Field Architecture for Traffic and Mobility Modeling in

Int. J. Ad Hoc and Ubiquitous Computing, Vol. 0, No. 00, 2004

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Field Architecture for Traffic and Mobility Modeling in Mobility Management Jingbo Sun, Hongbo Si, Yue Wang, Jian Yuan and Xiuming Shan Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China E-mail: {sjb06, shb07}@mails.tsinghua.edu.cn E-mail: {wangyue, jyuan, shanxm}@tsinghua.edu.cn

Ilsun You* School of Information Science, Korean Bible University, 16 Danghyun 2-gil, Nowon-gu, Seoul, South Kroea E-mail: [email protected] *Corresponding author Abstract: With the emergence of new technologies in mobile communication, users’ demand on service quality is growing aggressively, which makes mobility management a great challenge and opportunity. Aggregate mobility modeling, studying macroscopic rule of human movement, will make a huge contribution to the performance of mobility management. This paper is primarily focused on establishing an architecture based on field theory, using scalar and vector field to describe traffic and mobility respectively. Based on their temporal-spatial evolution, we try to discover the relationship between traffic field and mobility field. Moreover, Principal Components Analysis is adopted to decompose mobility field in order to find its typical patterns in both time and space domain. Meanwhile, we make use of complex analysis to describe the details of the inner movement for mobility field. This field architecture fits for mobility management in large temporal-spatial scale, since it not only benefits qualitative analysis of traffic and mobility in the perspective of field, but also provides a theoretical foundation and insight for issues in mobile communication. Keywords: mobility management; aggregate mobility modeling; field architecture. Biographical notes: Jingbo Sun is a Ph.D. candidate in the Department of Electronic Engineering at Tsinghua University. He received his B.S. degree from the School of Electronics and Information Engineering at Harbin Institute of Technology in 2006. His research interests include wireless networks and mobility management. Hongbo Si received his B.S. and M.S. degrees in Tsinghua University in 2007 and 2010 respectively. He is a Ph.D. student in the University of Texas at Austin at present. His research interests include wireless networks and information theory. Yue Wang received his Ph.D. degree from the Electronic Engineering Department of Tsinghua University in 2005. He is now an assistant professor at Tsinghua University. His research interests include computer networks and data fusion. Jian Yuan received his Ph.D. degree in electrical engineering from the University of Electronic Science and Technology of China, in 1998. He is currently an associate professor in the Department of Electronic Engineering at Tsinghua University, Beijing, China. His interest is complex dynamics of networked systems, and dependability of mobile networks. Xiuming Shan received his B.S. degree from the Electronic Engineering Department of Tsinghua University in 1970. He is the head and chair professor of Institute of Highspeed Signal Processing and Network Transmission, Electronic Engineering Department, Tsinghua University. His research includes radar signal processing, computer networks, and complex systems. IlsunYou received his MS and Ph.D. degrees in the Division of Information and Computer Science from the Dankook University, Seoul, Korea in 1997 and 2002, respectively.He is now an assistant professor in the School of Information Science at the Korean Bible University. His research interests include MIPv6 security, key management, authentication and access control. He is a member of the IEEK, KIPS, KSII, and IEICE.

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Int. J. Ad Hoc and Ubiquitous Computing, Vol. 0, No. 00, 2004

1 Introduction Mobility is believed to be one of the key features of current and future networks, which provides both challenge and opportunity for mobility management (Akyildiz et al., 2002; Yan et al., 2010; Bernardos et al., 2010; Kafle and Inoue, 2010). Mobility management aims to support seamless roam for a single user or a group of users in the network, and one basic issue is to explore the underlying rule of movement. However, this fact is not sufficiently considered in most of previous researches. Users are basically considered as moving randomly or in a simply way. But in fact, they are surely constrained by both behavior habits and geographic factors. This kind of deterministic rule underlying uncertain movement behavior needs exploration. Mobility modeling provides model parameters for system design and performance analysis. Since the end of last century, personal mobility modeling has been gradually paid attention to and exhibits more value for protocol design of mobility management (Liu and Maguire, 1995; Soh and Kim, 2001; Erbas et al., 2001). With the development of mobile communication, the technology of personal mobility modeling has been relatively mature and played a significant role in the performance improvement of handoff. However, another important category of mobility modeling, namely, aggregate mobility modeling, has been an unexplored area until recent years. The pioneering work by MIT Media Lab (Reades et al., 2007, 2009; Ratti et al., 2006) became a forerunner and representative in this field, and several continued researches developed from diverse points of view strengthened this topic and made it a hot topic. Real Time Rome project (Rojas et al., 2008) conducted by MIT and Italy telecommunication tried to explore the relationship between traffic history and user behavior. Through symmetrical gridding the whole coverage area of Rome city, it is discovered that the distribution of traffic load reflects users’ moving and living rules. Literature (Treuille et al., 2006) regarded aggregate mobility as the aggregation of individual movement. By introducing the concepts of hydrodynamics, moving model of singular person is established, and group mobility model is further constructed by assuming different practical scenarios. Literature (Brockmann et al., 2006) adopted a special media, bill, as the method of studying users’ movement trend. Through the analysis of bill currency among four cities, the researchers explored the rules of aggregate mobility in large spatial scale in the United States. This paper proposes a new perspective to study mobility management, field architecture, in which traffic distribution is considered as a scalar field, while mobility distribution as a vector field. So relative concepts and conclusions in field theory could be introduced into this architecture to describe traffic and mobility, which guarantees both the clarity of visual effect and the depth of theoretical analysis.

c 2009 Inderscience Enterprises Ltd. Copyright

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The rest of this paper is organized as follow. Section II introduces the basic framework of field architecture. Section III and IV describe traffic field and mobility field respectively. The last section makes a conclusion of the whole paper.

2 Field Architecture 2.1 Concepts Traffic and mobility are two key issues of mobility management, which are closely related and influenced by each other. Hence, we endeavor to establish a united architecture for their modeling. What we concern about is their spatial-temporal distribution, which may fall into the scope of field theory, since field itself is used to describe the spatial-temporal distribution of certain physical quantity. So field architecture is introduced to model traffic and mobility. Definition 1 (Traffic Field) For every point P in 2demensional space D at an appointed time t, if the traffic density at that point could be described as a scalar number, it is called a traffic field in this area, marked as T (P, t). Here traffic density means either the density of traffic’s occurring times or the one of traffic amount. From the definition, it shows that traffic field is a plane scalar field. Thus related concepts, such as isoline and gradient, can be adopted to describe traffic. Definition 2 (Mobility Field) For every point P in 2-demensional space D at an appointed time t, if the population velocity at that point could be described as a vector, it is called a mobility field in this area, marked ~ (P, t). as M Here population velocity means the amount of moving of users in unit time, which equals to population density multiply velocity at that point. From the definition, it indicates that mobility field is a plane vector field. Thus relative concepts, such as vector line, divergence and rotation can be employed to represent mobility accordingly. Vector line shows the spatial distribution of field, divergence and flux describe the diffusion ability of field, and rotation and circulation describe the circumgyration ability of field.

2.2 Relationship of Traffic and Mobility Field As the two key issues of mobility management, traffic and mobility play an importance role in mobile communication. Many researchers have devoted to exploring their relationship. Here, their relation is briefly revealed under the field architecture. In fact, there lies another field connecting traffic and mobility field. Population field describes the density distribution of users, and traffic field is just one kind of

3

Field Architecture for Traffic and Mobility Modeling in Mobility Management reflection of users by obeying certain traffic generating model. So traffic and population has some similarity to a certain extent. On the other hand, mobility field can be considered as the temporal variation of population field. In another word, the integral of mobility field on time reflects the difference of population field. So mobility is the radical reason of the change of population distribution. Thus in this paper, traffic field is paid less attention compared to with mobility field, and the latter is also investigated on the underlying evolving principle except for the intuitive behavior.

areas distribute near the transport hub and main roads. These result is consistent with common sense. Figure 2

Temporal evolution of traffic field.

(a) 5:00 a.m.

3 Traffic Field

(b) 6:00 a.m.

(c) 7:00 a.m.

3.1 Data Source and Pretreatment In this section, the database is from the empirical measurements of mobile communication system in one district of a northern city of China. It contains several quantities describing traffic of several thousands of base stations within a week, with temporal scale being one hour. The distribution of base stations is shown in figure 1(a). Here the Traffic CHannel (TCH) load is adopted as the field quantity in modeling, whose unit is Erl. Moreover, symmetrical gridding method is introduced to chop the background area into hundreds of grids, with the spatial scale being one kilometer, and traffic is distributed into every grid according to its area proportion of the cell. As an example, the traffic field at certain time is shown in figure 1(b). Figure 1

Data source of traffic field.

(d) 8:00 a.m.

(e) 9:00 a.m.

(f) 10:00 a.m.

3.3 Traffic hotspot In order to further explore the evolution of traffic field, the concept of traffic hotspot is introduced here. Definition 3 (Traffic Hotspot) For a fixed time t, points in traffic field T (D, t), whose field amount is larger than appointed threshold thr, are called hotspots. H = {P |T (P, t) > thr, P ∈ D, thr ∈ R}

(a) Base stations.

(b) Traffic field.

3.2 Temporal evolution of traffic field Figure 2 shows the evolution of traffic field at six different moments in one day. It is called the course of city revival in urban dynamics. It is clear that the intensity of traffic has also been multiplied as time evolves. And the traffic load in the area of downtown, located in the left middle part, changes most significantly. Moreover, based on the physical information of background, heavy traffic load

Here the threshold thr can be an absolute amount like 100 Erl/km2 , or a relative amount like 80% of the largest traffic load. Based on the definition of hotspot, a direct deduction is that with the increase of threshold the number of field hotspot will decline. What is more, distribution of hotspots at different times also explores the moving of intense traffic load, called hotspot excursion. Figure 3 shows the distribution of hotspots (marked in blue) in six different times in one day. The evolution process shows that, on 6:00 a.m., hotspots begin to emerge and distribute sparsely. From 7:00 a.m., they congregate to the center of city and concentrate there. This mode of traffic variation reflects the rule of mobility in the morning.

3.4 Applications of traffic field Traffic hotspot can be considered as a kind of regular and predictable outburst in network. This character is helpful for dealing with the problem of load balancing and resource optimization.

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author

Figure 3

et al., 2004). And their key issue is dealing with the traffic overload of isolated grid. Here a load range [Loadmin , Loadmax ] is adopted to determine three status of grid’s traffic load. If traffic load is less than Loadmin , the grid is defined as leisure status, marked in blue in figure 4(b); if more than Loadmax , defined as overload status, marked in red; otherwise defined as balanced status. Red grids with overload traffic are in need of balancing and those with no blue grids around, called isolated grid, should be paid more attention in the optimization of network resource.

Hotspot excursion of traffic field.

(a) 5:00 a.m.

(b) 6:00 a.m.

(c) 7:00 a.m.

4 Mobility Field 4.1 Data Source and Pretreatment (d) 8:00 a.m.

(e) 9:00 a.m.

(f) 10:00 a.m.

Traditionally, channel traffic load is used to measure the average distribution of traffic in network. In this field architecture, it is defined as channel traffic load =

traffic of grid number of channels.

In this section, the database is also from the measured data of mobile communication system, but in another southern city of China. It contains the population information of different grids (also adopting symmetrical modeling method) over one year, with the temporal scale of one hour and spatial scale of one kilometer. Figure 5

Data source of mobility field.

It is clear that distribution of channel traffic load can also be considered as a scalar field, describing the uniformity degree of traffic. Figure 4(a) shows the daily average channel traffic load field. It reveals that although the downtown of city accounts for most of the traffic load, the channel traffic load is not heavy there due to corresponding dense distributing of infrastructure. On contrary, some place in suburban may become in the risk of overload, which means lots of new calls will be blocked and the whole performance of network is degraded. Figure 4

(a) Daily average distribution of population field.

The field of channel traffic load.

(b) Example of ambulatory population field.

(a) Channel traffic load

(b) Classification status

of

load

Some methods like increasing the number of base station, adjusting coverage area of base station and channel borrowing, can be employed to solve the imbalance of traffic load distribution (Das et al., 1997; Karlsson and Eklundh, 1989; Du et al., 2003; Velayos

In this dataset, the only information provided is population. Comparing with mobility field, the valuable data is the temporal difference of population field, since it equals to the flux of mobility field along the boundary of grid. So before discussing the concrete character of mobility field, some pretreatment of data is performed and the new distribution of users is called ambulatory population field. A typical example is shown in Figure 5.

5

Field Architecture for Traffic and Mobility Modeling in Mobility Management

4.2 Temporal evolution of mobility field Ambulatory population field can partly reflect the character of mobility field, and its distribution in six different times in one day is shown in Figure 6. Here red represents for influx, while blue represents for effluence. It shows that in these hours users congregate from the whole city to downtown. According to field theory, it means the source of mobility field are mostly distributed at the suburban of city. Figure 6

 x11 x12 . . . x1n  x21 x22 . . . x2n    = . .. . . ..  ,  .. . . .  xm1 xm2 . . . xmn 

(1)

where m stands for the number of temporal sampling and n means the number of grids. Matrix XT X describes the autocorrelation of X, and its eigenvalues are all non-negative. In other words, {λi }ni=1 satisfies

Temporal evolution of ambulatory population field.

XT Xvi = λi vi

i = 1, . . . , n ,

(2)

and have the property of λi ≥ 0. If ranking them from large to small, we have λ1 ≥ λ2 ≥ . . . λr > λr+1 = . . . λn = 0 , (a) 5:00 a.m.

(b) 6:00 a.m.

(c) 7:00 a.m.

(d) 8:00 a.m.

where r is the rank of matrix X. Define p σi , λi i = 1, . . . , r , Xvi i = 1, . . . , r , ui , σi Vn×r , (v1 , v2 , . . . , vr ) , Um×r , (u1 , u2 , . . . , ur ) , Σr×r , diag(σ1 , σ2 , . . . , σr ) . By (2), it is simple to get

(e) 9:00 a.m.

xi =

(f) 10:00 a.m.

r X

vij · σj · uj

i = 1, . . . , n ,

(3)

j=1

Ambulatory population field, as the flux of mobility field, reflect the macro principle of population distribution and movement, but their underlying rules still need further discussion. Facing the complex structure of mobility field, its regular changing part is the foundation of modeling. So the decomposition of mobility field is first proposed in this section, and a familiar mathematical tool is adopted, Principal Component Analysis (PCA) (Hotelling, 1933). PCA is a linear dimension reduction technology, exploring the underlying structure of data. Essentially, it uses orthogonal transformation to seek for low dimension representation of a redundant space. Due to the symmetrical griding of background area, ambulatory population field at a certain time can be denoted as a matrix. When fixing a time, this matrix can be further rearranged into a column vector. If these vectors of different times are arranged into a new matrix, denoted as X, it will contain the information in both spatial and temporal domains. So the objective of using PCA is discovering the features of X: Xm×n , (x1 , x2 , . . . , xn )

which can be expressed in matrix form as X = UΣVT ,

(4)

In our measured data, n = 24 × 7 − 1 = 167 and m = 59 × 95 = 4655. So the eigenvalues of matrix XT X are distributed as Figure 7. Figure 7

Distribution of eigenvalues.

11

4

value of eigenvalues

4.3 Decomposition of mobility field

x 10

3

2

1

0

0

20

40

60 80 100 120 serial number of eigenvalues

140

160

It indicates that, approximately, the first five eigenvalues have much larger value, and this means their corresponding eigenvectors account for the main

6

author

body in the decomposition. So keeping only these eigenvectors can simplify the originally complex problem but maintain its primary character, as stated in the following equation: ˆi = x

p X

vij · σj · uj

i = 1, . . . , n p < r ,

(5)

j=1

where p is the the number of reserved principal components in the reconstruction process. The determination of a proper value of p is an important issue. From further analysis of data, it is explored that only first three components could meet the cumulate contribution rate of 85% (shown in Figure 8), and Figure 9 also indicates that reconstruction using only these three components really differs little from original field.

In (Sun et al., 2010), it is pointed out that the eigenvalues can be classified into three types, periodic, burst and noisy. In the case of mobility field, similar conclusion can be drawn from analysis, as shown in Figure 10. It shows that the first three components are all periodic, which also guarantees the modeling result reflects the primary character of mobility field. Figure 10

Periodic

Outburst

Noisy

0

Figure 8

Weights of first 10 components at 10 random selected times.

Classification of principle components.

20

40

60

80

100

120

140

160

Figure 11 shows the concrete distribution of these three components and their corresponding weights at different times. The periodic changing rule of weights is obvious as time evolves, which provides the three components an explanation of their physical meaning. They describe three different movement modes of group users in large scale respectively, but at a particular time these modes exhibits to different extents. The difference in extent is just embodied in the periodic changing weights. Figure 11

First three components and their weights. 0.5

Figure 9

Reconstruction of first three components.

0

−0.5 0

(a) First component

24

48

72

96

120

144

(b) Weights of first component 0.5

0

−0.5 0

(c) Second component (a) Original field.

24

48

72

(d) Weights component

96

120

of

144

second

0.5

0

−0.5 0

(e) Third component

(b) Reconstruction field.

24

48

(f) Weights component

72

96

of

120

144

third

So by adopting PCA to mobility field, the primary temporal-spatial character is discovered and the original complex problem is simplified to the

7

Field Architecture for Traffic and Mobility Modeling in Mobility Management research on relatively simple model of the three principal components.

Figure 12

Plane vector fields and their P´ olya fields.

4.4 Complex analysis description of mobility field In this subsection, the underlying structure of mobility field is further discussed by introducing complex analysis. The objective of this consideration is to reconstruct population velocity from its flux, but usually this target cannot be achieved by only grasping the knowledge of population information. So the assumption of conservative field is appended to mobility field, although in fact this assumption may not be available in every instance and the exceptions are excluded in the discussion of this paper. Then the assumption of conservative field brings several benefits to modeling mobility field, and one of them is introducing complex analysis theory into field architecture. In fact, complex function and plane vector field are radically consistent, i.e., two different views to considering a problem (Needham, 1997). Based on the definition of mobility field, it is essentially a mapping relation from D ⊂ R2 to R2 , while complex function itself is the mapping from D ⊂ C to C. So this consistency of these two concepts can help to construct relation between flux of mobility field and integral of complex function. For this, P´olya vector field is introduced (P´olya and Latta, 1974). Definition 4 (P´ olya Vector Field) If a plane vector field can be described as the form of f (z) = r · eiβ , then its P´ olya vector field can be defined as f¯(z) = r · e−iβ .

(a) f (z) = z

(b) P´ olya field of f (z) = z

(c) f (z) = z 2

(d) P´ olya f (z) = z 2

(e) f (z) = z −1

(f) P´ olya f (z) = z −1

Figure 13

field

field

of

of

P´ olya field and mobility field.

P´olya vector field is not a simple reflection by real axis (examples are shown in Figure 12), but it indeed maintains some key characteristics of original field, like zero and pole points. More significantly, it sets up a relation between flux of mobility field and integral of complex function. As shown in Figure 13, if assuming that dz = eiα dl f (z) = r · eiβ , then f¯(z) = r · e−iβ , so f¯(z)dz =r · ei(α−β) dl =r · (cos(α − β) + i sin(α − β))dl   ~ z dl , = f (z) · T~z + if (z) · N

which means Z Z Z ~ z dl . f¯(z)dz = f (z) · T~z dl + i f (z) · N L

L

(6)

the flux of mobility field. It is concluded that for certain form of field function f (z), the integral on tangential direction can be zero (its proof is shown later). Then (7) can be simplified as Z Z ¯ ~ z dl , f (z)dz = i f (z) · N L

(7)

L

So the term at the left hand side of (7) is a complex integral, while the second term at the right hand side is

which means Z

L

L

~ z dl = −i f (z) · N

Z

f¯(z)dz .

L

Based on Residue Theorem in complex analysis, this can be further changed into the form of residue at pole

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author

points: Z

~ z dl = 2π f (z) · N

L

n X

k=1

 Res f¯(z), zk .

(8)

So from the perspective of field theory, the non-zero character of mobility field’s flux means it contains influx or effluence, while from complex analysis it means field function has pole points. This is also another proof of the consistency of these two theories. Then in the modeling of mobility field, it can be assumed that the pole point lies on the center of grid, and it belongs to first-order pole point, then the field function can be described as f¯(z) =

n X

k=1

ak , z − zk

Here only consider the integral of function f (z) = 1/(¯ z − z¯0 ) on the boundary L of a certain grid, since mobility field is essentially the linear summation of it. For the convenience of proof, L is supposed as a unit square with center at origin of complex plane, as shown in Figure 15 and f (z) is marked as 1/(¯ z − z¯0 ). Proof of tangential integral.

Figure 15

(9)

where zk is the center of grid k and ak is a real coefficient. So  Res f¯(z), zk = lim (z − zk )f¯(z) = ak . z→zk

Then, combined with (8), we have R ~ z dl f (z) · N flux of mobility field ak = L = . 2π 2π

(10)

At first, the integral on L1 is considered, and at this time it can be assumed that z = x + 0.5i with x ∈ [−0.5, 0.5]. Then f (z) =

If mobility field’s P´olya vector field is selected as (9), the mobility field itself will obey the following form f (z) =

n X

k=1

Figure 14

1 (x − 0.5i) − r(cos θ − sin θi) (x − r cos θ) + i(0.5 − r sin θ) = , (x − r cos θ)2 + (0.5 − r sin θ)2 =

ak , z¯ − z¯k

whose physical meaning is very clear, as shown in Figure 14. Its vector line starts from the center of grid zk and extends radically. The direction of vector line depends on the sign of parameter ak . Cumulation of different grids finally shapes the whole mobility field and the absolute value of ak determines grids’ contribution.

1 z¯ − z¯0

So Z =

f (z) · T~z dl

L1 Z 0.5

−0.5 Z 0.5

−Re {f (z)}dx

(x − r cos θ) dx (x − r cos θ)2 + (0.5 − r sin θ)2 −0.5  0.5 1  = − ln (x − r cos θ)2 + (0.5 − r sin θ)2 2 −0.5 1 (0.5 + r cos θ)2 + (0.5 − r sin θ)2 . (11) = ln 2 (0.5 − r cos θ)2 + (0.5 − r sin θ)2

Illustration of mobility field function.

=

−

Similarly, for L2 , L3 and L4 (a) Simulation result of 1/(¯ z − z¯k )

(b) Sketch 1/(¯ z − z¯k )

map

of

This consideration and assumption has intuitive expression and explicit physical meaning. What is more, it meets all the preconditions of mobility field modeling like conservative field assumption, which makes this theory self-consistent. The only problem left behind is R the hypothesis of L f (z) · T~z dl = 0 on certain condition. Here gives the concrete proof.

Z

1 (0.5 − r cos θ)2 + (0.5 + r sin θ)2 f (z) · T~z dl = ln , 2 (0.5 + r cos θ)2 + (0.5 + r sin θ)2 L2 (12)

Z

1 (0.5 + r cos θ)2 + (0.5 + r sin θ)2 f (z) · T~z dl = ln , 2 (0.5 − r cos θ)2 + (0.5 + r sin θ)2 L3 (13)

Z

1 (0.5 − r cos θ)2 + (0.5 − r sin θ)2 f (z) · T~z dl = ln . 2 (0.5 − r cos θ)2 + (0.5 + r sin θ)2 L4 (14)

9

Field Architecture for Traffic and Mobility Modeling in Mobility Management Combining (11) (12) (13) (14), we have Z

f (z) · T~z dl =

L

4 Z X i=1

f (z) · T~z dl = 0 .

Li

So the mobility field architecture becomes selfconsistent and feasible. Further applying this conclusion into principle component of last part of this section, the expression of mobility field can be proposed as follow:   3 3  n(i) (i)  X X X a k ~ (z, t) = M . vti σi ui (z) = vti (i)  z¯ − z¯  i=1

i=1

k=1

k

(15)

Figure 16 gives an example of mobility fielding modeling and its local enlargements. Arrows in it represent the population velocity vectors but they have been normalized due to the diversity of vector value. Figure 16

Calculation of population fluxion.

Figure 17

Mobility field modeling using complex analysis.

to location update number and handoff number. So here gives the calculation method. Considering two breadthwise neighboring grids, as shown in Figure 17, their centers are denoted as x0 + y0 i and x0 + 1 + y0 i, so the points on their common boundary can be modeled as z = x0 + 0.5 + yi with y ∈ [y0 − 0.5, y0 + 0.5]. If further (i) (i) (i) define zk = xk + yk , then Z ~ (z, t) · N ~ z dl Nt = M L   Z X 3  n(i) (i)  X ak ~ z dl = vti ·N (i)  L i=1  z ¯ − z ¯ k=1 k   (i) Z 3  n (i)  X X ak ~ z dl = vti · N (i)   L z ¯ − z¯ i=1

=−

k=1

3 X

(

k

(i)

vti

n X

(i)

(i)

ak

arctan

y0 − yk + 0.5 (i)

x0 − xk + 0.5 !) (i) y0 − yk − 0.5 . (17) − arctan (i) x0 − xk + 0.5

i=1

(a) Mobility field

k=1

Similarly, population fluxion between lengthwise neighboring grids can be described as: Nt = −

3 X

(i)

vti

n X

(i)

(i)

ak

arctan

x0 − xk + 0.5 (i)

y0 − yk + 0.5 !) (i) x0 − xk − 0.5 − arctan . (18) (i) y0 − yk + 0.5

i=1

(b) Local enlargement 1

(

k=1

(c) Local enlargement 2

4.5 Mobility field prediction A benefit of using this mobility field architecture is the convenience to calculate the local population fluxion. In this modeling process, all the data source is about overall situation. But the mobility field provides a micro mechanism to describe users’ movement. So by applying integral along an appointed directed curve, any local population moving amount can be derived: Z ~ (z, t) · N ~ z dl . N= M (16)

Temporal stationarity is the foundation of mobility prediction. Adoption of PCA helps to establish a joint temporal-spatial model for aggregate mobility. So here we can use some results in principle components to study temporal stationarity. As shown in Figure 18, two matrices X1 and X2 with the same dimension but describing different temporal sequences may have some data overlapped. They can be reconstructed by their first three components respectively as:

L

In fact, the population fluxion between neighboring grids may be most valuable, since it is closely related

ˆ1 = X

3 X i=1

(1) (1) (1) T

σi ui vi

=

3 X i=1

(1) (1) T

X1 vi vi

,

(19)

10

author ˆ2 = X

3 X

(2) (2) (2) T

σi ui vi

i=1

=

3 X

(2) (2) T

X2 vi vi

5 Conclusion .

(20)

i=1

Figure 18 Data illustration of temporal stationarity analysis.

If utilizing character of X1 to describe X2 , which (2) (1) means substituting vi by vi , then we have ˆ ′2 = X

3 X

(1) (1) T

X2 vi vi

.

(21)

i=1

It can be proved that if X1 and X2 are only different in the arrangement of vectors they will share the same principal components. Then the two ways of reconstructing X2 are the same, which is just the ideal situation of temporal stationarity. But in fact, it cannot reach this precision and the error between them can be evaluated as ˆ′ −X ˆ 2 k2 kX 2 Err = . (22) ˆ 2 k2 kX Figure 19 shows the changing of error with the growing of the nonoverlapped part of two matrices, and it is clear to see the temporal stationarity becomes worse. When nonoverlapped part is less than 30 hours, ˆ ′2 and X ˆ 2 is acceptable, which the difference between X means the relative error is no more than 10%. So in these circumstances, principal components can be considered as invariant and this conclusion of short temporal stationarity can be adopted as the foundation of mobility field prediction. Figure 19

Temporal stationarity analysis.

1

10

0

Relative error

10

−1

10

−2

10

−3

10

0

20

40

60 80 100 Temporal interval of overlap (hour)

120

140

160

Due to PCA’s essence of exploring the regular changing part, this mobility field architecture can be adopted to perform large temporal scale prediction of population. Specifically, mobility field has been decomposed into three principle components with their weights changing periodically. So methods of time-series analysis can be further employed, like ARMA and ARIMA (Kohn and Ansley, 1986), then the prediction results can be utilized in the reconstruction of future mobility field.

This paper proposed a field architecture for mobility management in large temporal and spatial scale, which combine traffic and mobility into a uniform theoretical framework. The main contributions are intuitive expression benefitting qualitative analysis of characteristic quantities, and theoretical derivation combining with matrix theory and complex analysis to provide support for further discussion. Based on the different attributes of traffic and mobility, they are modeled as scalar field and vector field, respectively. By qualitative analysis of traffic field in temporal and spatial evolution, some conclusions consistent with common sense of daily life have been drawn. Moreover, traffic hotspots reflect the asymmetry of traffic distribution in network, and treating it as a kind of regular outburst lies a foundation for load balance and resource optimization. On the other hand, mobility field describes the rule of aggregate mobility. Some changing patterns were discovered by analyzing the flux of mobility field, adopting the mathematical tool of PCA. This joint temporal and spatial analysis method helped further concrete description of its underlying structure using complex analysis. All the definitions, conclusions and methods mentioned above collectively form the field architecture of group mobility model. This modeling method benefited lots of practical application areas in mobility management.

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