A Network Approach for Discovering Spatially Associated Objects

Abstract

Discovering spatially associated objects involves measuring objects’ similarities and retrieving associated objects. The integration of spatial topology and network models for discovering associated objects remains largely unexplored. Here, the concept of a maximum topological accessibility path was developed to quantify objects’ similarity attenuation. Considering the topological accessibility and spatial feature similarity of network nodes, an approach named the Weighted Similarity measure method considering Topological Accessibility (WSTA) is proposed to measure object association. The WSTA can capture both spatial interaction patterns and topological relationships in complex urban environments, thereby improving the accuracy of spatially associated object discovery. The proposed approach is validated using real-world point-of-interest (POI) datasets from Beijing city. The results suggest that integrating topological relationship approaches yields significant accuracy improvements in existing baseline methods, thereby enriching geospatial data retrieval in the era of big geospatial data.

1. Introduction

Along with the big geospatial era, discovering spatially associated objects is a crucial analysis method for linked data [1], geospatial variables [2], and geospatial entities [3,4]. Spatial association can be observed universally and applied in various scenarios across many domains [5], such as spatial prediction and interpolation. Linking data among multi-source data and discovering spatially associated objects are two well-known domains. The former aims to provide universal access and linkage among multi-source data, while discovering spatially associated objects aims to retrieve the associated objects by calculating the similarity of the objects.

We note that discovering spatially associated objects is an intrinsically different task to linking data. Linking data are used to establish links to access data with a harmonized description or feature relations [1]. The semantic web [6] and the uniform resource descriptor [1,7] are common approaches used to link other data. However, it is challenging to discover spatially associated objects owing to their complexity and implicit nature [8]. For example, as shown in Figure 1, we need to discover entities of object A associated with objects B, C, and D. Herein, objects A and C lie in a block with a co-location spatial relationship. Traditional methods depend on semantic or geometric similarity between objects, and object B may be identified as the best associated one owing to it having the shortest distance. However, the co-location of geospatial topological map objects enhances spatial association. As a result, given the co-locations of objects A and C, they are the most associated entities. Therefore, geospatial topology allows for more precise discovery of spatially associated objects.

Although many studies have been devoted to mining association rules or patterns from geographic phenomena [9,10,11], discovering associated objects is different. The latter belongs to information retrieval, while mining association rules consider geometric or semantic information to mine relationships with objects. Associated objects are usually helpful for mining association rules, yet the association rule is not necessary to discover associated objects. Therefore, in this study, “association” refers to discovering associated objects instead of association rules or patterns.

Methods for discovering spatially associated object methods can be grouped into three categories based on principle: feature similarity [12], association relationship [13], and association network [14]. The former two methods focus on feature representation, such as a uniform resource descriptor [1,7] or geometric or semantic similarity measurement [12,15] to harmonize object inequalities. The network model is capable of representing complex object relationships and interactions [16,17]. However, owing to complex spatiotemporal and topographic characteristics, the network cannot be directly applied in geographic domains. Spatial topology should be given more attention regarding integration with networks. In this paper, “topology” refers specifically to the spatial topological relationships between objects, defined based on classical theories such as the Region Connection Calculus (RCC). These relationships qualitatively describe the spatial interactions between geographic entities and serve as a foundational element for measuring object association. Existing studies have integrated topology into networks with two approaches. One is topological classification, using quantitative encoding as a property of network nodes or edges [13,14,18]. The other approach is integrating topology with other spatial analysis methods, such as spatial clustering [19]. These methods employ topology as a simple similarity measurement. However, the underlying impact on association from spatial topology (such as how co-location can strengthen object association) was ignored. Moreover, we do not know how the attenuation degree of object association changes with spatial topology relationships. More attention should be paid to this issue.

To address the aforementioned issues, in this study, we integrate spatial topology into the network model to discover associated objects. Existing methods based on feature similarity or direct associations often fail to account for multi-step path relationships and topological dependencies inherent in spatial data. In contrast, network models can explicitly encode both direct and indirect connections, enabling a more comprehensive analysis of spatial associations. Moreover, networks allow for the incorporation of topological accessibility, which reflects how similarity attenuates along spatial paths. Based on the network data model, topological accessibility was used to measure objects’ similarity attenuation. Then, an approach named the Weighted Similarity measure method considering Topological Accessibility (WSTA) was proposed to fuse spatial feature similarity and topological similarity between nodes. Using the WSTA approach, the ordered associated objects were retrieved. The Beijing city point of interest (POI) was used to validate our work. The results demonstrate that the accuracy was improved effectively. Specifically, the contributions of this paper are as follows:

• The maximum topological accessibility path was developed to quantify objects’ similarity attenuation along topological lines.
• The WSTA approach was proposed by integrating topological accessibility into the network model.
• This study demonstrates the effectiveness and suitability of WSTA by validating it using the Beijing POI dataset. The results show that WSTA leads to a significant improvement in accuracy when considering spatial topology.

The remainder of this article is arranged as follows: Section 2 investigates related studies. Section 3 presents the case study data and the benchmark dataset. Section 4 presents the approach proposed in our study. Section 5 denotes the results of our experiments. A detailed discussion of the results is provided in Section 6. Finally, Section 7 concludes this study.

2. Related Studies

2.1. Discovering Spatially Associated Object Methods

Traditional methods for discovering spatially associated objects focus on their geometric or semantic features’ similarity, such as the geometric shape of entities [15,20,21], the spatiotemporal features of entities [22], granularity or scale [3,23], and the semantic information of objects [24,25,26]. These widely applied methods face limitations in representing multidimensional features. The reliance on single-feature similarity fails to accurately capture intricate associations due to the multidimensionality of data, resulting in an incomplete representation of object relationships. To address this issue, scholars have employed object relationships as a metric to measure association. Using object semantic relationships extracted from properties or entities is intuitive, and such methods include studying the occurrence frequency of different objects in news or textual information [27], mutual information sharing between spatial objects [28], and the geographical context [29]. Compared to feature similarity, these approaches can map physical distribution and real association between objects in line with human cognition, However, these methods often consider only direct connections, i.e., objects’ own descriptive information or spatial distribution measures, and lead to a weak ability to mine underlying topological associations. Performing a comprehensive spatial analysis or employing other objects as media may be a potential approach for identifying topological association [30].

Integrating geospatial information into graph networks is beneficial for explicitly expressing the complex relationships and interactions between spatial objects, with useful approaches including community detection [31], human mobility [32], and similarity estimation [33]. The network model can better represent object relationships and discover associated objects by considering the relationship with network edges [16,17]. Therefore, integrating network models with comprehensive spatial analysis is a potential approach, with others including co-occurrence frequency [34,35] and assessing the clustering relationships of objects [19]. Many outstanding methods have been developed. Xu introduced a spatially aware graph relation network that automatically identifies key semantic and spatial relationships. This facilitates the association of objects within large-scale scenes [36]. Zhang and others proposed a geographic reference graph model for geospatial data matching, constructing a graph network using the centroids of geographic polygons as nodes, and introduced an algorithm for identifying many-to-many matching relationships [37]. The results show that this geographic reference graph model can effectively integrate context and spatial and formative distances in geospatial data matching, achieving efficient matching across different datasets. However, these graph models are generally complex, with close connections between nodes, lacking direct explanations of whether objects are associated. Furthermore, scholars have also explored a path as a definition of object association. A bibliographic network was implemented for performing object similarity searches in semantic networks in Ref. [38]. However, such methods cannot be directly applied to geospatial object scenarios, especially considering the complex spatial topological relationships involved. To explore their potential applications in the geospatial field, a graph network using Chinese cities as nodes was designed with the graph network and community detection method in Ref. [34]. Based on the path search algorithm, a geospatial entity ranking method was proposed and applied in Top-K recommendations. However, this method did not take into account the impact of geospatial factors on associated object discovery.

2.2. Involvement of Spatial Topology in Discovering Associated Objects

Topological theory is the foundation of spatial analysis [39,40]. Many studies have demonstrated its important role in many fields, making it a worthy research area. For example, extracting spatial topological features from geospatial data can portray data-rich information and enable its visualization [41,42]. By understanding topological relationships, compactness, and connections can be effectively identified and clustered [43,44]. Topology can improve the efficiency of spatial relationship indexing [45,46].

An increasing number of studies are integrating topology and graph networks owing to their capability to depict object topological relationships. On the one hand, integration is achieved through classifying, quantifying, and encoding topological relationships. By categorizing spatial topological relationships and combining association rule mining techniques to calculate the association of objects under different relationship categories, the richness and accuracy of association results have been effectively improved [18]. However, this method is affected by the rigid group of topological relationships, leading to a degree of association separation. By normalizing measuring spatial topological relationships, scholars have defined quantitative spatial associativity based on spatial topological relationships, which enhances the continuity and explainability of object associations [14]. To highlight the more subtle differences between objects, a topological spatial encoding method combining grid distance and direction association, which further refines the object association, was proposed in Ref. [13]. On the other hand, embedding spatial topology into other spatial analysis methods is also a useful approach. For instance, a novel spatial clustering method with topology was proposed to explore its impact on object association in view of object distance and distribution compactness [19].

In summary, existing research studies have integrated spatial topology and associated object discovery methods. However, spatial topology used to be employed as a direct measurement method for object association, which ignored the topologically deeper and quantitative impacts on object association. In particular, the association propagation attenuation measurement mechanism, along with spatial topology, was not manifested. Therefore, more attention should be paid to comprehensive research studies on the influence of spatial topology on association.

3. Study Area and Data

3.1. Study Area and Data Preprocessing

The Xicheng District of Beijing, China, was selected as the study area (Figure 2). It has a high population density and diverse industries [47], covering numerous and various points of interest (POIs) and areas of interest (AOIs) such as traffic facilities, tourism attractions, and buildings. The spatial patterns and spatial distributions of the POIs and AOIs provide a comprehensive understanding of urban planning and development.

The study data were collected from the Gaode Map published in September 2023. We developed a data collection module using Gaode API (https://restapi.amap.com accessed on 1 October 2023) and used the Xicheng boundary as a spatial filter. The module included 23 types of POI and AOI data. Each record contains many properties, such as name, geographic coordinates, and category information. To ensure accuracy and consistency, the original data were preprocessed with deduplication, data cleaning, and other operations, resulting in a spatial object dataset with 1430 records. The spatial distribution of the dataset is illustrated in Figure 2.

3.2. Benchmark Dataset

In this study, the benchmark dataset served as the ground truth for accuracy evaluation. The benchmark dataset was created based on real road distances. To avoid a large manual workload and individual subjective bias when evaluating the results, we used the real road walk navigation distance as the primary measure and manual verification as a supplementary measure. Employing the real road distance as a metric was not only in line with human spatial cognition but also facilitated establishing a unified measurement to enhance evaluation objectivity for big geospatial data. The benchmark dataset was compiled based on Gaode Map navigation Walking Path Planning API (https://restapi.amap.com/v5/direction/walking accessed on 1 October 2023), which we used to calculate the Gaode navigation distances between two objects. This dataset covered the entire Xicheng District of Beijing, with a total of 1,021,735 records.

4. Methodology

4.1. Design Principles and Considerations

The approach was proposed based on two theoretical hypotheses. Firstly, the network model based on spatial topology and spatial feature similarity allowed for a natural and clear representation of potential object association compared to a traditional two-dimensional relational table. Secondly, inspired by the first law of geography, the association was attenuated, along with the increase in the topological path and topological hierarchical level, when the number of connected nodes increased.

Thus, based on spatial topological theory and spatial feature similarity, an approach named the Weighted Similarity measure method considering Topological Accessibility (WSTA) was proposed to discover associated objects. The framework is shown in Figure 3. Firstly, spatial feature similarity networks and topology networks were created with objects as nodes and spatial feature similarity and topological relationships as edges, respectively. The spatial feature similarity and topological relationships were extracted from data with existing algorithms and methods. Secondly, WSTA was designed with a breadth-first search (BFS)-based accessible path mining module and a weighted association calculation module considering spatial topology. Here, the BFS-based accessible path mining module effectively depicted the spatial topological accessibility between objects; meanwhile, the weighted association calculation module accurately quantified the impact of topological accessibility on object association. Finally, the objects were obtained and sorted, with the Top-K-associated objects being extracted. Detailed information on each module will be introduced in the following subsections.

4.2. Construction of Network Model

Based on the aforementioned theoretical hypothesis, a spatial feature similarity network and a topology network were generated based on network construction principles [48]. The former aimed to portray the degree of spatial feature similarity between objects, while the latter was designed to portray object association attenuation as the topological connectivity length increased.

4.2.1. Constructing a Spatial Feature Similarity Network

The spatial feature similarity network is a network model with objects as nodes and spatial feature similarity as edges. In this study, we formalized the similarity network in the triplet form$G=(V,E,S)$, where $V\in R^N$ is the set of object nodes, N is the node number, and $S\in R^{N\ast N}$ is the similarity matrix in which each element denotes the object similarity of two spatial objects. The similarity was calculated based on spatial features such as the position, shape, scale, and boundaries of objects. $E\in R^{N\ast N}$ denotes the set of edges in the similarity network, which was determined via the following Equation (1) [49].

$$E_{ij}=\left\{\begin{array}{cc}1,& S_{ij}>0\\ 0,& S_{ij}=0\end{array}\right.$$

(1)

where $E_{ij}$ and $S_{ij}$ are specific elements in the edge set E and the similarity matrix S, respectively. The former manifests if the node $V_i$ is associated with node $V_j$, with a value of 1 indicating an association and 0 indicating no association between them. The latter is a similarity quantity matrix for each pair of nodes $V_i$ and $V_j$, whose values range from 0 to 1.

4.2.2. Construction of a Topology Network

The topology network is constructed using spatial objects as nodes and topological relationship types as edges, described as $G\_r=(V,E\_r)$. Here, $V$ and $E_r$ are sets of nodes and edges, respectively. The set of edges $E\_r=(T,C)$, where T is the set of spatial topological relationship types between the object nodes and C, represented the topological relational constraints for establishing network edges.

In our study, the spatial topological relationships in the network were defined by classical spatial topology theories. Considering the POI distribution and pattern of the study area, the boundary concept in topology was ignored; instead, only internal and external concepts were considered. To further simplify the modeling process and improve computational efficiency, we selected only four fundamental topological relationships—Disjoint, Intersects, Contains, and Equals—for constructing the topological network. These four topological relationships can clearly distinguish different levels of spatial association and are sufficient to capture the essential spatial associations among POIs and AOIs in our study area while avoiding unnecessary complexity introduced by more fine-grained or boundary-dependent relations. This design aligns with practical application needs, where hierarchical containment and proximity are most relevant, and reduces the risk of introducing noise from less frequent or context-specific topological interactions. Then, four types of topological relationships were proposed (Figure 4), which were Disjoint, Intersects, Contains, and Equals.

According to existing research [14,18], the ranking of association strength of the four topological relationships, from high to low, is $R_{equals},R_{contains},R_{intersects},R_{disjoint}$. Considering the real pattern and distribution of POIs, the “intersects” type was set as the topological relationship threshold; that is, modes with “intersects”, “contains”, and “equals” relationships were retained in the relational network. In particular, the thresholds were often determined according to practical applications instead of theoretical or empirical parameters. Taking the object set $\left\{V_1,V_2,V_3,V_4,V_5,V_6\right\}$ in Figure 5a as an example, the topological relational network with the “intersects” constraint condition was constructed, as shown in Figure 5b. In Figure 5a, the object $V_1$ is spatially disjoint from the objects $V_3$,$V_5$, and $V_6$; thus, no direct connection edge exists between object $V_1$ and these objects in Figure 5b. However, since the four objects are all contained by object $V_4$, spatial topological associations between $V_1$ and other nodes are shown in Figure 5b.

4.3. Weighted Similarity Measure Method Considering Topological Accessibility

Considering the impacts of potential connections on the association, an approach named the Weighted Similarity measure method considering Topological Accessibility (WSTA) was proposed. The method was based on the idea of higher similarity with closer topological accessibility. Moreover, the accessible topological path clearly represented underlying object associations and could precisely portray propagation attenuation along the topological path. Finally, the weighted similarity was obtained by summing the spatial feature similarity and maximum path similarity. This detail is depicted in Part 2 of Figure 3.

4.3.1. Maximum Topological Accessibility Path

The accessible path refers to a path from the starting node to the ending node in a relational network, which denotes the topological accessibility between nodes in the network model. The accessible path is expressed by an ordered list of non-repeating nodes $P_{ij,k}^h=\left\{V_i,V_{i+1},\dots ,V_j\right\}$. Here, $h$ is the association hop, indicating the number of edges passed by the accessible path, and $k$ is the unique identifier for distinguishing different accessible paths. Thus, the topological accessibility path can be expressed as a product of the multiple $P_{ij}$. The mining of accessible paths involves using the BFS approach [50], which systematically explores and expands layers to obtain all accessible paths from the starting node to other nodes. For example, the task could be to mine the accessible path from the starting node $V_1$ to the ending node $V_3$, as shown in Figure 5b. The procedure and results of accessible paths are shown in Figure 6.

In this study, we employed accessible paths in the relational network to characterize the potential topological connections between objects and further measure the impact on association from topological connections. Some studies have indicated that the similarity or association of nodes would be attenuated by an increase in the topologically connected path length [34,51]. Based on this, this study introduced the concept of path similarity to represent the decay of association and quantified it using the continuous product of spatial similarity between nodes on accessible paths. The specific calculation formula can be found in Equations (2) and (3):

$$\beta \left(P_{ij,k}^h\right)={\prod }_{x=1}^h{S}_{x,x+1},h\ge 2$$

(2)

$${\beta }_{ij}=\max \left(\beta \left(P_{ij,0}^h\right),\beta \left(P_{ij,1}^h\right),\dots ,\beta \left(P_{ij,k}^h\right)\right)$$

(3)

where $\beta \left(P_{ij,k}^h\right)$ represents the path similarity of the accessible path $P_{ij,k}^h$, with values ranging from 0 to 1, and $S_{x,x+1}$ represents the similarity between the x-th node $V_x$ and the next node $V_{x+1}$ in the accessible path $P_{ij}^h$.

There are multiple accessible paths between nodes; thus, the maximum similarity value is chosen to define the maximum accessible path. The maximum path similarity with the symbol ${\beta }_{ij}$ represents the topological relationship similarity between the first and last nodes and quantifies the impact of spatial topological accessibility on object association. The calculation formula for ${\beta }_{ij}$ is presented in Equation (3), where $\mathrm{0,1},\dots ,k$ serve as unique identifiers for accessible paths.

4.3.2. Weighted Association Calculation

A weighting mechanism was introduced to combine the impacts of the spatial feature similarity and topological relationship similarity. The weighted association calculation formula is shown in Equation (4):

$$A_{ij}=W_1\times S_{ij}+W_2\times {\beta }_{ij}$$

(4)

where $A_{ij}$,$S_{ij}$ and ${\beta }_{ij}$ denote the association, spatial feature similarity, and topological relationship similarity between nodes $V_i$ and $V_j$, respectively, while $W_1$ denotes the weight for spatial feature similarity and $W_2$ denotes that for topological relationship similarity. The sum of the two weights is 1.0. Considering their relatively independent influence, each weight is set to 0.5. Figure 7 illustrates the concept of weighted association for $A_{13}$.

To obtain multiple spatial associated objects, this approach was designed to employ the Top-K algorithm [52] to retrieve more potentially associated objects. Existing studies usually use the character “K” to indicate the resulting number determined by the application requirement. This is called the Top-K-associated result.

5. Experiments and Results

5.1. Experimental Parameter Configuration

5.1.1. Network Construction Parameters

The key parameters for network construction include topological relationship constraints and accessible path association hop. The former indicates the topological relationships between types of nodes; meanwhile, the latter denotes the topological connection path length. A longer length implies greater information attenuation and weaker object association. These two parameters collaboratively determine the degree of association between objects.

Topological relationship constraints are often configured according to data. Figure 8 visualizes a topology network with all objects in the case study area and extracted subnetworks with colorful topological relationships. Orange indicates the Contains relationship. The subnetworks in Figure 8b,c show that the majority of edges in the subnetwork are for the Contains relationship. Therefore, $R_{contains}$ is set as a topological constraint parameter for constructing topological relational networks in this study.

The accessible path association hop is a key parameter for searching for the maximum topological accessibility path. The bigger the hop parameter, the greater the attenuation of node similarity along the path and the smaller the probability of becoming the maximum topological accessibility path. The number of accessible paths grows exponentially as the number of hops increases. This leads to a rapid decrease in the ratio of effective accessible paths to all accessible paths. As shown in Figure 9, the ratio value sharply changes from two to three and gradually stabilizes at around four. Therefore, we preset the maximum value of the hop parameter $h_{max}$ to four.

5.1.2. Baseline Methods

To evaluate the performance of WSTA, four spatially associated object discovery methods were selected as baseline methods. The detailed information is shown below:

• Baseline 1 [18]. This method is based on spatial topological relationship, and we fully considered its influence on the object spatial association measure. In our study, the association factor parameter in this method is set to one.
• Baseline 2 [14]. The normalized topological relationships and metrics were employed to express the degree of object spatial association. Its advantage is its multiple-scale capability.
• Baseline 3 [13]. Adaptive topological relationships and metric thresholds according to object types are directly used to measure object spatial association. Considering the spatial scale in our study, 500 m was selected as the spatial filter unit parameter.
• Baseline 4 [19]. The principle of this method is spatial clustering. The associated objects were retrieved from clusters with specific conditions. The advantages of this method are its efficiency and accuracy. Based on the literature results, the DBSCAN spatial clustering algorithm was used in our study.

In our study, baseline methods were employed to obtain spatial feature similarity, then integrated into WATRA to discover associated objects. To ensure a concise experiment, the integrated approaches with baseline methods were named Baseline 1+, Baseline 2+, Baseline 3+, and Baseline 4+, respectively.

5.1.3. Evaluation Methods and Indicators

This study evaluated both the Top-K result using our approach and the corresponding Top-K objects in the benchmark dataset by comparison. The results from the benchmark dataset were obtained manually according to the Gaode navigation walk distance. The navigation distance was selected to improve evaluation objectivity. An example of the accuracy evaluation workflow with object $V_i$ is presented in Figure 10.

In this study, the evaluation was focused on the accuracy of “what” and “where”. The former denotes whether the correct associated objects were discovered. Meanwhile, the latter refers to the order of associated objects in Top-K results.

Inspired by an existing study [53], precision and Spearman’s rank correlation coefficient (Spearman’s ρ) were used. Precision was used to evaluate the proportion of correct results among Top-K association results using Equation (5) [54]. Spearman’s rank correlation coefficient is a metric for evaluating the ordering consistency of Top-K-associated object results, which were obtained in this study by measuring the ordering change between the Top-K-associated object results and the benchmark results using Equation (6) [55]:

$$Precisio{n}_k={\displaystyle \frac{T{P}_k}{T{P}_k+F{P}_k}}$$

(5)

$${\rho }_k=1-{\displaystyle \frac{6\sum d_i^2}{k\left(k^2-1\right)}},1\le i\le k$$

(6)

where $k$ is the number of association results, while $Precisio{n}_k$, ranging from 0 to 1, is the precision of the Top-K association results. A higher value indicates more accurate association results. ${\rho }_k$ represents Spearman’s rank correlation coefficient, with a range from −1 to 1. When ${\rho }_k$ is greater than 0, it indicates a positive correlation in the sequence of association results; otherwise, a negative value indicates a negative correlation. The higher ${\rho }_k$ is, the closer the sequence of results is to the benchmark. $T{P}_k$ represents the number of matched objects between the result and benchmark, while $F{P}_k$ represents the number of mismatching ones. The $d_i$ denotes the rank difference of corresponding elements between result and benchmark sequences.

5.2. The Top-K Results

5.2.1. Case Study for Associated Object Discovery

A POI object named Beijing Books Building was selected as an example to discover its associated objects in the case study area. Thus, it was input with objects of four baseline methods and four improved approaches; then, the Top-10-associated objects were discovered.

Table 1. The Top-10-associated objects discovered with methods for the object “Beijing Book Building”.

Method	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6	No. 7	No. 8	No. 9	No. 10
Benchmark Results	A	B	C	D	E	F	G	H	I	J
Baseline 1	A	E	C	D	G	B	H	F	I	L
Baseline 2	A	C	D	G	B	H	F	I	E	J
Baseline 3	A	C	D	G	B	H	F	E	I	J
Baseline 4	A	C	D	G	B	H	F	I	J	K
Baseline 1+	A	E	C	D	G	H	F	L	I	J
Baseline 2+	A	D	C	B	E	G	I	F	H	J
Baseline 3+	A	D	C	B	E	G	F	I	H	J
Baseline 4+	A	D	C	B	G	F	I	H	J	K

Note: A is Xinhua Bookstore (Beijing Books Building), B is HanGuang Department Store, C is Teld Charging Station (THE New Xidan Geng Xinchang), D is Teld Charging Station (Teld THE New Xidan Geng Xinchang), E is Teld Charging Station, F is 2nd Cross Alley, G is Hushang Ayi (HanGuang Department Store), H is Dior (HanGuang Department Store), I is CHongshang GAVIN STYLE (HanGuang), J is Qingheyuan Vegetable Dish Restaurant (Gaodeng Mansion Shop), K is World Fashion Xiaofu Sour Vermicelli (Xidan Mingzhu Shopping Center Shop), and L is Xiaodu Zaijia Unicom Experience Branch.

shows the benchmark and result objects. No. 1 to No. 10 denotes the ordering in sequence. Object A to object L are the 12 discovered objects. The benchmark objects listed in the first line were manually determined from the benchmark dataset according to the Gaode navigation walk distances, which are listed in

Table 2. The navigation walking distance between the case object “Beijing Books Building” and the associated objects’ results.

Object ID	Name of Associated Objects	Walking Route in Gaode APP	Distance (Meters)
A	Xinhua Bookstore (Beijing Books Building)	Head southwest 15 m to reach the destination.	15
B	HanGuang Department Store	Head 57 m and turn right, walk 72 m and turn right, walk 64 m north along the 2nd cross alley and turn left, walk 10 m and turn left, walk 82 m to reach the destination	285
C	Teld Charging Station (THE New Xidan Geng Xinchang)	Head 57 m and turn left, walk 42 m and turn right, walk 21 m north and turn left, walk 63 m west and turn right, walk 128 m to reach the destination	311
D	Teld Charging Station (Teld THE New Xidan Geng Xinchang)	Head 57 m and turnm left, walk 42 m and turn right, walk 21 m north and turn left, walk 63 m west and turn right, walk 132 m to reach the destination	315
E	Teld Charging Station (Teld THE New XidanGeng Xinchang)	Head 57 m and turn left, walk 42 m and turn right, walk 21 m north and turn left, walk 63 m west and turn right, walk 138 m to reach the destination	321
F	2nd Cross Alley	Head 57 m and turn right, walk 72 m and turn right, walk 206 m north along the 2nd cross alley to reach the destination	335
G	Hushang Ayi (HanGuang Department Store)	Head 57 m and turn right, walk 72 m and turn right, walk 156 m north along the 2nd cross alley, turn left, walk 64 m west along Xiaoshihu Hutong to reach the destination	349
H	Dior (HanGuang Department Store)	Head 57 m and turn right, walk 72 m and turn right, walk 64 m north along the 2nd cross alley, turn left, walk 10 m and turn left, walk 123 m and turn right, walk 35 m to reach the destination	361
I	CHongshang GAVIN STYLE (HanGuang)	Head 57 m and turn right, walk 72 m and turn right, walk 64 m north along the 2nd cross alley, turn left, walk 10 m and turn left, walk 123 m and turn right, walk 61 m to reach the destination	387
J	Qingheyuan Vegetable Dish Restaurant (Gaodeng Mansion Shop)	Head 57 m and turn right, walk 72 m and turn right, walk 164 m north along the 2nd cross alley, walk to the left, walk 109 m to reach the destination	402
K	World Fashion Xiaofu Sour Vermicelli (Xidan Mingzhu Shopping Center Shop)	Head 57 m and turn right, walk 72 m and turn right, walk 278 m north along the 2nd cross alley, turn left, walk 56 m to reach the destination	463
L	Xiaodu Zaijia Unicom Experience Branch	Head 57 m and turn left, walk 42 m and walk to the right, walk 30 m southwest along the 2nd cross alley, walk to the right, walk 143 m west along West Chang’an Avenue, walk straight, walk 13 m west along Fuxingmen inner Street Building, walk to the right, walk 34 m, walk 165 m to reach the destination	484

. To provide a more visualizable appearance, Figure 11 shows the 12 objects and their distances.

Table 1 shows that both the baseline methods and the improved methods can accurately identify Top-10-associated objects. However, the sort result of associated objects is weak. The improved methods are slightly better than the original baseline methods for sequence ordering, showing higher ordering accuracy. In particular, Baseline 4+ is outstanding compared with Baseline 4, depicting the right ordering objects from No. 1 to No. 4.

In view of the benchmark results, the sorted result is also compared in the columns in Table 1. In the first column, object A is always accurately discovered and listed in the right place. For objects in other columns, the results are not satisfactory.

5.2.2. Accuracy Comparison Results of Methods

The comparison of method accuracy is shown in

Table 3. Average accuracy of Top-10 results identified by associated object discovery methods.

Top-K	Top-1	Top-2	Top-3	Top-4	Top-5	Top-6	Top-7	Top-8	Top-9	Top-10
Method	Precision	Precision	Precision	Precision	Precision	Precision	Precision	Precision	Precision	Precision
	ρ	ρ	ρ	ρ	ρ	ρ	ρ	ρ	ρ	ρ
Baseline 1	0.2727	0.4242	0.5556	0.5379	0.5576	0.5808	0.6147	0.6477	0.6801	0.7152
	0.2727	0.2727	0.1515	0.2364	0.2	0.1965	0.145	0.1003	0.0768	0.0454
Baseline 2	0.5758	0.5606	0.6162	0.553	0.5818	0.6061	0.6494	0.6591	0.6835	0.7242
	0.5758	0.5758	0.303	0.2909	0.2636	0.2571	0.1948	0.158	0.1505	0.0935
Baseline 3	0.5758	0.5758	0.6263	0.5606	0.5879	0.601	0.6494	0.6553	0.67	0.6909
	0.5758	0.4545	0.1818	0.2303	0.2364	0.2797	0.2013	0.1674	0.149	0.0938
Baseline 4	0.5758	0.5455	0.5556	0.5076	0.5333	0.5556	0.5758	0.5833	0.588	0.5977
	0.5758	0.3333	0.2121	0.2121	0.1727	0.1273	0.079	0.0144	0.0274	0.0003
Baseline 1+	0.2727	0.2879	0.2828	0.3106	0.3515	0.4141	0.4372	0.4811	0.5118	0.5485
	0.2727	0.0303	0.197	0.1879	0.1333	0.103	0.0703	0.0382	0.0364	−0.0171
Baseline 2+	0.5758	0.6061	0.6061	0.5985	0.6182	0.6313	0.6623	0.6705	0.697	0.7152
	0.5758	0.697	0.4394	0.3152	0.2848	0.3004	0.2511	0.1854	0.2056	0.1368
Baseline 3+	0.5758	0.6061	0.6465	0.5833	0.6	0.6061	0.6537	0.6629	0.6801	0.6818
	0.5758	0.5152	0.2121	0.2727	0.2152	0.2346	0.1591	0.1277	0.1429	0.1074
Baseline 4+	0.5758	0.5758	0.5758	0.5227	0.5455	0.5606	0.5801	0.5871	0.5913	0.5977
	0.5758	0.3939	0.2727	0.2667	0.197	0.1325	0.0563	−0.0087	0.0052	0.0058

Note: Underlining grid indicates the best precision of the baseline methods; bolding indicates the best precision of original and improved meth.

, along with the metric precision and Spearman’s ρ. Following existing studies on the sample selection principle [56,57], this experiment randomly selected 50 input objects and calculated the average accuracy to mitigate the random error influence of individual objects. K was set from 1 to 10 to validate how many results were suitable for this method; then, the Top-K results’ average accuracy was obtained, as shown in Table 3. Figure 12 plots the accuracy change trends in line charts. The same color denotes the original and improved baseline method, and the dot symbol represents the original baseline method, while the triangle symbol indicates the improved baseline method.

Figure 12a illustrates the precision trends for each method. The improved Baseline 1+ method is worse than the original method, whereas the other three methods show significant improvements compared to the original ones. In particular, Baseline 2+ consistently has higher precision from K = 4 to 10.

Figure 12b shows the ranking trend of the methods in Spearman’s ρ metric. The general trend is continuously decreasing, which indicates that the ranking of the associated objects is changing frequently compared with the benchmark result. That is, the accuracy of ranking decreases with the increase in the K value. The maximum values of correlation coefficients belong to improved methods, which indicates the effectiveness of the improvements. The detailed accuracy values can be seen in Table 3.

In view of the method comparison, Figure 12a shows that the precision gradually improves with the increase in K, which means that a bigger K value has higher precision only when the associated object result without ranking is considered. This is in line with common sense. However, a small number of results with higher accuracy for associated objects are significant in some specific applications, such as emergency scenarios. In Figure 12b, it is noticeable that when K = 3, the attenuation rate of correlation coefficient accuracy slows down.

6. Discussion

6.1. Suitability Analysis

According to theory, the suitability of WSTA is determined by the research scenario. The aim of this approach is to find one or more topological accessibility paths. Therefore, dense networks with more nodes or edges provide more accessible paths to discover associated objects. However, in a scenario with sparse relationship networks, due to numerous null nodes or independently connected edges, WSTA would encounter challenges in establishing topologically accessible paths and discovering their associations, leading to a decrease in its discovery accuracy and efficiency.

Moreover, in terms of technique, topological constraints and the hop are key parameters of WSTA. Tailoring parameters for different scenarios enables a more comprehensive interpretation of spatially associated object discovery, thereby refining the precision and insight of spatial analysis. For example, the intersection constraints are favored as topological parameters as they provide a suitable framework for expressing the intricate relationships among these location entities. However, for spatial distribution or pattern application, the spatial topology Contains relationship likely serves as a topological parameter for expressing the spatial structure and patterns inherent among these entities.

For the association hop, a larger value indicates a longer accessible path and decreased similarity. Previous studies have also demonstrated that longer paths can lead to reduced association [38,58]. Consequently, the approach outlined in this study focuses on scenarios characterized by compact, short-connected networks, and its applicability to more complex relational networks needs further exploration and analysis.

To further discuss the impacts of the two key parameters (topological constraints and the number of hops) on the accuracy of the method, comparative experiments were conducted on the best-performing Baseline 2+ method. Nine different combinations of these two parameters were selected as experimental groups, whose average accuracy results are shown in Figure 13.

As depicted in Figure 13, models constructed with the topological Contains constraints and a hop of two generally exhibit better accuracy in associated object discovery. The experiment was conducted with urban AOI and POI objects in the study area. This result further underscores the significance of topological accessibility paths for the applicability of WSTA. Specifically, the use of topological Contains constraints can achieve higher accuracy because the Contains relationship can better express the hierarchical relationship between spatial entities. In contrast, topological Intersection constraints introduce noise into the analysis due to the excessive number of constructed paths, thereby affecting the overall accuracy and effectiveness of the analysis. In the comparison of the orange, purple, and gray parameter settings depicted in Figure 13, the orange scenario (with a parameter value of two) demonstrates superior accuracy and a higher Spearman ρ. This suggests that the average precision of association peaks at a hop of two. As the number of hops increases, the similarity between entities decays gradually through connection transfer, resulting in reduced accuracy.

6.2. Performance Analysis Integrating WSTA

The spatial topological relationship provided a means to better depict the true distribution of spatial objects and enhance the accuracy of associated object discovery compared to using Euclidean distance. For example, two POI points may be separated by a block, the straight-line distance between these points fails to adequately capture the spatial relationship, and utilizing spatial topological relationships can offer a more comprehensive understanding of the spatial configuration.

Although integrating topological relationships can enhance associated object discovery accuracy, the measurement of impact degree needs further exploration. Therefore, we conducted an experiment to evaluate the performance of models integrated with topological relationship accessibility. Figure 14 depicts the average accuracy difference of each improved method in comparison to its original method across various Top-K cases for Baseline 2+, Baseline 3+, and Baseline 4+ (Since Baseline 1+ had modest observed enhancement, further discussion will be undertaken in subsequent sections). Here, each color represents a method, so the vertical axis represents the different values of K taken for Top-K and gradually increasing from K = 1 to K = 10, and the horizontal axis represents the stacked values of the improvement level of multiple association methods. The result is the same between the original and improved methods when K = 1. In ranking accuracy, the Baseline 2+ has an accuracy improvement of up to 12% and 14% at Top-2 and Top-3, respectively, as shown in Figure 14b. The improved methods exhibit higher precision and Spearman’s ρ values, particularly for the Top-1 to Top-4 results. This indicates that the improved methods are particularly well suited for smaller K values, which are of greater concern in practical applications. It is in line with existing work that users have higher expectations for the first top associated objects [59].

Given the specificity of object topological relations and the algorithm principle of baseline methods, the performance of an integrated WSTA approach has variability in the microscale. Taking the experimental results in Section 5.2.1 as an example, it is evident that the actual distance between the case object “Beijing Book Building” and object A is just 15 m (see Table 2). Consequently, the spatial similarity between them is greater than that of other objects, leading to accurate identification across all original and improved methods. However, for the remaining objects (B–J), abnormalities in association results are observed across all methods: (1) Baseline 1 exhibits rigid differentiation in spatial topological relationships, resulting in a higher degree of similarity between case objects and object L. However, in Baseline 1+, the weighting exacerbates the consequences of misidentifying object L, consequently resulting in object L having a forward ranking. This indicates the diminishing of the average accuracy in Baseline 1+. (2) Baseline 4 employs density clustering to group-related objects. However, this method may lead to objects that are originally spatially neighboring being classified into different clusters, leading to the misidentification of objects in both Baseline 4 and Baseline 4+. (3) Despite the associated objects not being outside of the Top-10 benchmark results, Baselines 2 and 3 still exhibit errors in the ranking of Top-10 objects. Moreover, the algorithms of Baseline 2 and Baseline 3 guarantee a relatively continuous spatial similarity, allowing them to be efficiently used to compute topologically accessible path similarity and to further improve the performance of the association method by introducing the WSTA approach.

7. Conclusions

Discovering spatially associated objects plays a crucial role in geographic information science and spatial analysis applications. Compared to traditional similarity measurement methods, network analysis methods are effective for identifying complex topological relationships. However, existing network analysis methods ignore spatial topology relationships among objects, leading to similarity attenuation along the topological accessibility path. To address this issue, we proposed an approach named the Weighted Similarity measure method considering Topological Accessibility (WSTA) to determine the impacts of topological relationships on object association, which is integrated and strengthened as a novel approach for discovering associated objects. This approach is based on spatial similarity and explores topological accessibility paths in the network model by considering the spatial topological relationship. Considering the distribution patterns of POI data, only internal and external relationships were retained. From these, four core topological relationships (Disjoint, Intersects, Contains, and Equals) were extracted to construct the topological network. These four relationships are sufficient to capture the key spatial associations between point objects within the study area while avoiding the complexity introduced by finer-grained or boundary-dependent topological relationships. Our study was validated by analyzing Beijing city POI data via four existing baseline methods. The results indicate that this approach effectively improves the accuracy of associated object discovery, achieving an average precision improvement of 12% and a Spearman’s ρ increase of up to 14% over baseline methods, particularly in Top-3 results; experiments in the Discussion section further confirm the effectiveness of integrating topological accessibility, showing consistent improvements across all Top-K settings.

However, this study still has limitations in computational efficiency for complex networks. While the offline pre-generation of network models facilitates consumption, the computation requirement for calculating topological accessibility remains substantial. By excluding boundary-based or line-region relationships, the model may lose the ability to fully represent complex spatial interactions. In particular, dynamic computation is essential when the number of network nodes fluctuates, presenting additional challenges to computational efficiency. Therefore, we will explore efficient computation methods for spatial topological accessibility in future studies. Potential areas of study may include but are not limited to community algorithms, spatial indexing, and other high-performance computation algorithms, such as deep graph learning techniques.