BGM: Demand Prediction for Expanding Bike-Sharing Systems with Dynamic Graph Modeling

1 Introduction

Bike-sharing systems have gained popularity worldwide, offering short-term bike rentals through networks of stations distributed across urban areas. These systems provide an eco-

gestion, and promote sustainable urban mobility [Macioszek et al., 2020]. To amplify these benefits, providers have rapidly expanded their bike-sharing networks, as seen in Figure 1(a) and Figure 1(b), which illustrate the network expansion in New York City from January 2018 to January 2024 [Mahajan and Argota S´anchez-Vaquerizo, 2024]. While network expansion amplifies the environmental and societal benefits of bike-sharing systems, it also introduces critical societal challenges. One pressing issue is ensuring equitable access to mobility across socioeconomically diverse neighborhoods, as underserved areas often face limited access to transportation infrastructure. Another challenge lies in resource allocation-determining where and when to deploy bikes and docking stations to balance supply and demand effectively. Addressing these challenges requires accurate demand prediction, particularly for new stations, to optimize operations and support the sustainable and inclusive expansion of bikesharing systems, particularly in the absence of historical data for new stations. Moreover, new stations will also change usage patterns across the network, creating complex spatiotemporal dependencies. Therefore, it is crucial to model the dynamic relationships between new and existing stations to ensure equitable access and efficient resource allocation.

∗Corresponding author.

Various methods have been developed to address these challenges. Traditional regression-based models leverage historical data to identify demand patterns, providing a foundational understanding of user behavior [Chen et al., 2015; Liu et al., 2015]. Functional zone-based approaches improve this by incorporating diverse urban characteristics, such as population density and land use, allowing for localized and precise analyses of demand [Liu et al., 2017]. Machine learning models, such as random forests and support vector machines, further enhance prediction accuracy by integrating multiple external factors, including temporal features and points of interest [Kou and Cai, 2021]. Recently, advanced graph neural networks have been employed to capture complex spatial dependencies and mitigate data scarcity, enabling more robust and reliable predictions for new station deployment [Chen et al., 2020]. However, these methods assume the relationships between new and existing stations are static, failing to capture how demand shifts with expansion. This can lead to inefficient station deployment and inequitable service distribution, particularly in underserved areas. Addressing these challenges requires advanced models that adapt to evolving demand patterns, supporting more inclusive and data-driven urban mobility planning.

To tackle these challenges, we design a novel and unified Demand Prediction Framework in Bike-Sharing Systems Expansion with Dynamic Graph Modeling (BGM), which dynamically adapts to the evolving network structure and diverse contextual factors. Specifically, BGM leverages a dynamic graph architecture that continuously updates its structure and node attributes to reflect the evolving relationships between stations, such as spatial proximity, functional similarity, and temporal demand patterns. This dynamic modeling enables the framework to capture the shifting demand patterns. Building upon this, it incorporates two key modules: i) a knowledge transfer approach that studies the transformation of embeddings across existing and new stations through a learnable orthogonal mapping matrix, ensuring the adaptation and alignment of spatio-temporal patterns to dynamic conditions; and ii) a gated selecting vector-based feature fusion mechanism that selectively combines the transferred embeddings with the intrinsic features of stations, generating enriched and context-aware embeddings for precise predictions. The contributions of our work are summarized as follows:

• We present a unified dynamic graph framework that updates its structure and node attributes to capture the evolving spatial and temporal relationships in bikesharing networks.
• We propose a novel knowledge transfer approach that studies the embeddings transformation across existing and new stations by a learnable orthogonal mapping matrix to address data sparsity in newly added stations.
• We design a gated selecting vector-based feature fusion mechanism to seamlessly integrate transferred embeddings from existing stations with the intrinsic features of stations. This mechanism minimizes negative transfer while dynamically balancing contributions from both sources to ensure precise predictions.
• Extensive experiments demonstrate that our model outperforms other state-of-the-art methods on real-world public datasets (NYC's Citi-Bike), showcasing superior prediction accuracy and robustness for new station demand predictions.

2 Preliminaries

Expanding Bike-Sharing System. An Expanding bikesharing system consists of a set of stations V , including existing stations VA with historical data and new stations VB without historical data. A station v ∈ V is associated with a feature vector xv encoding its spatio-temporal and contextual attributes (location, Points of Interest, rental records) and external factors.

Bike Demand. Bike demand refers to the number of rentals at a station within a given time interval and is influenced by spatial and temporal factors. To model these variations, we represent bike-sharing systems as a dynamic graph, stations are represented as nodes vi ∈ V and time steps are t. The demand at station vi at time t is denoted as dti . The graph Gt = (Vt,Et,Wt) evolves, where edges (i, j) ∈ Et capture demand dependencies with weights wt ij reflecting spatial, proximity, and temporal correlations.

Problem Formulation. Given an expanding bike-sharing system with existing stations VA that have historical data and newly deployed stations VB, the problem of demand prediction is to accurately estimate the hourly demand for all stations in the bike-sharing network.

3 Methodology

To predict the bike demand in expanding networks, we propose a framework that combines spatio-temporal feature encoding, dynamic graph construction, embedding transformation, and feature fusion. Illustrated in Figure 2. The following subsections provide detailed explanations of each component.

3.1 Spatio-Temporal Feature Encoding

To effectively represent the spatio-temporal characteristics of bike-sharing systems, we encode spatio-temporal features of stations as the inputs for the graph-based learning framework. Spatial features Xt,spatial i include Points of Interest (POI) distributions, distance metrics, and road network connectivity, as depicted in the spatial feature extraction module in Figure 2. Edges between nodes capture interdependent relationships such as spatial proximity or functional similarity and are represented by an adjacency matrix At , which varies temporally over time to reflect dynamic changes. POIs within a fixed radius (e.g., 500 meters) are categorized and normalized into a weighted vector representing the station's functional environment. Distance metrics, modeled using a Gaussian decay function, emphasize closer station interactions, while road network connectivity is captured as binary indicators for links to key infrastructure.

Temporal features Xt,temporal i include normalized historical demand, weather attributes, and periodic time encodings. Hourly demand data is normalized across stations to ensure consistency, while weather features, such as temperature and precipitation, are processed similarly. The final feature vector for each station at time t is defined as:

where Taxiti denotes external mobility data, such as taxi trip records, associated with the station at time t.

This encoding process represents each station as a node embedding that encapsulates its spatial context, temporal variations, and external mobility influences, providing a robust foundation for downstream graph-based learning tasks.

3.2 Dynamic Graph Construction

Dynamic graph modeling captures evolving relationships between bike-sharing stations by dynamically updating nodes and edges over time, as represented in the dynamic graph construction layer of Figure 2. Unlike static graphs, dynamic graphs adapt to real-time changes in demand patterns and external factors, ensuring relationships remain relevant. Nodes and edges, as introduced in Section 2, form the foundation of this model. Each node represents a station, with features hti derived from Xt i as defined in Equation (1). Each edge weight atij is computed as:

where Sij is the cosine similarity of POI vectors, Dij is spatial proximity based on a Gaussian decay function, and T t ij is temporal demand similarity. Parameters α, β, and γ balance these factors. Here, i and j denote the indices of stations, and atij captures the relationship between station i and j.

At each time step, At is updated to reflect changes in demand or external factors. For example, connections between residential and business district stations strengthen during peak hours due to increased commuter trips. External factors, such as weather, further influence edge weights, dynamically adjusting relationships in real time.

New stations are integrated into the graph by initializing spatial features based on POI distributions, as shown in Figure 2. Edges for new stations are created based on functional similarity and spatial proximity:

where atinew represents the connection strength between an existing station i and a new station. Node embeddings hti are updated using a graph convolutional network (GCN):

where N(i) is the set of neighbors (including i itself), atij is the edge weight, W is a learnable weight matrix, and σ is a non-linear activation function such as ReLU. This ensures that node embeddings dynamically evolve to reflect both static and dynamic relationships.

3.3 Embedding Transformation

As illustrated in Figure 2, the knowledge transfer process leverages a learnable orthogonal mapping matrix to transform embeddings from existing stations (VA) to new stations (VB), facilitating the transfer of spatial and temporal knowledge across the stations. The process begins with identifying similarities between new and existing stations. For each time step t, the similarity between a new station i ∈ VB and an existing station j ∈ VA is measured using cosine similarity:

where hspatial i and hspatial j are spatial embeddings derived from POI distributions, road network connectivity, and taxi trip data. Cosine similarity is employed to capture proportional relationships in embedding spaces and adapt to dynamic temporal changes effectively.

Based on St ij , the top 3 existing stations with the highest similarity are selected for each time step t. Their embeddings are aggregated as follows:

To transfer and align the aggregated embeddings to the new stations, a learnable orthogonal mapping matrix X is applied:

where X is optimized under the orthogonality constraint X⊤X = I. This ensures the transferred embeddings preserve structural integrity and contextual info from existing stations while adapting to the unique characteristics of new stations.

The knowledge transfer approach leverages transformed embedding to capture shared patterns of new stations in the context of the existing network. By dynamically identifying and aligning spatio-temporal demand similarities, this transformation mechanism effectively mitigates the challenges posed by data sparsity and evolving station relationships.

3.4 Feature Fusion for Expanded Stations

As illustrated in Figure 2, the feature fusion mechanism integrates transferred embeddings, existing features, and temporal dependencies to generate representations for new stations. This process dynamically balances contributions from different feature sources, ensuring the final representation adapts to both spatial and temporal heterogeneity.

The fused feature f t i dynamically balances transferred embeddings (uiB (t)) and original features (hiB ) through a gating mechanism. It is computed as:

where gt i ∈ Rd is the gating vector at time t, ⊙ denotes element-wise multiplication, and hiB includes spatial features such as POIs, road networks, and taxi trip records.

The gating vector gt i adjusts the balance between transferred embeddings and original features based on their joint features. It is computed as:

where [hiB ; uiB (t)] represents the concatenation of the two feature vectors, Wg is a learnable weight matrix, bg is a bias term, and σ is the sigmoid activation function.

A temporal attention mechanism dynamically assigns varying weights to features from different time steps to effectively capture temporal dependencies:

where qt and kt are the query and key vectors derived from temporal embeddings, and T denotes the set of time steps.

The final fused feature integrates spatial, temporal, and dynamic relationships, ensuring robustness in predicting demand at stations. The fused feature f t i serves as the input to the demand prediction module, combining transferred knowledge (uiB (t)) and station-specific features (hiB ). This representation effectively addresses data sparsity while preserving station-specific characteristics.

3.5 Loss Function

The total loss function consists of spatial alignment loss Ls, temporal alignment loss Lt, and prediction loss Lp.

The spatial alignment loss Ls aligns embeddings with similar demand patterns using a contrastive approach:

where zt i is the node embedding of station i, hk is a prototype vector, and τ is a temperature parameter.

The temporal alignment loss Lt ensures consistency across consecutive time steps:

where σ is the sigmoid function.

The prediction loss Lp minimizes the mean squared error (MSE) between predicted and actual demand:

where yt i is the actual demand and ˆ yt i is the predicted demand.

4 Experiments

In this section, we present the implementation details and dataset descriptions, followed by the evaluation metrics and baseline methods used for comparison. We then conduct extensive experiments to assess the performance of BGM, including comparisons with baseline models, ablation studies, and robustness analysis under different expansion scenarios.

4.1 Implementation Details

Training. The training process involves three steps: (i) computing embeddings using the dynamic graph, (ii) refining embeddings with spatial and temporal alignment losses, and (iii) predicting demand using a multi-layer perceptron (MLP). The spatial alignment loss (Ls) encourages stations with similar demand to share representations, while the temporal alignment loss (Lt) ensures consistency across consecutive time steps. The refined embeddings are then fed into an MLP for demand prediction at t + 1, represented as ˆxt+1,n = MLP(hn), where hn is the station embedding. The overall training objective is given by Ljoint = Lp + Ls + Lt. The model is trained using the Adam optimizer, with a batch size of 64, the learning rate of 1 × 10-3 , and 200 epochs with early stopping, running on an NVIDIA RTX 4090 GPU. The model implementation is available at: https://github.com/ YixuanColt/BSS-BGM.git.

4.2 Data Description

To evaluate our model, we collected five datasets 1 , integrating bike-sharing operational data with meteorology, POI, road network, and NYC taxi trip records. While these sources collectively provide essential input features for model training, our performance evaluation primarily focuses on bikesharing operational data from NYC Citi-Bike and Chicago Divvy-Bike, as presented in Table 2.

Bike-Sharing Data. The dataset includes operational records from the NYC Citi-Bike sharing systems over one month (January 1 to 31, 2018). The NYC Citi-Bike data consists of two primary components: i) trip records for rentals and returns at each station, and ii) station-level details, e.g., spatial and temporal information.

Meteorology Data. The meteorological data was collected from the NYC Mesowest database, providing historical daily weather records. Each record describes the weather conditions, categorized into four main types: sunny, overcast, rainy/snowy, and extreme weather conditions.

1 https://opendata.cityofnewyork.us/

POI & Road Network Data. The POI data were collected from online mapping service providers in NYC (Google Place API). Table 1 shows the details. Besides, road network data was collected from NYC Open Data for further analysis.

NYC-Taxi. The dataset contains detailed trip records of yellow taxis in New York City. Each trip record includes the pick-up and drop-off times and the corresponding latitude and longitude of these locations.

4.3 EvaluationMetrics and ComparedMethods

Evaluation metrics. There are two performance metrics employed for evaluating our model, the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). These metrics collectively provide a robust framework to assess our model's capability to capture subtle variations in the data.

Compared methods. We select seven methods for comparison in this study. These methods are described as follows:

• Linear Regression (LR) [Singhvi et al., 2015]: Predicts bike demand using taxi, weather, and spatial features with neighborhood-level aggregation.
• Spatial Regression (SR) [Faghih-Imani and Eluru, 2016]: Models spatial lag and error structures to capture spatio-temporal dependencies in demand.
• Function Zone (FZ) [Liu et al., 2017]: Clusters stations into functional zones and predicts zone-level transitions using Random Forest and ridge regression.
• GraphSAGE (SAGE) [Hamilton et al., 2017]: Aggregates neighborhood features inductively, enabling generalization to unseen stations.
• IGNNK (N2K) [Wu et al., 2020]: Learns spatial message passing via subgraph reconstruction to interpolate unsampled nodes and generalize across dynamic graphs.
• TrafficStream (TS) [Chen et al., 2021]: Integrates continual learning with spatio-temporal graph networks for adaptive model updating over dynamic traffic data.
• Spatial-MGAT (MGAT) [Liang et al., 2023a]: Captures spatial dependencies through multi-graph attention on proximity and built environment similarity.

4.4 Comparison with Baselines

The comparative analysis of demand prediction models for bike-sharing system expansion, using data from NYC CitiBike and Chicago Divvy-Bike, highlights the superior performance of advanced spatio-temporal models over traditional baselines. As shown in Table 2, traditional models like linear regression and spatial regression exhibit higher RMSE and MAE values at both existing and new stations, indicating limitations in capturing complex and dynamic bike-sharing demand patterns. The Function Zone model performs better by clustering stations based on spatial features, reflecting the importance of functional heterogeneity in urban demand. GraphSAGE improves upon traditional methods by enabling inductive learning over unseen stations, but limited temporal modeling yields suboptimal results. IGNNK enhances prediction accuracy via subgraph reconstruction and spatial message passing, showing strong performance especially at new stations. TrafficStream integrates continual learning with spatio-temporal graph networks, offering robust results across dynamic settings but limited transferability to newly added stations. Spatial-MGAT further advances performance by capturing fine-grained spatial dependencies through multigraph attention, especially under sparse training data. Overall, while each method contributes uniquely, none achieves the consistent accuracy and adaptability demonstrated by BGM across all scenarios.

The proposed BGM framework demonstrates the best overall performance, achieving the lowest RMSE and MAE values across all NYC and Chicago datasets. Its ability to dynamically model evolving network structures, integrate spatio-temporal interactions, and facilitate knowledge transfer between existing and new stations makes it a robust solution for enhancing transportation accessibility, optimizing bike-sharing resource allocation, and supporting sustainable urban growth. As shown in Table 2, BGM highlights the importance of embedding transformation, feature fusion, and dynamic graph modeling in designing intelligent, fair, and scalable urban mobility systems. By improving demand prediction, BGM supports equitable infrastructure expansion and better decision making for urban planners, addressing the broader societal goal of inclusive, data-driven, and futureready transportation development.

4.5 Ablation Studies

In this section, we conduct ablation studies to evaluate the contributions of key components in the BGMmodel by testing three variants: W/O DM (without dynamic graph modeling), W/O ET (without feature embedding transformation), and W/O FF (without feature fusion). By systematically removing each component, we analyze their impact on model performance. As shown in Figure 3(a) and Figure 3(b), BGM achieves the lowest RMSE and MAE values across all datasets, with a 13.8% improvement over W/O ET, confirming the critical role of embedding transformation in demand prediction. Among the variants, W/O ET exhibits the most significant performance drop, underscoring the crucial role of embedding transformation in refining feature representations and enhancing predictive accuracy. The W/O DM and W/O FF variants also show declines, highlighting the importance of dynamic graph modeling and feature fusion in capturing spatial-temporal dependencies and optimizing prediction performance. These results confirm that all three components are essential for maintaining robustness and precision.

4.6 Robustness Analysis

To evaluate the performance and robustness of the BGM model under different expansion scenarios, we designed two experiments focusing on growth rates and expansion patterns. This analysis not only evaluates how the model and baseline methods perform under different network growth speeds but also addresses a critical societal challenge: ensuring the scalability and sustainability of bike-sharing systems in the face of rapid urbanization. The first experiment investigates the impact of varying growth rates while maintaining a fixed expansion pattern. As shown in Figure 3(c) and Figure 3(d), BGM consistently outperforms all baselines across existing and new stations. For instance, BGM achieves an RMSE of 1.78 and an MAE of 0.90 at a 15% growth rate. Even at 30% growth, BGM maintains a low RMSE of 1.75 and 0.87, outperforming all baselines. These results confirm that BGM adapts effectively to varying expansion rates, ensuring stable demand predictions in dynamic urban environments.

The second experiment examines the impact of different expansion patterns, directly addressing a key societal challenge: ensuring equitable access to bike-sharing services across diverse urban areas. Two patterns are considered: localized expansion, as shown in Figure 3(e), where new stations are distributed near existing networks to enhance connectivity, and regional expansion, as shown in Figure 3(f), where new stations are introduced in disconnected areas to improve accessibility in underserved regions. Results in Figures 3(g) and 3(h) show that BGM achieves the lowest RMSE and MAE in both scenarios, demonstrating robustness. Specifically, for localized expansion, BGM records an RMSE of 1.51 and an MAE of 0.83, outperforming the second-best model. It still performs well under regional expansion, with RMSE 1.73 and MAE 1.10, where historical data is scarce. These findings highlight the model's ability to support equitable and efficient mobility resource distribution, advancing broader goals of urban sustainability and inclusion.

5 RelatedWork

Demand Prediction for Bike-Sharing Systems. In recent years, bike-sharing demand prediction has evolved significantly. Early studies using time-series models like ARIMA and SARIMA captured temporal patterns but neglected spatial dynamics [Li et al., 2019]. Cluster-based methods improved short-term forecasts by grouping stations with similar demands, incorporating geographic and historical features [Chen et al., 2016; Lahoorpoor et al., 2019]. Station-level models integrated external data such as weather and events but faced challenges at data-sparse locations [Yu et al., 2023]. Advanced approaches, including spatio-temporal graph convolutional networks (STGCNs), effectively modeled spatialtemporal dependencies [Xiao et al., 2021; Ma et al., 2022; Tang et al., 2021], while TrafficStream introduced streaming GNN frameworks with continual learning for large-scale forecasting [Chen et al., 2021]. Reinforcement learning optimized operations in real-time settings [Demizu et al., 2023]. Bike-Sharing Systems Expansion. There is a substantial body of research dedicated to modeling the expansion of bike-sharing systems, such as planning optimal locations for new stations [Li and Zheng, 2020; Liang et al., 2024] or enhancing the capacity of existing stations [Liang et al., 2023b; Liang et al., 2023a]. However, much of this existing work assumes that the demand patterns for new stations will closely resemble those of existing stations, which differs significantly from our proposed approach. For example, the method in [Liu et al., 2017] proposed a zone-based hierarchical demand model to estimate average demand at newly added stations during different stages of expansion.

Dynamic Graph Modeling. Spatio-temporal graph networks capture dynamic spatial and temporal station relationships. RNN-based methods [Yu et al., 2017; Kapoor et al., 2020; Roy et al., 2021; Ghosh et al., 2020] use recurrent and graph convolutions but suffer inefficiency and gradient issues. CNN-based approaches [Mohamed et al., 2020; Zhang et al., 2022] enhance efficiency via graph and 1D convolutions yet risk oversmoothing. Inductive GNNs like GraphSAGE [Hamilton et al., 2017] and IGNN [Wu et al., 2020] enable scalable learning on large or incomplete graphs. Adaptive graph learning [Zheng et al., 2023] and transformer architectures [Jin et al., 2023] further improve modeling of dynamic and long-range dependencies.

6 Conclusion

In this paper, we proposed a novel framework BGM, designed to address the challenges of demand prediction in expanding bike-sharing systems. Built on dynamic graph modeling, BGM captures evolving spatio-temporal dependencies and enhances knowledge transfer to new stations through embedding transformation. The gated feature fusion mechanism optimally integrates transferred and intrinsic features, reducing negative transfer and ensuring accurate predictions with minimal historical data. Beyond technical advancements, BGM directly addresses societal challenges such as optimizing resource allocation, promoting equitable access to mobility, and supporting the sustainable expansion of urban transportation networks. Experiments on real-world datasets demonstrate that BGM outperforms existing methods, providing actionable insights for urban mobility planning. Future work will explore extending BGM to other urban systems and incorporating multi-modal data, further enhancing its societal impact in dynamic and diverse urban contexts.