research article the analysis of the properties of bus

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 694956 6 pageshttpdxdoiorg1011552013694956

Research ArticleThe Analysis of the Properties of Bus Network Topology inBeijing Basing on Complex Networks

Hui Zhang1 Peng Zhao1 Jian Gao1 and Xiang-ming Yao12

1 School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China2 State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Hui Zhang zhanghui3052gmailcom

Received 1 January 2013 Accepted 27 February 2013

Academic Editor Yi-Kuei Lin

Copyright copy 2013 Hui Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The transport network structure plays a crucial role in transport dynamics To better understand the property of the bus networkin big city and reasonably configure the bus lines and transfers this paper seeks to take the bus network of Beijing as an exampleand mainly use space L and space P to analyze the network topology properties The approach is applied to all the bus lines inBeijing which includes 722 lines and 5421 bus station In the first phase of the approach space L is used The results show that thebus network of Beijing is a scale-free network and the degree of more than 99 percent of nodes is lower than 10 The results alsoshow that the network is an assortative network with 46 communities In a second phase space P is used to analyze the property oftransfer The results show that the average transfer time of Beijing bus network which is 188 and 998 percent of arbitrary two pairnodes is reachable within 4 transfers

1 Introduction

Complex networks have been successfully used in manyreal complex systems since the researches of small-worldnetworks and scale-free networks [1 2] Many real complexsystems have been well studied which include the actornetwork [1ndash3]WWW[2 4] protein networks [5] and powergrid [1 3] During the last few years transport networkssuch as subway network [6] airport network [7ndash9] and streetnetwork [10] have been studied by the complex approach

As an important part of urban transport systems and atrip mode to alleviate the traffic congestion bus network hasbeen studied by an increasingly large number of researchersSienkiewicz and Holyst studied the public transport in 22Polish cities and found that the degree distribution of thesenetwork topologies followed a power law or an exponentialfunction [11] Xu et al analyzed the three major cities ofChina and they found that there is a linear behavior betweenstrength and degree [12] Soh et al contribute a complexweighted network analysis of travel routes on the Singaporerail and bus transportation systems [13]

In this paper we investigate the Beijing bus network(BBN) with 722 lines and 5421 nodes The data can

be achieved from the Internet (httpwwwmapbarcomsearch) In the network the nodes stand for bus stationsand edges (links) are the bus line connecting them alongthe route In order to analyze the static properties of BBNwe use the so-called space L and space P to represent theBBN The space L is mainly used to analyze the properties ofdegree distribution cluster the average shortest path degreecorrelation and community structure The space P is mainlyused to analyze the average transfer time

This paper is organized as follows In Section 2 we givethe description of the construction of the network Section 3analyzes the main properties of the BBN with space LIn Section 4 we analyze the average transfer time of theBBN and use space P to calculate the smallest transfertimes between any nodes in the network Discussion andconclusions are given in Section 5

2 The Construction of the Network

For easy calculation there are some assumptions followed

(1) The station name is the unique identification in thenetwork Do not account for the condition that some

2 Mathematical Problems in Engineering

1

2

3

4

5

11

12

7

8

9

106

13

Line 1

Line 2

Line 3

(a)

1

2

3

4

5

6

7

109

8

11

12

13

(b)

Figure 1 Illustration of space L (a) and space P (b)

stations have identical names but different parkingplace

(2) The stations have slight difference between upstreamline and downstream line In this paper the construc-tion of the undirected network is based on the stationsof the upstream lines

(3) This paper does not consider the number of linksbetween two stations and the frequency of bus thatis it does not consider the weight of the network

The bus network is usually represented by space L spaceP [11] space B [14] space C [15]This paper will study and theBeijing bus network based on space L and space P Space Lconsists of nodes which stand for the bus stations and a linkbetween two nodes exists if they are consecutive stops on theroute Space P is a link formed between any two nodes of aline Figure 1 gives a schematic representation of space L andspace P

Table 1 The stations of Beijing bus network with larger degree andthe value of degree

Serial number Bus station degree1 Sanyuanqiao 212 Liu Li Qiao Dong 203 Beijing Xi station 194 Liu Li Qiao Bei Li 185 Beijing Zhangdong 166 Madian Qiao Xi 167 Xi Dao Kou 168 Chong Wen Men Xi 159 Guang An Men Nei 1410 Zuo Jia Zhuang 1411 Qianmen 1412 Dabei Yao Nan 1413 Tianqiao 1314 Ma Dian Qiao Nan 1315 Bei Tai Ping Qiao Xi 1316 Si Hui station 1317 Deshengmen 1318 Xi Bei Wang 1319 Xiyuan 1320 Xin Fa Di Qiao Bei 13

Figure 2 The graph of Beijing bus network topology

3 The Main Properties of the Beijing BusNetwork under Space L

Here we represent network as a graph 119866 = (119881 119864) where 119881 isthe set of nodes and119864 is the set of edges (links)119866 is describedby the 119873 times 119873 adjacency matrix 119890

119894119895 If there exists an edge

between nodes 119894 and119895 119890119894119895= 1 otherwise 119890

119894119895= 0 119873 is the

number of nodes in the network Figure 2 gives the Beijingbus network topology graph with 5421 nodes and 16986 links

31 Degree The degree 119896119894of node 119894 is defined as the number

of nodes that connected with the node 119894 which reflects theimportance of the node 119894 In this paper it refers to the numberof bus stations with direct bus connecting with the currentbus station After calculation the largest degree of the BBNis 21 the smallest is 1 and the average degree of all nodes is313 which means one station averagely connects 3-4 stationsin Beijing bus network Table 1 shows partly the bus stationsof larger degree of BBN

Mathematical Problems in Engineering 3Pr

opor

tion

100

101

102

103

(deg)

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

119891(119896) = 1914 lowast Γ(119896)Γ(119896 + 247)

Figure 3 The degree distribution of Beijing bus network

Here we studied the proportion of the stations with thedegree from 1 to 21 Figure 3 shows the degree distribution ofthe stations in the BBN and we found that it follows a shiftedpower law distribution119891(119896) = 1914 lowast Γ(119896)Γ(119896 + 247) 119896 gt2 119896 is the degree and the number of stations with degree 1accounts for 448 It also shows that the degree of 99 of allstation is smaller than 10

32 The Average Shortest Path The average shortest path isthe property to reflect the efficiency of information circulat-ing on the network It is defined as 119897 = 2 lowast sum

119894gt119895119889119894119895(119873 lowast

(119873 minus 1)) 119897 is the average minimum shortest number of stepsbetween all pairs of nodes and119889

119894119895is the shortest path between

node 119894 and node119895

33 Cluster Coefficient Cluster coefficient is an importantproperty of characterizing the local cohesiveness of the cur-rent node or the extent to which the nodes in the network areclustered together In the BBN clustering coefficient reflectsthe ease of the bus transport among the neighboring busstations of the current one It is defined as 119862

119894= 119860119894(119896119894(119896119894minus

1)2) where 119862119894is the cluster coefficient of node 119894 119860

119894is the

actual number of links between the neighbor nodes of thecurrent node and 119896

119894is the degree of node 119894 The cluster

coefficient of the network is 119862 = sum119894isin119866

119862119894119873

34 Efficiency The efficiency is the property to characterizethe capacity of traffic and it can be calculated with theformula 119864 = 2 lowast sum

119894gt119895(1119889119894119895)(119873 lowast (119873 minus 1)) where 119889

119894119895is the

shortest path between node 119894 and node 119895 [16]

35 Degree Correlation Degree correlation reflects the rela-tionship between the degrees of nodes Nodes with highdegree tending to be connected with nodes with high degreeare called assortativity In contrast nodes with high degreewhich have the tendency to be connected with low degree are

Table 2 The value of the main properties of Beijing bus network

Network parameters ValueNumber of nodes (119873) 5421Number of links (119860) 16986Number of lines (119871) 722Average degree (Ak) 313Average shortest path (119897) 2003Cluster coefficient (119862) 0142Efficiency (119864) 0066Correlation coefficient (119903) 0185Number of communities (119885) 46Modularity (119876) 0905

called disassortativity It can be calculated with the formula[17]

119903 =

119872minus1sum119894119895119894119896119894minus [119872minus1sum119894(12) (119895119894 + 119896119894)]

2

119872minus1sum119894(12) (119895

2

119894+ 1198962

119894) minus [119872minus1sum

119894(12) (119895119894 + 119896119894)]

2 (1)

where 119903 is correlation coefficient and 119895119894and 119896

119894are the

degrees of the nodes at the ends of the 119894th links with 119894 =

1 119872

36 The Community Structure It is usually found that thereare many communities in one complex network withinthe community there are many links but between thecommunities there are fewer links [18] Newman and Girvan[19] gave ameasure119876 calledmodularity For a divisionwith119892communities then define a 119892times119892 matrix 119890whose component119890119894119895is the fraction of edges in the original network that

connects nodes in community 119894 to those in community 119895The modularity is defined to be 119876 = sum

119894119890119894119894minus sum119894119895119896119890119894119895119890119896119894

=

Tre minus 1198902 where 119909 indicates the sum of all elements of

119909 It can be achieved that 0 le 119876 le 1 119876 = 0 indicatesthe community structure is not stronger than it would beexpected by random chanceThe larger the modularity is thestronger the community structure is

Table 2 shows the main properties of BBN We can seethat the average shortest path is 2003 which means peoplecan reach destination by averagely taking 20 stops and it isobvious that the BBN exhibits a small-world property Thecluster coefficient of BBN is 0142 which means the BBN isa sparse network Furthermore the correlated coefficient is0185 which validates the result in the paper [m6] that whenthe number of nodes is larger than 500 the network is usuallyassortative In addition the community structure of BBN is soobvious and the modularity is 0905 with 46 communities

4 The Transfer Property of Beijing BusNetwork under Space P

The transfer capacity is an important index to evaluate theperformance of a bus network and travelers always expectthat they can reach the destination through the least numberof transfers In this paper the averageminimum transfer time


Table 3 The specific network of Figure 1

1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

11988110

11988111

11988112

11988113

1198711

1 1 1 1 1 1 0 0 0 0 0 0 01198712

0 0 1 0 0 0 1 1 1 1 0 0 01198713

0 0 0 0 1 0 0 0 0 0 1 1 1

1

2

3

4

5

6

7

109

8

11

12

13

(a)

1

2

3

4

5

6

7

109

8

11

12

13

(b)

Figure 4 The illustration of transfer network under space P

is used to evaluate the performance of the transfer capacityUsually travelers cannot reach the destination without trans-fer for a long distance trip and the minimum transfer timebetween any two nodes is specific The average minimumtransfer time is the average among all pair nodes

1198661015840= (119871 119881) is used to represent the specific bus network

where 119871 is the line 119881 is the bus station and 1198661015840 is a 119871 lowast 119873matrix 119887

119894119895 119887119894119895= 1 if line 119894 stops at station 119895 otherwise 119887

119894119895=

0Table 3 shows the specific network given in Figure 1 We

can achieve the minimum transfer time using Table 3 Forexample (1) search 119881

6rarr 1198817 Because there is no directed

line between the node 1198816and 119881

6 it needs transfer From

Table 3 we can see that 11988716

= 11988713

= 1 through 1198711 and

11988723

= 11988727

= 1 through 1198712 so we can achieve 119881

6

1198711

997888997888rarr

1198813

1198712

997888997888rarr 1198817 that is travelers need transfer at119881

3from 119871

1to 1198712

(2) Search 11988110

rarr 11988113 We can get the path 119881

10

1198712

997888997888rarr 1198813

1198711

997888997888rarr

1198815

1198713

997888997888rarr 11988113

through two transfersUsing the aforementioned method we can get the mini-

mum transfer time between any two nodes and calculate the

Table 4 The stations that have most line and the line numbers

Serial number Station Line numbers1 Sanyuanqiao 472 Liu Li Qiao Bei Li 403 Beijing Xi Zhan 394 Zuo Jia Zhuang 385 Liu Li Qiao Dong 346 Liu Li Qiao Nan 337 Dongzhimen Wai 318 Gong Zhu Fen Nan 319 Bei Da Di 3010 Bei Tai Ping Qiao Xi 2911 Jing An Zhuang 2912 Xiajia Hutong 2913 Qing He 2914 Xiyuan 2915 Beijing Zhangdong 2816 Xi Bei He 2817 Xiju 2818 Si Hui Zhan 2819 Liangmaqiao 2820 Mu xi yuan qiao dong 2721 Yan Huang yishu Guan 2622 Yuquanying Qiao Xi 2623 Dongwu Yuan 2524 Guang An Men Nei 2525 Qianmen 2526 Wanshou si 2527 Mu Xi Yuan Qiao Xi 2528 Kandan Qiao 25

average minimum transfer time But it becomes very hardwhen the scale of network is becoming huge In this paperwe use the space P to solve the problem Firstly we need toconstruct the network under space P where the weight ofthe network is 1 Secondly the Floyd algorithm is used toachieve the shortest path between any twonodesThe shortestpath value is the minimum line number that needs to useand the transfer time is the needed line number minus 1Figure 4 gives the illustration of the aforementioned exampleFigure 4(a) shows 119881

6can reach 119881

7by using the two dotted

lines Figure 4(b) shows 1198816can reach 119881

7by using the three

dotted linesHere we study the BBN Table 4 shows the stations that

have most lines It is found that the station that owns mostlines is San yuan qiao which has 47 lines

Figure 5 shows the distribution of the proportion of thestations and the amount of line number the result shows thatit follow the exponent distribution 119891(119909) = 079 lowast 119890

minus067119909where 119909 is the number of lines It is found that most stationsof BBN own less than 10 lines and only 9 stations own morethan 30 lines

In this paper the transfer time of BBN is studied by usingspace P From Table 5 we can see that the most pair nodes

Mathematical Problems in Engineering 5

Table 5 The proportion of the transfer time of Beijing bus network

atr 0 1 2 3 4 4 more unreachableProportion 00168 0283 0553 0141 00047 000013 000137

0 10 20 30 40 500

01

02

03

04

05

The amount of lines

Prop

ortio

n

Figure 5 The distribution of a stationrsquos amount of lines

are reachable through one or two transfers and 9985 percentof the pair nodes is reachable within four transfers and theaverageminimum transfer time atr = 188 Usually the largerthe atr is the worse the performance of the bus networkis In general atr cannot be more than 2 otherwise we canconsider that the performance of the bus network is bad andtravelersrsquo trip is inconvenient The transfer time of BBN is alittle large and there is a room for improvement

5 Conclusion

In this paper space L and space P are used to analyze the staticproperties of Beijing bus network Space L is used to researchthe main topology properties of the Beijing bus networkThe results show the Beijing bus network has small clustercoefficient scale-free feature and assortative correlation andthe community structure is obvious Moreover we researchthe transfer property using space P The result shows thatthe accessibility of the Beijing bus network is good andthe average minimum transfer time is 188 which is alittle large A convenient bus network needs less transfersand high performance and how to reduce transfer timeand enhance the bus network dynamical performance is avaluable research

Acknowledgment

This paper is supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(20120009110016)

References

[1] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquo Nature vol 393 pp 440ndash442 1998

[2] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999

[3] E Ravasz and A L Barabasi ldquoHierarchical organization incomplex networksrdquo Physical Review E vol 67 Article ID 0261127 pages 2003

[4] L A Adamic and B A Huverman ldquoPower-law distribution ofthe World Wide Webrdquo Science vol 287 no 5461 p 2115 2000

[5] S Maslov and K Sneppen ldquoSpecificity and stability in topologyof protein networksrdquo Science vol 296 no 5569 pp 910ndash9132002

[6] V Latora and M Marchiori ldquoIs the Boston subway a small-world networkrdquo Physica A vol 314 no 1ndash4 pp 109ndash113 2002

[7] A BarratM Barthelemy R Pastor-Satorras andAVespignanildquoThe architecture of complex weighted networksrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 101 no 11 pp 3747ndash3752 2004

[8] R Guimera S Mossa A Turtschi and L A N Amaral ldquoTheworldwide air transportation network anomalous centralitycommunity structure and citiesrsquo global rolesrdquo Proceedings of theNational Academy of Sciences of theUnited States of America vol102 no 22 pp 7794ndash7799 2005

[9] T Jia and B Jiang ldquoBuilding and analyzing the US airportnetwork based on en-route location informationrdquo Physica Avol 391 no 15 pp 4031ndash4042 2012

[10] B Jiang ldquoA topological pattern of urban street networksuniversality and peculiarityrdquo Physica A vol 384 no 2 pp 647ndash655 2007

[11] J Sienkiewicz and J A Holyst ldquoStatistical analysis of 22 publictransport networks in PolandrdquoPhysical ReviewE vol 72 ArticleID 046127 11 pages 2005

[12] X P Xu J H Hu F Liu and L S Liu ldquoScaling and correlationsin three bus-transport networks of Chinardquo Physica A vol 374no 1 pp 441ndash448 2007

[13] H Soh S Lim T Zhang et al ldquoWeighted complex networkanalysis of travel routes on the Singapore public transportationsystemrdquo Physica A vol 389 no 24 pp 5852ndash5863 2010

[14] K A Seaton and L M Hackett ldquoStations trains and small-world networksrdquo Physica A vol 339 no 3-4 pp 635ndash644 2004

[15] C von Ferber T Holovatch Y Holovatch and V PalchykovldquoPublic transport networks empirical analysis and modelingrdquoThe European Physical Journal B vol 68 no 2 pp 261ndash2752009

[16] V Latora and M Marchiori ldquoEfficient behavior of small-worldnetworksrdquo Physical Review Letters vol 87 Article ID 198701 4pages 2001

[17] M E J Newman ldquoAssortative mixing in networksrdquo PhysicalReview E vol 89 Article ID 208701 4 pages 2002

[18] M E J Newman ldquoDetecting community structure in net-worksrdquoThe European Physical Journal B vol 38 no 2 pp 321ndash330 2004


[19] M E J Newman and M Girvan ldquoFinding and evaluatingcommunity structure in networksrdquo Physical Review E vol 69Article ID 026113 15 pages 2004

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


1

2

3

4

5

11

12

7

8

9

106

13

Line 1

Line 2

Line 3

(a)

1

2

3

4

5

6

7

109

8

11

12

13

(b)

Figure 1 Illustration of space L (a) and space P (b)

stations have identical names but different parkingplace

(2) The stations have slight difference between upstreamline and downstream line In this paper the construc-tion of the undirected network is based on the stationsof the upstream lines

(3) This paper does not consider the number of linksbetween two stations and the frequency of bus thatis it does not consider the weight of the network

The bus network is usually represented by space L spaceP [11] space B [14] space C [15]This paper will study and theBeijing bus network based on space L and space P Space Lconsists of nodes which stand for the bus stations and a linkbetween two nodes exists if they are consecutive stops on theroute Space P is a link formed between any two nodes of aline Figure 1 gives a schematic representation of space L andspace P

Table 1 The stations of Beijing bus network with larger degree andthe value of degree

Serial number Bus station degree1 Sanyuanqiao 212 Liu Li Qiao Dong 203 Beijing Xi station 194 Liu Li Qiao Bei Li 185 Beijing Zhangdong 166 Madian Qiao Xi 167 Xi Dao Kou 168 Chong Wen Men Xi 159 Guang An Men Nei 1410 Zuo Jia Zhuang 1411 Qianmen 1412 Dabei Yao Nan 1413 Tianqiao 1314 Ma Dian Qiao Nan 1315 Bei Tai Ping Qiao Xi 1316 Si Hui station 1317 Deshengmen 1318 Xi Bei Wang 1319 Xiyuan 1320 Xin Fa Di Qiao Bei 13

Figure 2 The graph of Beijing bus network topology

3 The Main Properties of the Beijing BusNetwork under Space L

Here we represent network as a graph 119866 = (119881 119864) where 119881 isthe set of nodes and119864 is the set of edges (links)119866 is describedby the 119873 times 119873 adjacency matrix 119890

119894119895 If there exists an edge

between nodes 119894 and119895 119890119894119895= 1 otherwise 119890

119894119895= 0 119873 is the

number of nodes in the network Figure 2 gives the Beijingbus network topology graph with 5421 nodes and 16986 links

31 Degree The degree 119896119894of node 119894 is defined as the number

of nodes that connected with the node 119894 which reflects theimportance of the node 119894 In this paper it refers to the numberof bus stations with direct bus connecting with the currentbus station After calculation the largest degree of the BBNis 21 the smallest is 1 and the average degree of all nodes is313 which means one station averagely connects 3-4 stationsin Beijing bus network Table 1 shows partly the bus stationsof larger degree of BBN


opor

tion

100

101

102

103

(deg)

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

119891(119896) = 1914 lowast Γ(119896)Γ(119896 + 247)




119894gt119895119889119894119895(119873 lowast





119894= 119860119894(119896119894(119896119894minus


119894is the




119862119894119873



119894119895is the






119903 =


2

119872minus1sum119894(12) (119895

2

119894+ 1198962


119894(12) (119895119894 + 119896119894)]

2 (1)


119894are the


1 119872



119894119890119894119894minus sum119894119895119896119890119894119895119890119896119894

=








1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

11988110

11988111

11988112

11988113

1198711

1 1 1 1 1 1 0 0 0 0 0 0 01198712

0 0 1 0 0 0 1 1 1 1 0 0 01198713

0 0 0 0 1 0 0 0 0 0 1 1 1

1

2

3

4

5

6

7

109

8

11

12

13

(a)

1

2

3

4

5

6

7

109

8

11

12

13

(b)






119894119895=







= 11988713


11988723

= 11988727


6

1198711

997888997888rarr

1198813

1198712


3from 119871

1to 1198712

(2) Search 11988110


10

1198712

997888997888rarr 1198813

1198711

997888997888rarr

1198815

1198713

997888997888rarr 11988113






6can reach 119881



7by using the three









0 10 20 30 40 500

01

02

03

04

05

The amount of lines

Prop

ortio

n



5 Conclusion


Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










opor

tion

100

101

102

103

(deg)

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

119891(119896) = 1914 lowast Γ(119896)Γ(119896 + 247)




119894gt119895119889119894119895(119873 lowast





119894= 119860119894(119896119894(119896119894minus


119894is the




119862119894119873



119894119895is the






119903 =


2

119872minus1sum119894(12) (119895

2

119894+ 1198962


119894(12) (119895119894 + 119896119894)]

2 (1)


119894are the


1 119872



119894119890119894119894minus sum119894119895119896119890119894119895119890119896119894

=








1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

11988110

11988111

11988112

11988113

1198711

1 1 1 1 1 1 0 0 0 0 0 0 01198712

0 0 1 0 0 0 1 1 1 1 0 0 01198713

0 0 0 0 1 0 0 0 0 0 1 1 1

1

2

3

4

5

6

7

109

8

11

12

13

(a)

1

2

3

4

5

6

7

109

8

11

12

13

(b)






119894119895=







= 11988713


11988723

= 11988727


6

1198711

997888997888rarr

1198813

1198712


3from 119871

1to 1198712

(2) Search 11988110


10

1198712

997888997888rarr 1198813

1198711

997888997888rarr

1198815

1198713

997888997888rarr 11988113






6can reach 119881



7by using the three









0 10 20 30 40 500

01

02

03

04

05

The amount of lines

Prop

ortio

n



5 Conclusion


Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

11988110

11988111

11988112

11988113

1198711

1 1 1 1 1 1 0 0 0 0 0 0 01198712

0 0 1 0 0 0 1 1 1 1 0 0 01198713

0 0 0 0 1 0 0 0 0 0 1 1 1

1

2

3

4

5

6

7

109

8

11

12

13

(a)

1

2

3

4

5

6

7

109

8

11

12

13

(b)






119894119895=







= 11988713


11988723

= 11988727


6

1198711

997888997888rarr

1198813

1198712


3from 119871

1to 1198712

(2) Search 11988110


10

1198712

997888997888rarr 1198813

1198711

997888997888rarr

1198815

1198713

997888997888rarr 11988113






6can reach 119881



7by using the three









0 10 20 30 40 500

01

02

03

04

05

The amount of lines

Prop

ortio

n



5 Conclusion


Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 10 20 30 40 500

01

02

03

04

05

The amount of lines

Prop

ortio

n



5 Conclusion


Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the analysis of the properties of bus

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