Sep 18, 2018 - Fundamentals on Base Stations in Cellular. Networks: From ... The basic information of these eight ..... The real PDF and the fitted ones are presented in Fig. 4 .... munications (ICT), Thessaloniki, Greece, May 2016. [4] Z. Zhao ...
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Fundamentals on Base Stations in Cellular Networks: From the Perspective of Algebraic Topology
arXiv:1809.07681v1 [cs.SI] 18 Sep 2018
Ying Chen, Rongpeng Li, Zhifeng Zhao, and Honggang Zhang
Abstract—In recent decades, the deployments of cellular networks have been going through an unprecedented expansion. In this regard, it is beneficial to acquire profound knowledge of cellular networks from the view of topology so that prominent network performances can be achieved by means of appropriate placements of base stations (BSs). In our researches, practical location data of BSs in eight representative cities are processed with classical algebraic geometric instruments, including α-Shapes, Betti numbers, and Euler characteristics. At first, the fractal nature is revealed in the BS topology from both perspectives of the Betti numbers and the Hurst coefficients. Furthermore, lognormal distribution is affirmed to provide the optimal fitness to the Euler characteristics of real BS deployments.
I. I NTRODUCTION Prompted by the significance of base station (BS) deployment issues, substantial researchers have been working on the relevant subjects over decades [1]–[5]. However, the majority of related documents studied BS deployments only by means of analyzing BS density distributions. In our works, unprecedentedly, several principal concepts in algebraic geometry field, i.e., α-Shapes, Betti numbers, and Euler characteristics [6], are merged into the analyses of the BS topology of eight representative cities around the world. In essential, BSs can be abstracted into a discrete point set in the 2-dimensional plane, and connections between BSs can be specifically established according to the construction of α-Shapes, so α-Shapes are capable of reflecting topological features of BS deployments. Moreover, the topological information can be extracted from the Betti numbers and the Euler characteristics as well, because the corresponding α-Shapes can be expressed in terms of these two features. Due to the space limitation, interesting readers could refer to [7] to find the details of these three topological notions and their relationships. To be specific, this letter is devoted to searching for essential topological features in BS deployments, which transcend geographical, historic, or culture distinctions, and these properties stand a good chance to provide valuable guidance for designers Y. Chen, R. Li, Z. Zhao, and H. Zhang are with College of Information Science and Electronic Engineering, Zhejiang University. Email: {21631088chen ying, lirongpeng, zhaozf, honggangzhang}@zju.edu.cn This work was supported in part by National Key R&D Program of China (No. 2018YFB0803702), National Natural Science Foundation of China (No. 61701439, 61731002), Zhejiang Key Research and Development Plan (No. 2018C03056), the National Postdoctoral Program for Innovative Talents of China (No. BX201600133), and the Project funded by China Postdoctoral Science Foundation (No. 2017M610369).
of cellular networks. For this purpose, this letter offers two novel contributions as listed below: • Firstly, the fractal nature is uncovered in BS configurations based on the Betti numbers and the Hurst coefficients; • Secondly, log-normal distribution is confirmed to match the Euler characteristics of the real BS location data with the best fitting among all candidate distributions. The organization of this letter is arranged as follows: The actual BS location data are briefly introduced in Section II. The fractal phenomenon in BS deployments for either Asian or European cities is presented in Section III. The log-normal distribution is affirmed to offer the best fitting for the Euler characteristics of all the eight cities in Section IV. Lastly, conclusions are summarized in Section V. II. DATASET D ESCRIPTION A great deal of real data downloaded from OpenCellID community (https://community.opencellid.org/) [8], a platform to provide BS’s latitude and longitude data all over the world, are analyzed to guarantee the validity of results. Substantial BS location data of eight cities, including four representative cities in Asia (i.e., Seoul, Tokyo, Beijing, and Mumbai) and Europe (i.e., Warsaw, London, Munich, and Paris) respectively, have been collected from this platform. The basic information of these eight cities is provided in Table I. III. F RACTAL NATURE IN C ELLULAR N ETWORKS T OPOLOGY As a vital property of complex networks, the fractal nature has been revealed in a plenty of wireless networking scenarios [9]–[13]. In terms of the Betti numbers and the Hurst coefficients, the fractal feature in the BS topology is verified in this section. A. Fractal Features Based on the Betti Curves The random and fractal point distributions can be distinguished by their Betti curves as depicted in Fig. 1 (for details see [14]). Fig. 1(a) displays the practical point diagrams for both the random and fractal point deployments. For the left part of Fig. 1(a), the horizontal and vertical coordinates of each point are randomly designated according to Poisson point distribution, whereas the fractal point deployment in the right part of Fig.
2
Random
50
Fractal
15
TABLE I: The basic information of 8 selected cities.
10
40
5 30 0
Area
City
20
(
-5 10
10
20
30
40
-15 -15
50
-10
-5
0
5
10
15
(a) Points deployment diagrams: (left) random; (right) fractal. β for Random 0
2500
β1 for Random
200
β1
β0 2000
Europe
150
β
β0
1
1500 100
7327
14.31
21110
13.38
Munich 310.43
6837
22.02
50
512
Paris
12012 161753
Seoul
605.77
9202
15.19
Tokyo
2188
46755
21.37
Beijing
16410 138620
8.45
Mumbai
603.4
23.04
Asia
1000
No. of Density of BSs BSs ( )
London 1577.3
Warsaw
-10
0 0
)
13.47
500 0 0
0.5
1
1.5
α(/km)
0 0
2
0.5
1 α(/km)
1.5
(b) Betti curves for random: (left) β0 ; (right) β1 . β0 for Fractal
600
5000 4500
5000
4000
4000
3500 0.1
β0
0.2
0.3
1500
1000 0 0
α(/km)
City Warsaw
100
1000 0.5
0.5
State
300 200
2000
2000
β1
400 2500
3000
TABLE II: The positions of the ripples and peaks in the Betti curves.
β1 for Fractal
500
1
6000
β
β0
7000
13900
2
0.6
1
0.7
1.5
0 0
0.5
α(/km)
1
1.5
London
(c) Betti curves for fractal: (left) β0 ; (right) β1 . Europe
Fig. 1: Comparison between the Betti curves of random and fractal point distributions.
Munich
Paris
1(a) is realized by the hierarchical partitions of the area, which brings the most principal aspect of fractal features, i.e., selfsimilarity [15], into the point distribution. Fig. 1(b) illustrates the Betti curves for the random pattern, while Fig. 1(c) for the fractal pattern. Instead of the clearly monotonous decease in the β0 curve and the single peak in the β1 curve in the random situation, the fractal nature is manifested by the distinctive features of multiple ripples and peaks in the β0 and β1 curves, respectively [14], where a ripple is formed due to the distinct slope change as highlighted by the amplified blue subgraph in Fig. 1(c). In summary, the fractal property can be characterized by the hierarchical patterns, i.e., the multiple ripples and peaks of the Betti curves [14]. In our works, the fractal property in the BS topology is verified in Fig. 2 for European cities and Fig. 3 for Asian ones, respectively. Beyond the geographical constraints, it is extremely astonishing to find out the consistent fractal nature for all the aforementioned eight cities. Moreover, the entirely different positions of the multiple ripples and peaks can be evaluated on the basis of the crossover point of two straight lines with quite different gradients, as shown by the β0 curves in Fig. 2 and Fig. 3. As listed in Table II, it is clear that the peaks always come after the corresponding ripples, which makes sense because of the larger size of loops than that of components in the relevant topologies. B. Fractal Features Based on the Hurst Coefficients The Hurst coefficient between [0,1] is widely used as an evaluation index of the fractal property of data series, and
Seoul Tokyo Asia Beijing
Mumbai
Index
β0
β1
1st
1.880
2.620
2nd
4.180
4.290
1st
1.840
2.090
2nd
4.240
4.360
1st
0.990
1.940
2nd
3.820
3.920
3rd
6.320
6.570
1st
2.055
6.405
2nd
4.890
8.175
3rd
8.340
10.330
1st
1.530
4.515
2nd
2.970
4.815
1st
1.725
2.850
2nd
7.725
7.800
1st
1.920
2.060
2nd
5.360
5.800
1st
1.350
1.725
2nd
5.310
6.090
the indication of fractality increases gradually as the Hurst coefficient approaches to 1 [16]. For the sake of verifying the fractal nature in the BS topology, the rescaled range analysis (R/S) method [17] is adopted to compute the Hurst coefficients of all the selected cities in this letter. Due to the space limitation, the details about the calculation of the Hurst coefficients will not be provided here, and all the relevant information can be found in [7]. As listed in Table III, the fractal nature is evidently affirmed because all the Hurst coefficients turn out to be very close to 1. IV. L OG - NORMAL D ISTRIBUTION OF THE E ULER C HARACTERISTICS The Euler characteristics can be calculated from Betti numbers according to Euler-Poincare Formula [6]. Since a clear heavy-tailed property is demonstrated in the probability density functions (PDFs) of the Euler characteristics of the real BS location data, three classical heavy-tailed statistical
3
2
London
500
2
β0
12000
β 400
10000
10
1.5
5
8
4
3
6000
β0
2
7
1
3 1
2
3
β
0
β1
4000 3000
2
4
6 3
200
4000
Paris
5000
×10 4
6
9
300 1
1
β1
β0
6000
×10 4
1
8000
1.5
Paris
×105
1
London
×104
β
2.5
×10 4
2000
2 2000
0.5
0
0 0
5
0.5
100 3
4
5
10
0
6
15
α(/km)
0 0
20
5
10
15
α(/km)
0 0
20
1000
1
5
10
15
6
8
20
α(/km)
10
25
(a) London Munich
350 β0
2000
3000
6000
1500
2
3
4
5
1000
150
2000
500
100
1000
3000
5
5
6
10
7
8
15
α(/km)
30
1
2
3
4000
3000
200
2000
2000
100 1000
0 0
20
25
300
50 0
0 0
20
β1
3500
6000
200
3000
15
α(/km)
Warsaw
400
1
β0
500
1.5
β1
1
β
0
2000 0.5
4000
10
β0
4000
250
1000
Warsaw
8000 β
300
2500
5000
5
(c) Paris
β1
Munich
7000
0 0
30
5
10
15
α(/km)
0 0
20
5
3
4
10
5
15
α(/km)
0 0
20
(b) Munich
5
10
15
α(/km)
20
(d) Warsaw
Fig. 2: The Betti curves of the practical BS deployments in European cities. Beijing
×104
β
0
3
150
5500
1
2
2.5
4
2
6000
×10 4
3
3
6
4000
1
0 0
10
5000
0
2
4 5000
4000
2000
1.5
2
6000 0
8
4000
2000
3500 4
5
6
20
7
30
α(/km)
0 0
40
10
20
30
α(/km)
0 0
40
5
10
50 2
15
α(/km)
4
6
20
25
β0
5
2
4 1500
300
500
0 0
0
5
10
3
15
α(/km)
5
6
20
3
2.5
0
2
30
30
5
10
15
20
α(/km)
25
30
(b) Mumbai
0 0
Tokyo β
10000
1
1000
8000
500
6000 4000
0 0
25
4
7
25
20
1500
1
100 4
15
α(/km)
β0
200
4000
10
2000
×10 4
2
1000
2000
Tokyo
4
β0
0
β1
β0
4000
400
6000
×10
4 3.5
β1
500
6000
10000
5
(c) Seoul Mumbai
600
8000
12000
0 0
30
β1
Mumbai
14000
100
4500
(a) Beijing
8000
β1
β0
6000
4
Seoul
200
6500
8000
β1
β0
Seoul
10000 β1
8000
5
10
Beijing
10000
×10 4
β1
6
12
β
14
10
20
α(/km)
4
30
6
8
10
40
50
0 0
10
20
α(/km)
30
40
50
(d) Tokyo
Fig. 3: The Betti curves of the practical BS deployments in Asian cities. TABLE III: The Hurst coefficients for all the 8 cities. City
Hurst
City
Warsaw 0.96912 Europe
London 0.99591 Munich 0.93708 Paris
0.93917
Asia
TABLE IV: The candidate distributions and their PDF expressions.
Hurst
Seoul
0.94064
Distributions
Tokyo
0.90492
Poisson
PDF
Beijing 0.98367 Weibull
Mumbai 0.91981 Lognormal
distributions and widely-used Poisson distribution are selected as the candidates to match the PDFs. The candidates and their PDF formulas are given in Table IV. The real PDF and the fitted ones are presented in Fig. 4 for European cities and in Fig. 5 for Asian ones, respectively. In order to verify the best fitness for the Euler characteristics, the root mean square error (RMSE) between the real distribution and every candidate is listed in Table V, and it is obvious that the RMSE between log-normal distribution and the real PDF is almost one-order of magnitude smaller than
Generalized Pareto(GP)
the others in the same row. As a result, an astounding but well-grounded conclusion can be drawn as follows: regardless of the geographical boundaries, the culture differences and the historical limitations, the Euler characteristics of both Asian and European cities entirely conform to log-normal distribution.
4
PDF of the Euler Characteristics in London
0.1
Fitted Poisson Distribution Fitted Generalized Pareto PDF
Real PDF
TABLE V: The RMSE between each candidate distribution and the practical one.
Fitted Weibull PDF
0.1
Fitted Log-normal PDF
0.06
PDF of the Euler Characteristics in Paris
Fitted Generalized Pareto PDF Fitted Weibull PDF
0.08
PDF
0.15
Fitted Poisson Distribution
Fitted Log-normal PDF
PDF
0.12
Real PDF
Distributions
0.05
0.04
0 100
101
102
103
104
0 100
105
102
104
(a) London
(c) Paris
PDF of the Euler Characteristics in Munich
0.14
Fitted Poisson Distribution
0.14
Fitted Weibull PDF
PDF
PDF
Real PDF 0.08
0.08 0.06
0.06
0.02
0.02 101
102
103
0 100
104
101
Euler Characteristics
102
103
104
PDF of the Euler Characteristics in Beijing
0.14
PDF of the Euler Characteristics in Seoul Fitted Poisson Distribution Fitted Generalized Pareto PDF Fitted Weibull PDF Fitted Log-normal PDF Real PDF
0.12
Fitted Poisson Distribution Fitted Generalized Pareto PDF
0.1
0.1
Fitted Log-normal PDF
PDF
PDF
Fitted Weibull PDF 0.08
Real PDF 0.06 0.04
0.16
0.08 0.06 0.04
0.02
0.02
0 100
102
104
0 100
106
101
(a) Beijing
(c)Seoul 0.16
Fitted Poisson Distribution Fitted Generalized Pareto PDF Fitted Weibull PDF Fitted Log-normal PDF Real PDF
Fitted Log-normal PDF
0.1
Real PDF
PDF
PDF
0.12
Fitted Weibull PDF
0.08
0.08
0.06
0.06
0.04
0.04
0.02 0 100
104
PDF of the Euler Characteristics in Tokyo
0.14
Fitted Generalized Pareto PDF
0.1
103
Euler Characteristics
Fitted Poisson Distribution
0.12
102
Euler Characteristics
PDF of the Euler Characteristics in Mumbai
0.14
0.02 101
102
103
Euler Characteristics
(b) Mumbai
London
1.378
0.161
0.402
0.525
Munich
3.150
0.189
0.839
1.265
Paris
0.032
0.005
0.014
0.025
Seoul
2.302
0.098
0.582
0.841
Tokyo
1.109
0.085
0.286
0.418
0.129 0.817
Beijing
0.604
0.083
0.133
Mumbai
2.136
0.138
0.546
(d) Warsaw
Fig. 4: The comparison between the practical PDF and the fitted ones for the Euler characteristics of BS locations in European cities.
0.12
1.039
Euler Characteristics
(b) Munich
0.14
0.355
Asia
0.04
0.04
0 100
2.433
Europe
Fitted Poisson Distribution Fitted Generalized Pareto PDF Fitted Weibull PDF Fitted Log-normal PDF Real PDF
0.1
Fitted Log-normal PDF
0.1
1.189
Warsaw
PDF of the Euler Characteristics in Warsaw
0.12
Fitted Generalized Pareto PDF 0.12
106
Euler Characteristics
Euler Characteristics
0.16
Possion Lognormal Generalized_Pareto Weibull
City
0.02
104
105
0 100
101
102
103
104
105
Euler Characteristics
(d) Tokyo
Fig. 5: The comparison between the practical PDF and the fitted ones for the Euler characteristics of BS locations in Asian cities.
V. C ONCLUSION AND F UTURE W ORKS In this letter, several typical instruments in the algebraic geometric field, namely, α-Shapes, Betti numbers, and Euler characteristics, have been utilized in the discovery of inherent topological essences in BS deployments for Asian and European cities. Firstly, the fractal nature has been revealed in the BS topology according to both the Betti numbers and the Hurst coefficients. Moreover, it has been proved that lognormal distribution provides the best match for the PDFs of the Euler characteristics of the real BS location data in cellular networks among the classical candidate distributions. However, regardless of the topological discoveries above, some thought-provoking problems still need to be solved. For example, how can the fractal nature be applied to the design of cellular networks? How can log-normal distribution of the Euler characteristics facilitate BS deployments? All of these
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