A Large-scale Dataset and Benchmark for Commuting Origin-Destination Flow Generation
Authors: Can Rong
Jingtao Ding
Yan Liu
Yong Li
Venue: ICLR 2025 Poster (2025)
Keywords: Commuting
origin-destination flow dataset
urban computing
weighted graph modeling
Primary area: datasets and benchmarks
TL;DR: This paper provides a large-scale dataset including commuting origin-destination matrices of over 3000 areas in the United States for training and benchmarking origin-destination flow modeling methods.

Abstract: Commuting Origin-Destination~(OD) flows are critical inputs for urban planning and transportation, providing crucial information about the population residing in one region and working in another within an interested area. Due to the high cost of data collection, researchers have developed physical and computational models to generate commuting OD flows using readily available urban attributes, such as sociodemographics and points of interest, for cities lacking historical OD flows \textemdash commuting OD flow generation. Existing works developed models based on different techniques and achieved improvement on different datasets with different evaluation metrics, which hinderes establishing a unified standard for comparing model performance. To bridge this gap, we introduce a large-scale dataset containing commuting OD flows for 3,333 areas including a wide range of urban environments around the United States. Based on that, we benchmark widely used models for commuting OD flow generation. We surprisingly find that the network-based generative models achieve the optimal performance in terms of both precision and generalization ability, which may inspire new research directions of graph generative modeling in this field. The dataset and benchmark are available at https://anonymous.4open.science/r/CommutingODGen-Dataset-0D4C/.

Published as a conference paper at ICLR 2025
A LARGE-SCALE DATASET AND BENCHMARK FOR COMMUT- FLOW GENERATION
ING
Can Rong'
Jingtao Ding' Yan Liu Yong Li
Depaarmeett of Electronic Engineering, BNRist, Tsinghua University. Beijing, China *Comutter Science Department, University of Southern California, Los Angeles, CA, U.S.A.
rc20@mails. tsinghua. edu. cn, dingjt15@teinghus org. en, 1iyong07@teinghua edu.cn
ABSTRACT
Commuting Origin- Digiin-Destinationn (OD) flows are critical inputs for urban planning and transportation, providing crucial information about the population residing in one region and working in another within an interested area. Due to the high cost data collection, researchers have developed physical and computational models generate commuting OD flows using readily available urban attributes, such sociodemographies and points of interest, for cities lacking historical OD flows -commuting OD flow generation. Existing works developed models based dif- ferent techniques and achieved improvement on different datasets with different evaluation metries, which hinderes establishing a unified standard for comparing model performance. To bridge this gap. we introduce a large-scale dataset con- taining commuting OD flows for 3,333 areas including a wide range of urban en- vironments around the United States. Based on that, we benchmark widely used models for commuting OD flow generation. We surprisingly find that the network- based generative models achieve the optimal performance in terms of both pre- cision and generalization ability, which may inspire new research directions graph generative modeling in this field. The dataset and benchmark available at htttps://gitan /
com/ttinghua- fibb fiib-labb 1aallaaa/Caaaa Lan.b/oomtttiggDGennnnn-n -Dataset
INTRODUCTION
Commuting refers to the daily round-trip movement individuals from their homes to their work- places, which is important topic in fields like urban planning. transportation. environmental sci- ence, and economics (Batty, 2007: Gonzalez et al. 2008; Iqbal et al,, 2014 Liu al 2020). These movement between all pair of origins and destinations within the interested area can be effectively recorded as Origin- Destination (OD) flows. All OD flows across the entire area named the commut- ing OD matrix, where each element represents the number of people reside in one region and work in another. The commuting OD matrix can be naturally modeled as a directed weighted graph, i.e, com- muting OD network. where nodes represent regions and edges represent the commuting OD flows between regions (Saberi et all 2017; 2018). Understanding commuting OD flows at both the pair- wise and network-level allows urban planners analyze the structured mobility patterns, optimize the transportation system, and make informed decisions on urban development (Zeng et al, 2022: 2024; Imai al., 2021: Zhong al. 2014). However, collecting the data often costs a lot and raises Thus, researchers have developed both classic physical models (Zipf et al. 2012) and more recent..promiseeeeeeeeeeepppese promising data-driven approaches (Pourebrahim et al., 2019; Liu et all 2020; Simini et al,, 2021 Rong et al,, bo23c to model commuting OD flows and generate data for areas lacking historical flows. This task is named as commuting OD flow generation.
privacy
Simini
and
There are two main challenges lying two aspects: the lack of comprehensive dataset and the absence of unified and systematic evaluation. In details, existing works can be categorized in three types: physical models, classic machine learning models, and graph neural network models, Phys- ical models campare the OD flow to physical phenomenon such as the gravity model (Zipf, 1946 Barbosa al. 2018) and radiation model (Simini all, 2012). The physical models utilize sim- ple mathematical equations to capture the piir-wise relationships between origins and destinations, which have strong theoretical basis but are limited by the underfitting of the complex human mo- bility. Recent popular data-driven models (Rodriguez--Rus Ruedaa et al. 2021; Pourebrahim et al 2019
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Robinson & Dilkina, 2018; Simini al 2021; 1: Liu et a 2020; Rong et al 2023c) can cap- ture the complex relationships between urban attributes and commuting OD flows with sophisticated models. These works based on machine learning deep learning techniques learning from only one single or a few areas, have shown poor generalizablility to distinct urban environments. Despite the significant practical value of commuting OD flow generation, it has not gained widespread attention from the deep learning community. One key reason is the lack of unified benchmark based on a comprehensive dataset. Currently, studies use their own datasets from individual city scenarios for evaluation, making it difficult to compare and communicate insights between different model designs.
2018:
To address the above issue. we collect data from multiple sources and construct a large-scale dataset containing commuting OD matrices for 3,333 diverse areas around the whole United States (LargeCommuingOD) Thanks to the extensive spatial scale of the dataset, various urban environments are covered, including metropolitan areas, small cities, towns, and rural areas. For supporting better study of modeling, each area in the dataset has not only the commuting OD ma- trix but also regional sociodemographies and numbers point-of-interest (POIs) within different categories for all regions in the area. Specifically, each area is profiled with its boundary and the boundaries of regions within it, which are represented as polygons with detailed geographic coordi- nates, i.e., latitude and longitude. The sociodemographies include the population of different genders and age groups, the number of househoulds, and income levels, etc. The point-of-interess are cat- egorized into various types, such as restaurants, education, and shopping, etc. This dataset can be used to comprchensively study and evaluate the models for commuting OD flow generation.
Based on our dataset, we benchmark the existing widely used models for commuting OD flow genera- tion in a common framework. We utilize randomly selected areas in the dataset as the test set, which covers diverse urban environments, to comprehensively evaluate the models terms of both pre- cision and generalizablility. The remaining areas are leveraged to train the models. Existing works including physical models, classic machine learning models, and graph neural network models are all benchmarked. Besides, the generative models trained on the large-scale dataset emerge powerful per- formance. which has been, demonstrated not only in fields like natural language processing (Brown et al, 2020: Kaplan et al, 2020) and computer vision (Peebles & Xie, 2023) but also in spatial- temporal data modeling (Yuan et al. 2024; ii et al., 2023). We introduce a preliminary adaptation of the graph diffusion model to Weighted Edges Diffusion condition Attributed Nodes (WEDAN) into our benchmark. We surprisingly find that the network- based generative models perform the best in terms of both precision and generalization ability, which may call for new paradigm of graph generative modeling in this field.
In summary, the contributions of this work are follows:
We construct a large-scal scale dataset (LargeCommungOO) containing commuting OD flows for 3,333 diverse areas around the United States covering 9,372,610 km including wide range of urban en- vironments. Each area also includes sociodemographics and point-of-interest totaly 131 features as urban attributes for regions within it.
Based the LargeCommuingOD, we benchmark the existing widely used models for commuting OD flow generation. With dataset containing distinct areas, we can comprehensively evaluate the models in terms of precision and generalizablility.
We find that network- based modeling for commuting OD flow supported by our dataset gives a promising performance, which treats an area and the commuting OD flow within it as a network. Training on large number of commuting OD networks, generative models can capture the univer- sal and distinct mobility patterns at the city level, leading to better generalizablility
2 PRELIMINARIES
In this section, we introduce the definitions and problem formulation of commuting OD flow model- ing, followed by the existing works of this field.
2.1 DEFINITIONS AND PROBLEM FORMULATION
Definition 1. Regions. We divide the interested area into overllallap regions, represented as R. = {rali unique functions, indicated by their urban attributes X,,, which include sociodemographics and the distribution of oointe-of-interesss in different categories.
with N being the total count of the regions. Each region fulfills
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Table 1; Comparison of the proposed dataset and other dataset utilized in existing works.
Datuset
FArea
Area Type
Cover Area (km ) Metropoliaa
Kural
Curated & Public
Karimi al (0000
Central Distrset
Pourehruhim atall Taalin
Whole City
789
0| clal (M021
Whole City
789
x
X
Yo (202))
Central District
900
K
Aastuann < (1
Whole City
15,755
x
Kung et all 12 110220111
Whole City
25,954
x
Simini all (2122))
2,911
National Gridking Coverage
686,983
Ours
3,333
Census Area Coverage
9,372,610
Definition 2. Spatial Characteristics. The spaital charactcristics of an area CR are composed of urban attributes of each region (X,,/rv RR and the interactions, such distance, between all regions {deylr; rr and r E R].
Definition 3. Commuting OD Flow. The term commuting OD flow refers to the population Frar residing in Forg and working at rdst-
Definition 4. Commuting OD Matrix. Denoted by E RNXN, the commuting OD matrix includes commutings among all regions within the area. Fs, means the commuting from r; to r-
PROBLEM Commuting OD Flow Modeling. The problem aims to learn a model, given any area's spatial characteristics CR. generating their corresponding commuting OD matrices F that closely resemble those in the real world without any historical information.
2.2 ΕXISTIGG WORKS ON COMMUTING OD FLOW MODELING
Limitations of Dataset Used in Existing Works. As shown in Table 1. existing datasets used in commuting OD flow modeling have several major limitations. First, existing datasets utilized in the literature have limited spatial scale, usually focusing on single or few large cities, leading two very limited spatial coverage. For example. Karimi et al. (2020) and Yao et al. (2020) only consider a a.central district in a city, and Pourebrahim et all (2018: 2019). Liu et al. (2020). Lenormand et al. (2015), and Rong et al. (2023gg) only consider less than 8 large metropolitans, whose areas are less than 30,000 km Although Simini et al, (202 lh consider a national gridding coverage in the United Kingdom and Italy, the area is still limited to 686, 986,983 km? Besides, they do not provide the curated dataset for public use, which cannot be used for further research. In contrast, our dataset covers 3,333 areas around the United States, total area of 9.372,,610 km providing a much broader spatial scale, And our dataset is curated and publicly available. which can be found at https:: thub. com/ tsinghua- fib-lall 1ab/ htt-llaaammmmtnggeen CommutingODGen--Daaaaaaa
few
Second, with the limited spatial scale, existing datasets usually focus on a single type of urban environments, such as metropolitan areas, central districts, or whole cities, which cannot include a massive areas with high diversity in terms of size and structure. Models trained on such datasets may not be generalized to other areas with different characteristics, limiting their applicability on only similar areas. Our dataset covers metropolitan areas, towns, and rural areas around the United States, providing more comprehensive dataset for training and evaluating models. With the diversity of areas, models trained on our dataset can be more generalizable.
OD Flow Modeling Approaches. Existing works can be categorized into three types. The first is physical models, such as the gravity model ZZipl, 1946) and the radiation model (Simini et al 2012), which mimick the commuting OD flows as physical pheonomena and utilize simple mathe- matical equations to model the flows. Physicists dive into the mechanisms of individual mobility decisions and try to explain the phenomenon of commuting OD flows, The second is statistical learning models. such tree-based models (Robinson & Dilkina, 2018 Pourebrahim al. 2018 2019). SVR "Rodriguez-Ruete
et al 2021). 1). artificial neural networks (ANNs) (Sana et al 2018
Lenormand al. 2016 Simini al. 2021). which predict the OD flows between pairs of regions in data-driven schemes. The third is graph learning models. Liu et al (2020): Cai et al (2022) utilized GAT's to aggregate the neighbors' information to profile the regions better and improve the prediction accuracy. Yao et al. (2020) model the local spatial adjacenct structure of regions with graph convo- lutional networks and imputate the missing OD flows in a semi-supervised manner. RRng et al. ()023c introduce adversarial and denoising diffusion generative methods with graph transformers to model the commuting OD matrix generation as graph generation problem. Many researchers from urban planning and transportation have shown interest in data-driven models because of the better performance Barbosa et al. (2018); Luca et al. (2021): h): Rong et al. (2023a). But there lacks a large- scale dataset containing wide range of urban environments and unified benchmark for comparing
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ACS Data
demograppics
Socio-
clean
Conrattion
numpy. ndarray (N, Fars))
Regional Sacio-
demographics
correlation
LODES Data
Commuting Flows
selection
Commuting OD
Data processing
Flows numpy.ndarray (N,
County Boundaries
USCB
geo-malch
Census Tract
Census Geographies
Boundaries
Regions within Areas
Metropolitan Boundaries
shapefile
CBG Boundaries
geo-malch
Data Platform
Data Source
Comporee
Stallstii
clean
Raw
OSM
POIs Data
POIs Distribution
pprattion
numpy.ndarray
(N, Froe)
Dalaset
Data processing
Figure Discreption of the pipeline constructing our datasest.
the performance of different models, which hinders the development of more powerful models. Our dataset and benchmark can fill this gap and provide a common ground for evaluating the models.
3
LARGECOMMUINNGO: A LARGE-SCALE COMMUTING OD FLOw DATASET
3.1 DATA COLLECTION AND CURATION
The pipeline for constructing our dataset is shown in Figure I. As shown in the figure, the dataset contains four main componets: 1) boundaries of areas and regions 2) sociodemographies, 3) POIs distributions, 4) commuting OD flows. First of all, the boundaries of areas and regions are download from the U.S. Census Bureau, which include all counties, metropolitans, census tracts, and census block groups (CBGs). And we set the counties as the areas and census tracts as the regions for the county areas, and set the metropolitans as the areas and CBGs the regions for the metropolitan areas. The counties can be related to the census tracts by code of Federal Information Processing Stan- dards (FIPS). The CBGs belong the metropolitans, which is detected by the spatial relationship between the boundaries of CBGs and metropolitans, i.c., whether the CBG is inside the metropoli- tan. Then, the sociodemographics for each region can be accessed from the American Community Survey (ACS) on the website of the U.S. Census Bureau. For each indicators, we use regression analysis on the indicator and flow intensity to decide whether to choose the indicator into the urban attributes. The information not related to human mobility is excluded. And for each region, API of OpenStreetMap to get the number POIs in different categories. The POIs are divided into 36 categories, including restaurants, schools, hospitals, etc. The commuting OD flow is provided by the 2018 Longitudinal ties (LODES) dataset the website of the U.S. Census Bureau. The data orgagized in form of tables. Each table contains the commuting information of one state. Each row in the table represents the commuting flow between two specific census blocks. We aggregate the flow into census tract level and construct the OD matrix.
Dynamics Prigin-Destination Employment Statis-
3.2 DATA DESCRIPTION
We have collected data from a total count of 3,333 areas around the United States. There are two kind of spatial divisions in LargeConmuingOD: 1) 3,233 counties as the areas and census tracts inside each county as the regions, 2) 100 metropolitans, where the population is more than million, as the areas and census block groups CBGs inside each metropolitan as the regions. LargeCommuingO0 includes the following information: 1) regional urban attributes, including sociodemographics and POIs, 2) commuting OD flows, represented by OD matrices, which are aggregated commuting flows within areas. The counties are defined by the U.S. Census Bureau. Each county is a local government unit in the United States, and the counties should cover a similar number of households and popula- tion. The metropolitan boundaries are obtained from Topologically Integrated Geographic Encoding and Referencing (TIGER) dataset. The metropolitan areas exclude the rural areas, which do not have population and urban functionalities.
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Table 2: Summary of urban attributes used to profile a region.
Categories
Centents
#Features
Sociodemographics
Total population
Population with different genders and ages
56
Median age of people with different genders Median earnings
3
I
Ratio of different classes of jobs
5
Vehicle ownership
4
The number of houscholds with different types
4
Population with different education levels
21
Poverty with different genders
2
POIs
The number of POls in different kind.
34
Total
131
Regional Urban Attributes. Each region is characterized by sociodemographies and urban func- tionalities, derived from American Community Survey (ACS) (U.S. Census Bureau. 2012) by the U.S. Census Bureau and the distribution of POIs from OpenStreetMap (OpenStreetMap contribu- tors 2017). as shown in Table 2 Demographics include the population structure of region based on age, gender, income, education, and other factors, encompassing total of 97 dimensions. POIs are divided into 36 different categories. The distances between regions are calculated using the planar Euclidean distance between their centroids.
OD matrices. We construct commuting OD matrices for all areas using data on commuting patterns from the 2018 Longitudinal Employer-Househne Dynamics Origin-Destination Employment Statis- tics (LODES) dataset. These matrices represent aggregated commuting flows within areas. Each en- try in an OD matrix denotes the count of individuals residing in one region and working in another, effectively mapping commuting patterns of workers across different regions. The LODES dataset widely used in existing works (Liu et al, 2020; Pourebrahim et al., 2019: 2018). In LargeCom- muingOD, the commuting information is aggregated by the cooperation and other kind of work units, which is more reliable and accurate than the individual commuting data. Therefore, in the data collec- tion process, information has been ensured to be representative at national scale, thus eliminating sampling errors. The raw data provided is at the census block level, which is then aggregated to the census tract level for the county areas and to the CBG level for the metropolitan areas.
It is worth noting that the commuting OD flows within the 3,233 counties cannot carry the mobility across different counties, while the flows within metropolitans can. So LargeCommuingOD include both intra-county and inter-county flows.
3.3 DATA STATISTICS
We provide statistical analysis of CorgeCommuingOD to illustrate the diversity of the dataset. We analysis the dataset from two perspectives: area characteristies and mobility patterns. From Figure 2 we can see that the number of regions in each area varies significantly. which shows the heterogeneity
of the areas in LargeCommuingOD. Furthermore, cases in Figure 5 reveal the diverse structure of the areas, including monocentric, polycentric, and evenly distributed spreading. For analyzing the mobility patterns, we measure the average trip distances and the variance of the regional mobility intensity. The travel distances tend to be shorter but there are still long- long-distance trips, make the mobil- ity patterns complex. The variance of the regional mobility intensity is also diverse in a wide range, which indicates the heterogeneity of the mobility patterns. For commonalities among areas, we ana- lyze the distribution of OD flows and outflows in areas of different scales, as shown in Figure 4 We can observe that the heterogeneity exsits between different scales of areas. Yet, the commonalities also exist, i.c., the scaling behaviors are the same among areas. This demonstrates that LargeCom- muingOD is a comprehensive dataset that covers wide range of urban environments with diverse mobility patterns. To further intuitively understand the dataset, we provide the Visualization of the OD flows via heatmaps in Appendix A.1
3.4 DISCUSSION
Superiority. From the statistical amalysis, we can see that LargeCommuingOD is large-scale and comprehensive, covering a wide range of areas of different scales and mobility patterns, i.e., diverse urban environments. For learning. the sufficient scenarios in LargeComnuingOD can support the modeling research to capture the distinctness and commonalities of the mobility patterns in different
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200
0.9
130
100
100
Agaen
200
40
X8-053
2000
4200
0.5
100
200
300
(a) #Regions
(b) Distances.
(c) Variance.
Origin Destination
(a) Edge weights.
Figure 2: Statistical analysis of LargCommuingOD, including the distribution of a) the number of regions in each area, b) the average trip distance in each area, c) the variance of the in/out flow of each region in each area.
10
Pagon
12:50
10"
10
210m640
10
10"
10
10"
10"
10"
10"
Outflows
(b) Node degrees.
(a) Maricopa
(b) Alameda
(c) Contra Costa
Figure 4: Distributions of OD flows and outflows in areas of different scales. a) cumula- tive distribution function of edge weights, and b) probabilistic den- sity function at log scale of node degrees.
Figure 3: Visualization of the OD matrices of three areas with different mobility structure, a) monocentric (Maricopa in Ari- zona), b) polycentric (Alameda in California), and e) smoothly distributed (Contra Costa in California).
General Model
Large-scale Dataset including
Diverse Urban Environments
Any target cities
Figure 5: Superiority of LargeCommuingOD. The models trained on LargeConmuingOD can cap- ture the distinctness and commonalities of the mobility patterns in different areas.
areas, as shown in Figure For evaluation, the diverse urban environments in LargeCommuingOD can support the comprehensive evaluation of the models in terms of both precision and generalizabil- ity, which cannot be achieved by existing datasets.
Limitations. Despite the comprehensiveness of LargeCommuingOD. there are several limitations. First, the data is collected from single year, which may not fully capture the temporal changes of commuting patterns. Second, the data is limited the U.S., which may not generalizable to other countries with different characteristics and cultures. Third, the data only includes commuting OD flows, which may not fully represent the human mobility patterns in urban areas. This may limit the utility of our dataset in more fine-grad fine-grained mobility analysis tasks.
BENCHMARK
In this section, we investigate the precision and generalizability of benchmark models by answering the following questions:
For precision, how realistic the models in generating the commuting OD flows in terms of the flow values and network properties? (Section 4.2)
For generalizability. can the models capture the distinctness and commonalities of the mobility
pattern across different urban areas? (Section 4.3 and 4.4)
We further evaluated and discussed the models from three perspectives: interpretability, robustness,
and fairness. For details, please refer to Appendix B.1
B.2
and B.3

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Table 3: A comprehensive comparison of the benchmark models.
Paper
Model
Type
Perspective
Output
Barbosa et al. (2018)
GM
Physical Model
Pair-wise
OD flow
Rodriguez- Rueda et al. (2021)
SVR
Kernel-hased Model
Pair-wise
OD flow
Pourebrahim et al (2019)
RF
Tree-based Model
Pair-wise
OD flow
Robinson & Dilking (201
GBRT
Tree-based Model
Pair-wise
OD flow
Pair wise
Outflows to
Simini ct al
2021
DGM
MLP-based Model
for origin
all destinations
Transformer
Pair Wise
Outflows to all destinations
Iuo et al.
0024
TransFlower
bused Model
for origin
Liu (2020)
GMEL
GNN- Model
Pair-wise
OD flow
Rong (2023d
NetGAN
GAN-based Model
Network-wise
OD matrix
Rong 2023h
DiffODGen
Diffusion-based Model
Network-
OD matrix
WEDAN
Diffusion-based Model
Network-wise
OD matrix
4.1 EXPERIMENTAL SETUP
Benchmark Models. We utilize our propose dataset benchmark existing models. The existing models are in three categories: physical models, classical statistical learning approaches, and graph learning models. Besides, we also explore the feasibility of utilizing the graph generative modeing, which construct the fourth category (generative models).
Within the first category are two physical models:
Gravity Model with Power-law Decay (GM-P) Zipt. 1946; Barbosa et al., 2018) is inspired by the gravitation in physics, positing that the OD flow directly proportional to the populations of the origin and the destination, and inversely proportional to the distance between them. The power-law decay used model the distance decay effect.
Gravity Model with Exponential Decay (GM-E) Zipí, 1946; BBarbosa et al,, 2018) almost identical to GM-P, except that it uses an exponential decay function to model the distance decay
effect.
The second category encompasses classical statistical learning approaches tailored for OD flow mod- eling:
Support Vector Regression (SVR) (Rodriguez-Rueda al 2021 is a kernel-based machine learning algorithm that has been widely used in regression tasks. It is employed to predict the OD flow between two regions based on the urban attributes of the regions by Rodriguez-Ruus -Rueaa et al. (2021).
Random Forest (RF) (Pourebrahim al., 2019) stands out tree-based machine learning algo- rithm known for its robustness, demonstrating commendable results in generating OD flows. Gradient Boosting Regression Tree (GBRT) (Robinson & Dilkina, 2018) use boosting tech- niques to enhance the performance of decision trees, which has been applied predict the OD flow in the city.
Deep Gravity Model (DGM) (Simini et al., 2) use multi-layer perceptrons (MLPs) inspired by gravity models to calculate flows to different destinations by estimating the distribution proba- bilities. We have adapted this model to generate OD flow volumes directly.
TransFlower (Luo al. 2024) is a transformer-bast based model under the framework of DGM, which utilizes the transformer to model the spatial dependency of all destinations for each origin rather than MLPs. The model is also adapted to generate OD flows directly.
The third category includes approaches based on graph neural networks, which model the urban space commuting OD networks graphs:
Geo-contextual Multitask Embedding Learning (GMEL) eL 2020) leverages graph neu- ral networks (GNNs) to aggregate neighboring information for each region. This process enhances the spatial characteristic representation of the regions in a city, which contributes to the refinement of regional embeddings and augments precision.
NetGAN (Bojchevski et al., 2018) 8) is a GAN-style framework that recreates realistic network ar- chitectures by generating random walks that mirror the distribution of walks extracted from real networks. We have tailored it to construct directed and weighted graphs, i.c., OD matrices.
The last but not least category includes the generative models based ransformer-backbone models:

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DiffODGen (Rong et al, 2023b) employs cascaded diffusion model specifically for large cities, leveraging the sparsity of the mobility network to separately model the topology of the graph and the weights given edges, achieved SOTA results in large cities.
WEDAN is a preliminary try to adapt graph diffusion models model the joint distribution of all elements in OD matrices conditioned on urban attributes, which named WEDAN (Weighted Edges Diffusion condition on Attributed Nodes).. We use this model to explore the new paradigm for commuting OD flow generation. The details of the model are introduced in Appendix
Parameter Settings The graph transformer in diffusion models employs 4 layers with each hav- ing 32 hidden dimensions. We utilize 250 diffusion steps in diffusion models, following a cosine noise scheduler suggested by Nichol & Dhariwal (2021). Denoising networks are optimized using AdamW optimizer (Loshchilov & Hutter, 2017), with a learning rate set at le-3, Our method and DiffODGen both sample 50 times during generation and take the average final generated results.
For the gravity model, we adopt the approach outlined by Barbosa et al. (2018), which involves four fitting parameters. In the random forest algorithm, the number of estimators DGM (Simini et al., 1) is stacked by 10 layers with 64 hidden dimensions in each layer, while GNN-based models are designed with layers and 64 channels all. TransFlower is stacked by transformer encoder with 8 heads and 64 hidden dimensions in each head. The hyper-parameters for the denoising networks in two cascaded diffusion models of DiffODGen are aligned with our methodology.
100. The
All the selection of mance and computational resources.
is based on the validation set and trade off between the perfor-
Evaluation Metrics. We uniformly evaluate the performance based on widely adopted metrics from two perspectives: the error between the generated OD matrices and the corresponding real ones, and the distribution deviation
graph properties between the generation and the real data. The error metrics include Root Mean Square Error (RMSE), Normalized Root Mean Square Error (NRMSE) and Common Part of Commuting (CPC), while the distribution difference metrics include Jensen- Shannon Divergence (JSD) for inflow, outflow, and OD flow. These metrics calculated for each area and then averaged across all. The calculation formulas are shown as follows.
RMSE
Σ ER
FIg1
RMSE
NRMSE
(2)
Σ EM.TER |F/ Figl12
2 SER Fiin Fzz))
CPC
(3)
Em.ryeR Fúj + E EniYEER Fij
KL. Pp) + (LpPPP(P)))
JSD
2
where the denotes the mean of elements in OD matrix F, KL means Kullback -Lebllr divergence, and P denotes the empirical probability distribution. The inflow is determined by totaling all flows entering each region, while the outflow is calculated by summing up all flows leaving each region.
4.2 PERFORMANCE COMPARISON
The results are shown Table 1. All models utilize the ratio of 8:1:1 for dividing the data into training, validation, and test sets. We conducted experiments five times and averaged the results.
The exploration of the graph generative modeling, WEDAN, achieves the best performance. The OD matrix generated by WEDAN demonstrates superior realism, from both flow value and property distribution deviation perspectives. Notably, in comparison to the top-performing bascline, WEDAN reduces RMSE/NRMSE by more than 8.0% and improves the CPC over 11.5%. Further- more, the property distribution of the generated OD matrices closely matches the real ones, as evi- denced by the lowest JSD from all the perspectives.
The performance of data-driven approaches significantly outperforms the physical model. The Gravity Model, using only four parameters, attempts to fit the complex human mobility, leading to
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Table 4: Performance comparison on all existing models.
Flow Value
Property Distribution (ISD)
Model
CPCt
RMSEL
NRMSE,
inflow
outflow]
ODflow!
GM-P
0.321
174.0
2.222
0.668
0,656
0.409
GM-E
0.329
162.9
2,080
0.652
0,637
0,422
SVR
0,420
95.4
1.218
0.417
0.555
0.410
Pair-wise
RF
0.458
100.4
1.282
0.424
0.503
0.219
GBRT
0,461
91.0
1.620
0,424
0.491
0.233
DGM
0.431
92.9
1.186
0.469
0,561
0,230
TransFlower
0.488
1.249
0.356
0.337
0.269
GMEL
0.440
94.3
1.204
0.445
0.355
0.207
NetAN
0.487
89.1
1.138
0.429
0.354
0.191
DiffODGen
0.532
74.6
0.953
0.324
0.270
0.149
Network-based
0.593
68.6
0.876
0.291
0.269
0.147
WEDAN
(+11.5%)
(+8.04%)
(+8.04%)
(+10.2%)
(+0.96%)
(+1.34%)
inevitably underfitting. On the contrary, data-driven approaches, employing models with a multi- tude of parameters, go beyond by incorporating rich information such demographies and POIs. Therefore, they have shown significantly better performance.
Modeling the joint distribution of all elements in OD matrices from the graph perspective hold advantages. Modeling the dependency between the area's spatial space and the OD matrix globally, as opposed merely modeling human flows between two regions (i.e., origin and destination), results in more effective capture of the properties of the mobility networks, i.e., OD matrices.
Utilizing training data from various massive areas can enhance the performance. Existing mod- els based on graph generation have been designed only for large graphs, such as NetGAN and Dif- fODGen. contrast, WEDAN is more versatile, capable of adapting to areas/graphs, of various sizes, from small to large. Consequently, it achieves more outstanding results.
4.3 PERFORMANCE ANALYSIS ON HETEROGENEOUS AREAS
To further explore the heterogeneity handled by the models and the applicability in different urban scenarios, we conducted comparative experiments on the model's performance across areas with various sizes and structures. Typically, developed areas often larger and imply stronger attraction to populations. Conversely, underdeveloped areas are usually small in size. Areas of different sizes also exhibit distinct mobility patterns, especially in terms of the skewness of OD flow distribution. Larger areas typically indicate stronger heterogeneity in mobility patterns from both node and edge perspectives, with a more pronounced long-tail effect in flow distribution.
We divided the test areas into six groups based on the number of regions and into three groups based on structure, and the results are shown in Figure &. We find that when trained under the new paradigm. WEDAN can consistently achieve optimal performance across areas of all sizes and structures. Polycentric areas often have larger size and more complex pattern, as they develop satellite towns based on the original monocentric structure. Therefore, polycentric areas are more challenging to deal with. However, our model still achieved optimal performance in CPC. Larger areas tend to have more structured layouts, smaller areas mostly fall into the 'others' category, resulting in better metrics for this category. While DiffODGen is specifically designed for large also enhance its performance by 11.1% CPC and 33.3% on RMSE thanks
areas, our method to the various massive training data. Generative models demonstrate better adaptability to different structures of areas. And our method averagely improves the performance by 34 9% RMSE on the monocentric and polycentric areas. Further analysis is conducted in Appendix D.5
4.4 ANALYSIS THE COMMONALITIES CAPTURED ACROSS VARIOUS AREAS
We conducted in depth analysis of the dependencies captured across areas of different sizes and struc- tures (Xu et al., 2023) in the new paradigm. Specifically, we utilize areas with varying sizes and struc- tures to mutually serve as training and testing sets, thereby validating the capture of commonalities across areas. The results, as illustrated Figure 7 we find that there are commonalities across areas with different sizes and structures, allowing for a certain degree of mutual transfer between them. Experiments have shown that a performance transfer learning and applications. Large areas contains more information about flows. Therefore, achieving performance of 86.7% can be accomplished with only a small number of training large areas. Training the model with diverse range of areas enhance its generalizability. allowing it
89.7% can be achieved solely through cross-type

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~10
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14
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(a) Areas of different sizes on CPC.
(b) Areas of different sizes on RMSE.
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(c) Areas of different structures CPC.
(d) Areas of different structures RMSE.
Figure 6: Performance comparison across areas of different sizes and structures.
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Figure 7: Analysis the dependencies captured across areas with different sizes and structures. The small areas consist of less than 100 regions, and the large areas consist of more than 500 regions. The black dash line represents the performance of training with all types of areas.
to achieve good performance across various areas. This indicates the validity the novel paradigm. Extended analysis is conducted in Appendix D.6. To further explore the transferability of the model across even different countries, we conduct generation experiments on the United Kingdom, and the results shown in Appendix D.4
CONCLUSION
In this work, we introduce a large-scale commuting OD flow dataset (LargeCommuingOD) to sup- port sysmatical comparison of existing studies and to facilitate the development of more powerful models. The dataset contains 3,333 areas around the United States including diverse urban environ- ments. Besides, regions with each area profiled with urban attributes, such sociodemographics and POIs. Based on this dataset, we benchmark existing works with a common evaluation and find that network-based generative models may be a promising direction for future research, which could utilize the data collected from distinct areas to learn more generalizable model. The model should capture the universal and distinct mobility patterns the city level.
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ACKNOWLEDGMENTS
This work supported by the National Natural Science Foundation of China under Grant No. 62476152; and research grants from the Tsinghua-Toyota Joint Research Center and the Beijing Na- tional Research Center for Information Science and Technology (BNRist).
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A DETAILS OF THE DATASET
A.I VISUALIZATION OF THE COMMUTING OD FLOWS VIA HEATMAPS
(a) New York City.
(b) Los Angeles.
(c) Chicago.
(d) Etowah County.
(c) Jefferson County.
(f) Harris County.
Figure 8: Heatmaps of commuting OD flows for three large metropolitans and three counties in our dataset.
We plot the heatmaps of the commuting OD flows for three large metropolitans and three counties in our dataset in Figure S. The heatmaps show that the larger the city, the sparser the commuting OD flows are. This is because people tend to conduct closer commutes while the large city has more regions that far away from each other. In small cities, the regions all close to each other, leading to denser commuting OD flows.
B COMPREHENSIVE EVALUATION THE BENCHMARK: INTERPRETABILITY. ROBUSTNESS, AND FAIRNESS
B.I DISCUSSION ON THE INTERPRETABILITY OF BENCHMARK MODELS
We discuss the interpretability of the benchmark models in this part. Because different models have different structures and mechanisms, their interpretability varies. We dive into the details one by one.
Physical Models: Physical models are rooted in fundamental principles, providing strong inter- pretability through their clear and well-defined mathematical formulations.
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Gravity Model: The interpretability of the gravity model can be summarized from two key aspects: (1) it highlights the production at the origin and attraction at the destination, both modeled using population size, and (2) it incorporates travel costs between regions, repre- sented through distance decay functions (commonly power-law or exponential). The model includes three core variables: origin population, destination population, and distance, along with four parameters that control production, attraction, distance decay, and an overall scaling factor. This design makes the model intuitive and transparent in explaining how population and distance influence mobility. The formula of the gravity model is shown below.
(5) where Fij is the flow from region to region j. X is the scaling factor, fi and fj are the production and attraction functions, and fa is the distance decay function.
FgJ (f)
Radiation Model: The radiation model models OD (Origin-Destinatte Destinatiin) flow by mimicking the physical process where particles are released from the origin region and absorbed by the destination region. The release of particles in the origin region is function of the population, typically calculated as the total population multiplied by the proportion of the working rate. Whether the particles are absorbed by a destination region depends on the distance and the availability of its job opportunities. Specifically, the income associated with job opportunity in region is sampled independently from probability distribution p(z). The attractiveness of a region is quantified by the maximum job income available there, which determines its capacity to attract workers from other regions. Each individual also has an expected income threshold, which is defined as the maximum income they can earn their home region. The decision-makinn process for job selection involves two steps: (a) individuals (analogous to particles) are released from their home region; (b) they are absorbed by the nearest region offering job income higher than their expected income. This process mirrors the radiation mechanism, where particles move and are absorbed based on specific criteria. As such, the model leads to the following derived macro formula.
m,n
Pij >= T:P(1 nj, S:y) T. (m, + (m,) + ny + Si,)
(6)
the OD flow from region i to region j. T, is the outflow of region i, m, is the number of jobs in region nj is the total population of region j, and Sij is the number of jobs in the circle region between i and j.
where < Fii >
Statistical Models: Statistical models are inherently data-driven, which may reduce their inter- pretability compared to theoretically derived physical models. However, they still offer strong interpretability by providing comprehensive insights at the input level, even if a full explanation at the parameter level is not achievable.
SVR leverages all support vectors from the training process as references, enabling strong in- terpretability by assessing the similarity between the new prediction target and each support vector. For instance, if prediction sample has high similarity to a specific support vector, as determined by the kernel function, its predicted target will be closer to the target of that support vector. The contribution of this support vector to the prediction value will thus be more significant. In such cases, the features of the corresponding training sample can be refer- enced to explain the prediction target. For example, if certain features of a prediction sample closely resemble those of support vector, it can be inferred that these features have a similar influence on the predicted value. As shown in Figure the training samples are surrounded and profiled by the support vectors, which means that the distribution of the training samples is well captured by the support vectors.
Tree-based regression models (e.g., RF and GBRT) assess the importance of each feature by analyzing the conditions at each split and the proportion of data in the resulting subtrees. This allows the model to quantify the influence of each feature on the prediction results, providing degree of global interpretability. Specifically, Random Forest quantifies feature importance by calculating each feature's 's contribution to reducing impurity (e.g., Gini index or informa- tion gain) during splits. Features with greater importance are frequently used as split condi- tions across multiple decision trees, highlighting their global impact on the predictions. As shown in Figure 10, the feature importance of each feature is visualized, providing a clear understanding of the model's interpretability. Specifically, the distance provides the most significant contribution to the prediction, followed by the population and features that can denote job opportunities.
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Figure 9: Visualization of the training samples and the support vectors.
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(a) Feature importance of random forest.
(b) Feature importance of gradient boosting deci- sion trees.
Figure 10: Visualization of feature importance in tree-based models to access the global interpretabil- ity of the models.
NN-based predictive models generally have weak interpretability due to their complex architec- tures and large number of parameters, making it challenging to understand the specific meaning of individual parameters. However, certain techniques, such as SHapley Additive exPlanations (SHAP) and feature visualization, can provide a certain degree of interpretability. We combine these two approaches to discuss the interpretability of neural network-based predictive models in the benchmark.
DeepGravity: We utilize SHAP obtain the interpretability of this model according to Simini et al. (2021). As shown in Figure 11, the global SHAP values are visualized to provide an overview of the feature importance. The population working age and the economic activity index are the most influential features. It is interesting that the significant features are different from those in the tree- tree-based models. This may indicate that DGM can capture more complex patterns from another perspective.
Trowerlower: We visualize the relative position embedding following Luo et al. (024).. is important to note that TransFlower was originally designed for the setting where the outflow of each region is already given, generating OD flows toward a fixed number of destinations (256 in the work of Luo et al. (2024)). However, in our problem, the outflows are unknown, and we aim to model OD flows between all pairs of regions in the city. Limiting the number of destinations is therefore not applicable. As a result, the probability distribution of flows from a given origin to a fixed number of destinations. We adapt this model to generate flows directly, thus the attention to destinations cannot be obtained. The visualization of the relative position embedding is shown in Figure I2. The clustering of the relative position embedding indicates that the model can capture the spatial relationships between regions even under unregularized division of the urban area. We can see that the embedding under the Cartesian coordinate system exhibits clear circling patterns while the embedding under the polar coordinate system shows a fringe layer- like pattern. The regularity may not be as strong as Figure 3 in the original paper (Luo et al, 2024) because the original paper uses a grid-like division of the urban area, which is more regular than the unregularized division in our dataset. This demonstrates the strong interpretability of TransFlower in capturing the spatial relationships between regions.
cannot use model that predicts
GMEL: We visualize the attention weights in the graph attention networks within the model. These attention weights capture the similarity between regions, enhancing region's repre-
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High
M_Meiian Earnings (Dollars)
D_ Vehicles Available 1 Vehicle Available
D 30 34 Years Male
D_Total Population
Feature value
O Elementary: Grade to Grade
Vehicles Available 1 Vehicle Available
O Median Earnings (Dollars)
D_ Less Than Grade
D Vehicles Available 2 Vehicles Available
0_30 34 Years Male
Low
-0.002
0.000
0.002
0.004
(impact model output)
Figure 11: Visualization of the global SHAP values for the DeepGravity model.
blatttude
ALangitude
Angle (radians)
(a) Catesian Coordinate System
(b) Polar Coordinate System
Figure 12: Location embedding clustering the relative location encoder Harris County from trained TransFlower.
sentation by aggregating information from its similar neighbors. This provides degree of interpretability aligned with the First Law of Geography, as shown in Figure 13(a)
Graph Generative Models: Generative models typically aim fit the probability distribution of data, a challenging task that often results in highly complex model structures. As a result, they are generally the least interpretable class of methods. Additionally, generative models explicitly implicitly handle randomness and noise in the data, using probabilities to generate nodes and edges- pros---bbbbbliiies that are driven by random patterns in the data. Models like GANs and diffu- sion models inherently involve noise modeling: GAN generators often take Gaussian noise input, while diffusion models explicitly model small noise the diffusion process. This reliance on ran- domness and noise further reduces their intuitive interpretability. Moreover, generative methods are not well-suited for feature-level interpretability analysis using SHAP due to the high dimen- sionality of the conditional control variables, which scale N x f (where N is the number of nodes and fis the dimention node features). The number of features can also vary across samples visualizations of attention mechanisms and denoising dif-
with changes in N. Therefore,
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0.
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(a) Attention map of GMEL.
(b) Attntion map of NetGAN.
Figure 13: Visualization of the attention relationships in graph attention networks of GMEL and NetGAN,
fusion processes to discuss interpretability for NeIGAN, DiffODGen, and WEDAN. Specifically, the attention map NetGAN is shown in Figure 13(b)
(a) Desnoising diffusion process in DiffODGen.
Desnoising diffusion process in WEDAN.
Figure 14: Visualization of the denoising diffusion process in DiffODGen and WEDAN.
In summary, as model complexity increases, performance continues to improve, but interpretability declines correspondingly. This presents a critical challenge: how to trade off the performance gains brought by complexity with the need for interpretability. Alternatively, exploring techniques that enhance the interpretability of complex models erucial direction for future research, demon- strated by approaches like TransFlower.
B.2 DISCUSSION ON THE ROBUSTNESS Or BENCHMARK MODELS ON EDGE CASES
To evaluate whether the models in our benchmark demonstrate good robustness on edge cases, we designed experiments to assess their performance under extreme large OD flows. Specifically, measured the percentage of CPC on the top 5% largest OD flows relative to the overall CPC reported in Table 4. The results presented in Figure 15 From the results, observe that model complex-
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CPC (Normal Performance)
Edge Case Performance (% of Normal)
Extreme Scenario Performance
65%
0.4
0.6
55%
56%
53%
55%
43%
40%
0.2
27%
19%
0.1
GM-P
GM-E
SVR
RF
GBRT
DGM
TransFlower
GMEL
NetGAN
DiffODGen
WEDAN
Figure 15: Performance of benchmark models on the top 5% largest OD flows. The left y-axis represents the percentage of CPC on the top 5% largest OD flows, while the right y-axis shows the overall CPC.
ity increases, the ability to handle edge cases also improves, likely due to stronger nonlinear fitting capabilities. These models, during generation, continuously and smoothly model the distribution of commuting OD flows in urban spaces within the latent space. However, for edge cases, performance degradation still observed to some extent. This is partly due OD flows, where only a small number of extremely large flows are present, making it difficult to col- lect sufficient training data for these cases. Therefore, robustness on edge cases remains a challenge for such continuous modeling approaches in this field.
strong long-tailed distribution of
B.3 DISCUSSION ON THE FAIRNESS OF BENCHMARK MODELS
We utilize the median earnings of the regions as a proxy for the economic status of the regions. We then divide the regions into equal-sized two groups: low-income regions and high-income regions. We then adopt Demographic Parity (PD) for OD flow modeling (Wang et al, 2024) to evaluate the fairness of the benchmark models. Specifically, and compare the difference between the distributions the CPC values for the two groups. The results are shown in Figure 16 As formance, with the smallest difference the distributions of the CPC values between low-income and high-income regions. Graph diffusion- -aased models show a slightly higher performance for the high-income regions. The remaining models exhibit large DP values, but it seems like there is no obvious trend of modeling which group better. From the distribution differences shown in Figure 16 we can conclude that the distribution of the CPC values for low-income regions is more concentrated than that for high-ing high-income income regions.
calculate the CPC for every region in each group
see, the tree-based models exhibit the best fairness per-
The fairness performance is important but rarely studied in the field of commuting OD flow genera- tion. Our analysis is primary exploration of this topic, and we hope inspire more research in this direction in the future.
C ADDITIONAL INFORMATION ABOUT THE NEW PARADIGM
In this section, we give a detailed introduction to a new paradigm to solve the commuting OD flow generation supported by comprehensive dataset. In the paradigm, consider the whole area combined with its commuting OD matrix as an attributed directed weighted graph. Thus, the commuting OD flow generation problem can be formulated generating the weighted edges based
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Low Income Regions
income Reglons High-income Regions
High-income Regions
Norma Rormallized Reed Frequency
Noma Normalii
(a) Comparison of Demographic (b) Distributions of CPC values (c) Distributions of CPC values Parity Values for all models in the for low-income and high-income for low-income and high-income benehmark.
regions based Support Vector regions based on Random For- Regression (SVR).
est (RF).
High-nsoome Regions
Low Incomee Regions
Nomalzed Frequancy
Low income Regions Highvincome Regions
Noomalizea Proquancc
Low income Regions High-income Regions
Nomalized
CPC
(d) Distributions of CPC val-(e) Distributions of CPC values (f) Distributions of CPC values ues for low-income and high- for low-income and high-income for low-income and high-income income regions based on Deep- regions based on TransFlower. regions based on WEDAN. Gravity (DGM).
Figure 16: Analysis of fairness performance of benchmark models on regions with different income levels.
characteristics
destinations
pop
food
A
C
D
E
A
30
3
A
0
50
0
0
60
regions
B
40
10
origins
B
0
60
0
30
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50
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0
0
30
30
90
D
150
20
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10
E
200
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10
20
Figure 17: An example of construction of attributed directed weighted graph formated by the spatial characteristics and commuting OD matrix of the corresponding area consisting of 5 regions.
on the attributed nodes. this regard, we primarily adapt the graph generation model to the OD flow modeling task. And LargeCommuut DoreeoomuinnOOD containing diverse urban environments can support training on a large number commuting OD networks, which can capture the universal and distinct mobility patterns at the city level, leading to better generalizablility. The comparison of the traditional transfer paradigm and our novel generative paradigm is shown in Figure 5
To achieve better performance, adopt the advanced diffusion- based graph generation model to generate the weighted edges condition on the attributed nodes, which named WEDAN (Weighted Edges Diffusion condition Attributed Nodes). The framework of WEDAN is shown in Figure 18 We will introduce the relevant the graph construction, diffusion process, demoising network, and the training and generation process in detail next. The novelty of WEDAN shows in Appendix C.3
Graph Construction. As shown in Figure 17 we model an whole area as a graph G = (V,E). Specifically, each node E V on the graph represents region within that area, and the directed
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Pe (pt)
noise graph pr
raw graph Po
regions
regional characteristics
OD flows
Figure 18: The framework of WEDAN for network- generative commuting OD flow modeling.
edges Cty signify the commuting OD flows Fru.ry between regions. Herein, we let N | be the number of nodes in the graph, representing the number of regions, where denotes the cardinality of a set. Each edge corresponds to its unique origin node and destination node. The weight of each edge Wey, is the OD flow volume F.y- The graph is attributed with the spatial characteristics of each region r, which are represented as the node features X, of each node. The graph construction process is illustrated in Figure 17. Thus, the spatial characteristics of an area CR can be represented by a feature matrix X composed of the attributes of all nodes [urlr E RJ on the corresponding graph G, combined with the distances (dky]r, and ry E R) between all pairs of regions. Meanwhile, the commuting OD matrix F is equivalent to the set of all edges {ele } and their weights { (wele E E} on its graph G.
By constructing a conditional generative model PE(E, {wele E EHI, XR) that, given all nodes V and their attributes Xy of graph, generates all edges and the corresponding weights {we E on those edges, we can build an OD flow modeling model 0. The conditional distribution Po(E, {wele V, x mirrors Po (F(C).
Weighted Edges Diffusion Condition on Attributed Nodes. We will give a detailed introduction to the framework the weighted edges diffusion process, which models the conditional distribution (E, (wele E E Xx). As shown in Figure 18. the diffusion framework is composed of two parts: the forward diffusion process and the reverse denoising process pa. Both processes take place within the space of the edges & and the corresponding weights [wale E belong to the constructed attributed directed weighted graph.
Since OD matrices F E NNNN contain continuous flow values, forward diffusion process utilizes Gaussian noise to perform the diffusion process. The forward diffusion process is shown in Figure 18 from the right the left. It is important note that the noise perturbations applied to all edges independent. So the forward diffusion process can be described at the individual OD flow level by the following computational formula.
o ((F)) = -FFF VI F.Fl B.T),
T
(F.)..... F.J F II ((F
The reverse denoising process is the inverse of the forward diffusion process. In this context, the denoising process is facilitated by denoising neural network 0, which predicts the small noise to be removed based on the latent state of the noise space at step I, aiming to reach the noise state of step f - 1, in iteratively manner. Unlike the forward diffusion process, to ensure the modeling of the joint distribution of all elements in the OD matrix F, the noise to be removed for each edge needs to be determined based the entire state of the corresponding noisy data F'. Furthermore, to ensure the generation of OD matrices for new cities with given their spatial characteristics, we have designed the denoising process of OD matrices to be guided by the spatial characteristics of the corresponding cities, i.e., the nodes and their features. Therefore, the denoising step in reverse process can be represented as follows.
CR), Po (F F (T,CCR) = -FF ( (1-))))))11)))) (1))
(8)
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where
3. Vou. VI
po (o 10,,,, CR.
(F*
co(F CR)),
dz
Or 11-1--aa 3r, and oN -Ii-- 0g. Here, 5F(F", CR) is the noise predicted 0 based on the noisy state F", diffusion step and the spatial characteristics CR of the corresponding city.
The denoising network 0 is trained predict the noise Eo (F', t. CR. by minimizing the predictive errors. The parameterization and other detailed information of WEDAN is introduced in Appendix g such architecture of the denoising network, algorithms of training and generation processes.
To fully utilize the association between spatial interactions {dilrr and r, E R and the OD matrix F, have designed node and edge levels distance-based conditional guidance to direct the denoising generation. As shown in Figure 19. we perform spectral decomposition on the distance matrix to obtaín N Laplacian eigenvectors, which named distance- based Laplacian position encodings (d-LaPFs) are used to encode the specific position of each region in the planar urban space. Subsequently, the node features and edge features, before being inputted into each graph transformer layer, combined with the corresponding d-LaPEs and distances.
Distance-based guidance.
Log-Transform, Existing theoretical works have discovered scaling behaviors in human mobil- ity (Jiang et a 2016; Saberi et al, 2017: 2018 Li et all 2023b; Zhang et al, 2024), namely that many properties follow the power law distribution. To enable our model to better capture the het- erogeneity of OD flow distributions across different cities, we use log- org-transform to preprocess and post-process OD flows. The calculations are as follows.
F. log( Fig + 1),
(10)
F. exp(Fi,) 1.
where Fig is the log-transformed OD flow, which is used to train the denoising network. The gener- ated Pey, after inverse transformation, yields the real size of OD flows Fij.
C.1 DENOISING NETWORK
During each step in the reverse denoising process, the denoising network predicts the small Gaussian noise to be removed, based on the current noisy state. We adopt the transformer-basee neural network structure as the backbone, which has been proven to have strong learning and generalization capabilities across various domains.
As illustrated in Figure 19. the backbone of our denoising network is the graph transformer (Dwivedi & Bresson, 202(). It accepts inputs at both the node and edge levels, captures graph features, and then outputs noise predictions at the edge level. The characteristics of each region serve as node inputs, and the noisy OD matrix at the current state provides the edge inputs. They are processed separately through their respective Multilayer Perceptrons (MLPs) and then fed into the graph trans- former. The graph transformer consists of a series of layers. In each layer, every node computes attention weights with all other nodes through the self-attention mechanism and aggregates informa- tion from all other nodes based on these weights.
model the dependencies between nodes and edges, the weights computed through self-attention are fused with edge features using Feature-wiis Linear Modulation (FiLM) (Perez et al., 2018), resulting the final attention weights, Simultane- ously, the calculated attention information is also used to combine with the original edge features, serving as the new edge features for subsequent computations in the next layer denoising network. Moreover, after the aggregation of node and edge information, the data passes through ferd-forward network. The computations within each graph transformer layer can described by the following formula.
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1+T = O Σ
Ril 1+1 O (aij)
(1I)
Rd sof softmaz, (( (a:i)),
du
kJ Qhin K Khin dd
W e
dij
where hy and ely are the node and edge features the (-th layer, respectively. Qud,, K& and V are the query, key, and value matrices of the k-th attention head at the 1-th layer. Whd is the weight matrix of the k-th attention head at the (-th layer. O and O² are the output MLPs of the node and edge features at the I-th layer. dy the dimension of the query and key vectors. K the number of attention heads. Ne, is the set of neighbor nodes that are connected to node v,
After the layers, the final edge features are fed into the fully-connected layer to predict the noise.
C.2 TRAINING AND GENERATION
We the simple loss from DDPM (Ho et al, 2020) train the denoising networks in our attributed graph diffusion model. This involves minimizing the Mean Squared Error (MSE) between the noise predicted by the denoising network and the noise from the forward diffusion process. The calculation of this loss is as follows.
L = = ((,o) [|] - (F", C,CrR) C
(12)
where |i denotes the L - norm. The training algorithm shown Algorithm I The training and sampling methods are detailed in Appendix D.1
C.3 EXTENDED DISCUSSION ON RELATED WORKS OF WEDAN
WEDAN is novel and original model that applies denoising diffusion- -ased graph generation mod- els from network perspective to commuting OD flow generation. To our knowledge, this model is unique and has not been proposed elsewhere. The key novelties of WEDAN lies in two aspects:
It models all OD flows within a city as a directed weighted network, considering the entire OD network single data sample.
It utilizes the features of all regions (nodes) in the OD network guidance for the diffusion model, enabling the generation all edges and their corresponding weights.
It is worth noting that GraphMaker Li et (2023a)) also generates attributed graphs, but they differ significantly: WEDAN is specifically designed for the commuting OD flow generation task, empha- sizing that each OD flow is influenced by the attributes of its origin and destination nodes, resulting in continuous flow volumes. In contrast, GraphMaker focuses on generating large.sparse sparse graphs by determining the existence of edges between nodes. Additionally, other works Jo et al. (2022); Vi- gnac et al. (2022) generate both nodes and edge weights simultaneously, emphasizing the coupling between nodes and edges rather than using node attributes to guide edge generation.
D ADDITIONAL EXPERIMENTAL DETAILS
D.I TRAINING ALGORITHM Or GRAPH DENOISING DIFFUSION
The trained denoising network can be utilized in conjunction with the reverse denoising process to
generate the OD matrix for new cities, which lack any OD flow information, using their spatial charac- teristics. We adopt the sampling algorithm from Denoising Diffusion Implicit Models (DDIM) (Song
et al. 2020) to facilitate more efficient data generation. The sampling algorithm is shown in Algo- rithm
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Algorithm 1 Training of the Graph Diffusion Model
Input:
Graphs Gtrain that constructed from the data collected from the cities in training set
Output:
Learned noise prediction neural networks 8.
Sample a graph G from Grrain
2: Sample ( ~ (11, 2, T)..TT)
3: Sample e N (0, I)
4: loss = col - -1,,,,,,
5: optimizer.step(osss)
Algorithm 2 OD Matrix Generation through Trained Graph Diffusion Model
Input:
Spatial characteristics CR of a new city
Trained denoising network 0
Length of sub-sequence in DDIM sampling
Output:
OD matrix F of that new city.
1: Sample F" I0,
2: At
3: for to
T, At. ..., 1 do
4: F'-At + 1 F a-
ca (F'F (F*,t,CC) CRC)
var
V1-a.
5: end for
6: return FU
D.2 ARCHITECTURE OF DENOISING GRAPH TRANSFORMER
graph
distance matrix
edge
regional
[X,3
pt
noley
LaPE
distence
MLP
MLP
edge
H
E
distance guidance
Graph Transformer Layer
E
H
Graph Transfomer Layar
Graph Transformer
g4
Fully-sonnectedd Layer
nolea
Figure 19; The architecture of denoising neural network of our graph diffusion model.
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D.3 DETAILS FOR REPRODUCIBILITY
The computational resources we used to conduct the experiments are as follows: a server with Intel(R) (R) Platinum 8358P CPU @ 2.60GHz with 128 cores. The server is equipped with ITB of RAM and 8 NVIDIA A100 GPUs. For all the experiments in the paper, we run trials and report the average results.
D.4 ADDITIONAL RESULTS TRANSFERABILITY TO OTHER COUNTRIES
Model
CPCT
RMSEL
NRMSEJ
GM-P
0,240
101.6
1.752
RF
0.334
223.2
3.847
DGM
0,359
157.0
2.706
GMEL
0.362
149.1
2.570
NetGAN
0.331
198.9
3.429
WEDAN
0.485
72.68
1.253
Table 5: Transferability experiments of training models on the United States and generating OD flows for the United Kingdom.
We generated commuting OD flows for 326 Local Authority Districts (LADs) in the United Kingdom, where the flows among Middle Layer Super Output Areas (MSOA) within each LADs. The ground truth was obtained from the Office for National Statistics (ONS) of the UK. It is very difficult to obtain regional features in the UK with the same format and semantics as the US. Therefore, we used satellite images of regions to represent the input features consistently across these two countries. The experimental results are shown in Table 5. The results show that WEDAN outperforms other models in terms of CPC, RMSE, and NRMSE, demonstrating its strong generalization ability to other countries. This indicates that models trained on the US dataset exhibit some transferabilityy to other countries, particularly to developed countries like the UK. WEDAN benefits from graph generative modeling, achieving the best performance. However, the transferability cannot always be guaranteed, there may be significant differences between countries. We aim explore this direction in future work.
D.5 DETAILS OF PERFORMANCE ON THE HETEROGENEITY OF URBAN AREAS
From Figure 6(a), we observe that all models tend to perform better in smaller cities in terms of CPC, with performance declining as city size increases. This trend can be attributed to the increasing heterogeneity in OD flow distributions in larger cities. Smaller cities often have more homogeneous region-pairs with short-distance flows, making predictions relatively easier. In contrast, larger cities have both short-distance and long-distance commuting, leading to a long-tailed distribution OD flows and higher heterogeneity, which increases prediction difficulty. Figure 6(b) further illustrates that smaller cities tend to have higher RMSE values. This is because smaller cities typically exhibit higher flow volumes due a prevalence of short-distance commuting, which inereases the absolute prediction error. Conversely, larger cities often have sparser OD flows between many distant regions, with certain extreme flows, contributing large values but overall lower flow volumes, resulting in smaller RMSE. Figures b(c) and b(d) support similar conclusions for cities of varying structures. For larger monocentric and polycentric cities, models like DiffODGen, which incorporate hierarchical designs for large cities, perform well. However, DiffODGen struggles with the "others" category. typically smaller cities, where its performance less reliable. contrast, WEDAN, benefiting from large-scale training data, demonstrates robust performance across all city sizes and structures.
D.6 DETAILED ANALYSIS ON COMMONALITISS CAPTURED ACROSS URBAN AREAS
Figure reveals that cities of different types share certain common human mobility patterns (Liu et al. 2023; Zhou al, 2023). supporting the feasibility of using unified model to learn mobility
patterns across diverse cities. Modeling both commonalities and distinctions between cities helps enhance the model's generalization capability. Figures 7(a) and 7(b) show that both monocentric
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and polycentric cities achieve high performance during training, likely because these city types cover a wide range of human mobility patterns. However, Figures [cc and ((d) highlight that training solely small or large cities fails to achieve strong tramsferability across each other. This result demonstrates the existence of differences human mobility patterns across city types while also highlighting the value of training on diverse set of city types. Such diverse training data enables the model to effectively capture both the differences and the shared mobility patterns between cities.
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Reviews:
[{"review_id": "aZl4UUEFXF", "reviewer": "Reviewer_Zri5", "summary": "This paper introduces a large-scale dataset called LargeCommuingOD for commuting Origin-Destination (OD) flow generation and provides a benchmark for evaluating models in this domain. The key contributions are:\n\n1. Construction of a comprehensive dataset covering 3,333 diverse areas across the United States, including metropolitan areas, towns, and rural regions. The dataset spans 9,372,610 km² and contains commuting OD flows, sociodemographic data, and point-of-interest distributions for each area.\n\n2. Development of a benchmark framework to evaluate and compare different commuting OD flow generation models, addressing the lack of standardized evaluation in existing research.\n\n3. Benchmarking of 9 existing models, including physical models, classical machine learning approaches, and graph neural network models, using the new dataset.\n\n4. Introduction of a preliminary adaptation of graph diffusion models called WEDAN (Weighted Edges Diffusion condition on Attributed Nodes) for commuting OD flow generation.\n\n5. Analysis of model performance in terms of precision and generalizability across different urban environments.\n\nThe paper demonstrates that network-based generative models, particularly those leveraging graph diffusion techniques, achieve the best performance in both precision and generalization ability. This finding suggests new research directions in graph generative modeling for commuting OD flow generation.\n\nThe authors argue that their dataset and benchmark provide a valuable resource for researchers in urban planning, transportation, and related fields, enabling more comprehensive evaluation and development of commuting OD flow generation models.", "questions": "What do the authors see as the most promising directions for future research based on their findings? Are there specific areas where they believe the dataset and benchmark could be most impactful?", "limitations": "", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Originality:\n1. The creation of a large-scale, comprehensive dataset (LargeCommuingOD) for commuting Origin-Destination (OD) flow generation covering 3,333 diverse areas across the United States is highly original. This dataset significantly expands on previous efforts in terms of scale and diversity.\n\n2. The paper introduces a novel adaptation of graph diffusion models called WEDAN (Weighted Edges Diffusion condition on Attributed Nodes) for commuting OD flow generation, exploring a new paradigm in this field.\n\nQuality:\n1. The dataset construction process is rigorous, combining multiple reliable data sources including the U.S. Census Bureau, American Community Survey, and OpenStreetMap.\n\n2. The analysis is thorough, examining both precision and generalizability of the models across different urban environments.\n\nClarity:\n1. The paper is well-structured, clearly defining the problem, describing the dataset, and presenting the benchmark results.\n\n2. Figures and tables effectively illustrate the dataset characteristics and model comparisons.", "weaknesses": "1. **Limited Exploration of Model Interpretability**: The paper emphasizes model performance but lacks a thorough examination of model interpretability, particularly for the network-based generative models that achieve the best results. Understanding the reasons behind their success could offer valuable insights for urban planners and policymakers.\n\n2. **Insufficient Exploration of Edge Cases**: The paper would benefit from a more in-depth analysis of how the models perform in extreme or atypical urban environments within the dataset. This could reveal potential limitations or areas for improvement.\n\n3. **Lack of Discussion on Model Fairness and Bias**: Given the diverse nature of the dataset, an analysis of potential biases in the models' predictions across different urban environments (e.g., rural vs. urban, high-income vs. low-income areas) would be beneficial.\n\n4. **Basic Benchmark Methods**: Many of the selected benchmark methods are too basic. The paper should include more advanced models based on transformers and diffusion, such as those discussed in [this paper](https://arxiv.org/abs/2411.04453) and [this one](https://arxiv.org/abs/2402.15398).\n\n5. **Questionable Significance of Claims**: Are the claims regarding the significance of the research truly justified? Without this dataset, researchers in the field might only need about a week to conduct their own benchmarks. Additionally, there are already open-source frameworks, like LibCity, that have collected these benchmarks (see [LibCity](https://github.com/LibCity/Bigscity-LibCity)).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wdgeQT1Pec", "reviewer": "Reviewer_KbsU", "summary": "Commuting OD flows are critical inputs for urban planning and transportation, while the high data collection cost results in a lack of high-quality datasets. This paper introduces a large-scale dataset containing commuting OD flows for 3,333 areas including a wide range of urban environments around the United States. Based on this dataset, authors further benchmark widely used models for commuting OD flow generation. They find that, owing to the rich information contained in the constructed OD dataset, a generative method that considers each OD matrix collected from a county/city as a network and learns to generate it given conditional information can significantly outperform other methods. This may point to a new direction in this domain, which relies on collection of high-quality OD datasets.", "questions": "+ Can authors demonstrate that models developed based on this US dataset can manage to transfer to other countries?\n+ In Figure 1, why Commuting Flows are processed to generate Regional Socio-demographics?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "+ Commuting OD data is valuable in building smart cities, while the data is generally lacked in many areas around the world. The synthetic data generation technique is a promising solution for this problem, requiring high-quality and high-volume data for training ML models. This paper makes a good contribution towards this objective.\n+ According to Table 1, the constructed dataset is more useful than existing ones, in terms of size and richness. The data analysis is sufficient, as in Figure 2-4.\n+ The benchmark is designed in a reasonable manner. It covers a wide range of mainstreaming approaches in this field, and evaluate the model performance in a comprehensive way, including both flow generation accuracy and property distribution similarity.\n+ The observation based on benchmark results is insightful. Based on the constructed large-scale OD dataset that is unseen in previous works, authors demonstrate the powerfulness of graph generative modeling in terms of generating OD flows for diverse cities. This may point to a new direction in this domain.", "weaknesses": "+ The presented results are informative, but the explanations are rather insufficient. For example, Figures 6 and 7 should be explained in detail.\n+ Although authors discuss the limitations of the constructed dataset, the spatial scale (limited to the United States) and temporal scale (one year) can degrade its usefulness in applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2pdQbFMo2z", "reviewer": "Reviewer_Rw5X", "summary": "This paper constructs a large-scale dataset (LargeCommuingOD) containing commuting OD flows for 3,333 diverse areas around the United States covering 9,372,610 km2 including a wide range of urban environments. Based on the LargeCommuingOD, the authors benchmark the existing widely used models for commuting ODflow generation. These efforts make up for the lack of comprehensive dataset and the absence of unified and systematic evaluation. Furthermore, this paper find out that network-based modeling for commuting OD flow supported by their dataset gives a promising performance. Thus, training on a large number of commuting OD networks can help the generative models to capture universal as well as distinct mobility patterns at city level, and therefore enhance the generalization ability.", "questions": "Please refer to Weakness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1. This paper constructs dataset covers 3,333 areas around the United States, providing a much broader spatial scale comparing to existed datasets.\nS2. The dataset constructed in this paper covers metropolitan areas, towns, and rual areas, which is more comprehensive then existing datasets the focuses on usually a single type of urban environment.\nS3. Based on the LargeCommuingOD, the authors benchmark the existing widely used models for commuting ODflow generation, which makes up for the absence of unified and systematic evaluation.", "weaknesses": "W1. Please clarify whether WEDAN is a newly proposed model, or the idea of it has already been proposed somewhere else.\nW2. In Sec. 2.2, the link of the released dataset is given in plaintext, which may violate the anonymous policy.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KMv7STyo7b", "reviewer": "Reviewer_NxhM", "summary": "This paper introduces a large-scale dataset, LargeCommutingOD, which captures commuting origin-destination (OD) flows for 3,333 diverse regions across the United States, spanning 9,372,610 square kilometers and encompassing a variety of urban environments. Compared to existing datasets, LargeCommutingOD offers broader and more comprehensive coverage and is made publicly available. The authors benchmark several baseline models using a standardized evaluation protocol, uncovering that network-based generative models could be a promising avenue for future research in this domain.", "questions": "Please explain the questions.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The authors present a large-scale and comprehensive dataset, LargeCommutingOD, which includes commuting OD flows for 3,333 regions, covering 9,372,610 square kilometers across diverse urban environments in the United States. This dataset provides more extensive coverage than existing resources and is openly accessible, promoting transparency and reproducibility.\n2. By comparing LargeCommutingOD to previous datasets, the authors demonstrate that it captures a broader range of urban characteristics and types, further supporting its utility for various research purposes.\n3. The authors implement and evaluate several baseline models to demonstrate the feasibility and relevance of their dataset. Notably, they identify that network-based generative models show considerable potential, which opens avenues for further investigation.\n4. Upon reviewing the proposed code, I observed that the authors have implemented a comprehensive set of baselines, which could even serve as a robust library for OD flow generation. I hope the authors can commit to releasing this code if the paper is accpeted.", "weaknesses": "1. Given the dataset's coverage of 9,372,610 square kilometers and its focus on approximately 3,000 regions, the OD metrics may primarily reflect macro-level commuting patterns rather than fine-grained dynamics within individual cities. This limitation may restrict the dataset's applicability for studies requiring detailed intra-city analyses.\n2. The OD matrix, denoted as $\\mathbf{F} \\in \\mathbb{R}^{N \\times N}$ where $N$ exceeds 3,000, may be sparse. Providing a visual representation, such as a heatmap of the OD matrix, would clarify its density and structure, which could help readers better understand the data distribution.\n3. The authors included a non-anonymized GitHub link (https://github.com/tsinghuafib-lab/CommutingODGen-Dataset) in the submission, potentially compromising anonymity. And the authors have specified that only their names need to be anonymized.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)