The Time-Varying of Topological Characteristics: Analysis Based on

the Temporal Network on Public Bikes

Wang Li-Na

1,2 a

and Shen Jiang-Long

College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, China

Inner Mongolia Key Laboratory of Statistical Analysis Theory for Life Data and Neural Network Modelling,

Hohhot 010051, China

Keywords: Temporal Network, Multilayer Network, Complex System, Public Bike.

Abstract: Most networks evolve in time. We study the structure of interaction with time. Compared with the tradition-

al public transport, the flexibility of public bikes and the randomness of users' riding behaviours make the

riding route and riding time full of uncertainty. It is the task of scientific research to explore the regularities

behind these uncertainties. By mining the data of user's riding trajectories; we construct the temporal net-

work and the 24-layer multilayer network respectively. The topological characteristics of network presents

double peak. There is a strong correlation between the topological parameters, including positive and nega-

tive correlations. Furthermore, bike ridings among stations distribute heterogeneously and the hourly flow

of stations distributes heterogeneously. Transport system is a typical complex system. This research pro-

vides new evidence for empirical research on temporal network, multilayer network and transport network.

https://orcid.org/0000-0001-6422-5709

1 INTRODUCTION

In recent years, researches on transport networks

have been receiving close attention by the physics

community. Transport system is a typical complex

system. Scientists use the thoughts and methods of

network to study traffic problems. For example, traf-

fic congestion problems (Jang et al., 2019), invul-

nerability research (Zhang et al., 2018; Cats et al.,

2020) and key road identification (Feng et al., 2019).

Traditional public transport systems, composed

of buses and subways, have fixed routes, fixed

mileage, fixed running time, and fixed running

interval. We study the public transport system

composed of bikes in this paper. Compared with

traditional public transport systems, the randomness

of individual users' riding behavior makes the riding

routes, riding mileage and riding durations full of

uncertainty. It is the task of scientific research to

explore the regularities behind these uncertainties.

Traditional studies on complex network

consider time-independent structures, but most

networks evolve in time (Porter, 2020). In this paper,

we study the temporal network, structure of

interaction with time. The time-dependent nature of

the network reflects the nature of system, and these

time-dependent behaviors are manifestations of

human behavior. Citi Bike of New York is a public

bikes system. Based on user's riding data, the flow

information among stations and the time information

of user's riding are excavated. We analyze the flow-

weighted temporal network and the 24-layer

multilayer network respectively, to capture the

unobservable characteristics of networks under the

static model.

The paper is arranged as follows. We introduce

related works on transport networks in the second

part. In the third part, we introduce the public bike

system of Citi Bike, and the modeling method of the

temporal network and the definition of the 24-layer

multilayer network are given. Then the topological

characteristics of the networks are analyzed in the

fourth part. Finally, we summarize the conclusion.

2 RELATED WORKS

Recent years, the application of network ideas to the

study of public transport has become a research

422

Li-Na, W. and Jiang-Long, S.

The Time-Varying of Topological Characteristics: Analysis Based on the Temporal Network on Public Bikes.

DOI: 10.5220/0011949800003612

In Proceedings of the 3rd International Symposium on Automation, Information and Computing (ISAIC 2022), pages 422-427

ISBN: 978-989-758-622-4; ISSN: 2975-9463

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

hotspot. Sienkiewicz et al. established the public

transport network of Polish cities, and found the

small world and hierarchical characteristics (Sien-

kiewicz et al., 2005). Ferber et al. studied the public

transport systems in major cities such as Los Ange-

les, and found that: the networks show the character-

istics of small world or scale free (Ferber et al.,

2009). Derrible et al. analyzed the urban subway

system and found that most of the subway networks

are scale free (Derrible et al., 2010). Taking streets

of London and highways of American for instance,

Viana et al. applied multidimensional scaling meth-

ods to visualize the small world characteristics of the

network (Viana et al., 2011). Xu et al. analyzed the

public transport of Chinese cities and found that

these public transport networks have small world

features (Xu et al., 2013). Gallotti et al. studied the

public transport in Britain from the perspective of

multi-layer network (Gallotti et al., 2015). Using

methods of network, Bona et al. analyzed the public

transport of Curitiba in Brazil, and found the charac-

teristics of small world and scale free (Bona et al.,

2016). Using the network approach, Ren et al. ana-

lyzed the public transport of Shenyang and found

that the network is scale free (Ren et al., 2016).

Candelleri et al. analyzed the public transport of

Florence in Italy and Attica in Greece, and the net-

works were found to be potentially vulnerable (Can-

delleri et al., 2019). Yang et al. developed a network

analysis model to study the accessibility of public

transport (Yang et al., 2019). Using smart card data,

they examined the association between public

transport in Wuhan and urban accessibility. Ma et al.

established a bus-subway network, constructed a

vulnerability operator, and studied the impact of

rainstorm on urban public transport (Ma et al.,

2019). Based on the network model, Yu et al. stud-

ied the spatial and temporal distribution of the metro

passenger flow in Nanjing (Yu et al., 2020). Wang et

al. analyzed the bus systems in Hohhot by using

network modelling, and the network has the charac-

teristics of small world and robustness (Wang et al.,

2020). Cao et al. analyzed the public transport in

Changsha and found that the network has scale-free

characteristics but does not satisfy the small world

characteristics (Cao et al., 2020).

The researches on the above mentioned public

transport networks are mainly focused on the bus

and subway systems. These traditional public

transport systems have fixed routes, fixed mileage,

fixed running time, and fixed running interval. Un-

like them, bikes are more convenient. Public bikes

have recently entered the field of public transport as

a new sharing tool. The public bikes take on the

functions of public transport, enrich the types of

public transport, and enhance the efficiency of pub-

lic transport.

In the past years, scientists have used thoughts

of networks to study the public bike system in Lon-

don and some cities of China. Munoz et al. studied

the London public bike system from the perspective

of network (Munoz et al., 2018). Communities were

regarded as nodes. If there were public bike riding

trajectories between communities, the corresponding

nodes were connected. Saberi et al. regarded public

bike stations as nodes of the network. If there was

riding behaviour between stations, the nodes were

connected. They found that the cumulative degree

distribution of London public bike network is power

law (Saberi et al., 2018). Using the same modelling

method, Wei et al. studied public bike system of

Yixing city, and found that the degree distribution

and strength distribution of the network obeys nor-

mal distribution (Wei et al., 2019). Yao et al. con-

structed a public bike network of Nanjing, with sta-

tions as nodes and the number of rides between sta-

tions as the edge weight. They found that the degree

distribution of the network is power law (Yao et al.,

2019). In addition, Shi et al. used the same method

to build the public bike network of Hangzhou. They

divided the network community by different modu-

lar algorithm (Shi et al., 2019).

Citi Bike is the largest bike sharing program in

the United States. The existing literature on the Citi

Bike system is mainly on traffic flow prediction.

Based on clustering and geographically weighted

regression, Bao et al. constructed the relationship

between traffic flow and various factors. They found

that the split riding model gave a better prediction

(Bao et al., 2018). Wang et al. predicted bike de-

mands based on the feature model with contextual,

correlation and user features (Wang et al., 2018).

Using graph convolution neural network, Lin et al.

(Lin et al., 2018) and Yang et al. (Yang et al., 2018)

predicted the bike demands per hour.

3 DATA AND METHODS

3.1 Data

The Citi Bike is designed for quick trip, and it is a

fun and affordable way. From the Citi Bike official

website (https//www.citibikenyc.com), download the

user's riding trajectory data of October 3, 2017. The

format of the initial data in the Citi Bike system is

shown in Figure 1. Each column represents a com-

plete riding trajectory, including the riding duration,

The Time-Varying of Topological Characteristics: Analysis Based on the Temporal Network on Public Bikes

423

the start time and start location, and the end time and

location. The time information is accurate to sec-

onds, and the location information contains the lon-

gitude and latitude of the station. In addition, user

types include subscriber and non-subscriber. The

subscriber records gender characteristic, and gender

tags include 1 (male) and 2 (female). While the non-

subscriber does not record gender characteristic and

tag 0 is used to indicate.

For instance, the second column of Figure 1

shows a riding trajectory of a male subscriber. At

00:00:00 on October 1, 2017, he rented a public bike

with ID 30951 at station "9 Ave & W 45 St". After

riding for 457 seconds, he returned to the station "11

Ave & W 41 St". Delete trajectories with too short

riding durations, considering the abnormal ridings

caused by vehicle failure or other reasons. Delete the

trajectories with riding duration less than one mi-

nute. Delete trajectories with too long riding dura-

tions, considering the abnormal ridings caused by

vehicle theft, user forgetting or other reasons. Delete

the trajectories with riding duration longer than six

hours. After data preprocessing, 69066 valid data of

the riding trajectories are retained.

Figure 1: Examples for the data of user's riding trajectories

in Citi Bike.

3.2 Methods

Based on user's riding data, the flow information of

users among stations and the time information of us-

er's riding are excavated. The flow information will

be used to build the weight of the edge. The time in-

formation of user's riding will be the basis for con-

structing the time layer. Taking the riding time as

the hierarchical label, the flow-weighted temporal

network and the 24-layer multilayer network are

constructed respectively.

The network modelling method is illustrated by

taking the data of four riding trajectories as an ex-

ample. Assume that there are four riding trajectories,

as shown in Figure 2(a). There are two riding trajec-

tories from station A to station B. One riding trajec-

tory from station A to station C, and the other one

from station B to station C. The bike station is re-

garded as the node of the network. An edge will be

linked between the nodes if the riding behaviour oc-

curs between the stations. The direction of the edge

is from the start station to the end station. The traffic

flow between the stations is taken as the weight of

the edge. A flow-weighted directed network is con-

structed. According to the time information of riding

behaviour, the temporal network is established, as

shown in Figure 2(b).

(a)

(b)

Figure 2: The modelling diagram of temporal network.

When t = 6, there are two riding trajectories gen-

erated from station A to station B and to station C.

Thus, the network at t = 6 is composed of three

nodes A, B and C. While, at time t = 7, there is only

one riding trajectory generated from station B to sta-

tion C, thus the network at t = 7 consists of node B

and node C. Compared with the static network,

nodes in the temporal network change dynamically

with time. In addition, edges between nodes in the

temporal network are not persistent. For example,

there is no edge between nodes B and C at t = 6, but

directed edges exist between nodes B and C when t

= 7. The temporal network enables understanding of

network changes over time.

Through the above modelling method, each time

layer network can be obtained, and we can study

changes of the network over time. In addition, all

time layers are coupled into a network without con-

sidering the inter-layer links. That is, the adjacency

matrix of each layer is coupled in a hyper-adjacency

matrix. Suppose that the adjacency matrix of each

layer network is A

(t)

, t = 0,1,...,23. A

(t)

is an n

-order

square matrix. Where n

represents the number of

nodes per time layer network. They are different

from layer to layer. Without considering the inter-

layer links, the hyper-adjacency matrix is defined as

ISAIC 2022 - International Symposium on Automation, Information and Computing

424

()



















(1)

Hyper-adjacency matrix A is an n

-order square

matrix. It corresponds to a 24-layer multilayer net-

work.

4 NETWORKS

4.1 Temporal Networks

The temporal network dynamically displays the

changes in the system twenty-four hours a day. Ac-

cording to the above method, the temporal network

is established, as shown in Figure 3. Obviously, the

network is time-varying. At t = 3, the network is

sparse. However, at t = 8 and t = 17, the network is

obviously dense, and the naked eye can no longer

count the number of nodes and edges. In addition,

the hub nodes (red colour) have also changed with

time. Through the visualization of these 24 net-

works, we can intuitively find that the network is

denser during the day and sparse at night. The dy-

namic change of the network with time is related to

the law of human behaviour. During rush hours on

and off work, a large number of riding behaviours

occur, which change the topology of the network.

In temporal network, the number of nodes and

the number of edges are time-varying. The time se-

ries of changes is shown in Figure 4. On the left and

right sides of the ordinate, the number of nodes and

the number of edges are identified respectively. The

number of edges shows bimodal feature. The first

peak appeared at 8 o'clock and the second peak ap-

peared at 17 o'clock, which indicate that a large

number of riding trajectories occurred during these

two periods. This is related to the law of human be-

haviour. More users use public bikes during rush

hours. During the period from t = 6 to t = 20, the

number of nodes in the network is stable. In the sys-

tem, there are about 600 stations with bike rental.

When t = 3, the number of nodes and the number of

edges are the least. At this time, there are only less

than one hundred stations with bike rental.

Figure 3: The temporal network.

In a network, the number of connected edges is

called the degree of the node; in a weighted network,

the sum of the edges weights is called the strength of

the nodes. In our flow-weighted temporal network,

the degree k

of node i indicates that there are riding

trajectories between the station i and other k

stations,

and the strength s

represents the total traffic flow be-

tween station i and other k

stations. The time-series

changes of the topological characteristic parameters

of the temporal network are shown in Figure 5. The

The Time-Varying of Topological Characteristics: Analysis Based on the Temporal Network on Public Bikes

425

average degree <k>, the average strength <s>, and

the clustering coefficient C show obvious bimodal

characteristics. The two peaks appeared at 8 a.m. and

17 p.m. respectively. This is consistent with the bi-

modal feature of the number of edges in the temporal

network. When studying the time-varying nature of

temporal network, we find that some topological pa-

rameters are strongly correlated.

Figure 4: The number of nodes and edges of the temporal

network.

Figure 5: The topological characteristics of temporal net-

works: average degree, average strength, average smallest

path length and average clustering coefficient.

In fact, by analyzing correlations of topological

parameter time series, it can be found that: the num-

ber of edges, average degree, average strength and

clustering coefficient show strong positive correla-

tion (as shown in Figure 6); when the number of

nodes is stable (from t = 6 to t = 20), the average

shortest path length shows a strong negative correla-

tion with other topological parameters (as shown in

Figure 6(b)). When the number of nodes is fixed, the

average degree of the network will be larger if there

are more connected edges. More edges will bring

more total edge weights, so the average strength will

be greater. The existence of more edges will make it

easier to form triangles, and then the average cluster-

ing coefficient will be larger. In addition, more edges

increase the accessibility of the network, which re-

sults in a smaller average shortest path length. It can

be found that the average shortest path length of the

network is smaller at 8 a.m. and 17 p.m. Therefore,

from 6:00 to 20:00, the number of nodes is stable,

and the number of edges, the average degree, the av-

erage strength and the clustering coefficient show a

highly similar time-series trends. While the average

shortest path length exhibits the opposite trends.

Figure 6: Diagram for the correlation matrix of topological

parameters. The analysis object of (a) is the whole time

series of five topological parameters; and (b) is for frag-

ments of time series, from t = 6 to t = 20.

4.2 The 24-Layer Multilayer Network

Figure 7: The degree distribution p(k), the strength distri-

bution p(s) and the strength-difference distribution p(s

in-

out

) of the 24-layer multilayer network.

The time-varying characteristics of network topolo-

gy are found through the temporal network. On the

other hand, according to Equation (1), all the time

layers are coupled, and the 24-layer multilayer net-

work is used to analyze the Citi Bike system as a

whole. The network has 11988 nodes and 60786

edges. The degree distribution and the strength dis-

tribution are all power law distribution, as shown in

Figure 7(a) and Figure 7(b). Eighty percent of nodes

have a degree value less than 15. Only one thou-

sandth of the nodes has a degree value greater than

100. Bike ridings among rental stations distribute

heterogeneously. Furthermore, ninety percent of

nodes have a strength value less than 30. Nodes with

strength value greater than 100 account for three

thousandth. At bike rental stations, the hourly bike

flow distributes heterogeneously.

Strength-difference of a node is equal to the in-

strength minus the out-strength, which can measure

the net flow of information at the node. In the 24-

layer multilayer network, the strength-difference s

in-

out

of node i represents the net bike flow at the rental

station i. When s

in-out

≈ 0, the inflow and outflow of

bikes are in balance and the site is called a balanced

ISAIC 2022 - International Symposium on Automation, Information and Computing

426

station. When s

in-out

>0, the inflow is large and the

outflow is small. For this type of site, we need to

consider the transfer out of bikes when scheduling.

When s

in-out

<0, the inflow is small and the outflow

is large. Such site needs to transfer in bikes to meet

larger rental demands of bikes. The strength-

difference distribution p(s

in-out

) of the 24-layer multi-

layer network obeys Gaussian distribution, as shown

in Figure 7(c). The mean of the distribution is 0.045

and the standard deviation is equal to 2. In general,

at most sites, the number of rental bikes and the

number of returned bikes can maintain a balance.

5 CONCLUSIONS

Transport system is a typical complex system. We

study the structure of networks evolving with time.

In the temporal network, the number of edges, the

average degree, the average strength, and the cluster-

ing coefficient present obvious bimodal characteris-

tics. The two peaks appeared at 8 a.m. and 17 p.m.,

which is consistent with rush hours on and off work.

The time-dependent nature of the network reflects

the nature of system, and these time-dependent na-

ture are manifestations of human behaviour. In the

24-layer multilayer network, the degree distribution

is power law, the strength distribution is power law,

and the strength-difference distribution obeys Gauss-

ian. In the system, bike ridings among stations dis-

tribute heterogeneously and the hourly flow of the

station distributes heterogeneously. In most stations,

the number of rental bikes and returned bikes main-

tain balance. Furthermore, in temporal network, we

found strong correlations of topology parameters.

The research provides evidence for empirical re-

searches on temporal network, multilayer network

and transport network.

ACKNOWLEDGEMENTS

This work was supported by the Natural Science

Foundation of Inner Mongolia (Grant No.

2022LHMS01005) and the Fundamental Research

Funds for the Directly Affiliated Universities of In-

ner Mongolia (Grant No. JY20220095).

REFERENCES

Bao J, Shi X M, Zhang H, 2018. IEEE Access 6(11)

76049.

Bona D A A, Fonseca K, Rosa M O, Luders R, Delgado M

R, 2016. Mathematical Problems in Engineering

2016(6) 3898762.

Candelieri A, Galuzzi B G, Giordani I, Archetti F, 2019.

Public Transport 11(1) 27.

Cao N, Cao H Q, 2020. Chinese journal of Physics 65(3)

389.

Cats O, Krishnakumari P, 2020. Physica A 549(13)

124317.

Derrible S, Kennedy C, 2010. Physica A 389(17) 3678.

Feng H F, Bai F S, Xu Y J, 2019. Physica A 535(23)

122337.

Ferber C O, Holovatch T, Holovatch Y, Palchykov V,

2009. European Physical Journal B 68(2) 261.

Gallotti R, Barthelemy M, 2015. Scientific Data 2(1) 1.

Jiang X R, Wen X X, Wu M G, Song M, Tu C L, 2019.

Physica A 523(11) 364.

Lin L, He Z B, Srinivas P, 2018. Transport Research Part

C 97(12) 258.

Ma F, Liu F, Yuen K F, Lai P, Sun Q P, Li X D, 2019.

International Journal of Environmental Research and

Public Health 16(3) 329.

Munoz M F, Han k, Klemmer K, Jarvis S, 2018. ACM In-

ternational Joint Conference on Pervasive and Ubiq-

uitous Computing Singapore 1015.

Porter M A, 2020. Emerging Frontiers in Nonlinear Sci-

ence 32(NSCH) 131.

Ren T, Wang Y F, Liu M M, Xu Y J, 2016. Chinese Phys-

ics B 25(2) 5.

Saberi M, Ghamami M, Gu Y, Shojaei M H, Fishman E,

2018. Journal of Transport Geography 66(1) 154.

Shi X Y, Wang Y, Lv F S, Liu W H, Seng D W, Lin F,

2019. Journal of Visualization 22(6) 1177.

Sienkiewicz J, Holyst J A, 2005. Phys. Rev. E 72(4)

046127.

Viana M P, Costa L F, 2011. Physics Letters A 375(15)

1626.

Wang K L, Gulsah A, Chen Y J, 2018. Transport Re-

search Part A 116(10) 1.

Wang L N, Wang K, Shen J L, 2020. Physica A 545(9)

123498.

Wei S, Xu J G, Ma H T, 2019. Transport Planning and

Technology 42(3) 293.

Xu Q, Zu Z H, Xu Z J, Zhang W D, Zheng T, 2013.

Transport Systems Engineering Information Techno-

logy 13(1) 193.

Yang H, Xie K, Ma Y F, Wang Z Y, 2018. Transport Re-

search Record 2672(36) 92.

Yang R Q, Liu Y L, Liu Y F, Liu H, Gan W X, 2019. Sus-

tainability 11(20) 5634.

Yao Yi, Zhang Yifang, Tian Lixin, Zhou Nianxing, Li,

Zhilin, Wang Minggang, 2019. Sustainability 11(19)

5425.

Yu W, Ye X F, Chen J, Yan X C, Wang T, 2020. Sustain-

ability 12(3) 1113.

Zhang H, Zhuge C X, Zhao X, Song W B, 2018. Interna-

tional Journal of Modern Physics C 29(1) 1850004.

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