↑	$\mathit{K}=\mathbf{5}$↑	$\mathit{K}=\mathbf{80}$↑
SAR (1988) [1]	0.7038	0.6647	0.8056	0.8261	-	-
GCN (2016) [6]	0.5063 ± 0.0046	0.3849 ± 0.0140	0.6303 ± 0.0059	0.6122 ± 0.0078	0.3012 ± 0.0073	0.3126 ± 0.0086
GAT (2017) [8]	0.4989 ± 0.0218	0.5154 ± 0.0835	0.6440 ± 0.0036	0.6835 ± 0.0668	0.2404 ± 0.0112	0.3198 ± 0.0080
GraphSAGE (2017) [7]	0.6674 ± 0.0096	0.7396 ± 0.0102	0.7580 ± 0.0061	0.8469 ± 0.0063	0.3499 ± 0.0246	0.4414 ± 0.0397
MixHop (2019) [14]	0.7004 ± 0.0053	0.7742 ± 0.0041	0.8037 ± 0.0059	0.8898 ± 0.0038	0.3746 ± 0.0260	0.4963 ± 0.0230
H2GCN (2020) [16]	0.6363 ± 0.0149	0.7142 ± 0.0127	0.7289 ± 0.0132	0.8096 ± 0.0124	0.2670 ± 0.0167	0.3910 ± 0.0490
PNA (2020) [19]	0.6649 ± 0.0093	0.6764 ± 0.0305	0.7573 ± 0.0084	0.8428 ± 0.0193	0.4491 ± 0.0166	0.5490 ± 0.0598
FAGCN (2021) [18]	0.6444 ± 0.0161	0.7220 ± 0.0131	0.7435 ± 0.0049	0.8389 ± 0.0070	0.2869 ± 0.0144	0.4145 ± 0.0257
SHGNN (2023) [22]	0.7013 ± 0.0018	0.7860 ± 0.0024	0.7604 ± 0.0025	0.8888 ± 0.0019	0.3044 ± 0.0055	0.6284 ± 0.0134
BD-GNN (ours)	0.7134 ± 0.0037	0.8117 ± 0.0053	0.8155 ± 0.0047	0.8958 ± 0.0018	0.4763 ± 0.0035	0.6599 ± 0.0211

Note: SAR results for the Singapore dataset are not reported because model estimation could not be completed due to memory constraints. Despite using a high-RAM environment (51 GB) on Google Colab, the spatial weight matrix required for SAR exceeded available memory at both K = 5 and K = 80.

presents the $R^2$ results across all datasets and connectivity levels. BD-GNN consistently outperforms all baseline methods, with performance gains particularly evident at higher connectivity ($K=80$) where inter-boundary heterophily is strongest. For $K=80$, the model achieves $R^2=0.8117$ in Bangkok, $0.8958$ in King County, and $0.6599$ in Singapore. Relative to the best-performing baselines, these correspond to improvements of 3.26%, 0.67%, and 5.01% respectively. Pairwise t-tests at the 95% confidence level (p-value < 0.05) confirm significance across all dataset–connectivity combinations, underscoring the model’s ability to explain more variance in fragmented administrative environments.

For GER, the cumulative distributions in Figure 6 and the tabulated values in

Table 9. Cumulative General Error Rate (GER, %) at error thresholds of 5–20% for $K=5$ and $K=80$ across three datasets. Higher GER values indicate a larger share of predictions within each threshold. Bold entries denote the best overall performance, and underlined entries indicate the best baseline methods. BD-GNN consistently achieves the highest GER across all settings.

Dataset/Model	GER at $\mathit{K}=5$				GER at $\mathit{K}=80$
	<5%	<10%	<15%	<20%	<5%	<10%	<15%	<20%
Bangkok
GCN (2016) [6]	16.94	33.64	48.19	60.64	13.71	28.00	41.31	53.07
GAT (2017) [8]	17.13	33.01	47.54	60.72	17.19	33.55	48.22	60.96
GraphSAGE (2017) [7]	20.17	38.69	55.63	69.56	22.05	42.83	61.26	75.61
MixHop (2019) [14]	21.10	40.13	58.12	72.23	23.65	45.69	64.71	78.61
H2GCN (2020) [16]	18.74	37.02	53.59	67.37	21.19	40.70	58.36	72.92
PNA (2020) [19]	20.01	38.78	55.87	69.79	20.25	39.09	55.79	70.63
FAGCN (2021) [18]	19.31	37.97	54.06	67.70	22.11	42.49	60.30	74.06
SHGNN (2023) [22]	21.40	41.52	59.12	72.06	25.95	48.39	67.36	80.33
BD-GNN	21.42	42.00	59.55	73.51	27.66	51.70	70.46	83.03
King County
GCN (2016) [6]	13.76	27.14	39.36	50.72	13.75	27.06	39.69	50.80
GAT (2017) [8]	13.17	26.62	39.15	50.50	14.76	29.29	42.97	55.03
GraphSAGE (2017) [7]	16.17	31.48	45.75	58.92	21.43	41.05	58.06	71.15
MixHop (2019) [14]	18.21	35.57	51.17	64.38	27.31	50.61	68.58	80.73
H2GCN (2020) [16]	14.44	28.89	43.05	56.05	18.91	36.91	52.70	66.05
PNA (2020) [19]	14.86	29.71	44.32	58.03	20.71	40.49	58.15	72.22
FAGCN (2021) [18]	15.30	30.38	44.35	57.20	20.36	39.78	56.47	69.75
SHGNN (2023) [22]	17.10	33.05	47.01	59.43	27.21	50.50	68.13	80.48
BD-GNN	19.68	38.02	53.71	66.59	29.64	53.78	70.83	81.79
Singapore
GCN (2016) [6]	18.04	35.80	51.48	65.05	18.02	35.35	51.34	65.09
GAT (2017) [8]	17.69	34.39	49.58	62.17	18.36	36.20	52.61	66.32
GraphSAGE (2017) [7]	19.89	38.36	54.47	67.27	22.46	42.68	59.53	72.21
MixHop (2019) [14]	20.56	39.53	55.69	68.35	22.67	43.74	61.31	74.64
H2GCN (2020) [16]	18.26	35.77	51.38	64.26	20.79	40.02	56.68	69.69
PNA (2020) [19]	23.13	44.04	60.60	72.87	26.39	49.14	66.48	78.73
FAGCN (2021) [18]	18.88	36.71	52.46	65.22	21.59	41.28	57.94	70.73
SHGNN (2023) [22]	20.00	38.27	54.11	66.83	27.30	51.28	69.46	81.83
BD-GNN	23.74	44.98	61.90	74.20	28.98	53.61	71.51	83.08

paint a consistent picture. BD-GNN places a larger share of predictions within low error ranges across all datasets and thresholds. At $K=80$ and a 10% tolerance, it reaches 51.70% in Bangkok, 53.78% in King County, and 53.61% in Singapore, corresponding to improvements of 3.31, 3.28, and 2.33 percentage points over the best baselines. The comparison between $K=5$ and $K=80$ shows that BD-GNN maintains stable performance even when neighborhood size increases and cross-boundary connections become more prevalent. This effect is especially evident in Bangkok and Singapore, where larger neighborhoods introduce substantial boundary-induced heterogeneity, yet the model continues to deliver robust predictive accuracy.

Across all three datasets and both neighborhood sizes, BD-GNN consistently achieves the lowest MAPE, outperforming all baseline methods in Bangkok, King County, and Singapore. This advantage holds under both localized neighborhoods ($K=5$) and extended spatial interactions ($K=80$), indicating robustness to neighborhood scale.

The strongest baseline differs by dataset and neighborhood size. In Bangkok, the most competitive baselines are SHGNN and MixHop, while in Singapore, PNA is strongest at $K=5$ and SHGNN becomes strongest at $K=80$. In King County, SAR provides the best baseline performance at $K=5$, whereas at $K=80$ the best baseline shifts to MixHop, suggesting that global linear spatial dependence is already informative at small neighborhoods, while multi-hop mixing becomes more effective as spatial connectivity expands.

Despite these strong baselines—including cases where a classical spatial model (SAR) is most competitive—BD-GNN remains best in every dataset–connectivity setting. These results indicate that explicitly distinguishing spatial proximity from boundary-related effects, and adaptively weighting their contributions, improves predictive performance relative to both classical spatial models and existing GNN approaches.

5. Discussion: Boundary Dynamics, Adaptation, Boundary Robustness, Interpretability, Practical Trade-Off, and Potential Extension to Spatiotemporal

5.1. Interpreting $\alpha$: Spatial–Boundary Dynamics Across Cities

The adaptive gating parameter $\alpha$ measures how the model balances spatial and boundary information, serving as an interpretable signal of each city’s underlying structural regime. Its distribution provides insight into how different urban morphologies mediate the interaction between spatial proximity and administrative segmentation.

As shown in Figure 7 for Bangkok, $\alpha$ values remain below 0.5, increasing slightly from 0.43 to 0.46 as the neighborhood size (K) expands. This pattern indicates that boundary effects continue to dominate even at broader spatial scales. The trend reflects Bangkok’s village-based urban structure, where private housing developments are internally uniform but separated by distinct price gaps along their boundaries. Bangkok therefore represents a boundary-aligned regime, where spatial continuity exists but remains secondary to administrative or community segmentation.

As shown in Figure 8 for King Country, $\alpha$ averages around 0.44 across scales but shows greater dispersion as K increases, suggesting diverse local dynamics. Suburban areas depend more strongly on spatial proximity, while central zones are shaped by zoning and regulatory boundaries. The coexistence of these conditions produces a hybrid regime, in which spatial and administrative influences interact differently depending on the local context.

As shown in Figure 9 for Singapore, $\alpha$ decreases markedly from 0.51 to 0.33 as K grows, showing that boundary effects intensify with neighborhood expansion. The narrow distribution indicates structural uniformity across the city, consistent with Singapore’s centrally planned housing environment dominated by Housing and Development Board (HDB) estates. These estates exhibit strong internal homogeneity, with prices organized primarily by block-level boundaries. Singapore thus represents a boundary-dominant regime.

Across the three regimes, represented by boundary-aligned Bangkok, hybrid King County, and boundary-dominant Singapore, the results show that $\alpha$ effectively reflects key variations in urban structure. It operates not merely as a model parameter but as a diagnostic indicator of spatial–administrative balance, offering a more interpretable understanding of how governance and morphology shape property markets.

5.2. Boundary Robustness Under Weak and Artificial Segmentation

To assess the robustness of BD-GNN under imperfect boundary definitions, we conduct a controlled boundary-perturbation experiment on the Bangkok dataset, where villages act as explicit boundaries. We evaluate boundary robustness by selectively perturbing village labels for houses located near village borders, while keeping the underlying kNN graph topology unchanged. The perturbation level is increased from 0% to 60% to emulate progressively weaker or partially misaligned segmentation specifically in the most ambiguous boundary regions. Noise levels are limited to 60% because perturbations beyond this threshold no longer represent plausible boundary uncertainty in real-world valuation settings and instead correspond to near-random segmentation, which falls outside the intended scope of this robustness analysis.

Figure 10 (left) shows that prediction error increases gradually as border noise rises for both neighborhood sizes. This pattern indicates that BD-GNN is sensitive to boundary degradation in the expected direction, yet does not exhibit abrupt failure under moderate boundary perturbations. The degradation is smooth and limited in magnitude, suggesting that the model retains useful predictive structure even when boundary labels are partially corrupted at border areas.

Figure 10 (right) reports the behavior of the adaptive gating parameter $\alpha$. Overall, $\alpha$ tends to move upward under higher noise levels, indicating a shift toward relying more on spatial continuity when boundary information becomes less reliable. At the same time, the response is not strictly monotonic, particularly for larger neighborhoods ($K=80$), where broader connectivity increases interactions across villages and introduces more heterogeneous signals. This non-monotonic pattern reflects the fact that the gating mechanism is learned from data and adaptively balances spatial and boundary information under uncertainty, rather than following a fixed or deterministic rule.

Taken together, these results address the concern of weak or imperfect boundaries. BD-GNN does not assume boundary labels to be error-free; instead, it treats segmentation as a soft structural signal whose contribution is adaptively regulated. When border-level boundary information is degraded, the model maintains stable behavior: prediction error increases smoothly, while $\alpha$ adjusts to re-balance boundary-based and spatial aggregation. This supports the robustness of BD-GNN to boundary uncertainty in realistic settings where boundary regions are most prone to ambiguity or misclassification.

5.3. Model Adaptability and Transferability

The variations in $\alpha$ across cities highlight the importance of adaptive modeling. Conventional GNNs typically apply a single message-passing strategy regardless of spatial or administrative context, which can lead to systematic bias when these influences vary widely. In contrast, BD-GNN dynamically adjusts its balance between spatial and boundary pathways through the adaptive gating mechanism.

When $\alpha$ approaches 1, inter-boundary connections dominate the message-passing process, whereas when $\alpha$ approaches 0, intra-boundary relationships prevail. This flexibility allows BD-GNN to respond automatically to local urban morphology without requiring manual tuning or prior zoning assumptions. The model therefore learns the appropriate structural emphasis directly from data, aligning its internal representation with the city’s actual spatial organization.

From a practical perspective, $\alpha$ also functions as a diagnostic tool for model transferability. Before applying BD-GNN to a new context, analyzing $\alpha$ on a validation subset can reveal the city’s dominant regime:

• Hybrid regimes (for example, King County) may require stronger spatial aggregation to capture inter-boundary diffusion.
• Boundary-dominant regimes (for example, Singapore) benefit from a higher weighting on boundary pathways to reflect institutional segmentation.
• Boundary-aligned regimes (for example, Bangkok) represent intermediate conditions in which both spatial and boundary signals exert comparable influence.

This flexibility allows BD-GNN to adapt effectively across cities with different spatial and administrative configurations. Rather than training a separate model for each area, the same architecture can adjust its $\alpha$-based balance between spatial continuity and boundary segmentation. As a result, BD-GNN functions not only as a predictive tool but also as an adaptive framework for multi-city property valuation.

5.4. Policy and Planning Relevance

Understanding how boundaries influence property values is crucial for designing equitable and efficient urban policies. The $\alpha$-based regimes identified above offer practical guidance for aligning planning instruments with the structural organization of each area.

Boundary-dominant regimes, such as in Singapore, indicate that property values are largely determined within administrative units. In such cases, block-level taxation, estate-specific development controls, or targeted subsidy programs are likely to be most effective. Hybrid regimes, such as in King County, exhibit mixed spatial interactions, suggesting the need for inter-boundary coordination, including regional infrastructure planning, inter-municipal zoning, or coordinated service delivery. Boundary-aligned regimes, such as in Bangkok, emphasize the internal coherence of residential clusters, where neighborhood-scale interventions, such as localized land-use regulations or community-level infrastructure investments, can yield significant impacts.

Rather than applying uniform, city-wide valuation rules, policy design should align with the dominant spatial–administrative regime. BD-GNN supports this regime-aware formulation by revealing how spatial and boundary signals interact across scales. This integration of analytical insight and governance logic bridges predictive modeling with practical decision-making, enabling data-driven interventions that reflect the spatial realities of urban morphology. In this way, BD-GNN transforms boundary effects, traditionally seen as constraints, into measurable and actionable spatial processes.

5.5. Explainability and Local Insights

While $\alpha$ provides a global perspective on spatial–boundary dynamics, understanding individual predictions requires local interpretability. BD-GNN integrates the GNN Explainer to identify the most influential neighbors and structural attributes that contribute to each property’s estimated value.

Figure 11 shows a representative example from Bangkok. For a selected property, the model identifies key neighbors, two of which lie across administrative boundaries, with a gating value of $\alpha =0.757$. It also ranks the most influential structural features, including building area, number of floors, age, and land size. This node-level interpretation clarifies how spatial and physical attributes interact in determining the predicted price.

The combination of global interpretability (via $\alpha$) and local transparency (via GNN Explainer) allows BD-GNN to move beyond black-box prediction. The global signal captures citywide structural regimes, while the local explanation exposes the reasoning behind individual predictions. Together, they enhance transparency, user trust, and the interpretive value of spatial AI in real-world valuation practice.

5.6. Practical Trade-Off: Accuracy and Computational Cost

The computational cost of BD-GNN should be understood in relation to its modeling objective rather than as a direct efficiency comparison with classical spatial econometric models. Spatial regression models such as SAR are designed primarily for explanatory analysis under fixed linear specifications and are estimated through a single model-fitting procedure. However, their computational cost increases rapidly with sample size due to the construction and manipulation of large spatial weight matrices.

As shown in

Table A2. Theoretical computational complexity of spatial econometric and graph-based models. For GNNs, complexity is reported per training epoch under full-batch training with a directed KNN graph ($E\approx NK$). For spatial econometric models, complexity is reported per model fit.

Model	Architecture Used in this Study	Dominant Time Complexity
SAR	Exact maximum likelihood estimation	$O\left(n^3\right)$ (dense) or sparse-dependent
GCN	2-layer graph convolutional network	$O\left(2(NFH+EH)\right)$
GAT	2-layer, $h=4$ attention heads	$O\left(2\left(NF\right(hH)+E(hH\left)\right)\right)$
GraphSAGE (mean)	2-layer mean aggregator	$O\left(2(NFH+EH)\right)$
MixHop	2-layer, hop powers $\left|P\right|=3$	$O\left(2\left|P\right|(NFH+EH)\right)$
H2GCN	2-layer hop-separated aggregation (0-, 1-, and 2-hop)	$O\left(2(NFH+3EH)\right)$
FAGCN	2-layer low-/high-frequency decomposition	$O\left(2(NFH+2EH)\right)$
PNA	2-layer, $A=4$ aggregators and $S=3$ scalers	$O\left(2(NFH+(A·S\left)EH\right)\right)$
SHGNN	Sector–ring graph decomposition with multi-view fusion and gating	$O\left(C(NFH+EH)\right),C\gg 1$
BD-GNN (ours)	Dual-path (spatial + boundary) GCN, $3+3$ layers with node-wise gating	$O\left(6NFH+3EH+N(F+2H)H\right)$

Note: For SAR, dense exact ML has cubic complexity, while sparse ML implementations (as used in this study) rely on sparse matrix factorization or iterative solvers, with complexity depending on sparsity and the number of likelihood evaluations.

, SAR exhibits high computational and memory demands, especially under large and dense spatial graphs. In practice, this limitation becomes critical for large-scale datasets. In our experiments, SAR could not be estimated for the Singapore dataset ($n=177,171$) at either $K=5$ or $K=80$, even when using a high-RAM environment (51 GB) on Google Colab, highlighting its limited scalability in nationwide or city-scale mass appraisal settings.

For graph-based valuation models, neighborhood size (K) plays a central role in determining predictive performance. While small neighborhoods preserve local homogeneity, larger neighborhoods increasingly mix information across administrative boundaries. Standard GNNs aggregate neighbor information uniformly, which often leads to signal dilution when boundary-induced heterogeneity is strong.

BD-GNN addresses this issue by explicitly separating spatial proximity and boundary-based neighborhood structure, and by learning how to balance them through the adaptive gating parameter $\alpha$. This allows the model to maintain stable and improved accuracy at larger neighborhood sizes, particularly in cities where administrative segmentation strongly shapes property values.

In practice, the trade-off is therefore between modeling simplicity and structural fidelity. SAR remains suitable for moderate-sized datasets where fast estimation and global interpretability are prioritized. BD-GNN is most appropriate when accurate valuation under boundary-driven heterogeneity is required, and where a moderate and predictable computational overhead is acceptable.

5.7. Potential Extension to Spatiotemporal Property Valuation

This study focuses on a static property valuation setting in order to isolate the structural effects of administrative boundaries on spatial price formation. This design choice allows the boundary-induced heterogeneity to be examined without interference from temporal market fluctuations, thereby enabling a clearer interpretation of the proposed boundary-aware learning mechanism.

Nevertheless, temporal dynamics are undeniably important in real estate markets, where property values evolve over time in response to macroeconomic conditions, policy changes, and local development processes. Extending BD-GNN to a spatiotemporal setting therefore represents a natural and meaningful direction for future research. Such an extension would require additional modeling considerations, including constructing graphs that evolve over time, ensuring consistent boundary definitions across periods, and using transaction data with adequate temporal coverage.

Importantly, the dual-path architecture of BD-GNN provides a flexible foundation for incorporating temporal information. The explicit separation between spatial proximity and boundary-based relations could be combined with temporal graph neural networks or sequence-based learning frameworks, allowing spatial boundary effects and temporal price dynamics to be modeled jointly rather than conflated. For example, temporal encoders could be applied to node representations while preserving the boundary-aware aggregation mechanism at each time step.

While such spatiotemporal integration lies beyond the scope of the current study, the results presented here demonstrate that explicitly modeling administrative boundaries yields clear benefits even in a static setting. This suggests that boundary-aware spatial learning is a complementary component, rather than a substitute, to temporal modeling, and highlights the potential of BD-GNN as a building block for future spatiotemporal property valuation frameworks.

5.8. Summary of Discussion

This study shows that property values in urban areas are shaped not only by spatial proximity but also by administrative boundaries that organize markets into distinct segments. These boundaries function as socio-economic separators. This means that locations that are geographically close may still differ substantially in value if they lie within different governance or development contexts.

BD-GNN incorporates this segmentation directly by separating intra- and inter-boundary message passing and learning the relative importance of each through the adaptive gating parameter ($\alpha$). The learned $\alpha$ values indicate stable spatial–administrative regimes that reveal where price continuity holds and where boundary-driven differentiation is structurally reinforced. This provides a transparent mechanism for understanding how local market identities form.

Explainable GNN analysis shows that neighborhood features, access to amenities, and local development history work together with boundary segmentation to shape price patterns. In contrast, conventional GNNs tend to diffuse information uniformly and therefore overlook these distinctions.

Although BD-GNN is formulated under a static valuation setting in this study, extending the model to incorporate temporal dynamics is a promising direction for future work. Real estate markets are inherently time-dependent, with major macroeconomic events and policy interventions unfolding over time and influencing price formation. The dual-path, boundary-aware architecture of BD-GNN provides a natural foundation for integration with temporal graph or sequence-based models, enabling future research to jointly model spatial segmentation and temporal market dynamics.

All in all, the findings highlight that structured heterophily arising from governance, zoning, and administrative boundaries is a critical yet underrepresented factor in property valuation. Explicitly modeling this structure improves predictive accuracy and clarifies how urban morphology shapes price identities. In practice, this positions BD-GNN as a suitable choice when valuation accuracy and structural insight are prioritized, rather than as a replacement for faster but more rigid modeling approaches. This establishes a stronger foundation for decision support in urban planning, real estate governance, and valuation practice.

6. Conclusions

This study proposed BD-GNN, a boundary-aware dual-path graph neural network for property valuation in complex urban environments. By explicitly separating intra-boundary and inter-boundary message passing and adaptively balancing spatial proximity and administrative segmentation, BD-GNN addresses a key limitation of existing valuation models that uniformly aggregate neighborhood information and overlook boundary-induced price discontinuities.

Empirical evaluations across three heterogeneous urban datasets demonstrate that BD-GNN consistently improves predictive accuracy and stability in boundary-driven markets, reducing MAPE by up to 7.9% and increasing $R^2$ by up to 5.0% compared to strong baselines, particularly at larger neighborhood scales where standard GNNs tend to oversmooth local price signals. Additional robustness and scalability analyses further show that BD-GNN maintains linear scalability with respect to graph size and adapts gracefully to uncertainty or misalignment in administrative boundaries.

Beyond predictive performance, BD-GNN contributes a form of structural interpretability through its adaptive gating mechanism, enabling insight into the relative importance of spatial continuity versus boundary-induced segmentation in different urban contexts. This makes the framework particularly relevant for applied valuation tasks where transparency, boundary effects, and regulatory compliance play a central role, including mass appraisal, property taxation, and urban planning.

While this study examines static valuation settings to isolate boundary effects, the dual-path architecture provides a natural foundation for incorporating temporal dynamics. Extending BD-GNN to spatiotemporal valuation represents the most immediate direction for future research, while incorporating alternative or learned boundary definitions and integrating transformer-based architectures offer longer-term extensions. We believe that boundary-aware graph modeling provides a useful foundation for more realistic, interpretable, and policy-relevant property valuation in complex urban systems.